## Thursday, September 19, 2019

### Notes on Compensation of VNA Calibration Standards

I was wondering how the imperfections of "real-life" VNA calibration standards (e.g. short, open, and load, also known as "SOL") were compensated for within a VNA, and what sort of equations would be required if implementing them, say, in an app written for the NanoVNA.

Some googling took me to this Keysight application note, Specifying Calibration Standards and Kits for Keysight Vector Network Analyzers, and its terminated transmission-line model for SOL standards (one-port, not two):

From this model and a set of givens, one can arrive at the application note's equation 1.4, shown below:

A Simplified Model...

Keysight's equation 1.4 is a fairly complex equation.  Can we simplify it?

If we assume that we are working at frequencies in which loss is negligible (i.e.  α essentially equals 0), and if we assume Zc = Zr (i.e. Γ1 = 0), then equation 1.4 reduces to:

Γi = e-2jβl ΓT

Much simpler!

Note that β has units of radians/meter and l has units of meters.  Therefore βl has units of radians (of delay).  And the factor of 2 in the exponent represents, from the perspective of the standard's input, a wave first traveling the distance l down the line, and then its reflection traveling back that same distance, for a total travel-distance of 2l.

But -- the delays of SOL standards are typically defined in terms of time (e.g. picoseconds), not length.  How do we convert this equation into one that is a function of time?

First, recognize that β = 2π/λ (where λ is the wavelength, in the transmission-medium, at the frequency of interest -- i.e. λ = c*VF/f, where c is the speed-of-light (m/s), VF is the Velocity Factor (unitless), and f is frequency in cycles-per-second).  Thus λ's units are meters/cycle.

Making this substitution for lambda into the equation for beta, we get:

β = 2πf/(c*VF)

Now, note that, if the transmission-medium's relative permeability is 1, then VF = 1/√εr, where εr is the relative permittivity of the transmission-medium.

We can now rewrite the equation for beta as:

β = (2πf√εr )/c

Thus:

βl =  (2πf√εr )l/c

Next, note that the Keysight app note (referenced above) defines "Offset-delay" as:

offset-delay = √εr / c     (Keysight's equation 1.5)

Substituting this into our equation for βl, we get:

βl =  2πf*(offset-delay)

And thus our earlier equation for Γi, expressed in terms of βl, becomes (when expressed in terms of frequency and offset-delay):

Γi = e-2j(2πf)*(offset_delay) ΓT

So now we can calculate Γin terms of frequency, the specified offset-delay, and ΓT.

But what is ΓT?

For an Open standard:

ΓT =  (Zopen - Zr)/(Zopen + Zr)

where Zr is our system's reference impedance (50 ohms), and Zopen is defined to be:

Zopen = -j/(2πf*(C0 + C1*f + C2*f2 + C3*f3))

given C0, C1, C2, and C3 coefficients specified by the open-standard's supplier.

Similarly, for a Short standard:

ΓT =  (Zshort - Zr)/(Zshort + Zr)

where

Zshort = j2πf*(L0 + L1*f + L2*f2 + L3*f3)

and given L0, L1, L2, and L3 coefficients specified by the short-standard's supplier.

An example:

Let's look at the cal kit definitions for the Keysight's 85033D/E 3.5mm standards (per Keysight PNA 85033DE -- note that values for all the standards can be found here: Keysight Cal Kit Page).

For their Open Standard (male or female):

C0 = 49.433e-15
C1 = -310.13e-27
C2 = 23.168e-36
C3 = -0.15966e-45
Delay_Open = 29.2e-12
Loss_Open = 2.2E9
Offset_Zo_Open = 50

And for their Short Standard (male or female):

L0 = 2.077e-12
L1 = -108.5e-24
L2 = 2.171e-33
L3 = -0.01e-42
Delay_Short = 31.8e-12
Loss_Short = 2.36e9
Offset_Zo_Short = 50

Using the following Matlab code I can calculate their reflection-coefficients at 900 MHz (the upper-frequency limit of the NanoVNA, at this time):

f = 900e6;  % Frequency

divisor = (2 * pi * f * (C0 + C1 * f + C2 * f^2 + C3 * f^3))
Zop = -1i / divisor
gammaOpen = ((Zop/50) - 1) / ((Zop/50) + 1)
delayedOpenGamma = gammaOpen * exp(-1i * 2 * 2 * pi * f * Delay_Open)
mag_delayedOpenGamma = abs(delayedOpenGamma)
ang_delayedOpenGamma = angle(delayedOpenGamma)*180/pi

Zsh = 1i * (2 * pi * f * (L0 + L1 * f + L2 * f^2 + L3 * f^3))
gammaShort = ((Zsh/50) - 1) / ((Zsh/50) + 1)
delayedShortGamma = gammaShort * exp(-1i * 2 * 2 * pi * f * Delay_Short)
mag_delayedShortGamma = abs(delayedShortGamma)
ang_delayedShortGamma = angle(delayedShortGamma)*180/pi

The results are:

Gamma(open), without delay compensation:
= 0.9996 - j0.0278
(magnitude = 1.0, angle = -1.5931 degrees)

Gamma(open), with delay compensation:
= 0.9366 - j0.3504
(magnitude = 1.0, angle = -20.5147 degrees)

If I now force C1, C2, and C3 to be zero and leave C0 unchanged, Gamma(open) is:
= 0.9365 - j0.3506
(magnitude = 1.0, angle = -20.5231 degrees)

Gamma(short), without delay compensation:
= -1.0000 + j0.0004
(magnitude = 1.0, angle = 179.9743 degrees)

Gamma(short), with delay compensation:
= -0.9359 + j0.3524
(magnitude = 1.0, angle = 159.3679 degrees)

If I now force L1, L2, and L3 to be zero and leave L0 unchanged, Gamma(short) is:
= -0.9359 + j0.3524
(magnitude = 1.0, angle = 159.3667 degrees)

And if all four L-terms are set to 0, Gamma(short) is:
= -0.9360 + j0.3519
(magnitude = 1.0, angle = 159.3936 degrees)

Several conclusions can be made from the above calculations (run at 900 MHz):
1. Offset Delay (e.g. the values Delay_Short and Delay_Open, above) has the greatest impact on the standards' compensated Gamma, compared to the effect of the terms C0-C3 or L0-L3.
2. In fact, the total impact of all of the L-coefficients (L0-L3, combined) is insignificant -- their total effect is a clockwise-rotation of Gamma on the Smith Chart of only about 0.03 degrees.
3. Of C0-C3, C0 has the most effect at 900 MHz.  In the example above, if C1-C3 are forced to zero but C0 left at its original value, the difference in the angle of Gamma is less than 0.01 degrees, compared to Gamma calculated with the original non-zero values of C1-C3.
Note:  The results above assume my simplified terminated transmission line model is valid at 900 MHz,.  Let's verify...

The Effect of Loss and Offset Zo on Gamma:

I'll use the published values of Loss and Offset Zo, along with the appropriate equations in Keysight's document, to verify my assumption that, for the NanoVNA's maximum operating frequency of 900 MHz, the spec'd Loss and Offset Zo values have (in my opinion) an insignificant impact on the compensated Open and Short Gammas.

In the image below I have summarized the important equations forming a "complete" terminated transmission-line model of an Open or Short standard, from the Keysight paper, referenced above:

And here's my Matlab code for calculating gamma for both my simplified model and for Keysight's complete model, per the equations in the image, above:

zr = 50;    % reference impedance
f = 900e6;  % frequency

% *****************************
% Calculations for the Open Standard
% *****************************

% First,  calculate the Short's Gamma assuming
%  no Loss and Offset_Zo = 50 ohms.
divisor = (2 * pi * f * (C0 + C1 * f + C2 * f^2 + C3 * f^3));
Zop = -1i / divisor;
GammaOpen = ((Zop/50) - 1) / ((Zop/50) + 1);
abs(GammaOpen);
angle(GammaOpen)*180/pi;
delayedOpenGamma = GammaOpen * exp(-1i * 2 * 2 * pi * f * Delay_Open);
mag_delayedOpenGamma = abs(delayedOpenGamma);
ang_delayedOpenGamma = angle(delayedOpenGamma)*180/pi;

% Now, let's repeat the calculation for the Open's Gamma,
% but now assuming loss and specified Offset_Zo
% per equation 1.10 in Keysight's "Specifying Calibration Standards
% and kits for Keysight Vector Network Analyzers".

al_op = ((Loss_Open*Delay_Open)/(2*Offset_Zo_Open))*sqrt(f/1e9);
Bl_op = 2*pi*f*Delay_Open + al_op;
zc_open = Offset_Zo_Open + (1-1i)*(Loss_Open/(4*pi*f))*sqrt(f/1e9);

% From the givens of eq. 1.1:
gammaEl_op = al_op + 1i*Bl_op;

% Next, calculate a common exponential term...
e2gammaEl_op = exp(-2*gammaEl_op);

% Now, calculate Gamma-sub-1 and Gamma-sub-i
% (the latter equation 1.4 in Keysight doc)
Gamma_1_op = (zc_open-zr)/(zc_open+zr);

Open_Gamma_i = (Gamma_1_op*(1-e2gammaEl_op-Gamma_1_op*GammaOpen)+(e2gammaEl_op*GammaOpen)) / ...
(1-Gamma_1_op*(e2gammaEl_op*Gamma_1_op + GammaOpen*(1-e2gammaEl_op)));

full_open_rho = abs(Open_Gamma_i);
full_open_theta = angle(Open_Gamma_i)*180/pi;

% *****************************
% Repeat for the Short Standard
% *****************************

% First, calculate the Short's Gamma assuming
% no Loss and Offset_Zo = 50 ohms.
Zsh = 1i * (2 * pi * f * (L0 + L1 * f + L2 * f^2 + L3 * f^3));
GammaShort = ((Zsh/50) - 1) / ((Zsh/50) + 1);
abs(GammaShort);
angle(GammaShort)*180/pi;
delayedShortGamma = GammaShort * exp(-1i * 2 * 2 * pi * f * Delay_Short);
mag_delayedShortGamma = abs(delayedShortGamma);
ang_delayedShortGamma = angle(delayedShortGamma)*180/pi;

% Now, let's repeat the calculation for the Short's Gamma,
% but now assuming loss and specified Offset_Zo
% per equation 1.10 in Keysight's "Specifying Calibration Standards
% and kits for Keysight Vector Network Analyzers".

al_sh = ((Loss_Short*Delay_Short)/(2*Offset_Zo_Short))*sqrt(f/1e9);
Bl_sh = 2*pi*f*Delay_Short + al_sh;
zc_short = Offset_Zo_Short + (1-1i)*(Loss_Short/(4*pi*f))*sqrt(f/1e9);

% From the givens of eq. 1.1:
gammaEl_sh = al_sh + 1i*Bl_sh;

% Next, calculate a common exponential term...
e2gammaEl_sh = exp(-2*gammaEl_sh);

\% Now, calculate Gamma-sub-1 and Gamma-sub-i
%  (the latter equation 1.4 in Keysight doc)
Gamma_1_sh = (zc_short-zr)/(zc_short+zr);

Short_Gamma_i = (Gamma_1_sh*(1-e2gammaEl_sh-Gamma_1_sh*GammaShort)+(e2gammaEl_sh*GammaShort)) / ...
(1-Gamma_1_sh*(e2gammaEl_sh*Gamma_1_sh + GammaShort*(1-e2gammaEl_sh)));

full_short_rho = abs(Short_Gamma_i);
full_short_theta = angle(Short_Gamma_i)*180/pi;

Examples.

Here are the Matlab results for various cal-kit standards:

Notes:

1.  The "Simple" model uses C0-C3 and L0-L3, but not (offset) Loss nor (offset) Zo (i.e.  α essentially equals 0).  Also, Zc is assumed to equal Zr (refer to Keysight equations).
2.  The "Very Simplified" model is the "Simple" model, but now with C1-C3 and all the L terms zeroed-out.
3.  The "Keysight" Model is the "full" model I've created using the equations in their app note.
4.The very last "cal kit" represents the NanoVNA's compensation, as cleaned by looking at its firmware (note that I set Loss to 0 ohms/S to force α to be 0).

Conclusion:

In the above results, the worst case Gamma differences between the "Full" and "Very Simplified" models, calculated at 900 MHz, are:
• rho (magnitude):  Less than 0.003
• theta (degrees):  Less than 0.2 degrees.
In my opinion, these differences are insignificant.

Other Notes:

Similar equations can be found at these URL's:

Effect of Loss on VNA Calibration Standards, Raul Monsalve

One Port Direct/Reverse Method for Characterizing VNA Calibration Standards, Raul Monsalve, et al. (see Appendix at end of paper).

Calibration values for all the Keysight standards can be found here: Keysight Cal Kit Page

Standard Caveat:

As always, I might have made a mistake in my equations, assumptions, drawings, or interpretations.  If you see anything you believe to be in error or if anything is confusing, please feel free to contact me or comment below.

And so I should add -- this information is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.

## Thursday, September 12, 2019

### N Connectors for the NanoVNA

The NanoVNA is a cute little handheld VNA at an incredibly low price (given its performance).  One small issue, though, is that it uses SMA connectors, which can be a bit fragile.

Thinking that this might become an issue, I thought I would change the connector type to N by using N-to-SMA adapters.

But I wanted to ensure that the weightier N-connectors (and whatever might be attached to them) were not physically supported only by the NanoVNA’s SMA connectors themselves.  That could increase the chances of future SMA connector failure.

So I purchased two N-to-SMA bulkhead adapters, and, after they arrived, I made a simple chassis from PCB stock onto which to mount both my NanoVNA and the two adapters:

I used two pieces of copper-clad PCB stock.  One piece (double-sided) serves as the bottom plate onto which the NanoVNA attaches (using 4 screws -- these screws are longer than the stock ones used to attach the NanoVNA’s back cover).

The original back cover is still attached (so that I wouldn’t lose it) – you just cannot see it.

The second piece of copper-clad is single-sided, and upon it the two N-to-SMA bulkhead adapters mount.  I chose single-sided stock because, on the NanoVNA PCB itself, the grounds for the two SMA connectors split apart, and I wanted to continue this “separate-ground” approach out to the N adapters and their mounting hardware.

Thus, if you look at the next two pictures, you will see how each N-connector has its body and mounting hardware isolated from the other N-connector as well as from the ground-plate holding the NanoVNA.

No conductive copper on the front side:

And mounting hardware isolated on the backside. (I used a Dremel tool with an engraving bit to create the copper "islands", but you could use an Xacto knife, too.  It’s just a bit more work.)  Note that only two screws hold each adapter to the front plate.

If you look closely at the photo, above, you’ll see the solder-seam connecting the two boards.  There is an identical seam on the other side of the bottom plate, as you can see in the photo, below:

Solder on both sides makes this joint mechanically rigid, relieving any mechanical stress on the NanoVNA's SMA connectors.

By the way, the only change I made to the NanoVNA, itself, was to use slightly longer screws (I believe the thread size is M2.5, but don’t quote me!).

That's it!  Hopefully this will spur your own ideas!

Standard Caveat:

As always, I might have made a mistake in my equations, assumptions, drawings, or interpretations.  If you see anything you believe to be in error or if anything is confusing, please feel free to contact me or comment below.

And so I should add -- this information is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.

## Tuesday, July 30, 2019

### 1:4 Transmission Line Transformers: Coax Impedance versus Length

I have been working on an HF Power Amplifier project that uses a Guanella 1:4 impedance transformer in its output stage to transform a load of 50 ohms down to 12.5 ohms across the drains of the push-pull FETs in the PA.

These sorts of broadband impedance transformers (Guanella, Ruthroff, and others) are created with transmission lines (usually coax, but sometimes twisted pair), and the rule for determining the transmission line's Characteristic Impedance (Zo) is that it should equal the geometric mean of the impedances at the output side and at the input side of the impedance transformer.

In other words...

Zo = √(Zin*Zout)

In the amplifier design, above, the goal is to transform an output load of 50 ohms down to 12.5 ohms, so the coax cable's Zo should be 25 ohms (i.e.  √50*12.5).

A note regarding the impedance transformation being classified as 1:4 rather than 4:1...  Some authors use 4:1 as the impedance transformation of a transformer whose impedance at its output is four times the impedance of its input (e.g. Ruthroff), while others call this a 1:4 transformer (e.g. Davis/Agarwal).  I prefer the latter classification, 1:4, which is the one I will use in this blog post.

Although this equation represents the general rule for selecting the transformer's transmission line Characteristic Impedance, it seemed to me that if the length of coax were very short compared to the wavelength of the highest frequency of the transformer's operation, then transmission line impedance would not be critical and I could use whatever coax was at hand.  (As a rule of thumb I generally consider a circuit to be "lumped element" (i.e. we can ignore transmission line effects) when distances are no more than, say, 1/20th of the shortest wavelength being used).

And so I got to wondering...how does the input impedance of a 1:4 broadband transformer, be it a Ruthroff or a Guanella implementation, vary with transmission line Zo and line length?

Ruthroff''s original article, "Some Broad-Band Transformers" contains loop-equations used to derive the input impedance of his 1:4 impedance transformer as a function of load impedance, line length, and line characteristic impedance.  Assuming a lossless line, his equation for input impedance is:

(where β is the phase constant of the line and l is line length; i.e. βl is the phase delay from one end of the line to the other.)

But the source of the cosine and sine terms in the above equation were not obvious to me (although clearly they were related to delay), and, also, I couldn't find a corresponding equation for the Guanella 1:4 balun, so it was time to do some digging and deriving...

Some Fundamental Transmission Line Mathematics:

Consider an electrically long transmission line:

And let us define a voltage V1 and current I1 at its "left" end and voltage V2 and current I2 at its "right" end:

What is the relationship between the voltages and currents at the two ends of this line?

The voltage at any point on this transmission line can be considered to be the sum of a forward moving wave and a backward moving wave at that point.  That is:

V(@point) = V+(@point) + V-(@point)

where V+ is the forward moving voltage wave  (left to right) and V- is the backward moving voltage wave (right to left).

Similarly, the current at any point of this transmission line can be considered to be the difference of the forward moving current wave and the backward moving current wave at that point.

I(@point) = I+(@point)  -  I-(@point)

where I+ is the forward moving current wave and I- is the backward moving current wave.

Because the current at any point on the transmission line equals the voltage at that point divided by the transmission line's characteristic impedance, Zo, we can rewrite this last current equation as:

I(@point) = V+(@point) / Zo  -  V-(@point) / Zo

Let's now consider the voltage and current at the left-hand side of the transmission line (i.e. the side we would normally drive).  Dropping my "(@point)" subscript, I'll use the equations above to define this voltage and current to be:

V1 = V+ + V-

I1 = V+ / Zo  -  V- / Zo

We can rearrange these to become equations for V+ and V-.

V+ = (V1 + I1*Zo) / 2      (equation 1)

V- = (V1 - I1*Zo) / 2      (equation 2)

This left-hand side of the transmission line will be my reference point.

