## Monday, February 19, 2018

### "Tandem Match" Transformers and Calculating their Maximum Flux Density

This blog post discusses the "Tandem Match" directional-coupler topology and how to calculate the flux-density in the cores of this directional-coupler's two transformers (voltage-sense transformer and current-sense transformer).

(Note:  In the circuit above the values of the two resistors labeled "R" should be the same as the characteristic-impedance of the transmission-line system being analyzed.  E.g. each R should be 50 ohms if the transmission-lines are 50 ohms.  Also, the load should really be represented by a complex impedance, "Zload".  I am using a resistance "Rload" for simplicity.  And for the same reason I am assuming ideal transformers.)

Why is calculating the cores' flux-density important?

Selecting toroid cores for a Tandem-Match directional coupler depends in large part upon the maximum anticipated flux-density in each transformer's core (the core for the current-sense transformer and the core for the voltage-sense transformer).  Too small of a core and it might burn up from excessive core loss, and too large a core means you are probably wasting money and certainly wasting space.

(In addition to heating failures, cores can also fail from saturation.  But the saturation flux density is significantly higher than the flux-density limit for heating.  Never the less, it should be verified, too).

An important note!  Every tandem-match coupler implementation I've seen uses the same core size for the current-sense transformer and for the voltage-sense transformer.  This need not be!  In fact, the current-sense transformer's core can be significantly smaller than the voltage-sense transformer's core, as I explain in part 8 of my Automatic Antenna Tuner post (please go there for more info).

What are the flux-density limits?

With respect to core heating, iron-powder cores and ferrite cores have, essentially, the same maximum flux limits versus frequency (these are the limits that the DL5SWB Mini Ring Core Calculator uses and which are based on (and interpolated from) Amidon's recommendations.):

But cores can also saturate.  Per Amidon, saturation is a secondary cause of core failure, and the saturation flux-density is typically 2000 gauss for ferrite cores and 5000 gauss for powdered-iron cores.

So the saturation flux-density limit is significantly higher than the "core-heating" maximum flux density limits shown in the table, above.

Calculating Flux Density:

Flux density is a function of the voltage across the "driven winding" of a transformer and follows the relationship below:

(Flux-density equation derivation can be found here.  Note that the 10^8 factor in the numerator is  because Ae is in cm^2, not m^2, and we are calculating Gauss, not Teslas (1 gauss = 0.0001 tesla).)

Note: when calculating flux-density for core-saturation purposes, I would recommend using Vpk rather than Vrms for voltage in the Bmax equation.  In other words, for saturation calculations:

Bmax(saturation) = (Vpk * 10^8)/(4.44 * f * N * Ae)

And Vpk should be calculated under worst-case conditions.  I use maximum forward power on the transmission line with an infinite SWR.

Determining the voltage:

The voltage (Vrms or Vpk in the Bmax equations, above) is easily determined for the voltage-sense transformer -- the voltage across its driven-winding is simply the voltage seen at the directional-coupler's output port.  In the circuit, above, it is Vo.

But the voltage across the "driven winding" of the current-sense transformer isn't as obvious.  However, it can be derived by "reflecting" the impedance connected to current-sense transformer's n-turn secondary into the transformer's primary, and then calculating the primary's voltage by multiplying the primary's current by this reflected impedance.

But what is this reflected impedance?

Back when I designed the Directional Coupler for my Automatic Antenna Tuner (see Tuner, Part 5 and Tuner, Part 8, I made the assumption (through analysis that was simplified to the point of error) that this reflected-impedance was 25/(n^2) ohms.

With this blog post I am revisiting this assumption -- with a bit more analysis I discovered that the current-sense transformer's reflected-impedance is only 25/(n^2) in one case -- when the output load (Rload, in the circuit above) is shorted.  For all other loads, this impedance is greater than 25/(n^2).

I'm going to derive this reflected impedance in this post, and I will also derive the reflected impedance seen at the primary of the voltage-sense transformer, although this latter value isn't important because we already know the voltage across this transformer's driven-winding.

Deriving the Reflected-impedance of the Voltage-sense Transformer:

I'll first derive the reflected-impedance of the voltage-sense transformer's secondary as seen at this transformer's n-turn primary.  I'll use "Loop Analysis" for this derivation.

