Tuesday, April 28, 2015

Notes on Antenna Tuners: the T-Network

Previously I'd looked into L-networks (see this post: K6JCA, L-Networks).  I thought I'd now look into T-networks and how they compare to L-networks.  (I'll use Smith charts from time to time.  If you're rusty, here's a quick tutorial:  K6JCA, Smith Chart Tutorial).

L-networks were easily analyzed because they can be defined with a set of equations that, for a given topology, generate a unique set of component values for a given Zload.  And I was able to write an EXCEL spreadsheet to perform these calculations.

In contrast, a T-network with its three variables (the two series components in the arms and the shunt component) will have a range of settings that generate a 1:1 SWR, not a single unique setting.  What I wanted to know: which setting would be the best?

So I decided to use a "brute-force" approach and analyze a T-network by stepping through all of the possible component values (within a range that I'd select) while searching for those combinations that generated results that met a criteria I've selected, such as lowest loss with an SWR equal-to or better-than a target SWR (e.g 1.2:1).

I have no idea how to do this sort of iterative analysis in EXCEL, so I decided to pick up a copy of Matlab and see if I could create an analysis tool using it.  Fortunately, Matlab is available in an inexpensive Home version: Matlab Home Version  (\$149 as of 23 April 2015).  I also used their Smith Chart routine, available in their RF Toolbox  (an additional \$45).

Here's the GUI I created for my analysis tool:

(click on image to enlarge)

This tool allows me to do the following:

1.  Enter in a Zload and manually adjust the network component values (via sliders or typed-in values) for a match.  The program automatically calculates SWR and network power loss.

2.  For the same Zload, press the "Autotune" button and have the program automatically calculate the component values to satisfy user-entered criteria (such as minimal loss).

3.  For a given SWR, step around a Smith chart's "circle of constant SWR" in discrete steps, calculating the component values which satisfy the user-defined criteria and, after stepping around the entire circle, plot the component and power loss values and also store them in a .CSV file.

But before I get into T-networks, let's briefly look at L-network component values from my previous post...

Specifically, let's look at the maximum component values required to match to any load with an SWR of 10:1 at 3.5 MHz:

Note that the maximum values for the lowpass L-networks (LsCp/CpLs) are much less than the values required for the highpass L-networks.

Here's a plot of the component values for the lowpass versus the highpass.  You can really see how the highpass component values skyrocket at certain reflection-coefficient angles.

(click on image to enlarge)

Well, I'm not likely to design a tuner with variable capacitor whose maximum value is 95,000 pF and with a 238 uH variable inductor.  Instead, let's suppose I limit the max capacitance value to, say, 3000 pF and the max inductance to 12 uH:

(click on image to enlarge)

Clearly I could design a low-pass L-network that covers the entire range of 10:1 reflection coefficients.  The highpass L-networks, on the other hand, has some gaps because the values are no longer large enough.

So if I have 3000 pF variable caps, the lowpass L-network is clearly the way to go.

But even 3000 pF is a lot of capacitance, if it's variable.  Yes, I could have a set of capacitors whose values increase in powers of 2 and which could be switched in parallel with each other via relays.  But controlling the relays, although very do-able, becomes more complex than just turning a couple of knobs.

Suppose instead I just use 500 pF variable transmitting caps?  How effective would these L-networks be?

Here's a plot of how both types of networks would handle the range of 10:1 reflection coefficients at 3.5 MHz.

(click on image to enlarge)

Note that now the highpass CsLp/LpCs networks match a wider range of reflection coefficient angles than do the lowpass LsCp/CpLs networks (although there's also a very wide range that cannot be matched by either).

So let's take the fact that the highpass network with 500 pF caps has a wider reflection-coefficient "angular" range than the lowpass network and see if we can improve upon it.  What would happen if we add a second series capacitor (also 500 pF) and make the highpass L into a highpass T network?  Let's look...