Next, let's assume that the transmission line is being driven with a sinusoidal waveform.  Therefore, at any point on the transmission line, both V+ and Vare sinusoids in the form of Vcos(ωt - φ), where φ is the phase delay from the left-hand side of the transmission line to any point on the transmission line to its right.

Given this sinusoidal form, we can simplify our calculations by using phasors (see this reference:  http://www.ee.ic.ac.uk/hp/staff/dmb/courses/ccts1/01800_LinesB.pdf) in lieu of cos(ωt-φ).  (Note:  transmission line is assumed to be lossless!)

So, using the left-hand side of the transmission line as our "reference" point, the voltage and current at the right-hand side of the transmission line (V2 and I2) can be expressed as the left-hand side voltage and current (see the equations for V1 and I1, above) with a phase-delay factor applied to them.

V2 = V+e-jβl + V-ejβl           (equation 3)

I2 = (V+/Zo)e-jβl  - (V-/Zo)ejβl         (equation 4)

Where l is the distance between the left side and the right side of the transmission line, and β is the transmission line's phase constant.

You will see a form similar to the above in the Wikipedia entry for "Telegrapher's Equation":

Note that the exponents are expressed using the term jkx rather than jβl.  The two forms are equivalent, as I will explain:

The units for βl are radians, as are the units for kx.  x and l are equivalent -- they are simply length (or distance), in meters.  And thus k and β are identical, as are their units (radians per meter).

Note also that k equals ω/u, (or, alternately, ω√LC) where ω is frequency of the driving sinusoidal signal (radians/sec) and u is the propagation speed of waves traveling within the transmission line medium (meters/sec).  And therefore u equals 1/√LC.

Note that the negative exponent represents the phase delay for a wave moving from left to right with respect to our left-hand transmission line reference point, while the positive exponent represents the phase delay for a wave moving from right to left, again with respect to our left-hand transmission line reference point.

Continuing...recall Euler's formulas...

ejβl = cos(βl) + jsin(βl)

e-jβl = cos(βl) - jsin(βl)

We can apply these to equations 3 and 4 to replace the complex exponentials with sines and cosines.  The result is:

V2 = V1*cos(βl) - j*I1*Zo*sin(βl)           (equation 5)

I2 = I1*cos(βl) - j*(V1/Zo)*sin(βl)           (equation 6)

It is important to note that we can also derive equations for V1 and I1 in terms of V2 and I2 (i.e. using the right-hand side of the transmission line as our reference point instead of the left-hand side -- note that this will flip the sign of the exponents, and thus the signs of the sine terms (Davis/Agarwal)).  The resulting two equations are:

V1 = V2*cos(βl) + j*I2*Zo*sin(βl)           (equation 7)

I1 = I2*cos(βl) + j*(V2/Zo)*sin(βl)           (equation 8)

If you refer to the Ruthroff article, you will see that these last two equations are two of the four loop equations specified in his Appendix A:

Let's use these equations to derive some input impedances!

Input Impedance of the Guanella 1:4 Impedance Transformer:

Let's first derive the input impedance of a Guanella 1:4 Impedance Transformer as a function of load impedance, transmission line characteristic impedance, and line length.

Here is a schematic representation of a Guanella 1:4 transformer.  It consists of two separate 1:1 transmission-line transformers, each wound with transmission line having a characteristic impedance of Zo and a length l.

Let's annotate this drawing with voltages and currents for both 1:1 transformers:

From inspection, it is clear that:

V1 = V3 = Vin

I2 = I4

But does V4 equal V2, and does I3 equal I1?  It would seem so from symmetry, but it's always a good idea to verify mathematically.  How do we do this?

We can write out the input voltages of the two transformers using equation 7:

V1 = V2*cos(βl) + j*I2*Zo*sin(βl)

V3 = V4*cos(βl) + j*I4*Zo*sin(βl)

Knowing that V1 = V3 and that I2 = I4, we can make these substitutions into these two equations and quickly prove that V4 = V2.

Similarly, we can express the input current of each transformer using equation 8:

I1 = I2*cos(βl) + j*(V2/Zo)*sin(βl

I3 = I4*cos(βl) + j*(V4/Zo)*sin(βl

We already know that I2 = I4 and V4 = V2, and using this information we can quickly simplify these two equations to show that I1 = I3.

So the Guanella currents and voltages simplify to the following:

Our goal is to derive an equation for Zin.  Inspecting the diagram above, we know that:
• Zin = Vin / Iin
• Iin = 2*I1
• Vin  = V1
Now, recall equations 5 and 6:

V2 = V1*cos(βl) - j*I1*Zo*sin(βl)         (equation 5)

I2 = I1*cos(βl) - j*(V1/Zo)*sin(βl)           (equation 6)

Substituting these two equations (for V2 and I2) into the equation I2*Zload = 2*V2 and then replacing V1 with Vin and I1 with Iin/2, the result is:

Rearranging...

Given Zin = Vin/Iin, it is a quick step to our solution:

Here is the equation again, in a (perhaps) more readable form...

An interesting check is to see what Zin becomes when the line-length is 0 (i.e. cos = 1, sin = 0) and when the line-length is a quarter wavelength (cos = 0, sin = 1).  The results are:

1.  Line-length = 0:

And so, if Zload = 200 ohms, Zin equals 50 ohms.

Note that this result does not depend upon the transmission line's characteristic impedance.

2.  Line-length = λ/4  (i.e. βl  = 90 degrees) and  Zo = 50*200):

Given Zo = 100 ohms and Zload = 200 ohms.  Zin still equals 50 ohms.

An Additional Note:  After I derived the above equation for Zin and wrote this blog post, I found the following equation in Jerry Sevick's book, Transmission Line Transformers, Fourth Edition, for a 1:4 Quanella impedance transformer:

Assuming that the imaginary term in the numerator is j*(Zload/2)*tanβl and not j*Zload/(2*tanβl), Sevick's equation 1-1 is identical to my equation for Zin of a Quanella 1:4 impedance transformer.

Guanella 1:4 Impedance Transformer implemented with transmission lines with various characteristic impedances and lengths:

So, with either a transmission line-length of 0 or a line-length of a quarter-wave, the Guanella 1:4 impedance transformer acts as a 1:4 impedance transformer, assuming the transmission line's characteristic impedance is Zo = √(Zin*Zout) .

What about transmission line-lengths between 0 and a quarter wave-length?  And what about other coax impedances?

1.  Input impedance for transmission lines of characteristic impedance that are half or double the ideal characteristic impedance:

If I define Zload to be 200 ohms, the ideal Zo is 100 ohms.  If I then use MATLAB to plot the SWR of the Guanella 1:4 Impedance Transformer's input (50 ohm reference for SWR), for coax lengths from 0 to a quarter wavelength, and for three coax characteristic impedances: Zo = 50, 100, and 200 ohms, the plots look like this:

Clearly, for Zo = 100 ohms (i.e. Zo = √(Zin*Zout)  = 50*200 ), Zin is 50 ohms irrespective of line length -- a great result!

But if I were to use, for example, a 50 ohms transmission line rather than a 100 ohm one, SWR can quickly rise.  SWR becomes 1.58:1 at a length of only 1/20th of a wavelength, and 4:1 if line length is a quarter of a wavelength!

So, if I wanted to use 50 ohm coax to wind a Guanella transformer to match a 200 ohm load to 50 ohms, I would need to make sure that the length of coax was very short!  At a length of only 3 percent of a wavelength, SWR is already about 1.3:1

Here's a plot of showing how the Zin impedance changes as the transmission line length is increased from 0 to a quarter wavelength, for non-ideal transmission lines of 50 ohm and 200 ohm characteristic impedance and a load of 200 ohms.  Note that the 50 and 200 ohm plots start in the center of the circle (Gamma = 0) when length is 0 and move away from that ideal match as the transmission line is lengthened):

In the case of my PA (see start of this blog post), in which the Guanella transformer is transforming a load impedance of 50 ohms to 12.5 ohms, if I were to wind this transformer with 50 ohms coax in lieu of the preferred 25 ohm coax, I would get the same result as the above plot for 200 ohm coax, as you can see below:

2.  Characteristic impedance with an error 10 percent or 20 percent from the ideal characteristic impedance:

Suppose the transmission line characteristic impedance is only a little off, say, for example, 10% or 20%, from our ideal characteristic impedance.  What happens?

Here are the SWR plots for transmission lines with characteristic impedances of 80, 90, 110, and 120 ohms, as well as our "ideal" characteristic impedance of 100 ohms, for transmission line lengths from 0 to a quarter wavelength:

A 10 percent error in Zo results in impedances that are still probably acceptable even if the line were a quarter wavelength long (i.e. SWR between 1.2:1 and 1.3:1).

A 20 percent error becomes more of a problem, as SWR could rise to almost 1.6:1 (although that, too, might still be acceptable, depending upon application).

Below is a Smith Chart containing plots of the 1:4 transformer's Reflection Coefficient for transmission lines with characteristic impedances of 80, 90, 110, and 120 ohms, plotted as the transmission line length increases from 0 (in which Gamma = 0 in all cases) to a length equal to a quarter wavelength.

3.  Transmission line lengths greater than a quarter wavelength:

Transmission line length was limited to 1/4 of a wavelength in the above discussion.  Suppose line length extends further?

Again, assuming lossless line, here are some plots for line lengths up to one wavelength long...

First, given a 200 ohms load, here is a plot of the magnitude of the 1:4 transformer's input impedance for transmission line characteristic impedances of 50, 100, and 200 ohms:

And SWR versus line length for transmission lines of various characteristic impedances:

And finally, the Reflection Coefficient of the transformer's Zin versus line length:

Clearly, if Zo = √(Zin*Zout), then Zin does not vary with transmission line length.

4.  Guanella 1:4 Impedance Transformer Conclusions:
• The ideal transmission line impedance is Zo = √(Zin*Zout)
• A 10 percent error in transmission line Zo can result in an input SWR of about 1.2:1 when the line length approaches a quarter wave.
• A 20 percent error in Zo can result in an input SWR of about 1.6:1 when the line length approaches a quarter wave.
• Don't use 50 ohm coax to wind a transformer whose ideal transmission line Zo is 25 ohms, unless the coax length no more than about 3 to 5 percent of a wavelength long at the highest frequency of operation.  (At 3% the SWR is about 1.3:1, while at 5% of a wavelength, SWR has risen to about 1.6:1).

Input Impedance of the Ruthroff 1:4 Impedance Transformer:

Let's look at the topology of a Ruthroff 1:4 impedance transformer:

I'll add in voltages and currents (using the same V and I labels and current directions as were used by Ruthroff):

Ruthroff, in his paper, gives an equation for Zin (his equation 7).  I rederived it myself, using the equations I introduced above, but the intermediary equations become very long and the math, although straight-forward, becomes tedious.  So rather than list all the steps in detail, I'll summarize my method:

First, from inspection of the transformer circuit above, recognize the following:
• Zin = Vin / Iin
• Iin = I1 + I2
• Vin = V1
• V1 + V2 = I2*Zload
And let's also recall the following equations, from earlier in this blog post:

V2 = V1*cos(βl) - j*I1*Zo*sin(βl)         (equation 5)

I2 = I1*cos(βl) - j*(V1/Zo)*sin(βl)           (equation 6)

Because Zin equals Vin / Iin, I want to end up with an equation in terms of Iin and V1 (because V1 = Vin).  To do this, I'm first going to create an equation expressing I1 in terms of Iin and V1 by using the equation Iin = I1 + I2 and substituting equation 6 into the I2 term.  This will give me:

I1 = Iin / (1+cos(βl)) + j*(V1/Zo)*sin(βl) / (1+cos(βl))

Next, substitute this new equation for I1 into the I1 terms of both equation 5 and equation 6.  Equation 5 becomes an equation expressing V2 in terms of Iin and V1 (let's call this equation "A"), and equation 6 becomes an equation expressing I2 in terms of Iin and V1 (let's call this equation "B").

Now take the equation "V1 + V2 = I2*Zload".  Into its V2 term substitute our new equation "A", and into its I2 term substitute our new equation "B".

We now have a very long equation that is solely in terms of Iin and V1 (the latter being equivalent to Vin).  Gather the Iin terms on one side of the equation and the V1 terms on the other side, divide to create the relationship V1/Iin (i.e. Vin/Iin), and then reduce.  (After much pencil pushing), the final equation will be:

And this equation is exactly Ruthroff's "equation 7" from his paper!

Again, as a check of performance, let's look at the input impedance when the length of the transmission is either 0 or a quarter wavelength.

Line-length  = 0:

The cosine terms are 1, the sine terms are 0, and the equation reduces to:

Excellent!  It is a 1:4 transformer!

Line-length = λ / 4:

Phase shift is 90 degrees, so cosine terms are 0 and sine terms are 1.  The equation for Zin becomes:

Zin = Zo*(j*Zo*sin(βl) / (2*Zo + j*Zload*sin(βl))

Hmmm...Zin is now a complex impedance with real and imaginary parts no longer simply 1/4 of Zload!

So, unlike the Guanella transformer, which maintained its 1:4 impedance relationship irrespective of line length (assuming Zo was correctly chosen), the Ruthroff transformer only has an ideal 1:4 impedance transformation when the transmission line is very short, irrespective of Zo!

Let's examine this relationship more closely...

Ruthroff 1:4 Impedance Transformer implemented with transmission lines with various characteristic impedances and lengths:

Let's take a look at how the Ruthroff 1:4 impedance transformer's Zin various with transmission line length for transmission lines of various characteristic impedances.

Zload will be 200 ohms, and so the transformer should ideally present an impedance of 50 ohms at its input.

Given the 200 ohm load and the resulting 50 ohms at the transformer's input, the ideal Zo is 100 ohms.  I'll look at SWR and I will also plot Gamma on a Smith Chart for this Zo, as well as for characteristic impedances of 50 and 200 ohms (i.e. halving or doubling the ideal 100 ohm Zo), as well as characteristic impedances that differ by 10 percent and 20 percent from 100 ohms.

Here's a plot of SWR as the lengths of these seven different transmission lines are increased from 0 to a quarter wavelength:

And here's a plot of the Reflection Coefficient (i.e. Gamma), again as the transmission line lengths are increased from 0 to a quarter wavelength.  (Note that when any transmission line's length equals 0, its Gamma is 0, representing a perfect match to 50 ohms).

Transmission line lengths greater than a quarter wavelength:

Transmission line length was limited to 1/4 of a wavelength in the above discussion.  Suppose line length extends further?

Again, assuming lossless line, here are some plots for line lengths up to one wavelength long...

First, SWR for transmission line lengths from 0 to one wavelength long, for various transmission line characteristic impedances:

Note that at 1/2 of a wavelength SWR goes to infinity!

Here is a plot of the associated Reflection Coefficients:

At a transmission line length equal to 1/2 of a wavelength, the Reflection Coefficient is 1 + j0 (i.e. Zin = infinity).  You can verify this by plugging 180 degrees into the sine and cosine terms of Ruthroff's equation for Zin (i.e. sin() = 0 and cos() = -1) -- the equation's denominator goes to 0 and therefore Zin goes to infinity.

Ruthroff 1:4 Impedance Transformer Conclusions:
• The ideal transmission line impedance is Zo = √(Zin*Zout)
• The transformation ratio of 1:4 is only perfectly 1:4 when the transmission line length is 0!

Final Notes...

1.  Modeling versus Testing...

This post uses derived mathematical equations to show how the input impedance of both Guanella and Ruthroff 1:4 impedance transformers vary with transmission line Zo and line length.  The next step would be to verify these simulations by building actual transformers using lines of different characteristic impedances and lengths, but I have not done this.

Do not assume my conclusions are correct.

Verify for yourself!

2.  Use of cos(βl) and jsin(βl) versus cosh(γl) and sinh(γl):

In the discussion above, I followed Ruthroff's method of defining voltages and currents along a transmission line in terms of cos(βl) and jsin(βl).

Other references use the terms cosh(γl) and sinh(γl) in lieu of cos(βl) and jsin(βl).  For example, the Wikipedia entry for Telegrapher's Equations expresses the voltage and current at the input of a Transmission Line, given the voltage and current at its output, as:

V1 = V2*cosh(γl) + I2*Zo*sinh(γl)

I1 = (V2/Zo)*sinh(γl) + I2*cosh(γl)

Compare these hyperbolic trigonometric forms to equations 7 and 8 defined earlier in this blog post:

V1 = V2*cos(βl) + j*I2*Zo*sin(βl)

I1 = j*(V2/Zo)*sin(βl) + I2*cos(βl)

These two sets of equations are actually identical.

To prove this to yourself, all you need to do is prove that cosh(γl) = cos(βl) and sinh(γl) = jsin(βl), which is pretty straightforward.

First, recognize that for a lossless line, γ = jω√LC. (You can verify this by setting the loss terms in the Wikepedia post's equation for γ(s) (see the very last diagram in the Wikipedia post -- it's of the equivalent circuit -- and you will be left with the equation γ = jω√LC).

And because β = ω√LC (per earlier discussion, above), therefore γ = jβ.

Next, there are two important Hyperbolic Trigonometric Identities:

cosh(x) = cos(jx)

sinh(x) = -jsin(jx)

Using the identity for the cosh term and given γl = jβl, then cosh(γl) = cosh(jβl) = cos(j*jβl) = cos(-βl) = cos(βl).

The sinh term is similar:  sinh(γl) = sinh(jβl) = -j*sin(j*jβl) = -j*sin(-βl) = j*sin(βl).

And thus the two forms of expressing transmission line voltages and currents are equivalent.

Resources:

"Some Broad-band Transformers", C.L. Ruthroff, Proceedings of the IRE, August, 1959

"New Method of Impedance Matching in Radio-Frequency Circuits", G. Guanella, The Brown Boveri Review, September, 1944

"Transmission Line Transformers," Chapter Six of Radio Frequency Circuit Design, W. Alan Davis, Krishna Agarwal, John Wiley & Sonse, Inc. 2001

"A novel topology of Broad-band Coaxial impedance transformer", Centurelli, Piatella, Tommasino, Trifiletti, Proceedings of the 40th European Microwave Conference"

Phasors and Transmission Lines

And viewing...

Transmission Line YouTube Series, Professor Gregory D. Durgin, Georgia Tech.  Starting with this video:  https://www.youtube.com/watch?v=7Oz1sazpekM

Standard Caveat:

As always, I might have made a mistake in my equations, assumptions, drawings, or interpretations.  If you see anything you believe to be in error or if anything is confusing, please feel free to contact me or comment below.

And so I should add -- this information is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.

## Wednesday, February 6, 2019

### An Arduino Version of Brooks Shera's GPSDO

Introduction:

This blog post is a continuation of my two earlier GPSDO blog posts.  The first one (from a few years back) details a simple Frequency-Locked Loop GPSDO design, based around an Arduino processor.  The second (more recent) blog post discusses simulating Brooks Shera's GPSDO algorithm (from the July, 1998 issue of QST) using The MathWorks Simulink program.

This third blog posts describes my modification of my original Frequency-Locked Loop (FLL) GPSDO to be a Phase-Locked Loop (PLL) GPSDO, and it includes the hardware schematics, Simulink models, and the Arduino code I wrote to implement Brooks Shera's GSPDO algorithm on an Arduino processor.