First, let me define the loops I will use:

Here are the equations for these four loops:

Loop 1:  Vi - Va - Vo = 0

Loop 2:  Va*n - R*(i1/n + i4*n) = 0

Loop 3:  Vo/n - i4*n*r - R*(i4*n + i1/n) = 0

Loop 4:  -Vo + Rload*(i1 - i4) = 0

If I rearrange loop 4's equation to be an expression for i1 in terms of Vo and i4, and then if I substitute this result for the i1 term in the loop 3 equation, I will have an equation in terms of Vo and i4.  Rearranging, I can derive the following relationship for the impedance seen at the driven-winding of the voltage-sense transformer:

(Please feel free to check my work!)

Although this equation isn't important for calculating the voltage-sense transformer's flux density (because we already know the voltage across the transformer's driven-winding -- it is simply Vo), it is interesting to note that when Rload = R (e.g. when both are 50 ohms), this impedance is infinite.  In other words, no current flows through the transformer.

And this result makes sense, because when Rload = R, the directional coupler's output load is "matched" to its design-impedance, and the reflected power (represented by the voltage Vr in the circuit, above) should be 0.  And Vr is only 0 if there is no current flowing through the transformer.

But, although this equation isn't important for calculating the voltage-sense transformer's flux density, it is useful for deriving the reflected-impedance of the current-sense transformer's secondary as seen at its 1-turn primary...

Deriving the Reflected-impedance of the Current-sense Transformer:

Let's rearrange the Vo/i4 equation (just derived) to be an expression for Vo:

This equation for Vo can then be substituted into the loop 4 equation's Vo term.  We can then express i4 in terms of i1:

Substituting this equation for i4 into the loop 2 equation and rearrainging, we can derive the current-sense transformer's primary impedance (reflected from the secondary):

The primary's impedance is a function of Rload, R, and n.  Not a simple relationship, is it?

Now that we have this equation...

How Does Core Flux-Density Vary with SWR?

Voltage-sense Transformer Flux-density:

First, let's look at the voltage-sense transformer.

If we know the forward-power on a transmission line and its SWR, the maximum voltage across the voltage sense transformer can be calculated in a straightforward fashion:

|Vmax| = (1 + |Γ|)*|Vforward|

...where Γ (gamma) is the reflection coefficient (e.g. 0.5 for a 3:1 SWR).

And the voltage-sense transformer's flux density can be calculated as:

Bmax = (|Vmax| * 10^8) / (4.44 * f * N * Ae) gauss

...where |Vmax| = (1 + |Γ|)*|Vforward|.

(Note that Vmax in the equation above should be an RMS value when calculating heating flux-density and a peak-voltage value (i.e. 1.414 * Vrms) when calculating saturation flux-density.)

Current-sense Transformer Flux-density:

But how does the current-sense transformer's voltage vary with SWR?

Let's use my  recently-derived equation for Va/i1 to calculate the resistance reflected into the current-sense transformer's primary, and, assuming a Forward Power and an SWR, calculate the transmission-line currents (both min and max).  I will assume the load is placed directly at the directional coupler's output to keep the load (as seen at the directional coupler's output port) resistive and not reactive, for simplicity of calculation.

I'm going take three different loads as test cases and calculate the flux-density in the current-sense transformer for each:

1.  Rload = 50 ohms (SWR = 1:1) and Forward Power  = 200 watts.

In this case, Vforward on the transmission line is 100 volts RMS and the voltage across the load would also be 100 volts RMS.  And therefore, because the SWR is 1:1, the directional coupler's Vo would be 100 volts RMS and the current on the transmission line (and therefore through the current-sense transformer) will be 2A rms.

2.  Rload = 150 ohms (SWR = 3:1) and Forward Power = 200 watts.

Because the Forward Power hasn't changed, Vforward is still 100 Vrms.

In this case, assuming the load is connected directly to the directional coupler's output port, the voltage on the transmission line, and therefore |Vmax| at this point (the directional/coupler with load attached) can be calculated using

|Vmax| = (1 - |Γ|)*|Vforward|

So, in this case, |Vmax| is 150 Vrms and the current into the load (and thus through the directional coupler) will be 150V/150 ohms, or 1A rms.

3.  Rload = 16.67 ohms (SWR = 3:1) and Forward Power = 200 watts.

Again, because the Forward Power hasn't changed, Vforward is still 100 Vrms.

But because the load is smaller than 50 ohms, |Vmax| at the load is now:

|Vmax| = (1 - |Γ|)*|Vforward|

Note the change to the subtraction of |Γ|!

So |Vmax| at the load is now 50 Vrms and the current into the load, and thus through the directional coupler, is now 2A rms.