T-Networks:

We can think of a T-network as being two L-networks connected back to back:

(Yes, I know.  I'm stating the obvious).

As we saw above, the high-pass L-network required some incredibly large component values if it were to match all possible Zload combinations for, say, an SWR of 10:1 at 3.5 MHz.  But even though a high-pass T-network can be thought of as being two high-pass L-networks back-to-back, it doesn't have the same high component values -- they are much more reasonable.

In other words, changing a highpass L-network to a T network brings the peak component values down to something  much more reasonable.

So, rather than a highpass L-network with a 95,000 pF variable capacitor and a 238 uH variable inductor, let's say I have two much smaller 500 pF variable capacitors and a 10 uH variable inductor and I connect them as a C-L-C highpass T-network.  With these values I can match the entire Smith chart "circle of constant SWR" for a 10:1 SWR at 3.5 MHz.

Here's how the component values change as we rotate around the constant-SWR circle:

(click on image to enlarge)

With the T-Network, we can now cover the entire range of SWR 10:1 reflection coefficients with reasonably sized components (L-networks require higher component values).

But there's a tradeoff...

Power loss in the T-network (compared to the L-network) has increased!

Here is how network power loss (as a percentage of total power) varies for these same settings.  I've included the loss for the highpass L-networks for comparison (blue curve -- but again note that for the L-networks to meet this curve the maximum component values need to be very large).

(click on image to enlarge)

So, although the T-network now lets us match all of the load values with reasonable component values, the tradeoff is the significantly increased power dissipation in the T-network (which peaks between 20 and 25% in the graph above).

My Matlab analysis tool generated the T-network plot above by adjusting components to provide minimum loss along with a match..

But it's not really practical (if it's even possible) to actually tune a tuner for minimum loss.  And we can't just tune for minimum SWR, because a T-Network will tune to a 1:1 SWR over a range of different combinations of the C-L-C components, some of which (as we will later see) can cause significant amounts of power to be wasted within the tuner.

So -- given that tuning for minimum SWR is insufficient, let's add a qualifier...let's tune for minimum SWR with minimum inductance.

How does efficiency look if we include this minimum-inductance requirement?

(click on image to enlarge)

There's a bit more loss when tuning for minimal inductance, but it's not too bad.

It's important to note:

Minimum inductance does not necessarily correlate with minimum loss.

Some authors claim that when a T-network has been tuned for a 1:1 SWR with minimum inductance, this point also corresponds to minimum loss.  This is not true, and you can prove it using W9CF's on-line T-Network Tuner Simulator.

For example, enter into the W9CF simulator a load of 100 + j0 ohms, frequency 3.5 MHz, and set the Cmax values for each capacitor to 2000pF.  If you press the simulator's "Autotune" button, the algorithm will tune for minimum inductance.  The result is a network with an inductance of 3.2 uH and a power-loss of 2.2%.

But if I use the simulator's controls to manually adjust the component values, I can find a 1:1 match with an inductor of 3.7 uH.  And power-loss is now 1.3%, which is obviously lower than the "minimum-inductance" power loss of 2.2%.

You can see the Smith Chart plots of these two networks in the image below:

(click on image to enlarge)

Fortunately, although the hand-tuned network has less loss, it's not less by much, and so tuning for "minimum-inductance" seems to be a good compromise.

Let's look further into power loss.  Is there a way to improve upon the T-network's rather dismal power-loss performance?

Yes, there is.  Let's add two switches that will let us "devolve" from the T-network to one of the two L-network configurations by shorting out either the Cin or Cout capacitor.

(This is not a new idea.  See:  http://www.g3ynh.info/atu/mfj989c.html)

So I'll run my Matlab analyzer to include shorting-out either Cin or Cout (or both), if doing so improves network performance (the analyzer will "tune" for minimum inductance that a meets an SWR of 1.2 or better).

Here are plots of the component values.  Note that the red triangles signify whenever either of the two capacitors is shorted to devolve the T-network into an L-network.