My notes are here to organize them in one place (and not have them scattered across different folders, notebooks, and loose sheets of paper), and also so that others who want to construct or experiment with their own GPSDO designs might find my notes useful in furthering their own projects.

You should be able to find a copy of Brooks Shera's original July 1998, QST article at either the ARRL website or here:   https://www.qsl.net/n9zia/wireless/QST_GPS.pdf

System modeling can be a huge design time-saver, as it allows one to experiment with various circuit topologies and algorithms before writing actual code or designing actual hardware.

In my specific case, I first created a "Phase-Space" Simulink model of Shera's algorithm and hardware from his original QST article.

This model allows me to vary parameters and see how, for example, the VCO (and DAC input/output) respond to various "reference" (1 PPS) signal perturbations.  (For more information on this Shera Simulink model, go here:  http://k6jca.blogspot.com/2018/12/simulating-brooks-shera-w5ojm-gpsdo.html -- note, this Simulink model is for a 10 MHz VCO, but I also show the VCO response for a 5 MHz VCO (per Shera's original design) in the blog post)).

To investigate how I could adapt my Arduino hardware and code to match the Shera design, I then created a new Phase-Space Simulink model incorporating Shera's algorithm but modified for my particular FLL GPSDO hardware platform:

Using the same stimulus to drive both models simultaneously, I can then compare the responses of this new model to the responses created by Shera's original model:

Let's look more closely at my Arduino/Shera Simulink model.  In many respects it is similar to my model of Shera's original design, but there are differences that arise due to differences in my Arduino hardware compared to Brooks' design.  (I would recommend first familiarizing oneself with the Brooks Shera Simulink model at this blog post: http://k6jca.blogspot.com/2018/12/simulating-brooks-shera-w5ojm-gpsdo.html.)

Here, again, is the Arduino/Shera model:

Note that the different color traces and blocks refer to their time domains, as defined below:

For the most part the model should be self-explanatory.  But there are some subsystem blocks that we should click on to see what is going on inside them...

First, the phase-wrapper block.  This block is instantiated as a MATLAB function:

Next, here's the FIR "boxcar" filter that sums together (with equal weighting) 30 samples.

And here is the down-sampler that samples the FIR "boxcar" filter's output once every 30 samples:

Now let's look at the heart of Shera's algorithm, his IIR (Infinite-Impulse-Response) filtering...

Below is the IIR block as it is shown in the Simulink model's top-level.  If you compare the constants (in purple) with the model of Shera's original design, you will note that they are different.  I will explain why, later in this post.

If we click on the "Shera Software IIR Loop Compensator" block, we see the model for the IIR filter, itself:

If we return to the model's top-level and then click on the subsystem block representing the 16-bit DAC, we will see the following:

Note that the DAC's output goes from 0 to +5 for unsigned inputs from 0 to 65535.

Also, this subsystem has two outputs -- the lower output represents the DAC's voltage output, while the upper output is used solely for displaying (via Simulink) the DAC's output as a signed value.

There is an input function that "clips" the incoming value, which mimics what I do in the Arduino software when calculating values to send to the DAC.

If we return back to the model's top level and then click on the VCO subsystem block, we see the VCO model:

Note that the VCO itself is simply represented by the 1/s integrator (phase is the integral of frequency).

Functionally, my Arduino/Shera hardware differs from Shera's original as follows:
1. My phase detector, over 30 seconds, should have a maximum possible value of  24660 (i.e. 30*822).  Shera's phase detector has a maximum possible value of 2304 (i.e 30*76.8) for the same 30 second period of phase delta integration.
2. My DAC is 16 bits unsigned, with a 5 volt output range.  Shera's design uses an 18 bit signed DAC, with a 6 volt output range (-3 to +3 volts).
3. My Arduino hardware creates a -5 to +5 voltage from the DAC output (i.e. a gain of 2 with a DC offset -- this circuit is from my original FLL design, and I did not want to remove it).  Shera's design does not amplify his DAC output.
4. My VCO's Kv is -0.32 Hz/V.  Shera's design assumes a VCO Kv of 0.075 Hz/V at 10 MHz.
5. My design divides the VCO's output (to the phase detector) by 8 to create 1.25 MHz.  Shera's designed divides the VCO by 32 (if the VCO frequency is 10 MHz) to create 312.5 KHz.
In the model subsection below, the gray block represents the voltage-gain of +2 (plus DC offset) that exists in my hardware (implemented with two op-amps) to amplify and DC-shift the 16-bit DAC's output.

The yellow block contains the different compensations (implemented in hardware) that I use to compensate for each of the hardware differences that I have listed above.

Following the signal path in the yellow block, these compensations are:
1. Compensate for my hardware's external op-amp gain of two by multiplying by 0.5.
2. (Next, a "catch-all" attenuation (for experimentation), labeled ExtAtten).
3. Compensate for my DAC's 5V range (compared to Shera's 6 volt range) by multiplying by 6/5.
4. Compensate for my N value of 8 (i.e. VCO divider) compared to Shera's N value of 32 (for the Shera design if using a 10 MHz VCO in lieu of his original 5 MHz VCO).
5. Compensation for my different value of VCO Kv.
6. (Note that the compensation for the difference in phase-detector max values is done in software.  Refer to the model's top-level and you will see that the filter output (either IIR or Type 1 filter) is multiplied by the ratio of 2304/24660 prior to sending this value to the DAC input.)
As I mentioned above, these gain-compensations in the yellow block are all external to the software.  That is, they are combined and implemented between the DAC output and the VCO's EFC input as a resistive voltage-divider.

If we take into account all of the compensations listed above, this attenuator should attenuate the EFC signal to the VCO by a factor of 0.035 (i.e. 1/28 or 1/29).  Note that more attenuation (which decreases the overall loop gain), either in software or in hardware, will increase ringing on the VCO response and decrease its rise time to a step at the phase detector input.

By the way, any of these hardware compensations could be implemented, instead, in software.  I chose not to, because implementing these attenuations in software (prior to the DAC) would reduce the DAC's input with respect to its Full Scale value (after all, they are attenuations and not gains), and I wanted to retain the same DAC input values that the original Shera design uses.

Finally, let's look at the Simulations' input stimulus section...

There are two different stimuli available for testing.  One stimulus is a step-function (whose amplitude has been set to 400 nanoseconds):

And the second stimulus is an Excel file containing actual 1 PPS jitter data.  In the above example, this data is from a Trimble Resolution T GPS receiver, but there is also an Excel file with data from the Quectel L76 GPS receiver.  (Refer to this blog post for more info:   http://k6jca.blogspot.com/2018/12/simulating-brooks-shera-w5ojm-gpsdo.html)

The stimulus data (for whichever stimulus is selected) is then normalized to the appropriate phase-detector's period.  E.g. the stimulus for the Arduino phase detector is normalized to 800 ns, and thus, for the 400 nanosecond step function, its value should be 0.5 after normalization.

Similarly, the stimulus is normalized for the original Shera phase detector's period.  In this case, this period is 3200 ns, so a 400 nanosecond step normalizes to 0.125.

The normalized step function is then added to 0.5.  This 0.5 represents a phase that equals the setpoint, when normalized, so it should represent an ideal phase-delta error of 0 (i.e. when the setpoint is subtracted from the phase-delta).  In other words, at startup, the model starts with no phase perturbations and in lock.

Let's now look at simulation results...

Simulation Results:

These simulations all use the 400 nanosecond step-function stimulus.

Type 1 Filter Responses:

First, let's look at how the Type 1 filter responds to this step function.

Here's a plot of the responses for both the Arduino and the original Shera VCOs:

Note that these two responses are exactly the same.

And here is the DAC inputs for both models:

Again, the DAC inputs are exactly the same.

Note that, if I were exactly matching Arduino to original Shera, then the DAC input values of each design should have the same relation with respect to the DAC full-scale value.  For example, given the same stimulus driving both models, if the peak value in Shera's DAC were, say, 1/5 of DAC full scale value, then the peak value at my DAC's input should also be 1/5 of that DAC's full scale value.

Given that the Shera DAC is 18 bits and my DAC is 16 bits, then, to maintain the same "ratio-to-full-scale," my input values should be reduced by a factor of 4.

But dividing the 16-bit DAC's input by four essentially reduces its resolution.  Given that, for a positive 400 ns phase step, the 16-bit DAC's peak input is only about 9000 compared to a maximum positive input for this DAC of 32767, I decided I had plenty of headroom and that it would be a fairly safe gamble to make the input values to the two DACs equal, rather than scaling my inputs down by a factor of 4.

IIR Filter Responses:

Here are the VCO responses for the first Shera IIR filter (this is the filter selected when his filter select switch is set to '2'):

Again, the VCO responses for the two models are essentially identical.

Similarly, the DAC input values for the two models are also essentially identical.

Note that the maximum value is just less than 5000, implying that, for the same step-input, DAC input range is determined by the Type 1 filter (which had a larger transition at the DAC input), not the IIR filter.

The plots above are for Shera's "lowest" filter.  If we step sequentially through the higher filters, we see that, for each step "up", F1 increases by a factor of 2 and Kcpu decreases by a factor of 2.

The net effect on VCO response and peak DAC input (given the same step-function stimulus), is that, for each step up to the next filter:
• VCO response rise-time and settling-time doubles.
• Peak DAC input is halved.
(You can see this effect in my Brooks Shera simulation blog post: http://k6jca.blogspot.com/2018/12/simulating-brooks-shera-w5ojm-gpsdo.html)

Simulation Experiments:

Now let's do some experiments with our simulation model...

Experiment 1:  Increase Kcpu by a factor of 2 (i.e. double the overall loop gain):

Note that the rise time is a bit faster, but, compared to the original Kcpu response (blue line) there is less overshoot and it settles more quickly.  (And note that the peak DAC input (for IIR filters) is, essentially, doubled).

Experiment 2:  Increase Kcpu by a factor of 4:

The VCO response rises even more quickly, which might not be desirable.  And again, overshoot is lessened.  And note that full settling is at approximately 1500 seconds.

The peak DAC input is essentially quadrupled, from the original 4.9K to 18K, which is starting to get uncomfortably close (for me) to the maximum positive peak value of 32767.

Experiment 3:  Let's take the last experiment and "slow" the filter down by a factor of 2 (i.e. increase F1 by 2, from 2^8 to 2^9, and decrease Kcpu by 2, from 2^7 to 2^6, as we would do with Shera's original filter constants):

Not too bad!  The rise times are similar and the overshoot is less.  And the peak DAC input is about 9K, or twice the peak DAC input of Shera's IIR Filter "2".

In my opinion, increasing the loop gain by 2, or by 4 (with a doubling of F1 for the latter) produces step-function VCO responses that both seem, to me, to be better than the "stock" Shera filter's step-response, as shown below.

Sidebar:  A Mystery!
One point about my simulations puzzles me, and I haven't found a satisfactory explanation.
If you read Brooks Shera's article, he mentions that IIR filter 2 has a tau (time constant) of 1500 seconds.
As Shera notes, the time constant is "approximately the time required for the PLL to fully recover from a transient".
If you look at the plots for the lowest IIR filter (filter 2), it is close to complete recovery at abut 4000 seconds (although I might say it is fully recovered at 6000 seconds, which is four times the time constant specified by Shera).
Interestingly, increasing Kcpu by 2 seems to produce a settling time closer to 1500 seconds (i.e. 2000 seconds).
Why is there a difference?  I don't know.  I would almost believe that I've missed a gain of 2 or 4 somewhere in his hardware or software, but, if so, I cannot find it.
Oh well.  A mystery!

Below is a table summarizing the results of the above experiments, plus others, that I performed using the Simulink models:

IntAtten and ExtAtten are in the Simulink model.  The former is meant to be a software-only attenuation, prior to the DAC, and the latter is meant to be a hardware-only attenuation, after the DAC.

Note that the first four entries show how I progressed from Shera's original values for F1, F2, and Kcpu (e.g. 2^11, 64, and 2^10, respectively, for Filter 2, the minimum IIR filter) to the values I've used in my Arduino model (F1 = 2^8, F2 = 8, Kcpu = 2^5).  Note that they produce the same VCO response (and DAC input) for the Arduino model that Shera's original values produce for the "original Shera" Simulink model.

Regarding the last table entry:  these are values I would use if I changed the phase-delta measurement period from 800 ns to 1600 ns (i.e. the period of 10 MHz / 16).  Note that the external analog attenuation network would not change.  It would still be 1/28 (approx.).  Because, although I've added a new attenuation of 0.5 (i.e. ExtAtten), the attenuation compensation due to mismatched 'N' VCO-division factors (between Arduino and Shera designs) decreases by a factor of 2 with the new Arduino 'N' of 16.  Thus, the increase in ExtAtten is cancelled by the decrease in N-compensation attenuation.

Hardware:

The hardware is based upon my FLL hardware, with some small changes to incorporate a Phase Detector and an EFC attenuator.

The original FLL hardware is described in more detail in my original blog post:   http://k6jca.blogspot.com/2016/02/an-arduino-based-gps-disciplined.html.  I would suggest first reading it if the functionality of the circuitry below does not seem clear.

First, the main board with the Arduino NANO subassembly.

There are only a few changes to this board from the description found in this post:   http://k6jca.blogspot.com/2016/02/an-arduino-based-gps-disciplined.html

These changes are:
• Change the serial port (J5) to be bidirectional,  rather than TX-only to PC.
• Add the 10-pin header plug, P2, as the interface with the new PLL phase detector circuitry.
• Add 1-pin header J8 as an input pin for the 1PPS_3V3 signal (from the Quectel receiver via newly added 1-pin plug P3).  J8 allows me to drive the Arduino with other 1 PPS sources (e.g. a Trimble Resolution T GPS receiver).
(Note:  the earlier blog post, http://k6jca.blogspot.com/2016/02/an-arduino-based-gps-disciplined.html, also includes the modifications necessary to incorporate the Quectel GPS receiver into the design.)

The following schematic is the additional circuitry I added to incorporate a PLL phase detector into my design.

Notes on the PLL Phase Detector circuitry:
• This circuitry attaches to the Arduino board via an 10-pin header (J4).
• The Quectel L76's 1 PPS output is spec'd with a VOL(max) = 0.42V and VOH(min) = 2.4V.  This 1 PPS needs to go to an 'LVC74 flip flop operating at 5V.  But the 'LVC74 VIH(min) is 3.5V -- significantly greater than Quectel's VOH(min).  So U3, with its input specs of VIH(min) = 2V and VIL(max) = 0.8V, performs the signal-level translation from Quectel's 3.3V "1 PPS" signal to one with appropriate signal levels for the 'LVC74 operating at 5V.
• D flip-flops U1B, U6B, and U6A divide the 5 MHz clock (VCO clock divided by 2 on the Arduino board) by a factor of 8 to 1.25 MHz.  (I originally started out with only a divide-by-two to create 2.5 MHz, but I discovered that its 400 ns period was too short to handle, for example, thermal drift when the Shera IIR filter is set to long time constants.  So I added another 'LVC74 package to give me an additional divide-by-two, with a spare D flip-flop in case 800 ns should prove insufficient.  So -- if starting from scratch, one could probably take all of this into account and use a different IC (or IC's) in lieu of my "growth by accretion" choices).
• Various jumpers are on the board solely for debugging purposes.
The D flip-flop U1A, in conjunction with the NAND gate U2 and the tri-state buffer U4 (with its output RC network) are the heart of the phase detector.  Here's a closer look at its operation (shown in block diagram form)...

More Phase Detector notes:
• The Tri-state buffer's output is enabled on the rising edge of the 1 PPS pulse, which starts the 1 nF capacitor charging through the 4K ohm resistor.
• The rising edge of the 1.25 MHz clock latches the logic state of the 1 PPS signal into the D flip-flop which, if the 1 PPS signal is high, sets the Q-not output low and puts the Tri-state buffer back into tri-state, stopping the charging of the 1 nF cap.
• Therefore, the 1 nF cap charges for the time period between the rising edge of the GPS 1 PPS pulse and the rising edge of the VCO 1.25 MHz clock.  This period can range from 0 to 800 ns, depending upon the phase difference between these two edges.
• This capacitor connects to one of the Arduino's ADC inputs.  The reference voltage for this ADC is 1.1V, and the values of the RC charging circuit (4K ohms and 1 nF) have been selected so that, if the phase difference between the clocks were 800 ns (the maximum amount), the capacitor would charge to about 4/5's of the ADC's full-scale value.
• The rising edge of the 1 PPS pulse also generates an interrupt at the Arduino (although, because there is an NPN inverter between the 1 PPS signal and the Arduino pin, this is a falling-edge interrupt), which initiates a read of the ADC.  Note that it takes about 114 microseconds to complete the ADC read instruction, so there should be no danger of reading the ADC while the cap is charging (800 ns << 114 us).
• The 10 Meg ohm resistor (plus inherent leakage currents at U3's output and the Arduino ADC's input) discharge the capacitor during the remainder of the 1 second period, before the arrival of the next 1 PPS pulse.
• Note that the RC charging characteristic isn't a straight-line, it is an exponential curve.  However, as I've shown in the graphs above, we are only using a small portion of this exponential curve (0 to about 0.9 volts in the graph), and for our purposes it is essentially linear.
• But, although "essentially" linear, it isn't exactly linear.  I've shown the deviation from linear of this part of the exponential curve in the inset graph (plotting over 800 ns).  The red line is the tangent to the exponential curve at T = 400 ns, which is the ideal spot that the PLL wants its Phase Delta to sit at (i.e. it's the Setpoint).  Ideally, if the response were linear, we would move along this tangent line as we deviate away from the setpoint.  But actually, as we move away from the setpoint, the actual voltage (along the exponential curve) deviates more and more from the ideal "linear" line.  Fortunately, this deviation isn't much.  In the ideal case, if the 1-second phase delta were to be at 300 ns or 500 ns (i.e. +/- 100 ns, or  +/- 25% from the setpoint) then there is an error of about 0.4 %.  Given the large amount of 1 PPS jitter, this error is, in my opinion, insignificant.  (Note:  this 0.4% is the ideal error -- it will change due to output stage non-linearities, etc.).
• The inset-graph above shows that the capacitor should charge to 0.9V over the full period of 800 ns.  The ADC count for 0.9 V (given the ADC's 1.1V reference) should be 837 (=1023*0.9/1.1).  The measured maximum ADC is 822.  The difference between these two is most likely due to RC component tolerances and voltage drop across the output of the tri-state buffer.
By the way, the idea of a phase-detector using the charging of a capacitor and reading its value via a processor's ADC is not a new one.  Here are a couple of links to similar designs (that inspired me to try the same thing!):  Lars GPSDO, and Kasper Pedersen GPSDO.

• For simplicity's sake I have not added any software temperature compensation.
• I used a tri-state gate to control the charging of C1 (similar to Pedersen's design) in lieu of a gate driving a diode (Lars design), because I believe the former, consisting of only one IC's output stage between +5VDC and the RC charging network, will have charging-voltage that is less sensitive to voltage variation with temperature, compared to the latter design, which consists of an IC's output stage in series with the temperature-sensitive Von drop of a diode.  Note, though, that I haven't bothered to verify this belief.