(Again, note:  I am assuming that the load is connected directly to the directional coupler's output.  In other words, the load and the directional coupler can be treated as a lumped-element circuit, and thus the current through the directional coupler (i1 in the drawing, above) is, in essence, the same as the current being delivered to the load.  (I am assuming for this discussion that i4 is negligible compared to i1)).

Assuming the core is an FT50, type 43 core (Ae = 0.13 cm^2) and the frequency is 3.5 MHz, I can use these currents and Rload values to calculate the current-sense transformer's flux-density, and we will note a very interesting result:

The current-sense transformer's Flux Density is unchanged, regardless of Rload (i.e. regardless of SWR), assuming Pforward (and thus Vforward) has not changed.

An important caveat!  I have not verified this statement for complex loads.  But it certainly seems to be true for resistive loads.

And regarding the note that Pforward is unchanged, this really means that for the three examples, above, Vforward (the voltage on the transmission line attributable to Pforward) is unchanged, and that |Vmax| and |Vmin| on the transmission line are calculable according to the following equations:

|Vmax| = (1 + |Γ|)*|Vforward|

|Vmin| = (1 - |Γ|)*|Vforward|

So what does this result mean for our current-sense transformer's Flux Density calculation?

Rather than go through the complicated calculation of calculating Va/i1 for the max and min loads, we can instead simply use the 1:1 SWR load (i.e. Rload = R = 50 ohms), which reduces the transformer primary's voltage equation to:

Va = i1 * R/(n^2)

...where i1 is the transmission line current when Rload = R = 50 ohms (for a 50 ohm system).

And, given this value Va, the Flux Density in the current-sense core is:

Bmax = (Va * 10^8) / (4.44 * f * N * Ae) gauss

(Per the equation earlier in this post.)

Combining the two:

Bmax = ((i1 * R/(n^2)) * 10^8) / (4.44 * f * N * Ae) gauss

...where i1 is the current through the current-sense transformer's primary assuming a 1:1 SWR.  In other words, i1 = |Vforward|/Zo, where Zo is the characteristic impedance of the transmission line.

Note that if we design the voltage sense transformer so that it has a high turns ratio (and so that its inductance is high compared to our Rload values), then i4 should be much smaller than i1, and the current through the transformer is essentially the current into the load presented to the directional-coupler's output port.

Conclusion:

When designing a "Tandem-match" directional coupler, the voltage-sense and current-sense transformer cores should each be sized appropriately for their respective anticipated maximum flux-density with respect to heating (calculate assuming long-term average-power usage) and with respect to saturation (calculate assuming peak-power (and thus peak-voltage) under worst-case conditions usage).  For more information (and an example) on determining the conditions to use when calculating these cases, see part 5 of my Automatic Antenna Tuner posts for the conditions I had used.

The flux-density in each transformer can be calculated as follows:

Voltage-sense transformer flux-density:

Bmax = (|Vmax| * 10^8) / (4.44 * f * N * Ae) gauss

...where |Vmax| = (1 + |Γ|)*|Vforward|.

Current-sense transformer flux-density:

Bmax = ((i1 * R/(n^2)) * 10^8) / (4.44 * f * N * Ae) gauss

...where i1 is the current through the current-sense transformer's primary assuming a 1:1 SWR and R represents the value of the resistors terminating the directional-coupler's Vf and Vr ports (typically this value is 50 ohms).

In other words, i1 = |Vforward|/Zo, where Zo is the characteristic impedance of the transmission line.

Note that Vmax and i1 in the equations above should be RMS values when calculating heating flux-density and peak-voltage values (i.e. 1.414 times the RMS value) when calculating saturation flux-density.

And a final note:  I've ignored effects such as winding inductances for the calculations in this post, assuming, for example, that their impedances are sufficiently greater than any resistances in parallel with them.  But a good designer will check and verify these, too.

Links to my Directional Coupler blog posts:

Notes on the Bruene Coupler, Part 2

Notes on the Bruene Coupler, Part 1

Notes on HF Directional Couplers

Building an HF Directional Coupler

Notes on the Bird Wattmeter

Notes on the Monimatch

Notes on the Twin-lead "Twin-Lamp" SWR Indicator

Calculating Flux Density in Tandem-Match Transformers

And some related links from my Auto-Tuner posts:

Part 5:  Directional Coupler Design

Part 6:  Notes on Match Detection

Part 8:  The Build, Phase 2 (Integration of Match Detection)

Standard Caveat:

I might have made a mistake in my code, designs, equations, schematics, models, etc.  If anything looks confusing or wrong to you, please feel free to comment below or send me an email.

Also, I will note:

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.