(click on image to enlarge)

Here's the new loss curve (with the previous one for comparison).  Note that we've improved power loss for almost 40% of the SWR circle.

(click on image to enlarge)

You're probably wondering what's happening at 215 degrees where both the Cin and Cout bypass-switches are switched on.  As it happens, at this point the load lies on (or very close to) the Smith chart's "G = 0.02 mhos" circle.  And so the load can be matched with a single shunt inductor, as shown below:

(click on image to enlarge)

For comparison, let's also look at power loss for an SWR circle of 4:1 and compare it to our 10:1 SWR results.

(click on image to enlarge)

Note that the power-loss, although significantly better at the lower SWR, has a smaller angular-range of improvement when devolving to an L-network, compared to the 10:1 SWR.

Here are the component values at an SWR of 4:1:

(click on image to enlarge)

We've been looking at performance with the maximum capacitance set to 500 pF.  What happens if we make Cmax 2000 pF?

Here are the plots of the component values (SWR is still 4:1).  Note that we devolve to the more efficient L-networks much more often.

(click on image to enlarge)

And network power dissipation has also improved significantly:

(click on image to enlarge)

Let's now keep Cmax at 2000 pF but change the SWR back to 10:1.

Component plots:

(click on image to enlarge)

And power-loss plot:

(click on image to enlarge)

Better!

So...

The greater the maximum values of Cin and Cout, the more often we can "devolve" to the more efficient L-network by shorting out either Cin or Cout.

Now let's take a slight digression.  A number of T-tuners use tapped-inductors (tap selected with a switch) rather than continuously-variable inductors, such as roller inductors.

Let's take a look at T-network power loss if inductance were fixed, not variable.

I'll use as an example the EZ-Tuner, a T-network tuner designed by W8ZR and which appeared in QST in 2002.

W8ZR recommends that, for 80 meters, one of these three values of inductance should be used: 10, 4.6, or 3.1 uH.  His design uses two variable caps with a Cmax of about 500 pF.

Let's again assume an SWR of 10:1 at 3.5MHz and look at matching with these fixed values of inductance.  (I don't know what the Q is for the EZ-Tuner components, but let's assume they're the same numbers I've been using:  capacitor Q is 2000 and inductor Q is 100).

Here's the calculated power loss for the three different inductor values:

(click on image to enlarge)

Note that, although the 10 uH inductor allows the network to be matched over the entire range of "10:1 SWR" loads, internal power loss can almost reach 50 percent (assuming inductor Q is 100)!

(Of course, from the plots above, the user should use the 3.1 uH inductor if the Gamma angle is between about 30 and 180 degrees, the 4.6 uH inductor from 180 degrees to about 290 degrees, and the 10 uH inductor everywhere else. But if the user isn't paying attention to the inductance they've selected, a large amount of power could be lost in the tuner!)

If inductor Q were changed from 100 to 250, power-loss (for 10uH) is roughly halved.  Much better, but still not great.

(click on image to enlarge)

So, the points of this digression are several.  If using a high-pass (C-L-C) T-network tuner:

1.  You may find find a match yet have terrible power loss within the tuner if you choose the wrong value of inductance to match with.

2.  Rule of thumb -- use the smallest value of inductance possible to provide the match.

3.  If the Q's are equivalent, go with a continuously-variable inductor rather than a tapped-inductor (but note that a roller inductor might have significantly worse Q).

OK, end of the digression.  Back to T-networks...

Note that the value of the 2000 pF caps used in the analyses above is not that far from 3000 pF.  What's significant about 3000 pF?  Well, if Cmax is 3000pf, we can skip T-networks entirely and instead use a simple lowpass L-network:

Here are its component values (we saw these at the beginning of this post, too).

(click on image to enlarge)

Here's its power dissipation, compared against a T-network (with shorting-switches and with 3000 pF, not 2000 pF, caps).

(click on image to enlarge)

Not too different, but...