The output reference-distribution schematic is unchanged from the original FLL design:

There is a change to the power-supply / VCO Module interface schematic.  I have added a resistive attenuator to attenuate the VCO's EFC voltage and a method for adjusting the EFC's offset.

The four resistors that perform this function (pots R1 and R4, resistors R2 and R3) are very similar to Brooks Shera's design described in his article.

Note:  I implemented R1 with a 10 turn pot because I wasn't sure how sensitive this adjustment might be in terms of  "ohms per turn" resolution.

R1 (in conjunction with R2 and the 5V from U10) is used to set the VCO's frequency to be very close to 10 MHz when the DAC is set to its midpoint (i.e. 0x8000).  Note that this design only offsets the EFC voltage in a positive direction, away from 0V.

R4 and R3 (in conjunction with R1 and R2) form external resistive divider that I described above, in the Simulation model section.  Note that if the values of R1 (when adjusted) plus R2 are large with respect to R3 (100 ohms) then R1 and R2 will have little impact on the value of R3 (the latter being only 100 ohms), and the amount of attenuation can be approximated to be equal to:

Av = (R3/(R4+R3)) * (R1/(R2+R1))

I used a 5K pot for R4, which gives me some flexibility in adjusting the amount of attenuation.

Procedures for adjusting these pots are described later in this blog post.

As an optional GPS receiver in lieu of the Quectel L76 GPS receiver (per testing described later in this post), here's a schematic incorporating a Trimble Resolution T GPS receiver into my GPSDO design:

The design should be fairly self-explanatory, but here are some notes regarding the Trimble Resolution T GPS Receiver:
• The Trimble receiver has a bidirectional UART interface.  I've connected this to a stereo jack (ring = TX out, tip = RX in) so I can connect it to my laptop using an FTDI serial-to-usb cable.  (I use a "5V" serial-to-USB cable, thus the attenuation via R1 and R2 to drop the incoming 5 volt RXD signal down to 3.3V.  (Note that the FTDI cable's Vin has a maximum threshold of 1.5V, so it can work with either a 3.3V or 5V TX signal arriving from the outside world).
• The Trimble receiver's 1 PPS signal is 3.3V.  This goes to the NPN inverter on the main Arduino board (via that board's J8 1-pin header) to convert it to 5V for the Arduino.

As I mentioned, an LDO regulator provides the 3.3V for the Trimble receiver.  Rather than use this LDO, I could have converted directly from +12V to 3.3V, rather than take the two-steps of converting first from 12V to 5V, and then 5V to 3.3V.

Why use two steps?

The Arduino NANO's 5V regulator (which is powering most of the circuitry in this system) is, in my opinion, very poorly heat sunk, and I've long considered running a separate linear +5V regulator's output to the +5V pin of the Arduino NANO to take the load off of this on-board regulator.

So the Arduino's 5V regulator doesn't need to power the Trimble receiver, plus, should I want, I could drive the Arduino's 5V pin with this regulator (it can source more current and it has a better heat sink), thus, in essence, replacing the Arduino's on-board 5V regulator with this one.

Is there an issue driving the Arduino's 5V pin with an external 5V linear regulator?

Keep in mind that the NANO uses a UA78M05 linear regulator whose output pin is driven by an NPN emitter.  Ditto for the LM7805 I've added.  These emitters can be connected together and, whichever one has the higher output voltage will reduce the base-emitter junction's forward voltage of the other regulator's output, thus reducing its contribution of current.

And note that the UA78M05 regulator has a 0.6 ohm series resistance between its driver's emitter and its output pin, while the LM7805 has 0.25 ohms of resistance.  This means that, as more current is drawn from the UA78M05, its output will drop more quickly and the LM7805 should take over sourcing current.  (Unless, of course, the UA78M05's output is appreciably higher than the LM7805's output).

However -- I've discovered in practice that the added LM7805 regulator has a measured output voltage of 5.00 VDC, while the NANO's UA7805 has an output of 5.2 VDC.  So the NANO's regulator, because it has a higher output voltage, supplies the majority of the current to the system 5V bus, and as a result, the NANO's regulator becomes very HOT.

How do I know it is hot?

The regulator lies on the other side of the NANO board from the silkscreened NANO emblem on the top side of the PCB.  If you touch this area of the board with your finger, and if it feels HOT to the touch, you can safely assume that the regulator is dissipating significant power.

Touching this area of my system's NANO board with my fingertip -- it was HOT!

Clearly simply connecting the two regulators in parallel was not sufficient, and instead I should reduce the current load on the NANO's regulator.  The simplest solution would be to power the NANO board, and only the NANO board, from its on-board 5V regulator, and power all other system 5-volt circuitry from the newly-added LM7805 regulator (see above, regarding the Trimble receiver).  In this way, I would not overtax the NANO's regulator.

Here's the circuit for the external 5V regulator (it's also in the schematic, above, showing the Trimble GPS receiver).  This regulator powers all 5V circuitry that is not on the NANO board, itself.

After adding this regulator, the NANO board now is much cooler.

Arduino Software:

And below is a listing of this same code (revision 190207).  For easier reading, I would recommend copying and pasting into an app where you do not get the line wrap-around.

//----------------------------------------
//  Arduino GPSDO PLL Controller based
//  upon Brooks Shera's GPSDO Algorithm
//
//  Author:  Jeff Anderson, K6JCA
//
//  To echo Brooks Shera (in his PIC
//  code):
//    "This program may be used only
//     for non-commercial
//     purposes, and carries no warranty
//     of any kind, including any implied
//     warranty of fitness for
//     any particular purpose."
//
//----------------------------------------

#include <LiquidCrystal_I2C.h>
#include <SoftwareSerial.h>
#include <PWM.h>
#include <Wire.h>
#include <stdlib.h>
#include <EEPROM.h>

// Compiler Directives
//#define PRINT_CSV             // In FLL mode, print FLL parameters
//#define PRINT_DELTA           // In FLL mode, print Delta, second by second
//#define HOLD_DAC              // In FLL mode, hold the DAC at its initialized valu
//#define PRINT_PHASEDELTA      // Print second-by-second phase-delta
#define PRINT_PLL             // print PLL parameters every 30 seconds - VERY USEFUL!!!

#define SW_REV 190207  // SW Revision!  (yr,mo,day)

#define MAX5217  0x1C  // DAC I2C address

// NANO Pin Defines
#define INT0_PIN    2  // NANO pin, INTERRUPT 0
#define GPS_RST     7  // NANO pin, reset GNSS-2 receiver
#define TX_IN      10  // NANO pin, software UART RX
#define ICP1        8  // NANO pin, T1's ICP1 input
#define T1_CLK      5  // NANO pin, T1's clock input
#define SERIAL_IN   6  // NANO pin, software uart, RX from pc.
#define SERIAL_OUT 11  // NANO pin, software uart, TX to pc
#define DIG9        9  // NANO pin, Digital In (or Out, for testing)

// Starting Values for Frequency-Locked Loop algorithm
#define MAX_DAC_STEP     2816   // 16-second DAC step
#define GAIN             15.0
#define START_DAC        0x8000
#define OSC_PPB_DRIFT_PER_HOUR      0.0     // DON'T CORRECT FOR AGING
#define PPB_CORRECTION_PER_DAC_STEP 0.0044  // from osc. measurements
#define IDEAL_DELTA 19264   // for a 1 second period.

// gpsdo Frequency-Lock Mode states:
#define WAIT   0
#define START  1
#define SETTLE 2
#define GO     3
#define SETTLE_2 4
#define PLL_SETTLE_1 5
#define PLL_SETTLE_2 6

// run states
#define HOLD 0
#define RUN 1

// constants for Phase-Locked Loop operation
#define D_INIT        30    // for phase-delta filter
#define F1_INIT      256    // F1 for Shera's "fastest" IIR Filter
#define F2_INIT        8    // F2 for Shera's "fastest" IIR Filter
#define KCPU_INIT     32    // Kcpu for Shera's "fastest" IIR Filter
#define KCPU_INIT     64    // Kcpu for Shera's "fastest" IIR Filter
#define KCPU_T1_INIT   8    // kcpu for Shera's Type 1 Filter
#define N_INIT         8    //  = 10MHz / 1.25 MHz
#define SHERA_MAX_CNT 2304  // max phase-delta count in shera's design
#define KV_INIT     -0.32   // 10 MHz Kv for my HP oscillator module.
#define RECIP_EXT_ATTEN 29   // Reciprocal of extern_atten.
//                              Use 29 for K6JCA HW gains (and N=8).
//                              Note:  larger = more ringing)

// constants for auto filter selection
#define MIN_IIR_INIT       2    // minimum IIR Filter
#define MAX_IIR_INIT       5    // maximum IIR filter
#define SETTLING_INIT     2000  // Settling time, in seconds (3000 chosen for minimum IIR filter)
#define DELTA_LIMIT_INIT 100*30 // To switch filters up, PD error must be less than this.
#define DROPBACK_LIMIT    3000  // drop to lower filter if |phase delta error| exceeds this value.

#define FLL  0   // Frequency-lock mode
#define PLL  1   // Phase-lock mode

#define AUTO 1   // Auto IIR filter select
#define MAN  0   // Manual IIR filter select

#define TYPE1 0  // Shera Type 1 filter
#define IIR   1  // Shera IIR filter

#define ADDR_DATA_VALID    0   // 1 byte.  '1' = eeprom contents valid.  Else = invalid
#define ADDR_LOCK_MODE     1   // 1 byte.  Either FLL or PLL
#define ADDR_F1_ROOT       4   // 2 bytes  F1 for IIR Filter 2
#define ADDR_F2            6   // 2 bytes  F2
#define ADDR_KCPU_ROOT     8   // 2 bytes  Kcpu for IIR Filter 2
#define ADDR_KCPU_T1       10  // 2 bytes  Kcpu for Type 1 Filter
#define ADDR_KV            12  // 2 bytes  VCO Kv (Hz/V, signed)
#define ADDR_REC_EXT_ATTEN 14  // 2 bytes  The RECIPROCAL of attenuation between DAC & VCO
#define ADDR_DAC_START     16  // 2 bytes  Initalized DAC to this value
#define ADDR_SETTLING_ROOT 18  // 2 bytes  Filter settling time for IIR Filter 2
#define ADDR_MIN_FILTER    20  // 2 byte   For PLL mode, Auto Filter select: Min filter
#define ADDR_MAX_FILTER    22  // 2 byte   For PLL mode, Auto Filter select: Max filter
#define ADDR_FILT_SEL_MODE 24  // 1 byte   Auto Filter Select mode (in PLL mode), Manual or Auto
#define ADDR_MAXMIN_LIMIT  25  // 2 bytes  *** not used ***
#define ADDR_FILTER        27  // 2 bytes  PLL filter selected at Initialization

// set the LCD address to 0x27
LiquidCrystal_I2C lcd(0x27, 16, 2);

// Define software UART, Rx pin 6, Tx pin 11
SoftwareSerial mySerial(SERIAL_IN, SERIAL_OUT);

boolean       first_lcd_update; // identifies first LCD update after initialization
boolean       one_pps_flag;     // signals leading-edge of one_pps_pulse

// frequency-locked loop variables (from original FLL design)
unsigned int  prev_t1_count;
unsigned int  t1_count;
unsigned int  t1_delta;

byte          gpsdo_FLL_state;

long          dac_delta;

int           dac_direction;   // +1 = positive change, -1 = neg. change, 0 = no change.

char          tick_count;
boolean       display_countdown;
unsigned int  prev_countdown;

unsigned long zero_count;
unsigned long prev_count;
boolean       one_flag;
boolean       minus_one_flag;

float         float_zero_count;

int           last_direction;
long          run_error;

float         drift_corr_per_hour;  // drift correction, per hour (not used)

// FLL and PLL variables
unsigned int  dac;            // value sent to DAC
unsigned long long_dac;       // DAC value, in LONG format

// phase-locked loop and Shera variables
unsigned int phase_delta;    // 1-second phase delta value from ADC
unsigned int prev_phase_delta; // previous phase delta
unsigned int dac_start;      // Initialize the DAC output to this value
unsigned int d;              // D, Number of seconds of phase-detector aggregation
float        kv;             // VCO Kv (Hz/V)
float        setpoint;       // setpoint for 30 second sample

float        f1;             // Shera's F1
float        f2;             // Shera's F2
float        kcpu;           // Shera's Kcpu
float        f1_root;        // F1 for the lowest IIR filter (i.e. Filter 2)
float        kcpu_root;      // Kcpu for the lowest IIR filter
unsigned int kcpu_t1;        // Kcpu for Shera's Type 1 filter
unsigned int second_count;   // count seconds to D (30 seconds)
boolean      aggregate_rdy;  // set true when reach D sec.

unsigned int running_sum;    // running sum of inputs
unsigned int d_sample;       // sum of aggregated samples after D counts (e.g. 30 seconds)
float        in0;            // input sample: in(0)
float        in_m1;          // in(-1)
float        out0;           // output sample: out(0)
float        out_m1;         // out(-1)

float        extern_atten;   // external analog attenuation (following "compensated" DAC)
float        pd_error;       // phase-detector error (+/- delta from setpoint)

float        dac_less_offset; // DAC value without the DC offset when using usigned DAC chip

unsigned int max_pd;          // maximum 1-sec phase delta (in a 30 sec aggregated phase delta)
unsigned int min_pd;          // minimum 1-sec phase delta (in a 30 sec aggregated phase delta)
unsigned int mid_pd;          // Average 1-sec phase delta (in a 30 sec aggregated phase delta) (not used)

unsigned int filter;          // IIR filter ID (per Shera, 2 to 7)
unsigned int init_filter;      // Initialization filter, 1-7 (1 = Type 1, 2-7 = IIR)

int          filter_type;     // IIR or TYPE 1 filter.  See Shera article.

// Automatic Filter Select Mode stuff...
byte         filter_select_mode;// AUTO or Manual (via serial port) filter select mode
unsigned int wraparound_count;  // counts how many times PLL has dropped its filtering to a lower filter due to phase-detector wraparound
long         settling_count;    // Count seconds that of a filter's settling after filter has been changed.
long         settling_root;     // settling time for the lowest IIR filter
long         settling_limit;    // settling root scaled for selected filter
unsigned int max_filter;        // maximum filter to shift up to
unsigned int min_filter;        // minimum filte to shift down to
float        maxmin_limit;      // limits phase-detector error must be within when up-selecting filter
boolean      clear_err_counts;  // true to clear wraparound_count;
float        upper_wraparound_lim;    // phase-detector error limit for wraparound test
float        lower_wraparound_lim;    // phase-detector error limit for wraparound test
bool         wraparound_error;  // true if second-by-second phase detector value is flipping between high and low limits
unsigned int dropback_count;    // counts how many times |pd_error| is > DROPBACK_LIMIT

// misc variables
unsigned long accum_seconds; // count seconds as a measure of elapsed time.
byte          run_state;     // RUN or HOLD (DAC value)
byte          lock_mode;     // PLL or FLL
int           serialByte_In; // rcvd character from mySerial software UART
long          long_value;    // temporary "long" value
String        inString;      // temporary string (for building strings from mySerial sw UART rx data)
boolean       pll_to_serial; // flag to print seconds, PD error, and DAC to serial port every 30 seconds
boolean       pd_to_serial;  // flag to print PD every second.

// temporary variables
int          int_temp;
unsigned int uint_temp;
float        temp_float;

void setup()
{

Serial.begin(57600);  // init serial port
lcd.init();           // initialize the lcd
InitTimersSafe();     //initialize all timers except for 0, to save time keeping functions

pinMode(INT0_PIN, INPUT);
pinMode(TX_IN, INPUT);
pinMode(ICP1, INPUT);
pinMode(T1_CLK, INPUT);
pinMode(GPS_RST, OUTPUT);

pinMode(DIG9, OUTPUT);      // I/O pin used for testing

analogReference(INTERNAL);  // set analog reference to 1.1V.

// Reset for 15 msec
digitalWrite(GPS_RST, HIGH);
delay(15);
digitalWrite(GPS_RST, LOW);

// Set Digital Pin 9 to 0:  (*** Used as output for testing timing
digitalWrite(DIG9, LOW);

// pull up the sw uart's RX pin
// (so that it doesn't float if unconnected to external TX signal)
digitalWrite(SERIAL_IN, HIGH);

//  Print a message to the LCD...
//
lcd.init();
lcd.clear();
lcd.backlight();
lcd.print("  Hi, I'm your");
lcd.setCursor(0, 1);
lcd.print("  K6JCA GPSDO!");

mySerial.begin(9600);
mySerial.println("");
mySerial.println("Hello Jeff");
mySerial.println("");

delay(2000);
lcd.clear();
lcd.print("SW Rev: ");
lcd.print(SW_REV);
delay(1500);
lcd.clear();

lcd.print("  Waiting for   ");
lcd.setCursor(0, 1);
lcd.print(" 1 PPS Pulse... ");

// if eeprom has valid contents, load
// the pll and fll initialization parameters
// from eeprom...
// dump eeprom contents
mySerial.println(F("Initializing from EEPROM... "));
lock_mode = PLL;
}
else {
lock_mode = FLL;
}
filter = init_filter;
// if in auto mode, start at min filter
filter_select_mode = AUTO;
filter_type = IIR;
}
else {
// if in manual filter select mode, start with init_filter from eeprom
filter_select_mode = MAN;
if (filter == 1) {
filter_type = TYPE1;
}
else {
filter_type = IIR;
}
}
f1_root = (float) uint_temp;
f2 = (float) uint_temp;
kcpu_root = (float) uint_temp;
kcpu_t1 = (float) uint_temp;
kv = ((float) int_temp)/1000;
extern_atten = 1/((float) uint_temp);

// *** add maxmin_limit  (below line is temporary)
maxmin_limit = (float) DELTA_LIMIT_INIT;

filter = min_filter;
}
else {
// nothing in eeprom.  Initialize with constants as follows...
mySerial.println(F(" EEPROM not loaded.  Initialize from program "));
kv        = KV_INIT;
dac_start = START_DAC;
max_filter = MAX_IIR_INIT;
min_filter = MIN_IIR_INIT;
filter = min_filter;
init_filter = filter;
filter_select_mode = AUTO;
f1_root = (float)F1_INIT;
f2 = (float) F2_INIT;
kcpu_root = (float) KCPU_INIT;
kcpu_t1 = (float) KCPU_T1_INIT;
extern_atten = 1/((float) RECIP_EXT_ATTEN);
//  lock_mode = FLL;  // init to frequency-lock mode
lock_mode = PLL;  // init to phase-lock mode
settling_root = SETTLING_INIT;
maxmin_limit = (float) DELTA_LIMIT_INIT;
filter_type = IIR;
}
f1 = f1_root;
kcpu = kcpu_root;
settling_limit = settling_root * ((int) (pow((float)2,(float) (filter - min_filter))+0.5));

dac       = dac_start;
d         = D_INIT;
setpoint  = d * adc_max / 2;    // *** store in eeprom?