1.  The lowpass L-network only requires one 3000 pF variable cap, not two.

2.  The lowpass L-network will tune to a unique solution.  This is very important.  There's no worry about tuning to a match with a value of inductance that causes a significant amount of power to be lost within the tuner.

So,in my opinion, given a choice between highpass T and low-pass L networks, choose the lowpass L network, if you can manage to get enough capacitance.

Here are the lowpass L-network's maximum component values required to match loads with 10:1 SWRs on 160, 80, and 10 meters:

Conclusions:

If designing an antenna tuner, my first choice would be a lowpass L-network that's switchable between LsCp and CpLs configurations.  Assuming I could get the appropriate component values, this network minimizes loss and tunes to a single, unique, matching solution.

My second choice would be a highpass T-network:
• I'd want its capacitors to be as large as possible.  And high-Q, of course.
• I'd want either a continuously-variable inductor or, if a tapped-inductor, one with many taps.  And again, maximizing inductor Q (especially the inductor's Q) is very important!  (Note that roller-inductors can have low Q.  Check out the discussion here:  http://www.w8ji.com/antenna_tuners.htm )
• There should be switches to selectively short-out Cin or Cout, thus "devolving" the T-network to a more efficient L-network whenever possible.

The T-network should be tuned so that the desired SWR is reached with minimum amount of inductance, to ensure that the power-loss within the network is near its minimum (note that minimum inductance does not necessarily equate to minimum loss, but it's often close).

Other Thoughts and Comments:

1.  Component Q's:

Please note that the component Q's I used for my calculations are for illustration only.   Inductor Q's might be higher; capacitor Q's might be lower.

For example, at 3.5 MHz, a 2500 pf cap would need to have an ESR of 0.009 ohms (!!!) if it were to have a Q of 2000.  And this resistance could include variable-capacitor wiper contacts, or relay (or switch) contact resistance.  So care must be taken in the tuner design to maintain a high capacitor Q.

Most likely, though, inductor Q will be significantly less than capacitor Q.  In this case, inductor dissipation will dominate the total power dissipation for worse-case loads.  It is only when capacitor Q (for the case of L-networks) approaches inductor Q that the worst-case dissipation in each component become equivalent, as you can see in the figure below, which plots L-network power dissipation for various capacitor Q's, all with the inductor Q fixed at 100.

(click on image to enlarge)
(Note that the plots above are for a highpass L-Network.  A lowpass L-Network performs similarly as capacitor Q is changed).

2.  Current and Peak Voltage in a Lowpass L-Network.

If I were to build a lowpass L-network Antenna Tuner, I was curious what would be the RMS-Current-Through and Peak-Voltage-Across the inductor and capacitor.

Here's a plot for loads of SWR = 10.  Note that Power Into the network is assumed to be 1000 watts.

(click on image to enlarge)
Note that peak voltage, worst-case, is just a bit more than1000 volts.  And RMS current, worst case, is just a bit more than 15 amps.  These values will be lower for loads with lower SWR's.

Resources, articles:

"The E-Z Tuner", James C Garland, W8ZR, QST Apr 2002 p40-43. May 2002 p28-34, Jun 2002 p33-36.

"Understanding the T-tuner (C-L-C) Transmatch" William E Sabin, W0IYH, QEX, Dec. 1997, p13-21.
Suggests that shorting out C1 and C2 will improve efficiency in some situations.

"Save Your Tuner for Two Pence", Tony Preedy, G3LNP, Rad Com, May 2000, p20-25.  Another article which selectively shorts the T-networks capacitors.

"Getting the Most Out of Your T-Network Antenna Tuner", Andrew Griffith, W4ULD, QST, Jan. 1995, p44-47.    http://www.arrl.org/files/file/Technology/tis/info/pdf/9501046.pdf

T-network sites:

W9CF T-network Simulator:  http://home.sandiego.edu/~ekim/e194rfs01/jwmatcher/matcher2.html    (note that the simulator does not always result in lowest power dissipation)

W8JI site:
http://www.w8ji.com/antenna_tuners.htm (note discussion on roller inductor Q)

G3YNH site (lots of great information):
http://www.g3ynh.info/atu/mfj989c.html

T-network equations: http://home.earthlink.net/~w6rmk/math/wyedelta.htm  (I'm not a big fan of equations such as these.  They presume resistive source and load.  The first is often unknown, the latter usually reactive.  But some may find them useful).

Network synthesis site: http://home.sandiego.edu/~ekim/e194rfs01/jwmatcher/matcher2.html  Note that T-network solutions require that you also input a Q value.

VK5BR:  http://users.tpg.com.au/users/ldbutler/Approach_Ant_Tuning.htm

My Related Posts:

A quick tutorial on Smith Chart basics:
http://k6jca.blogspot.com/2015/03/a-brief-tutorial-on-smith-charts.html

Plotting Smith Chart Data in 3-D:
http://k6jca.blogspot.com/2018/09/plotting-3-d-smith-charts-with-matlab.html

The L-network:
http://k6jca.blogspot.com/2015/03/notes-on-antenna-tuners-l-network-and.html

A correction to the usual L-network design constraints:
http://k6jca.blogspot.com/2015/04/revisiting-l-network-equations-and.html

Calculating L-Network values when the components are lossy:
http://k6jca.blogspot.com/2018/09/l-networks-new-equations-for-better.html

A look at highpass T-Networks:
http://k6jca.blogspot.com/2015/04/notes-on-antenna-tuners-t-network-part-1.html

More on the W8ZR EZ-Tuner:
http://k6jca.blogspot.com/2015/05/notes-on-antenna-tuners-more-on-w8zr-ez.html  (Note that this tuner is also discussed in the highpass T-Network post).

The Elecraft KAT-500:
http://k6jca.blogspot.com/2015/05/notes-on-antenna-tuners-elecraft-kat500.html

The Nye Viking MB-V-A tuner and the Rohde Coupler:
http://k6jca.blogspot.com/2015/05/notes-on-antenna-tuners-nye-viking-mb-v.html

The Drake MN-4 Tuner:
http://k6jca.blogspot.com/2018/08/notes-on-antenna-tuners-drake-mn-4.html

Measuring a Tuner's "Match-Space":

Measuring Tuner Power Loss:

Other interesting sites:

DJ0IP Site: http://www.dj0ip.de/antenna-matchboxes/

W0QE: Antenna Tunera Paper: http://www.w0qe.com/Papers/Antenna_Tuners.pdf

Standard Caveat:

I could have easily made a mistake in the above post.  If anything appears wrong or is confusing, please let me know.  Thanks!

- Jeff, k6jca

Tuesday, April 7, 2015

Revisiting L-Network Equations and Constraints

[ I first mentioned this topic within my much longer post on L-Networks.  Because it addresses a commonly-held misconception regarding the application of L-Networks, I thought it deserved a post of its own.

- Jeff, K6JCA  7 April 2015 ]

The selection of one L-network configuration over another (series-parallel versus parallel-series) is based upon a relationship between the load impedance and the "target" resistance that the load is to be transformed to.

This relationship typically assumes a resistive load, and thus the selection constraints are also based upon a resistive load, rather than being based upon a more universal "complex" load.

These constraints are commonly defined to be:
• Apply parallel-series networks if Rload < Rs
• Apply series-parallel networks if Rload > Rs
(Note that some authors use Zo in lieu of Rs as the target resistance.)

(click on image to enlarge)

But suppose the load is complex (as it often is)...the constraints really should be:
• Apply parallel-series networks if Rload < Rs
• Apply series-parallel networks if Gload < 1/Rs

Note that the first constraint has not changed.  But the second constraint is now in terms of load conductance, not load resistance.