upper_wraparound_lim = f_adc_max * 0.875; // i.e. within 12.5% of max 1-sec phase delta value
lower_wraparound_lim = f_adc_max * 0.125; // i.e. within 12.5% of min 1-sec phase delta value
wraparound_error = false;

print_current_values();

// How accurate are float calculations?
// calc_floating_point_reciprocals();

delay(2000);

one_pps_flag = false;

first_lcd_update = true;

tick_count = 0;
display_countdown = false;
prev_countdown = 0;

zero_count = 0;
accum_seconds = 0;
dac_direction       = 0;
gpsdo_FLL_state = WAIT;

dac_delta = 0;

one_flag = false;
minus_one_flag = false;

last_direction = 0;
run_error = 0;
prev_count = 0;

run_state = RUN;

inString = "";

long_value = 0;

drift_corr_per_hour = OSC_PPB_DRIFT_PER_HOUR / PPB_CORRECTION_PER_DAC_STEP;
long_dac = long(dac);
write_to_dac(dac);

// Shera PLL
second_count = 0;
aggregate_rdy = false;

running_sum = 0;
d_sample = 0;
in0 = 0;
in_m1 = 0;
out0 =  0;
out_m1 = 0;

phase_delta = adc_max/2;   // start phase delta near setpoint.
pd_error = 0;
max_pd = 0;
min_pd = 65535;

pll_to_serial = false;
pd_to_serial = false;

wraparound_count = 0;
settling_count = 0;
clear_err_counts = false;

dropback_count = 0;

// attach interrupt for falling edge of one_pps signal.
// Note that this falling edge is at the collector of an NPN
// invertor and it corresponds, in time, to the rising edge of
// the 1 PPS pulse from the GPS receiver.
attachInterrupt(digitalPinToInterrupt(INT0_PIN), one_pps, FALLING);

// Timer 1: external clock.  Neg. Edge capture.  No noise cancellation.
timer1_setup (0x00, -1, 0x00, 0x00, 0x00);

#ifdef PRINT_PLL
Serial.println(F("seconds,max_pd,min_pd,pd_error,filter,in(-1),out(0),out(-1),DAC-DC,DAC,settling"));
#endif

#ifdef PRINT_CSV
Serial.println(F("Total Seconds,DAC,DacDirection,ZeroCount,DacDelta"));
#endif

}

void loop()
{

// If 1 PPS has occured (and a phase-delta sample
//
if (one_pps_flag) {
one_pps_flag = false;
aggregate_samples();    //
if (run_state == RUN) {
// in RUN mode, so do either the
// FLL or the PLL calculations,
// depending upon which is selected...
gpsdo_state_machine();
}
else {
// GPSDO in HOLD state.  So bump the elapsed time counter...
accum_seconds++;    // inc seconds counter
}
// check if there's serial data from the serial port...
if (serialByte_In != -1) {
// Yes, there's data.  Get it and
// do something with it...
get_serial_in(serialByte_In);
}
update_lcd();
}
}

void update_lcd()
{
// first, get magnitude of run_error and
// magnitude of error
long mag_run_error;
if (run_error < 0) mag_run_error = -run_error;
else mag_run_error = run_error;

// first time into this call, clear the screen.
// Otherwise, just write w/o clearing.
if (first_lcd_update == true) {
lcd.clear();
first_lcd_update = false;
}
tick_count++;
if (tick_count == 6) {
tick_count = 0;
}

// Print first line of LCD
lcd.setCursor(0, 0);
if (lock_mode == FLL) lcd.print("FLL:");
else lcd.print("PLL:");
lcd.setCursor(4, 0);
if (run_state == RUN) lcd.print(" r");
else lcd.print(" h");
lcd.print(" DAC: ");
// Print DAC value and
// print leading zeroes (normally suppressed
if (dac < 4096) lcd.print("0");
if (dac < 256) lcd.print("0");
if (dac < 16) lcd.print("0");
lcd.print(dac, HEX);

if (lock_mode == FLL) {
// Print second line of LCD
lcd.setCursor(0, 1);
// Print current value Zero count
if (zero_count < 10000) lcd.print(" ");
if (zero_count < 1000) lcd.print(" ");
if (zero_count < 100) lcd.print(" ");
if (zero_count < 10) lcd.print(" ");
lcd.print(zero_count);

lcd.print(",");

// Now print the previous run count.
if (prev_count < 10000) lcd.print(" ");
if (prev_count < 1000) lcd.print(" ");
if (prev_count < 100) lcd.print(" ");
if (prev_count < 10) lcd.print(" ");
lcd.print(prev_count);

// Print magnitude of previous run-error.
if (mag_run_error < 1000) lcd.print(" ");
if (mag_run_error < 100) lcd.print(" ");
if (mag_run_error < 10) lcd.print(" ");
dac_direction_to_lcd();
lcd.print(mag_run_error);
}
else {
// PLL Mode...
// Because the PLL data is updated
// every 30 seconds, I'll do its LCD updating
// in the PLL state machine, further down...
}

}

void one_pps()
{
// This is the one pps interrupt routine

prev_phase_delta = phase_delta;   // store previous phase delta

digitalWrite(DIG9, HIGH);  // To test timing from 1pps leading edge...
digitalWrite(DIG9, HIGH);  // extend...
digitalWrite(DIG9, HIGH);  // extend...
digitalWrite(DIG9, LOW);
#endif
one_pps_flag = true;

#ifdef PRINT_PHASEDELTA
Serial.print(accum_seconds);
Serial.print(",");
Serial.print(phase_delta);
#endif

// if the following flag is true, print phase-delta
// to mySerial port (every second)
if (pd_to_serial == true) {
mySerial.print(accum_seconds);
mySerial.print(",");
mySerial.println(phase_delta);
}
}

void timer1_setup (byte mode, int prescale, byte outmode_A, byte outmode_B, byte capture_mode)
{
// Code for FLL (using Arduino's Timer 1)
// NOTE:  This code found at:
// http://sphinx.mythic-beasts.com/~markt/ATmega-timers.html

// enforce field widths for sanity
mode &= 15 ;
outmode_A &= 3 ;
outmode_B &= 3 ;
capture_mode &= 3 ;

byte clock_mode = 0 ; // 0 means no clocking - the counter is frozen.
switch (prescale)
{
case 1: clock_mode = 1 ; break ;
case 8: clock_mode = 2 ; break ;
case 64: clock_mode = 3 ; break ;
case 256: clock_mode = 4 ; break ;
case 1024: clock_mode = 5 ; break ;
default:
if (prescale < 0)
clock_mode = 7 ; // external clock
}
TCCR1A = (outmode_A << 6) | (outmode_B << 4) | (mode & 3) ;
TCCR1B = (capture_mode << 6) | ((mode & 0xC) << 1) | clock_mode ;
}

void gpsdo_state_machine()
{

if (lock_mode == FLL) {
// do for Frequency Lock Loop...
switch (gpsdo_FLL_state) {

case WAIT:
gpsdo_FLL_state = START;
break;

case START:
#ifdef PRINT_CSV
Serial.print(accum_seconds);
Serial.print(",");
Serial.print(dac);
Serial.println(",");
#endif
mySerial.print(accum_seconds);
mySerial.print(",");
mySerial.print(dac);
mySerial.print(",");

write_to_dac(dac);

gpsdo_FLL_state = SETTLE;
accum_seconds++;
break;

case SETTLE:
// DAC has been set.  Let's let it settle.
gpsdo_FLL_state = SETTLE_2;
accum_seconds++;
break;

case SETTLE_2:
// Let DAC settle for another second.
// Meanwhile, get current count and
// set to previous count.
t1_count = ICR1;
prev_t1_count = t1_count;
gpsdo_FLL_state = GO;
accum_seconds++;
break;

case GO:
accum_seconds++;  // another second has passed
prev_t1_count = t1_count; // prev. counter snapshot
t1_count = ICR1;          // new counter snapshot

// compensate for counter wraparound
if (t1_count < prev_t1_count) {
t1_delta = 65536 - (prev_t1_count - t1_count);
}
else {
t1_delta = t1_count - prev_t1_count;
}
#ifdef PRINT_DELTA
Serial.print("Seconds: ");
Serial.print(accum_seconds);
Serial.print(", zero_count: ");
Serial.print(zero_count);
Serial.print(", delta: ");
Serial.print(t1_count);
Serial.print(" - ");
Serial.print(prev_t1_count);
Serial.print(" = ");
Serial.print(t1_delta);
if (minus_one_flag) Serial.println(", minus-flag ");
else {
if (one_flag) Serial.println(", plus-flag ");
else Serial.println(", no-flag");
}
#endif
// If snapshot delta equals ideal delta, do
// nothing except increment the zero-run count.
// Also, very infrequently the program misses
// capturing a delta, so on the next 1-second
// pulse the value is TWICE what it should have been.
if ((t1_delta == IDEAL_DELTA) || (t1_delta == IDEAL_DELTA << 1)) {
zero_count++;

}
else {
// Check if the delta is greater than
// the ideal delta (or  greater than
// 2x the ideal delta, plus 1,
// in case a sample was missed).
if ((t1_delta == IDEAL_DELTA + 1) || (t1_delta == (IDEAL_DELTA << 1) + 1)) {
if (minus_one_flag) {
// previously saw a "negative" delta.
// +1 and -1 cancel so
// continue counting
dac_direction = 0;
zero_count = zero_count + 1;
minus_one_flag = false;
}
else {
if (!one_flag) {
// Previous delta was 0, set set flag
// that this delta is a 1 (i.e. postive).
// Don't stop counting zeroes yet.
one_flag = true;
zero_count = zero_count + 1;
dac_direction = 0;
}
else {
// Two positive deltas in a row.
// Run of zeroes is over!
// Reset flag and indicate direction to
// change dac.
// Positive deltas mean increase dac
// to lower the oscillator frequency.
dac_direction = +1;
one_flag = false;
}
}
}
else {
// same idea as above, but instead with a
// negative delta preceeding a pos. delta.
if ((t1_delta == IDEAL_DELTA - 1) || (t1_delta == (IDEAL_DELTA << 1) - 1)) {
if (one_flag) {
// -1 and +1 cancel, so continue with zeroes
dac_direction = 0;
zero_count = zero_count + 1;
one_flag = false;
}
else {
if (!minus_one_flag) {
minus_one_flag = true;
zero_count = zero_count + 1;
dac_direction = 0;
}
else {
// two minus deltas in a row.
// Run of zeroes is over!  Reset flag
// and identify direction to move DAC.
// Negative deltas mean decrease dac
// to raise the oscillator frequency.
dac_direction = -1;
minus_one_flag = false;
}
}
}
}
}
if (dac_direction != 0) {
// Run of zeroes has ended.  From run length
// determine dac correction factor (= run_error).
// Will also add to that value a "drift" offset
// (proportional to run length)
prev_count = zero_count;  // for lcd display purposes
float_zero_count = float(zero_count);
// Note that this next calculation now includes a gain factor, 1/extern_atten,
// to compensate for the additional attenuation added between DAC and VCO
// for PLL operation.  I.e. need to increase the dac value to compensate
run_error = long((1/extern_atten)* MAX_DAC_STEP * (GAIN / (float_zero_count + 1)));  // add one so don't divide by 0.
if (dac_direction == -1) {
run_error = -run_error;
last_direction = -1;  // For lcd display purposes
}
else {
last_direction = +1;
}
dac_delta = run_error;

#ifdef PRINT_CSV
Serial.print(dac_direction);
Serial.print(",");
Serial.print(zero_count);
Serial.print(",");
Serial.println(dac_delta);
#endif
mySerial.print(dac_direction);
mySerial.print(",");
mySerial.print(zero_count);
mySerial.print(",");
mySerial.println(dac_delta);

// Calculate new dac value.

#ifdef HOLD_DAC
dac_delta = 0;
#endif

long_dac = long(dac) + dac_delta;
if (long_dac > 65535) long_dac = 65535;
if (long_dac < 0) long_dac = 0;
dac = uint16_t(long_dac);

// dac updated.  Start counting 0's anew...
zero_count = 0;
dac_direction = 0;
gpsdo_FLL_state = START;
}
break;

default:
break;
}
// and keep pll stuff in sync, too, (in case we jump to PLL mode)
// by forcing the integrator's accumulator
// to follow the current dac value
out0 = (((float) dac) - 32768.0)*(f_adc_max*((float) d))/(kcpu*(-2304.0));
out_m1 = out0;
second_count = 0;
running_sum = 0;
aggregate_rdy = false;
max_pd = 0;
min_pd = 65535;
wraparound_error = false;
//  Serial.println(out0);

}
else {
// in Phase Lock Mode !!!
// first check if the second-by-second phase delta
// is flipping between the high and low limits.
// If so, and the filter is NOT the lowest filter,
// then want to drop back to the lowest filter.
if (((phase_delta >= ((unsigned int) upper_wraparound_lim)) && (prev_phase_delta <= ((unsigned int) lower_wraparound_lim))) || ((phase_delta <= ((unsigned int) lower_wraparound_lim)) && (prev_phase_delta >= ((unsigned int) upper_wraparound_lim)))) {
wraparound_error = true;
}
accum_seconds++;
settling_count++;
if (settling_count > settling_limit) settling_count = settling_limit;

if (clear_err_counts) {
wraparound_count = 0;
dropback_count = 0;
clear_err_counts = false;
}
// write to dac at 1pps
// and then calculate new dac
if (aggregate_rdy == true) {
aggregate_rdy = false;
pd_error = (float)d_sample - setpoint;  // subtract the setpoint from the aggregated 30-second sample.

if (filter_type == IIR) {
// Shera IIR Filter
in_m1 = in0;    // shift the last input "in(0)" value in in(-1).
out_m1 = out0;  // shift the last output "out(0)" value to out(-1)
in0 = pd_error; // New in(0) equals pd_error.

// Calculate the new filter output value, out(0), using Shera's formula:
out0 = ((out_m1 + in0 * (1 / f1 + 1 / f2) + in_m1 * (1 / f1 - 1 / f2)));

// multiply the filter output value by Kcpu.  Then scale, per the Simulink simulation
dac_less_offset = (out0 * kcpu * (((float)(-2304.0)) / (f_adc_max * ((float) d))));

// auto filter selection code

if (filter_select_mode == AUTO) {
// have we experienced a phase-detector wraparound during the 30-sec aggregation period?
// (if so, want to jump down to a lower filter so that we
// can try to get the phase-detector re-centered to the setpoint).
// Note that we do not care about wraparounds if the filter selected is the MINIMUM filter,
// because we cannot jump back to a lower filter.

if (wraparound_error == true) {
// a wraparound has occured!  First, reset the settling_count
// then, drop filter back to the minimum filter
settling_count = 0;
wraparound_count++;  // This count, displayed on the LCD, tells us how many wrap-arounds have occured
filter = min_filter; // Jump down to the minimum filter if there's a wraparound.

second_count = 0;
running_sum = 0;

// Changed the filter, so update filter values and out0 so that there isn't
// a discontinuity.
temp_float = (pow((float)2,(float) (filter - min_filter)));
settling_limit = settling_root * ((int) (temp_float+0.5));
f1 = f1_root * temp_float;
out0 = out0*kcpu/(kcpu_root/temp_float);  // adjust the 'dc' in the integrator to prevent jump
kcpu = kcpu_root / temp_float;
aggregate_rdy = false;
mySerial.println("wraparound!");

}
else {
// No wraparound has occurred, but is pd_error too large?
// If so, will want to backdown to a lower filter so that
// we recover more quickly back towards the phase delta setpoint.
//
if ((pd_error > DROPBACK_LIMIT) || (pd_error < -DROPBACK_LIMIT)) {
// big phase delta error, so backdown the IIR filter to a lower one
// to recover more quickly
dropback_count++;    // increment the backdown count (for LCD display)
settling_count = 0;  // Reset settling count timer.
filter = min_filter; // Simply jump back to the minimum filter if there's a backdown

second_count = 0;
running_sum = 0;

// Changed the filter (if we weren't at min filter,
// so update filter values and out0 so that there isn't
// a discontinuity in the DAC output.
temp_float = (pow((float)2,(float) (filter - min_filter)));
settling_limit = settling_root * ((int) (temp_float+0.5));
f1 = f1_root * temp_float;
out0 = out0*kcpu/(kcpu_root/temp_float);  // adjust the 'dc' in the integrator to prevent jump
kcpu = kcpu_root / temp_float;
aggregate_rdy = false;
mySerial.println("backdown!");
}
else {
// Don't backdown, so check if we should select a higher filter...
// test if we have settled since the previous filter switch
if (settling_count >= settling_limit) {
// (note that the settling time doubles
// for each filter step up.)
// yes, we've settled                                             // and add 0.5 so that int properly truncates
// Now -- is the phase delta error (which is the same as in0) close to zero?
// and are we not already at the highest (slowest) filter?

if ((in0 > -maxmin_limit) && (in0 < maxmin_limit)) {
// OK, we are near the setpoint, so we can upshift, but first
// check that we are not at the highest filter...
if (filter < max_filter) {
// OK, can select the next filter up
filter++;
temp_float = (pow((float)2,(float) (filter - min_filter)));
settling_limit = settling_root * ((int) (temp_float+0.5));
settling_count = 0;
second_count = 0;
running_sum = 0;
f1 = f1_root * temp_float;
out0 = out0*kcpu/(kcpu_root/temp_float);  // adjust the 'dc' in the integrator to prevent jump
kcpu = kcpu_root / temp_float;
aggregate_rdy = false;

}
}
else {
// Although we've reached the settling time,
// the Phase Delta isn't close enough to the setpoint
// (not within the maxmin_limit window).
// So do nothing and continue...
}
}
else {
// Haven't setled yet, so do nothing and continue...
}
}
}
}
}
else {
// Shera Type 1 Filter, not IIR filter
// calculate DAC input per Simulink model
dac_less_offset = pd_error * KCPU_T1_INIT * (((float)(-2304.0)) / (f_adc_max * ((float) d)));

// To minimize discontinuities when
// switching from Type 1 filter to an IIR filter,
// update the IIR filter's variables while
// in Type1 filter mode.
out0 = dac_less_offset * f_adc_max * ((float) d) / (kcpu * ((float)(-2304)));
out_m1 = out0;
in0 = pd_error;
in_m1 = in0;
settling_count = 0;

}
// Clip (i.e. limit) the DAC input (see Simulink model).
if (dac_less_offset > 0) dac_less_offset = dac_less_offset + 0.5; // round up if positive
if (dac_less_offset < 0) dac_less_offset = dac_less_offset - 0.5; // round down if negative
if (dac_less_offset > 32767) dac_less_offset = 32767;             // clip if too positive
else if (dac_less_offset < -32768) dac_less_offset = -32768;      // clip if too negative

dac = (uint16_t)(dac_less_offset + 0x8000);  // add dac midpoint because DAC is unsigned.
write_to_dac(dac); // write to the DAC!