These new constraints also allow us to define network usage slightly differently:
• If Rload > Rs:  Can only use series-parallel networks (LsCp or CsLp).
• If Gload > 1/Rs:  Can only use parallel-series networks (LpCs or CpLs).
• If Rload < Rs and Gload < 1/Rs:  Can use either series-parallel or parallel-series networks (LsLp, LpLs, CsLp, or CpLs if Xload is negative;  CsCp, CpCs, LpCs, or LsCp if Xload is positive).

Note, too, that Gload is defined to be:

Using this equation, we can do a quick "sanity" check.  If our load is resistive, then Xload  = 0, and the equation above reduces to Gload = 1/Rload.  If we then substitute 1/Rload for the Gload term in our constraint equation, Gload < 1/Rs, and rearrange, we end up with the original constraint: Rload  > Rs.  So, for the subset of loads that are solely resistive, our new, more general constraint, is completely compatible with the original constraint.

These new constraints are the result of a set of L-network design equations that can be used with complex loads.  These equations are not limited to resistive loads (unlike design equations utilizing Q).  And they are universally applicable to all eight L-network configurations:

Let's now look at these design equations and their derivation.  And in doing so, we will see how we derive the new constraints...

L-Network Design Equations

Parallel-Series Network Equations:

Let's first derive the equations for the parallel-series L-network.  We can draw a general form of this network with the series-element a reactance X and the parallel element a susceptance B.

The load is complex: Zload = Rload + jXload And we want to transform it to be Rs.

The impedance looking into the network, Zin, is:

Zin = 1 / (jB + (1 / (Rload + j(Xload + X))))

Let's expand this further...

Zin = (Rload + j(Xload + X)) / (1 + jBRload - B(Xload + X))

Our goal is to select B and X such that Zin = Rs.  In other words:

Rs = (Rload + j(Xload + X)) / (1 + jBRload - B(Xload + X))

We can expand this equation and then separate out the equation's real terms and the imaginary terms.  If we do, we get the following:

Equating the Real terms:

Rs - Rload = BRsXload + BRsX

Equating the Imaginary terms:

We can take that last equation, plug it into the X term of the previous equation, and solve for B.  The resulting equation will be in quadratic form and therefore has two solutions.  Each of these two solutions for B can then be plugged into the equation for X (the last equation above).  The final results are two B, X pairs:

Solve for B and X, then convert the resulting reactances and susceptances into actual inductors and capacitors.  Note:
• If B is negative, the component will be a shunt inductor
• If B is positive, the component will be a shunt capacitor
• If X is negative, the component will be a series capacitor
• If X is positive, the component will be a series inductor
Thus, the signs of B and X will determine what kind of network a particular B,X pair creates.  The choices are:
• shunt-inductor, series-inductor (LpLs)
• shunt-inductor, series-capacitor (LpCs)
• shunt-capacitor, series-capacitor (CpCs)
• shunt-capacitor, series-inductor (CpLs)

Deriving the constraint  Rload < Rs:

Note the term Rs-Rload under the square root sign.  This term must be positive for the answer to exist.

And thus, setting this term to be greater than 0 and then rearranging it, we have our constraint for parallel-series networks:

Let's now look at series-parallel networks...

Series-Parallel Network Equations:

In a fashion similar to our analysis of parallel-series networks, we can draw a general form of the series-parallel network, with the series-element a reactance X and the parallel element a susceptance B.

Again, Zload = Rload + jXload And again we want to transform the load impedance to be Rs.