#ifdef PRINT_PLL
Serial.print(accum_seconds);
Serial.print(",");
Serial.print(max_pd);
Serial.print(",");
Serial.print(min_pd);
Serial.print(",");
Serial.print(pd_error);
Serial.print(",");
Serial.print(filter);
Serial.print(",");
Serial.print(in_m1);
Serial.print(",");
Serial.print(out0);
Serial.print(",");
Serial.print(out_m1);
Serial.print(",");
Serial.print(dac_less_offset);
Serial.print(",");
Serial.print(dac);
Serial.print(",");
Serial.println(settling_count);

// update second line of LCD (PLL info)
lcd.setCursor(0, 1);
if (max_pd < 100) lcd.print(" ");
if (max_pd < 10) lcd.print(" ");
lcd.print(max_pd);
lcd.print(" ");

//      mid_pd = d_sample/d;
//      if (mid_pd < 100) lcd.print(" ");
//      if (mid_pd < 10) lcd.print(" ");
//      lcd.print(mid_pd);
//      lcd.print(" ");

if (min_pd < 100) lcd.print(" ");
if (min_pd < 10) lcd.print(" ");
lcd.print(min_pd);
lcd.print(" ");

// display wraparound count
lcd.print("w");
if (wraparound_count > 9) lcd.print("^"); // print ^ if count > 9
else lcd.print(wraparound_count);
lcd.print(" ");

// display dropback count
lcd.print("d");
if (dropback_count > 9) lcd.print("^"); // print ^ if count > 9
else lcd.print(dropback_count);
lcd.print(" ");

lcd.print("F");
lcd.print(filter);
#endif

if (pll_to_serial == true) {
// print to serial port every 30 seconds...
mySerial.print(accum_seconds);
mySerial.print(",");
mySerial.print(pd_error);
mySerial.print(",");
mySerial.print(filter);
mySerial.print(",");
mySerial.println(dac);
}

// reset our max_pd and min_pd values.
// We will compare the next 30 1-sec
// PD values against these to find
// max and min PD in a 30 second
// sample aggregation period.
max_pd = 0;
min_pd = 65535;

// clear out wraparound_error flag at the end of the D-sample calcs
wraparound_error = false;
}

// and hold these Frequency-locked loop
// terms constant:
zero_count = 0;
dac_direction = 0;
gpsdo_FLL_state = START;

}
}

void write_to_dac(unsigned int dac_value)
{
// writing 16-bit word to Maxim MAX5217

byte msbyte;
byte lsbyte;

msbyte = byte((dac_value & 0xFF00) >> 8);
lsbyte = byte(dac_value & 0x00FF);
Wire.beginTransmission(MAX5217);
Wire.write(0x01);  // control word
Wire.write(msbyte);  // MS Byte
Wire.write(lsbyte);  // LS Byte
Wire.endTransmission();
}

void dac_direction_to_lcd()
{
if (last_direction == +1) lcd.print("+");
else {
if (last_direction == -1) lcd.print("-");
else lcd.print(".");
}
}

long parseLong()
{
boolean endparse;
long long_input;
int inChar;

endparse = false;
inString = "";
long_input = 0;

while (endparse == false) {
while (mySerial.available() > 0) {
mySerial.write(inChar);   // echo back

if ((inChar != 0x0D) && (inChar != 0x0A)) {
// As long as the incoming byte
// is not a carriage return or a line-feed,
// convert the incoming byte to a char
// and add it to the string
inString += (char)inChar;
}
// if received CR or LF, terminate
// and print the string,
// then convert string's value to LONG:
// NOTE:  per Arduino Reference, .toINT
// actually returns a LONG, not an INT
// (as there is no .toLong StringObject
// function).
else {
mySerial.println("");
mySerial.print(F("Input string: "));
mySerial.println(inString);
long_input = inString.toInt();
inString = "";
endparse = true;
}
}
}
return (long_input);

}

{
mySerial.println("");
mySerial.println(F(" ********* GPSDO Command Menu *********"));
mySerial.println(F(" (Values should be entered as INTEGERS.)"));
mySerial.println("");
mySerial.println(F("   u:  dump variables"));
mySerial.println(F("   l:  erase EEPROM"));
mySerial.println(F("   n:  dump EEPROM"));
mySerial.println(F("   g:  To print 1-sec PD, type 'g1',"));
mySerial.println(F("       ...to print 30-sec PD_Error, Filter, DAC, type 'g2',"));
mySerial.println(F("       ...to STOP, type 'g'. "));
mySerial.println("");
mySerial.print(F("   r:  Toggle RUN/HOLD state (currently "));
if (run_state == RUN) mySerial.println("RUN)");
else mySerial.println("HOLD)");
mySerial.print(F("   p:  Toggle PLL/FLL mode (currently "));
if (lock_mode == PLL) mySerial.println("PLL)");
else mySerial.println("FLL)");
mySerial.println("");
mySerial.println(F("   d:  Write DAC, 0-65535"));
mySerial.println(F("   b:  Bump DAC, -16386 to 16386"));
mySerial.println("");
//  mySerial.println(F("   s:  set Setpoint,10-90 (%)"));
mySerial.println(F("   a:  set ADC Max, 0 to 1023"));
mySerial.println(F("   k:  set VCO Kv, +/-0 to 10000 milliHz/V"));
mySerial.println(F("   t:  set Reciprocal of external attenuation, 1 to 100"));
mySerial.println(F("   v:  set DAC Init, 0-65535"));
mySerial.println(F("   w:  set F1 (root), 1-32768"));
mySerial.println(F("   x:  set F2, 1-32768"));
mySerial.println(F("   y:  set Kcpu (root), for IIR Filters, 1-32768"));
mySerial.println(F("   z:  set Kcpu for Type 1 Filter, 1-32768"));
mySerial.println(F("   q:  set Settling Time (root), 1-10000"));
mySerial.println(F("   h:  set PLL's Initialization Filter (1-7)"));

mySerial.println("");
mySerial.print(F("   e:  Toggle PLL Filter Select Mode (Auto/Man, currently: "));
if (filter_select_mode == AUTO) mySerial.println(F("AUTO)"));
else mySerial.println(F("MANUAL)"));
mySerial.println(F("   i:    Min IIR FIlter, 2-max_filter)"));
mySerial.println(F("   j:    Max IIR FIlter, min_filter-7"));

mySerial.println("");
mySerial.println(F("   0:  set DAC to 0x8000"));
mySerial.println(F("   1:  select Shera Type 1 Filter"));
mySerial.println(F("   2-7:  select Shera IIR Filter"));
mySerial.println(F("   8:  set DAC to 0x0000"));
mySerial.println(F("   9:  set DAC to 0xFFFF"));

mySerial.println("");
mySerial.println(F("   c:  Clear error counts"));
}

void get_serial_in(int input_char)
{
// echo back character
mySerial.println("");
mySerial.write(serialByte_In);

switch (serialByte_In) {

case 'a':   // Set ADC Max value
case 'A':
long_value = parseLong();
if (long_value < 0) long_value = 0;
if (long_value > 1023) long_value = 1023;
mySerial.println("");
break;

case 'b':  //  bump the dac value
case 'B':
long_value = parseLong();
if (long_value < -16386) long_value = -16386;
if (long_value > 16386) long_value = 16386;
mySerial.println("");
mySerial.print(F(" DAC start = 0x"));
mySerial.println(dac, HEX);
mySerial.print(F("  DAC Bump = "));
mySerial.println(long_value);
dac = dac + long_value;
if (dac < 0) dac = 0;
if (dac > 65535) dac = 65535;
mySerial.print(F(" DAC end   = 0x"));
mySerial.println(dac, HEX);
second_count = 0;
running_sum = 0;
settling_count = 0;
write_to_dac(dac);
out0 = (((float) dac) - 32768.0)*(f_adc_max*((float) d))/(kcpu*(-2304.0));
out_m1 = out0;
in0 = 0;
in_m1 = 0;
break;

case 'c':   // clear any error counters
case 'C':
clear_err_counts = true;
mySerial.println("");
break;

case 'd':  // Set the DAC
case 'D':
long_value = parseLong();
if (long_value < 0) long_value = 0;
if (long_value > 65535) long_value = 65535;
dac = long_value;
write_to_dac(dac);
second_count = 0;
running_sum = 0;
settling_count = 0;
out0 = (((float) dac) - 32768.0)*(f_adc_max*((float) d))/(kcpu*(-2304.0));
out_m1 = out0;
in0 = 0;
in_m1 = 0;
write_to_dac(dac);
mySerial.println("");
mySerial.print(F(" DAC = 0x"));
mySerial.println(dac, HEX);
break;

case 'e':  // Toggle filter select mode (audo vs manual)
case 'E':
mySerial.print(F(" Filter Select Mode ="));
if (filter_select_mode == AUTO) {
filter_select_mode = MAN;
mySerial.println(F(" MANUAL"));
}
else {
filter_select_mode = AUTO;
mySerial.println(F(" AUTO"));

}
break;

case 'f':
case 'F':
break;

case 'g':   // enable/disable printing pll run data to mySerial port.
case 'G':
long_value = 0;
long_value = parseLong();
if (long_value == 1) {
pll_to_serial = false;
pd_to_serial = true;
mySerial.println(" 1-second print...");
}
else {
if (long_value == 2) {
pd_to_serial = false;
pll_to_serial = true;
mySerial.println(" 30-second print...");
}
else {
pd_to_serial = false;
pll_to_serial = false;
mySerial.println(" print OFF");
}
}
break;

case 'h':
case 'H':
long_value = parseLong();
if (long_value < 1) long_value = 1;
if (long_value > 7) long_value = 7;
init_filter = long_value;
mySerial.println("");
mySerial.print(F(" Init Filter = "));
mySerial.println(init_filter);
break;

case 'i':  // set minimum IIR filter
case 'I':
long_value = parseLong();
if (long_value < 2) long_value = 2;
if (long_value > max_filter) long_value = max_filter; // cannot be greater than max_filter
if (long_value > 7) long_value = 7;  // cannot be greater than 7
min_filter = long_value;
mySerial.println("");
mySerial.print(F(" Min IIR filter = "));
mySerial.println(min_filter);
break;

case 'j': // set maximum IIR filter
case 'J':
long_value = parseLong();
if (long_value < min_filter) long_value = min_filter; // max filter can't be less than min filter
if (long_value > 7) long_value = 7;
max_filter = long_value;
mySerial.println("");
mySerial.print(F(" Max IIR filter = "));
mySerial.println(max_filter);
break;

case 'k':  // set Kv for the VCO
case 'K':
long_value = parseLong();
if (long_value < -10000) long_value = -10000;
if (long_value > 10000) long_value = 10000;
kv = ((float)long_value) * 0.001;      // input as mHz/V
mySerial.println("");
mySerial.print(F(" VCO Kv(Hz/V) = "));
mySerial.println(kv,3);
break;

case 'l':   // erase eeprom
case 'L':
EEPROM.update(ADDR_DATA_VALID,0XFF);  // set Data-valid memory to invalid
mySerial.println(F(" EEPROM erased"));
break;

case 'm':   // print command menu
case 'M':
break;

case 'N':
mySerial.println(F(" EEPROM Contents:"));
// dump eeprom contents
mySerial.print(F(" Initialize with Lock Mode = "));
mySerial.println(F("PLL"));
}
else {
mySerial.println(F("FLL"));
}
mySerial.println(uint_temp);
mySerial.print(F(" Filter Select Mode = "));
filter_select_mode = AUTO;
mySerial.println(F("AUTO"));
}
else {
filter_select_mode = MAN;
mySerial.println(F("MANUAL"));
}
mySerial.print(F(" Settling Time (Root) = "));
mySerial.println(uint_temp);
mySerial.print(F(" Initialization Filter = "));
mySerial.println(uint_temp);
mySerial.print(F("   min filter = "));
mySerial.print(uint_temp);
mySerial.print(F(", max filter = "));
mySerial.println(uint_temp);
mySerial.print(F(" F1 (root) = "));
mySerial.println(uint_temp);
mySerial.print(F(" F2 = "));
mySerial.println(uint_temp);
mySerial.print(F(" Kcpu (root) = "));
mySerial.println(uint_temp);
mySerial.print(F(" Kcpu Type 1 Filter = "));
mySerial.println(uint_temp);
mySerial.print(F(" Kv = "));
mySerial.println(((float)int_temp)/1000,3);
mySerial.print(F(" 1/External_Attenuation = "));
mySerial.println(uint_temp);
mySerial.print(F(" DAC Starting Value "));
mySerial.println(uint_temp);

}
else {
// nothing in eeprom.
mySerial.println(F(" EEPROM does not contain valid contents"));
}
break;

case 'O':
EEPROM.update(ADDR_DATA_VALID,0X01);  // set Data-valid memory to valid
EEPROM.update(ADDR_LOCK_MODE,lock_mode);   // Note: sets initialization of lock mode to CURRENT mode

//      mySerial.println(extern_atten,6);
//      mySerial.println((unsigned int) (1/extern_atten));

break;

case 'p':  // toggle gpsdo fll/pll mode
case 'P':
if (lock_mode == PLL) {
lock_mode = FLL;
mySerial.println("");
mySerial.println(F(" FLL Mode"));

}
else {
lock_mode = PLL;
mySerial.println("");
mySerial.println(F(" PLL Mode"));
}
break;

case 'q':   // set Settling_root
case 'Q':
long_value = parseLong();
if (long_value < 1) long_value = 1;
if (long_value > 10000) long_value = 10000;
settling_root = long_value;
mySerial.println("");
mySerial.print(F(" Settling Time (root) = "));
mySerial.println(settling_root);

break;

case 'r':  // Put GPSDO into RUN mode (from HOLD)
case 'R':
if (run_state == RUN) {
run_state = HOLD;
mySerial.println("HOLD");
second_count = 0;
running_sum = 0;
settling_count = 0;

}
else {
run_state = RUN;
mySerial.println("RUN");
}
break;

case 's':
case 'S':
break;

case 't': // set reciprocal of external attenuation (see Simulink Model)
case 'T':
long_value = parseLong();
if (long_value < 0) long_value = 0;
if (long_value > 100) long_value = 100;
extern_atten = 1/((float) long_value);
mySerial.println("");
mySerial.print(F(" External Attenuation RECIPROCAL = "));
mySerial.println(1/extern_atten);
break;

case 'u': // Dump current operating values
case 'U':
print_current_values();
break;

case 'v': // set DAC initialization value
case 'V':
long_value = parseLong();
if (long_value < 0) long_value = 0;
if (long_value > 65535) long_value = 65535;
dac_start = long_value;
mySerial.println("");
mySerial.print(F(" DAC Initialization Value = 0x"));
mySerial.println(dac_start, HEX);
break;

case 'w': // set F1 (root)
case 'W':
long_value = parseLong();
if (long_value < 0) long_value = 0;
if (long_value > 32768) long_value = 32768;
f1_root = (float) long_value;
mySerial.println("");
mySerial.print(F(" F1 (root) = "));
mySerial.println(f1_root);
break;

case 'x': // set F2
case 'X':
long_value = parseLong();
if (long_value < 0) long_value = 0;
if (long_value > 32768) long_value = 32768;
f2 = (float) long_value;
mySerial.println("");
mySerial.print(F(" F2 = "));
mySerial.println(f2);
break;

case 'y': // set Kcpu (root)
case 'Y':
long_value = parseLong();
if (long_value < 0) long_value = 0;
if (long_value > 32768) long_value = 32768;
kcpu_root = (float) long_value;
mySerial.println("");
mySerial.print(F(" Kcpu (root) = "));
mySerial.println(kcpu_root);
break;

case 'z': // set Kcpu for Type 1 filter
case 'Z':
long_value = parseLong();
if (long_value < 0) long_value = 0;
if (long_value > 32768) long_value = 32768;
kcpu_t1 = (float) long_value;
mySerial.println("");
mySerial.print(F(" Kcpu (Type 1 Filter) = "));
mySerial.println(kcpu_t1);
break;

case '0':    // set DAC to 0x8000, hold mode
run_state = HOLD;
mySerial.println("   HOLD");
dac = 0x8000;
second_count = 0;
running_sum = 0;
settling_count = 0;
write_to_dac(dac);
break;

case '1':    // Shera's Type 1 Filter
second_count = 0;
running_sum = 0;
filter_type = TYPE1;
aggregate_rdy = false;
max_pd = 0;
min_pd = 65535;
filter = 1;
mySerial.println("");
mySerial.print(F(" filter = "));
mySerial.println(filter);
break;

case '2':    // first of Shera's IIR filters
second_count = 0;
running_sum = 0;
settling_count = 0;
f1 = f1_root;
out0 = out0*kcpu/(kcpu_root);  // adjust the 'dc' in the integrator to prevent jump (div old kcpu by new kcpu)
kcpu = kcpu_root;
filter_type = IIR;
aggregate_rdy = false;
max_pd = 0;
min_pd = 65535;
filter = 2;
settling_limit = settling_root * ((int) (pow((float)2,(float) (filter - min_filter))+0.5));
mySerial.println("");
mySerial.print(F(" filter = "));
mySerial.println(filter);
break;

case '3':
second_count = 0;
running_sum = 0;
settling_count = 0;
f1 = f1_root * 2;
out0 = out0*kcpu/(kcpu_root/2);  // adjust the 'dc' in the integrator to prevent jump
kcpu = kcpu_root / 2;
aggregate_rdy = false;
filter_type = IIR;
max_pd = 0;
min_pd = 65535;
filter = 3;
settling_limit = settling_root * ((int) (pow((float)2,(float) (filter - min_filter))+0.5));
mySerial.println("");
mySerial.print(F(" filter = "));
mySerial.println(filter);
break;

case '4':
second_count = 0;
running_sum = 0;
settling_count = 0;
f1 = f1_root * 4;
out0 = out0*kcpu/(kcpu_root/4);  // adjust the 'dc' in the integrator to prevent jump
kcpu = kcpu_root / 4;
filter_type = IIR;
aggregate_rdy = false;
max_pd = 0;
min_pd = 65535;
filter = 4;
settling_limit = settling_root * ((int) (pow((float)2,(float) (filter - min_filter))+0.5));
mySerial.println("");
mySerial.print(F(" filter = "));
mySerial.println(filter);
break;

case '5':
second_count = 0;
running_sum = 0;
settling_count = 0;
f1 = f1_root * 8;
out0 = out0*kcpu/(kcpu_root/8);  // adjust the 'dc' in the integrator to prevent jump
kcpu = kcpu_root / 8;
filter_type = IIR;
aggregate_rdy = false;
max_pd = 0;
min_pd = 65535;
filter = 5;
settling_limit = settling_root * ((int) (pow((float)2,(float) (filter - min_filter))+0.5));
mySerial.println("");
mySerial.print(F(" filter = "));
mySerial.println(filter);
break;

case '6':
second_count = 0;
running_sum = 0;
settling_count = 0;
f1 = f1_root * 16;
out0 = out0*kcpu/(kcpu_root/16);  // adjust the 'dc' in the integrator to prevent jump
kcpu = kcpu_root / 16;
filter_type = IIR;
aggregate_rdy = false;
max_pd = 0;
min_pd = 65535;
filter = 6;
settling_limit = settling_root * ((int) (pow((float)2,(float) (filter - min_filter))+0.5));
mySerial.println("");
mySerial.print(F(" filter = "));
mySerial.println(filter);
break;

case '7':   // ...last of Shera's filters
second_count = 0;
running_sum = 0;
settling_count = 0;
f1 = f1_root * 32;
out0 = out0*kcpu/(kcpu_root/32);  // adjust the 'dc' in the integrator to prevent jump
kcpu = kcpu_root / 32;
filter_type = IIR;
aggregate_rdy = false;
max_pd = 0;
min_pd = 65535;
filter = 7;
settling_limit = settling_root * ((int) (pow((float)2,(float) (filter - min_filter))+0.5));
mySerial.println("");
mySerial.print(F(" filter = "));
mySerial.println(filter);
break;

case '8':   // set DAC to 0x0000
run_state = HOLD;
mySerial.println("   HOLD");
second_count = 0;
running_sum = 0;
settling_count = 0;
dac = 0X0000;
write_to_dac(dac);
break;

case '9':    // set DAC to 0xFFFF
run_state = HOLD;
mySerial.println("   HOLD");
second_count = 0;
running_sum = 0;
settling_count = 0;
dac = 0XFFFF;
write_to_dac(dac);
break;

default:
break;