The impedance looking into the network, Zin, is:

Zin = jX +  1 / (jB + (1 / (Rload + jXload))

Again, I'll set Zin equal to our target resistance, Rs.  Expanding the above equation and then separating out the resulting real and imaginary terms, we get:

Real terms:

Imaginary terms:

The real-term equation can also be solved for X:

Through X, we can now equate this last equation to the previous one and solve for B.  Again, the results for B are in the form of a quadratic equation whose solutions are two roots, as shown in the equation below.  These two roots, when calculated, are then substituted into the second equation below (which is our equation for X, from above):

The results are again two pairs of susceptances and reactances that can then be converted into actual component values.  Note:
• If B is negative, the component will be a shunt inductor
• If B is positive, the component will be a shunt capacitor
• If X is negative, the component will be a series capacitor
• If X is positive, the component will be a series inductor
Thus, the signs of B and X will determine what kind of network a particular B,X pair creates.  The choices are:
• series-inductor, shunt-inductor (LsLp)
• series-inductor, shunt-capacitor (LsCp)
• series-capacitor, shunt-capacitor (CsCp)
• series-capacitor, shunt-inductor (CsLp)

Deriving the constraint Gload < 1/Rs:

Note the term under the second square root sign in the equation for  B

This term must be positive for the equations' results to exist.

And thus, setting this equation to be greater than 0 and then rearranging, we have our new constraint for parallel-series networks:

That's it for deriving design equations and the constraints defining when they can be used.  For the sake of brevity I've left out quite a few steps.  Here's what my derivations actually look like...

Sanity Check!

Let's do a quick test to verify if the new constraint for series-parallel networks is really true.  Let Zload = 25 + j50 and our target resistance Rs = 50 ohms.

Clearly, Rload (25 ohms) is less than Rs (50 ohms), so if the constraint "Rload > Rs" is to be believed, we shouldn't use a series-parallel network.  Yet, here's a series-parallel solution (LsCp) plotted on a Smith Chart:

(click on image to enlarge)

This same result can be verified with an on-line Impedance Matching Network Designer.

Other Fun Facts:

Given the constraint Rload < Rs, parallel-series networks can be used to match impedances in the yellow area of the Smith Chart below.

(click on image to enlarge)

And given the new constraint Gload < 1/Rs, series-parallel networks can be used to match impedances in the yellow area of the Smith Chart below.

(click on image to enlarge)

Note the similarity in form between the two charts above.  You do not get this symmetry if the constraint for series-parallel networks is limited to Rload > Rs.

This similarity in form is interesting given that the equations for series-parallel networks are so much more complex than the equations for parallel-series networks.  Which got me wondering -- is there a way to express the series-parallel equations with the same simplicity as the parallel-series equations?

I found that if we express the network in terms of admittances, then yes, there is.  Let's use the following diagram:

This drawing is equivalent to the series-parallel diagram earlier in the post, except now the load is expressed as an admittance.

And rather than writing an equation for Zin, the impedance looking into the network, I'll instead write an equation for Yin, the admittance looking into the network.

The equation for Yin is:

Yin = 1 / (jX + 1/(Gload + j(Bload + B)))

If I expand this equation and then equate the imaginary terms on either side of the "equals" sign and then do the same with the real terms in the equation, I get the following two equations:

Real terms:

Yin - Gload = YinX(Bload + B)

Imaginary terms:

If I plug this last equation into the previous and solve for X, we'll again get a quadratic equation.  The equations representing the two roots of this quadratic equation can then be plugged into the X term in the equation above for B.  The results will be two X,B pairs:

(Note the similarity in form with the parallel-series equations derived earlier!)

Recall that our target impedance for Zin was the resistance Rs.  Yin is just the reciprocal of Zin and so its equivalent target is the conductance 1/Rs -- when solving these equations, this is the value you'd substitute into Yin.

And again, for the solutions to exist, the term under the square root sign must be positive.  So the equations' constraint is:

Not surprisingly, this is the same constraint that we found for the earlier series-parallel network analysis.  But note how much simpler the design equations are.

If you'd like to verify that both sets of equations produce the same results, keep in mind that:

and

Using a spreadsheet, I did a quick test using Zload = 25 + j50 and a target resistance, Rs, of 50 ohms:

(click on image to enlarge)

As you can see, the two different sets of series-parallel equations return the same results.