}

}

void aggregate_samples()
{
running_sum = running_sum + phase_delta;
second_count++;
if (phase_delta > max_pd) max_pd = phase_delta;
if (phase_delta < min_pd) min_pd = phase_delta;
if (second_count == d) {
second_count = 0;
d_sample = running_sum;
running_sum = 0;
aggregate_rdy = true;

}
}

void write_2bytes_eeprom(uint16_t address, uint16_t int_data) {

int_data = int_data >> 8;

}

uint8_t byte1;
uint8_t byte2;

read_int = byte1 | (((uint16_t) byte2) << 8);

}

void print_current_values() {
mySerial.println(F(" Current Values:"));
mySerial.print(F(" DAC Init = "));
mySerial.println(dac_start);
mySerial.print(F(" D = "));
mySerial.println(d);
mySerial.print(F(" Setpoint = "));
mySerial.println(setpoint);
mySerial.print(F(" Lock Mode = "));
if (lock_mode == PLL) mySerial.println(F("PLL"));
else mySerial.println(F("FLL"));
mySerial.print(F(" Initialization Filter = "));
mySerial.print(init_filter);
mySerial.print(F(", Current Filter Type = "));
if (filter_type == IIR) {
mySerial.print(F("IIR, "));
mySerial.print(F(" filter: "));
mySerial.println(filter);
mySerial.print(F(" Filter Select Mode: "));
if (filter_select_mode == AUTO) mySerial.println("AUTO");
else mySerial.println("MANUAL");
mySerial.print(F("   Min Filter = "));
mySerial.print(min_filter);
mySerial.print(F(", Max Filter = "));
mySerial.println(max_filter);
mySerial.print(F(" Settling Time (Root): "));
mySerial.println(settling_root);
mySerial.print(F(" Settling Limit: "));
mySerial.println(settling_limit);
mySerial.print(F(" MaxMin Limit = +/-"));
mySerial.println(maxmin_limit);
mySerial.print(F(" Upper wraparound limit = "));
mySerial.print((unsigned int) upper_wraparound_lim);
mySerial.print(F(", Lower wraparound limit = "));
mySerial.println((unsigned int) lower_wraparound_lim);
mySerial.print(F(" Dropback limit = "));
mySerial.println(DROPBACK_LIMIT);
}
else {
mySerial.println(F("Type 1"));

}
mySerial.print(F(" F1 = "));
mySerial.println(f1);
mySerial.print(F(" F1 (root) = "));
mySerial.println(f1_root);
mySerial.print(F(" F2 = "));
mySerial.println(f2);
mySerial.print(F(" Kcpu = "));
mySerial.println(kcpu);
mySerial.print(F(" Kcpu (root) = "));
mySerial.println(kcpu_root);
mySerial.print(F(" Kcpu Type 1 Filter = "));
mySerial.println(kcpu_t1);
mySerial.print(F(" Kv = "));
mySerial.println(kv, 3);
mySerial.print(F(" 1/External_Attenuation = "));
mySerial.println(1/extern_atten);
mySerial.print(F(" DAC = "));
mySerial.println(dac);

}

void calc_floating_point_reciprocals() {
// just a test to see how accurate
// the reciprocals of f1 and f2 are,
// in floating point
float ff1;
float ff2;
float rec_f1;
float rec_f2;
float sum_f1f2;
float diff_f1f2;
float delta_f1f2;

ff2 = (float) 8;
rec_f2 = 1/ff2;

int i;
int ii;

for (i=0; i<=6; i++) {
ii = (int) (pow((float)2,(float) i) + 0.5);
ff1 = ((float)2048)*((float)ii);
rec_f1 = 1/ff1;
sum_f1f2 = rec_f1 + rec_f2;
diff_f1f2 = rec_f1 - rec_f2;
delta_f1f2 = sum_f1f2-diff_f1f2;

mySerial.print(i);
mySerial.print(",");
mySerial.print(ii);
mySerial.print(",");
mySerial.print(ff1);
mySerial.print(",");
mySerial.print(ff2);
mySerial.print(",");
mySerial.print(rec_f1,10);
mySerial.print(",");
mySerial.print(rec_f2,10);
mySerial.print(",");
mySerial.print(sum_f1f2,10);
mySerial.print(",");
mySerial.print(diff_f1f2,10);
mySerial.print(",");
mySerial.println(delta_f1f2,16);

}

}

(A note regarding the software -- I am not a programmer by trade, so I am sure my programming techniques will make other programmers wince.  If you use my code, please feel free to modify it to your heart's content!)

Regarding the Arduino code:

1.  The GPSDO Modes:

The GPSDO can operate in either its original FLL mode or in its new PLL mode.  The mode can be manually toggled between PLL and FLL via the external Serial Port (serial port command 'p', described later, below).

2.  NEMA Satellite Data:

The NEMA satellite data is no longer received and displayed -- the Quectel L76's  NEMA stream can get quite verbose with both GPS and GLONASS satellites being received, and I discovered that this serial data stream into the Arduino could delay the Arduino's reading of the voltage on the Phase Detector A/D input pin.

There should be no additional delay of the Arduino read of the A/D pin after the 1 PPS interrupt occurs -- it needs to be read before the capacitor discharges appreciably.  Because this (software) serial port was adding additional, random, delay to the A/D read, I decided to simply delete this section of the code.

3.   PLL Filtering:

Shera's algorithm has 7 different filters.  These are all discrete-time filters, and they consist of one "Type 1" filter and six IIR filters whose time constants step by a factor of 2.

Let's take a look at the equations for these 7 filters.  To do so, I will define a filter's input at discrete-time 'n' to be i(n) and its output to be o(n).

The equation for the Type 1 filter is simply:

o(n) = KcpuType1 * i(n)

where KcpuType1 = 32.

The equation for the discrete-time IIR filters all have the form:

o(n) = Kcpu* [o(n-1) + i(n)*(1/F1 + 1/F2) + i(n-1)*(1/F1 - 1/F2)]

where, for Shera's first IIR filter (filter selection = 2), F1 = 2^11, F2 = 64, and Kcpu = 2^10.

For each successive IIR filter (Shera filters "3" to "7")  from this "root" filter, F1 increases by a factor of 2 and Kcpu decreases by a factor of 2.  F2 remains unchanged.

4.  Automatic Switching of Filters:

In Shera's design the filter must be selected manually.  I decided to automate this process with a very simple algorithm (feel free to improve!).  The automatic filter selection works as follows...

In PLL mode, the GPSDO initializes with the filtering set to filter "2", the lowest (fastest settling) IIR filter of the six IIR filters.

The PLL then begins operating.  After a time period equal to the IIR filter's "settling time", the algorithm will then begin examining the magnitude of the phase detector's error.  If it is fairly close to the setpoint (within the limits defined by "maxmin_limit", then the algorithm deems the filter to have settled.  The algorithm then switches to the next filter and initializes a new settling time for this new filter.

If, on the other hand, the phase detector error is not close to the setpoint by the time defined by the filter's "settling time", the algorithm notes that the settling time has been reached but it continues PLL calculations with the current filter.  Then, when the phase detector error finally falls within the maxmin_limit, the algorithm will switch to the next higher filter and the new settling time will be set accordingly.

This process of successively moving up to higher and higher filters (i.e. lowering the filter bandwidth with each step up) continues until the algorithm reaches the highest filter that the algorithm allows to be selected (specified as "max_filter").  This "top" filter will be used until either the GPSDO is re-initialized or an error condition occurs (wrap-around or dropback -- these are described, below), at which point the filter is set back to the lowest filter.

5. Phase Detector Wrap-around Events:

What is a wrap-around event?

If the 1-second phase delta  into the ADC is near either the upper limit (822) or the lower limit (0), jitter on the 1 PPS pulse can cause it to flip back and forth across the limit because of the way the phase detector wraps around (see the phase-delta wrapper block in the Simulink model).

For example, if the phase delta is near 822, jitter can cause it can flip over to a low value (e.g. 10) and then back again.  Back and forth.

Similarly, if the phase delta is near 0, jitter can cause it to flip over to a high value (e.g. 820) and then back again.

At initialization wrap around events can occur because the phase difference between the VCO and the GPS 1 PPS initializes randomly, depending upon the phase difference between the two at power-up.

But these initialization "1-second" wrap-arounds are not an issue, because the VCO will eventually drift enough to bias the 30-second aggregated phased delta to be either near 0 nanoseconds or near 800 nanoseconds, and the PLL algorithm will start tugging the VCO so that the phase difference moves towards the midway setpoint.

Ideally, wrap-arounds of the 1-second phase deltas should not occur after initialization.  But they could happen.  If the Trimble Resolution T receiver loses reception of all GPS satellites, after a minute or so its 1 PPS pulse goes a bit wild and wrap-arounds can occur.  If this happens I want to force the filtering back to the lowest IIR filter (and reset the settling time) so that the PLL can more quickly force the phase delta back to its setpoint.

You might wonder -- why not just look at the magnitude of the 30-second phase deltas, rather than the 1-second phase deltas?

If the phase between the VCO and the 1 PPS signals were so far from its setpoint that wrap-arounds were occurring, the 30-second phase delta could mask this fact because it is an average of the 30 samples.  For example, suppose the 1-second phase deltas were flipping between 0 and 822.  Their 30 second average would be 12330, which is exactly the setpoint.  So, to the PLL it would look as though the phase delta was exactly where it was supposed to be, rather than actually being very far off from its setpoint goal.

Finally, whenever a wrap-around occurs, the software also increments a counter (wraparound_count) for each wrap-around event.  This is displayed on the LCD (and is resettable via the serial port), to inform me if wrap-arounds have occurred.

6.  Filter Dropback if Large Phase Delta Error:

In addition to wrap-around events, I also want to reset filtering if the 30-second phase delta moves too far from its setpoint.  Here are some examples why I do this...

While testing the GPSDO algorithm (with my Trimble Resolution T GPS Receiver), I noticed a large transition in the Phase-Delta Error (i.e. the 30-second phase delta minus the setpoint):

Two things caught my eye.  The first is the very slow recovery from this phase-delta-error transient (because the algorithm had been using IIR filter 4 which has an 8000 second settling time).  And the second thing is that the algorithm switched to IIR filter 5 well before filter 4 had settled, which greatly worsens the settling time.

Ideally, the phase-delta error should be fluctuating around 0.  But after the transient there is a huge offset from 0, which means that the VCO frequency is not near its ideal frequency for a very long time.

To speed up recovery I've added code that now compares the magnitude of the phase-delta error against a "DROPBACK_LIMIT".  The latter is currently set, in software, to 3000 (out of a peak value of about 12,300)).

If a delta greater than 3000 is detected (which is a large delta), then the algorithm drops back to the minimum filter (filter 2), the settling timer is reset, and a counter which counts dropback events is incremented.  This counter is displayed on the LCD (and is resettable via the serial port).
.
Just after I wrote this routine I ran a test and...a large phase-delta-error transient occurred!  The image below shows the dropback to the lowest filter followed by the recovery from the transient:

Note, too, that if there is a large phase-delta between the 1 PPS and the VCO clock at GPSDO initialization, then the phase delta error magnitude will start at a value larger than DROPBACK_LIMIT and the algorithm will register dropback events until it can bring the phase delta error under the DROPBACK_LIMIT value.

Here is a new capture.  You can see this effect starting at T = 0, below:

At the start the Phase-delta error is much larger (i.e. much more negative) than negative BACKDOWN_LIMIT.  In fact, the phase-delta error was larger than this limit for the first ten 30-second samples.  For this reason, the settling-time counter did not start incrementing until after these ten samples, when the phase-delta error finally rose above  -3000.

Thus, because the settling timer count was delayed (I purposely reset the settling timer any time there is a too-large phase-delta transient, while the lowest filter is selected, to give the transient time to settle), we will switch to filter 3, later, by the same amount of delay.

If there had been no delay, the filter change would have been at 2000 seconds (my defined settling time for filter 2).  But because the settling timer has been delayed by about 300 seconds due to the excessively large phase deltas at the start, we make the switch at 2300 seconds, instead.

Dropbacks can also occur if the selected IIR filter's response is too slow to keep up with, say, VCO frequency drift due to changes in ambient temperature.  Here's an example:

Even though my VCO has an oven, the graphs above shows that, in winter (it's Feb 5 with snow on the ground), filter 5 cannot keep up with the VCO's frequency change due to the overnight temperature gradient in my Nevada City lab.  So I should probably change the maximum filter that the automatic filter-selection routine uses from 5 to 4 to allow the GPSDO to more easily track this sort of temperature change.

7.  LCD:

Here's an annotated picture of the information displayed on the GPSDO's LCD in PLL mode:

Note that the phase delta values are the values read every second via the Arduino's ADC.  The Max and min values are updated every 30 seconds on the LCD, and they are from the latest 30-second sample that was used to calculate the displayed DAC value.

8.  Serial Port:

A software-implemented serial port (9600 baud) allows me to monitor and change GPSDO operation on-the-fly with my laptop using the terminal-emulator program (PuTTY, in my case) and a USB-to-serial adapter cable (FTDI-TTL-232R-5V-AJ).

This serial port allows me to monitor GPSDO operation and change, on the fly, various parameters without having to first plug an Arduino Development cable into the Arduino board (plugging this cable into the Arduino board causes the Arduino to re-initialize -- an action I usually don't want to have happen if I simply want to monitor the GPSDO while it's running).

Here's a snapshot of the serial-port's command menu (invoked by typing the command 'm' into the serial port):

For the most part, the menu items should be self explanatory.  Here are a few commands explained in greater detail...

Invoking command 'u' (dump variables) results in this display:

Commands '0', 8', and '9' (set DAC to 0x8000, 0x0000, and 0xFFFF) are used when setting R1 and R4.  Note that their invocation will place the GPSDO into HOLD mode.  Type command 'R' to return to RUN mode.

Commands 'd' and 'b' (write DAC, bump DAC) are there for experimentation purposes.  I use 'd' to  force the phase detector's output to "quickly" approach the setpoint.  That is, while monitoring phase at the PHASE_TP with an oscilloscope, I set the DAC to a value where the phase delta starts scrolling either left or right.  When it reaches midway (i.e. the setpoint), I quickly change the DAC to a value that slows the drift way down.  (Of course, I first need to determine what this latter value is).

Command 'c' clears the "wraparound event" counter and the "dropback event" counters.  (Because these counts are updated on the LCD only every 30 seconds, there can be a delay of up to 30 seconds between typing 'c' and seeing the counts go to zero on the LCD.)

Command 'g' will print PLL data, in an on-going fashion for real-time monitoring, to the serial port.  This data can be in two formats:

Typing 'g1' will print the following data every second:
• Time (in terms of accumulated seconds from GPSDO initialization).
• The 1-second phase delta (that is, the value read each second from the Arduino's ADC).
Typing 'g2' will print the following data every 30 seconds:
• Time (in terms of accumulated seconds from GPSDO initialization).  Thus each new entry will be 30 seconds greater than the last entry, because the DAC is updated every 30 seconds.
• The 30-second phase-delta "error" (relative to the setpoint).  A phase-delta error of '0' means that the phase-delta equals the setpoint (which is the target the PLL is adjusting the phase-delta to be equal to).
• The current filter (e.g. 2 to 7)
• The current DAC value (0 to 65535).
And typing 'g' (or 'g' followed by any other character) will stop the printing (if it had been invoked  by either commands 'g1' or 'g2').

Here is an example of these different print commands:

Note that this command can be used at any time to monitor GPSDO performance, rather than monitoring through the Arduino Development cable, which requires that this USB development cable first be connected  to the Arduino, forcing a (perhaps unwanted) system reset.

Commands 'l', 'n', and 'o' are all Arduino EEPROM commands:
• Command 'n' dumps the EEPROM's contents to the serial port.
• Command 'o' loads the EEPROM with the current parameters.  If you update any parameters via the serial port, you can load them into EEPROM using this command.
• Command 'l' erases the EEPROM (by simply writing to the EEPROM byte that identifies if the EEPROM contents are valid or invalid).
The EEPROM contains the following information (per the EEPROM addresses, below):

If the EEPROM's contents are valid (as identified by the first byte), the GPSDO will use these values for parameter initialization.  Otherwise, it will initialize with values hard-coded into the software.

Initial Setup:

There are two potentiometers in the design, R1 and R4.  These will need to be set before using the GPSDO.

Here's how I set the EFC's potentiometers R1 and R4 (see schematic).

NOTE:  Before setting up the VCO's frequency, I strongly recommend that the VCO and its oven be on for at least several hours (and preferably longer) to allow its temperature (and thus its frequency) to be (roughly) stabilized.

First, set R1 and R4 to their full resistance and set the DAC to 32768 ( (serial port command '0').  The GPSDO should be in HOLD, not RUN mode.

With these settings, if using a 10 MHz VCO, verify that the VCO frequency  is close to 10 MHz.  You should probably do this using a frequency counter or some other method of accurately measuring frequency.  Possibly you could do this by monitoring the phase waveform at the PLL Phase Detector's "PHASE_TP", but there is an issue with this technique -- it will show a lock at 10 MHz (good), but it will also show locks for frequencies that are spaced at 8 Hz increments from 10 MHz (bad).  If your VCO can be adjusted to be this far off frequency , then you should not use the PHASE_TP test point for this initial frequency adjustment.

However, if the VCO has a smaller range of adjustment, you can monitor its frequency offset via the PHASE_TP.  For example, assuming a 10 MHz VCO and a VCO division factor, N, of 8, if the phase-delta at the test point sweeps from its min (0 ns) to its max (800 ns) in 8 seconds (or from max to min in 8 seconds), the VCO frequency is within 1 Hz of 10 MHz.  A 16 second sweep time would correspond to a 0.5 Hz offset, 32 seconds to 0.25 Hz, and so on.

The longer the sweep time from min to max or max to min, the better.

If a VCO has a mechanical adjustment (such as my HP module has), try adjusting it so that the VCO is close to 10 MHz.  For my VCO module, which has a negative Kv, I want the frequency to be slightly above 10 MHz.  If my module had a positive Kv, I would want this frequency to be slightly below 10 MHz.  (Potentiometer R1 will only increase the VCO's EFC voltage above 0.  Therefore, for my VCO with its negative Kv, decreaseing R1 from its maximum value will raise the VCO's EFC voltage and drive the initial slightly positive frequency error towards 0.  I.e. towards exactly 10 MHz.)

How much above or how much below 10 MHz the VCO should be initially set depends upon the range that the frequency can be shifted with pot R1.  With my module, I should be able to shift the frequency by about 1.6 Hz using R1, so I want the frequency offset to be less than this amount.  A good target would be, say, no more than 1/5 of this (about 0.3 Hz in my case), and the smaller this initial error is, the better, as the DAC will have more "headroom" to allow it to compensate over time for, say, VCO aging.

After the initial VCO adjustment, R1 (the EFC DC offset) can be adjusted, as follows...

2.  Adjusting EFC DC Offset potentiometer R1:

With R4 set to its full resistance and the DAC still at 32768, adjust R1 to minimize, as best one can, the long-term phase-delta's drift, as measured at "PHASE_TP" test-point.

I would recommend adjusting the frequency with R1 so that the phase-error drift, as measured at the PHASE_TP, takes at least 108 seconds for the phase to move from 0 ns to 800 ns (or vice-versa), which corresponds to the frequency being within 7.5 ppb of 10 MHz (i.e. within 0.075 Hz of 10 MHz).

Note that the longer the drift is from phase-min to phase-max (or vice-versa), the better!

(Given Shera's sensitivity factor, S, of 7.5E-9/volt, a frequency offset of 0.075 Hz would need the DAC to move 1 volt from its "center" voltage to compensate.)

3.  Adjusting EFC Attenuation potentiometer R4:

R4 should be adjusted only after R1 (the DC offset) is adjusted.  Use this procedure:
1. Set the DAC to 0 (serial port command '8').
2. Measure and log the voltage at op-amp U2B pin 7 (See Arduino schematic).  Call this value VinLo.
3. Measure and log the voltage at the VCO's EFC pin (e.g. pin 6 of the HP 10811-60111 VCO Module).  Call this value VoutLo.
4. Set the DAC to 65535 (serial port command '9').
5. Measure and log the voltage at op-amp U2B pin 7.  Call this value VinHi.
6. Measure and log the voltage at the VCO's EFC pin (e.g. pin 6 of the HP 10811-60111 VCO Module).  Call this value VoutHi.
The amount of attenuation is calculated as follows:

Av = (VoutHi - VoutLo) / (VinHi - VinLo)

If the attenuation is not the desired value (e.g. 1/28), adjust R4 and repeat steps 1-6 until it is.

Note that one should only need to measure VinHi and VinLo once -- and in my case their difference equals 10 volts.  Therefore, in my design, for a desired attenuation of  0.036 (1/28), R4 should be adjusted until the difference between VoutHi and VoutLo equals about 0.36 volts.

4.  (Optional) Manually Adjusting Phase Difference:

Using the serial port commands, one can manually adjust the phase difference between the 1 PPS pulse and the VCO by changing the DAC values.  I find this useful for testing, when I want to quickly set the phase-detector's phase difference close to its target of 400 ns before starting an experiment.

The image below describes the basic technique:

For my "negative Kv" VCO, lowering the DAC value increases the VCO's frequency, while raising the DAC value decreases the VCO's frequency.

Therefore, if I want to reduce the phase difference, I lower the DAC (and thus increasing the frequency).

Performance Tests:

The following tests use, as the "root" filter parameters (i.e. parameters for the lowest IIR filter, Filter 2), the following:
1. F1 = 256 (i.e. 2^8)
2. F2 = 8
3. Kcpu = 64 (i.e. 2^6)
4. Settling time = 2000 seconds
Note that Kcpu is 2x the value Shera uses, per the results of my simulation experiments, above.

Overview:

Let's take a look at the hardware and algorithm's performance...

I'll measure the Arduino GPSDO's stability against my HP-106B Quartz Oscillator (that I use in my original Shera GPSDO configuration), using an HP 5370B Time Interval Counter whose measurements are captured, via GPIB, using a MATLAB program written by Dick Benson, W1QG, and available from the Mathworks File Exchange:

https://www.mathworks.com/matlabcentral/fileexchange/68914-gpib-control-of-hp-counters?s_tid=prof_contriblnk

The HP-106B GPSDO system will have its "disciplining" disabled  -- the HP-106B's EFC (Electronic Frequency Control) pin will be driven either by an external DC power supply, or its Shera GPSDO controller will be placed in DAC HOLD mode.  In other case, the VCO's control voltage will be constant.  This will prevent frequency-change characteristics due to HP-106B/Shera GPSDO disciplining process from masking the Arduino/Shera GSPDO PLL characteristics.

In other words, the HP-106B is, essentially, a stable frequency reference against which I will measure the Arduino/Shera GPSDO performance.

Note that the performance comparison is made by measuring the time interval between the rising edge of the HP-106B system's 5 MHz output and the rising edge of the Arduino/Shera GPSDO's 5 MHz output.

With the HP-106B "free-running" at a fixed frequency, then any short-term "error" artifacts should be due to the Arduino/Shera system.  The HP-106B should only be introducing long-term drift (i.e. temperature and/or aging), not short-term artifacts.  (And there will be low-level artifacts due to the HP 5370B counter's time-interval measurement uncertainty).

Performance with Quectel L76 GPS Receiver:

My original Frequency-Locked Loop GPSDO design uses a Quectel L76 GPS receiver.  Using this same receiver to provide the Arduino/Shera's 1 PPS reference pulse, here's a seven-hour capture of the time-interval between the rising edge of the HP-106B "free-running" 5 MHz signal and the rising edge of the Arduino/Shera GPSDO's 5 MHz output.

Over the seven hours, the maximum error span (when measured with a 30 second window) is roughly 50 parts-per-trillion, peak-to-peak.  This would be quite satisfactory for my purposes.

(Note:  this 7-hour segment was captured after the systems was running for many hours, so the IIR filter for this capture was filter 5, the maximum IIR filter that I've currently defined for my Arduino/Shera GPSDO).

Here's a look at the DAC value and the 30-seconds phase-delta error (i.e. the 30-second phase delta minus the setpoint), for the same period:

Performance with Trimble Resolution T GPS Receiver:

I was curious how a Trimble Resolution T GPS Receiver would work in lieu of the Quectel receiver.

For this test I used an external Trimble Resolution T GPS Receiver board and jumpered its 1 PPS signal (and ground) to my Arduino/Shera GPSDO in lieu of the Quectel 1 PPS.

Note that, using Trimble's GPS monitoring program (in this case, TrimbleMon_V1-05-0.exe), the Trimble receiver was set up as follows:
• Over-Disciplined Clock mode
• Stationary position
• 100% Position Fix (i.e. fix completed).
Note that, while the Trimble receiver is performing its self-survey, the 1 PPS timing will not be good.  Only after it finishes its self-survey will it enter into "Over-Disciplined Clock Mode," and the 1 PPS signal should become much more predictable.

Self-survey can take some time, depending upon how many satellites the Trimble receiver sees.  The Trimble Resolution T receiver needs at least 4 "usable" satellites to perform its self-survey.  Usable satellites show as "green" in the Trimble GPS Monitor program:

...and after the self-survey has been completed:

Below is a 19-hour plot of error using the Trimble receiver (already in "Over-Disciplined Clock Mode"), starting from GPSDO initialization (starting with filter 2 selected as the initial PLL filter at time T = 0.  2000 seconds later, then it switches to filter 3.  4000 seconds later the algorithm switches to filter 4, and finally, 8000 seconds later (total duration of 3.9 hours from T = 0), it switches to filter 5, the highest filter that I use.

Below is the phase-delta error and the DAC value for the same 19-hour period.

To compare against the Quectel receiver, let's examine the final seven-hours of the error of the Trimble capture (during which filter 5 is selected).

Note that the error looks much more well-behaved than the Quectel receiver's error.  In the case of the Trimble receiver, the error for this seven-hour period is on the order of 30 parts-per-trillion.

Here's a look at the performance of the two receivers, side-by-side.

If we compare the PPB error of the Quectel L76 receiver versus the Trimble Resolution T receiver, the Trimble receiver clearly has better performance.

But is this the full story?

With the Trimble RX installed in my GPSDO chassis (see picture, below), I took the GPSDO up to the second home in the Sierra Nevada foothills (Nevada City) for more testing.

(The Quectel receiver is still in the chassis (it would take much to much work to remove it), but I've removed the PWR SELA jumper on its PCB, so that it no longer is being powered by 5V.)

One issue I quickly discovered is that GPS satellite reception at the second house in Nevada City, CA, seems to be worse than reception at the Los Altos QTH.

Some potential issues I've noticed:

1.  If the number of satellites in use by the Trimble receiver drops to 0, its 1 PPS can go haywire (see the plot, below).

2.  The 30-second aggregated phase delta is the sum of 30 1-second phase deltas (read via the ADC).  If I find the minimum and maximum of the 1-second phase deltas that make up a 30 second block and then calculate the delta between these two, I will see that these deltas are larger at the second QTH in Nevada City compared to the deltas measured at the Los Altos QTH.

The Blue trace summarizes the Los Altos deltas, which are usually right around 40 counts (or less).  The Nevada City deltas are summarized by the orange trace.  The latter are appreciably higher, and sometimes huge!

Is this a problem?  I have no idea, as I don't have a second frequency reference at the Nevada City QTH against which I can compare this Arduino GPSDO's performance.

And why is there a difference in performance between the two residences?  One reason might be the many 100 foot (or taller) pine trees surrounding the Nevada City house -- if I monitor the GPS satellites being received by the Trimble receiver (using the Trimble GPS monitor program), there is a dance as satellites constantly enter and then leave the antenna's "view" (perhaps being blocked by trees?).  In contrast, the satellites in view in Los Altos change much less frequently.  Could this constant fluctuation of visible satellites account for the larger deltas between the max and min 1-second phase deltas?

Anyway -- because of the Trimble's potential reception issues, I've decided to keep the original Quectel receiver for use of this GPSDO at the Nevada City, CA site.

Here's a look at the GPSDO's performance (with Quectel receiver) over 38 hours at the Los Altos, CA location, measured against my HP-106B with its EFC controlled by an external power supply (set to -0.272V):

Note:  the maximum filter is 4 (rather than 5, which I had used in earlier tests), and we switch to this filter after 6000 seconds (1 hour, 40 minutes) after the start of the capture.  I changed the maximum filter from 5 to 4 because filter 5 has a more difficult time compensating for ambient temperature changes at the Nevada City location.

A closer look at the frequency error:

There is some bias 'up' in the values shown (roughly 14 parts-per-trillion), due to long term drift (see the blue line in the first plot), but I think it reasonable to call the maximum short-term error equal to +/- 50 parts per trillion when using Filter 4 with the Quectel receiver.

Here's a plot of the Phase-delta Error and the DAC values over the same period:

What's interesting is that you can see long-term drift here, too.  So the drift shown in the error plots is not entirely due to the HP-106B and/or DC EFC voltage, but also it is due, to some extent to drift in the Arduino GPSDO (ambient temperature changes?)

However, the slope-magnitude of the DAC values between 20 and 35 hours is less than its slope between 0 and 15 hours, whereas the time-interval difference (the blue line in the error plot) has a higher slope magnitude between 20 and 35 hours compared to its slope from 0 to 15 hours, implying that the time-interval slope between 20 and 35 hours is not simply due to the GPSDO's VCO, but to the HP-106B and/or the DC supply driving its EFC pin.

Frequency-Locked Loop (FLL) Performance:

Apart from removing the decoding of the NEMA stream, the only other change to the original Frequency-Locked Loop code is the addition of a gain factor (when calculating a new DAC delta) to offset the new 1/29 additional attenuation between DAC and VCO EFC pin.

To verify that the FLL code still works, I put the GPSDO into FLL mode (serial command 'p') just after initialization.  Here's a plot of error, measured against my HP-106B (the latter with its EFC voltage fixed at -0.271 volts):

Following initialization, the error is well under 0.1 parts-per-billion.  Here's how the error looks, zoomed in, for the last 11 hours:

Each large delta step up (and there are five in the chart, above) represents a correction made by the FLL algorithm to the DAC.

Worst case error for the 11 hours is a bit less than 60 parts per trillion, which I consider to be pretty good for such a simple design.  Note that by increasing the GAIN value (defined in the Arduino code), the correction could probably be improved (e.g. closer to 0 ppb), but I purposefully kept GAIN a bit low because I was concerned about oscillations, and a bias of error in the negative direction did not concern me for my application.

Other Notes and Thoughts:

1.  Shera's "S" Factor, Kv, and the Derivation of his Equation for R6:

In note 8 of Shera's QST article he defines an equation for calculating the value of R6 in his attenuator network.  This equation is:

R6 = R5*[(RA / (RA+RB)) * (S/(7.5*10-9)) - 1]

I was curious about the relationship between this network's attenuation, Shera's "S" factor, and oscillator Kv, so I thought I would do some equation-deriving myself.

Here is a schematic representing the resistor network between Shera's DAC output and his VCO's EFC (Electronic Frequency Control) input pin:

(Note: this schematic corrects ambiguities and mistakes in Figures 1 and 3 of Shera's original article.)

If we assume that the sum of RA + RB is much larger than R5, and if we recognize that +VDC does not affect the amount of the network's attenuation, it just adds a DC offset, then we can write the voltage gain of this networks as:

Av = (VCO_EFC / V_DAC_OUT) = (R5/(R5+R6))*(RA/(RB+RA))

(Note that the assumption that RA+RB is much larger than R5 means that RA and RB have an insignificant effect on R5, allowing us to treat the R5, R6, RA, RB network as two independent two-resistor voltage dividers).

Solving for R6 (as Shera did), the resulting equation is:

R6 = R5*[(RA/(RB+RA))*(1/Av) - 1]

Comparing this equation to Shera's equation, we see that:

Av = 7.5*10-9 / S

What is "S"?

Per Shera, S is the "change in the rate of drift" when the VCO's EFC voltage is changed by 1 volt.

That's a bit confusing to me, but he also notes:  "the goal is to obtain a relative frequency change, ΔF/F = 7.5*10-9, when the voltage applied by the DAC to R6 changes by 1 V."

That's clearer.  In other words, S = ΔF/F per volt, or ΔF/F/V

We can rearrange this equation to be:  S = (ΔF/V)/F, and therefore, given that the VCO's Kv is defined to be ΔF/V, we can rewrite S as:

S = Kv / F

(Note that S is not unit-less, but has units of Volts-1).

Also, given a target S of 7.5*10-9/V, then, for a 10 MHz VCO, Kv must be 0.075 Hz/V.

Let's go back to my equation for network gain in terms of S:

Av = 7.5*10-9 / S

My VCO has a Kv of 0.32 Hz/V.  (Note, here the Kv is the magnitude of my VCO's Kv -- the sign isn't important.)

Therefore, if I were to use my VCO with a Kv magnitude of 0.32 in lieu of Shera's ideal VCO (with a Kv of 0.075, i.e. Av = 1), then I need to compensate by attenuating the DAC's output by the following amount:

Av = 7.5*10-9 / S
= (7.5*10-9 /V)/ (Kv/F)
= (7.5*10-9 / V) / (0.32 Hz/V)/(10 MHz)
or
Av = 0.234375

One last note: my resistors R1 - R4 have the following relationship to Shera's resistors R5, R6, RA, RB in his schematic:
• R4 = R6
• R3 = R5
• R2 = RB
• R1 = RA

2.  Using other Filter Coefficients with this Algorithm:

As I pointed out in my Brooks Shera Simulation post, The z-domain transfer function of Shera's IIR filter topology can be expressed as:

If one would like to use a different filter whose coefficients are expressed as a0 and a1 instead of F1 and F2, recognize that a0 and a1 can be written into the filter algorithm using the Serial Port commands that write to F1 and F2.  To do so, we must first calculate the values of F1 and F2 in terms of the desired a0 and a1 values, as follows:

F1 = 2/(a0 + a1)

F2 = 2/(a0 - a1)

Then load these new values of F1 and F2 into the GPSDO via the serial port (using serial commands 'w' and 'x' -- see Serial Port description later in this post).

3.  False Locking:

Given the phase detector's 1 Hz signal at its "reference" input, the PLL will lock to any signal at its other input whose frequency can be expressed as an integer of 1 Hz.

The 10 MHz is divided by 8 to provide a signal of 1,250,000 Hz at this other input of the phase detector. Therefore, the PLL will also lock if, for example, this signal is 1 (or more integer counts) high or low from 1,250,000 Hz, e.g. 1,250,001 Hz or 1,249,999 Hz.

Will false-locking be a problem for my HP 10811-60111 in this GPSDO design?

Note that a 1 Hz delta at the phase detector's input equates to an 8 Hz delta at the VCO output (because N, the VCO division factor, is 8).  This means that the VCO frequency would need to be either 10,000,008 or 9,999,992 Hz.

My HP 10811-60111 oscillator module has a Kv of 0.32 Hz/V.  Plus, there is a large amount of attenuation between the DAC output and the VCO's EFC control pin.  So, if my module's "nominal" frequency has been set  to be very close to 10 MHz (either via potentiometer R1 or with the 10811-60111's screw adjustment), there is no way it can be skewed by the 8 Hz or so that would be required for a false lock.

In other words, worst-case frequency deviation of the VCO, when the DAC is placed at 0x0000 or 0xFFFF, is nowhere near 8 Hz.

(Note:  if the oven is cold the VCO frequency can be very far off, possibly 8 Hz or more .  Never the less, as the VCO's oven warms, the VCO's frequency is forced away from any extremes towards 10 MHz and the point where, irrespective of the value of the DAC's output, my VCO , if properly adjusted, cannot be driven far enough off frequency to create a false lock).

Software and Model Repository:

Arduino code and Simulink models can be found at the following GitHub repository:
https://github.com/k6jca/Shera-Arduino-GPSDO

My  GPSDO posts:

An Arduino-based GPSDO (frequency-locked loop):
http://k6jca.blogspot.com/2016/02/an-arduino-based-gps-disciplined.html

Simulating Brooks Shera Phase-Locked Loop GPSDO (his design that appeared in the July, 1998 issue of QST):
https://k6jca.blogspot.com/2018/12/simulating-brooks-shera-w5ojm-gpsdo.html

(This post) Implementing the Brooks Shera Phase-Locked Loop GPSDO on an Arduino Platform:
https://k6jca.blogspot.com/2019/02/an-arduino-version-of-brooks-sheras.html

Thanks!

Again, thanks to Dick Benson, W1QG, for his expert help and advice!

Standard Caveat:

As always, I might have made a mistake in my equations, assumptions, drawings, or interpretations.  If you see anything you believe to be in error or if anything is confusing, please feel free to contact me or comment below.

And so I should add -- this information is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.