Even More Fun Facts

Given the new constraint for series-parallel networks (Gload < 1/Rs), these equations allow the entire Smith Chart to now be analytically "matched" to a target impedance with just a pair of networks.

If physically realized, a simple toggle-switch can select the appropriate network for the appropriate Smith Chart region.  For example, a switch-selectable LsCp -- CpLs network pair can span the Smith Chart, as can a switch-selectable CsLp -- LpCs pair of networks:

Here are the regions covered by the CsLp -- LpCs pair.  One network spans the yellow region, the other spans the non-shaded region.

(click on image to enlarge)

And here are the regions covered by the LsCp -- CpLs pair:

(click on image to enlarge)

If I'd used the original constraints (specifically Rload > Rs for series-parallel networks) to select networks for, say, a computer-driven analysis of matching networks as the load is rotated around a Smith Chart's "circle of constant SWR" (as I did in this post: L-Networks ), there would be a large swath of the circle where I would be missing networks.

For example, suppose I wanted to analyze component values required for a CsLp -- LpCs switch-selectable network.  I won't have calculated matching networks for the entire bottom part of the chart because, if I use the original constraints, neither network will create a match for impedances within that area!

(Click on image to enlarge)

(Note the same issue exists for the LsCp/CpLs duo, but in this case the top part of the Smith Chart cannot be matched by the CpLs configuration.)

Of course, in actual life this isn't the case. If I built the networks, I would find that they covered the entire constant-swr circle.  The fact that an analysis missed part of the circle is solely due to the incorrect constraint used for the analysis.  And if we instead used the correct constraint, Gload < 1/Rs, our analysis would be fine.

References:

The following websites have similar derivations for the L-network design equations.

http://www.engr.uky.edu/~gedney/courses/ee523/notes/Set4_Matching%20Networks.pdf

http://whites.sdsmt.edu/classes/ee481/notes/481Lecture7.pdf

http://www.ece.msstate.edu/~donohoe/ece4333notes5.pdf

http://www.ittc.ku.edu/~jstiles/723/handouts/section_5_1_Matching_with_Lumped_Elements_package.pdf

Note however, all of these websites define the use of series-parallel L-networks only when Rload > Rs (or Zo in lieu of Rs, per author preference).

My Related Posts:

A quick tutorial on Smith Chart basics:
http://k6jca.blogspot.com/2015/03/a-brief-tutorial-on-smith-charts.html

Plotting Smith Chart Data in 3-D:
http://k6jca.blogspot.com/2018/09/plotting-3-d-smith-charts-with-matlab.html

The L-network:
http://k6jca.blogspot.com/2015/03/notes-on-antenna-tuners-l-network-and.html

A correction to the usual L-network design constraints:
http://k6jca.blogspot.com/2015/04/revisiting-l-network-equations-and.html

Calculating L-Network values when the components are lossy:
http://k6jca.blogspot.com/2018/09/l-networks-new-equations-for-better.html

A look at highpass T-Networks:
http://k6jca.blogspot.com/2015/04/notes-on-antenna-tuners-t-network-part-1.html

More on the W8ZR EZ-Tuner:
http://k6jca.blogspot.com/2015/05/notes-on-antenna-tuners-more-on-w8zr-ez.html  (Note that this tuner is also discussed in the highpass T-Network post).

The Elecraft KAT-500:
http://k6jca.blogspot.com/2015/05/notes-on-antenna-tuners-elecraft-kat500.html

The Nye Viking MB-V-A tuner and the Rohde Coupler:
http://k6jca.blogspot.com/2015/05/notes-on-antenna-tuners-nye-viking-mb-v.html

The Drake MN-4 Tuner:
http://k6jca.blogspot.com/2018/08/notes-on-antenna-tuners-drake-mn-4.html

Measuring a Tuner's "Match-Space":

Measuring Tuner Power Loss:

Standard Caveat:
I could have easily made a mistake anywhere in this blog post.  If something looks wrong, is unclear, or doesn't make sense, please feel free to contact me!