"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 will explain later in this post.

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).)

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

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.

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 the 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.

Let's derive this reflected impedance...


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

Below is a model of the "Tandem Match" directional coupler.  I am assuming that the coupled-inductors are ideal transformers.  And I assume that the load is at the output port of the directional-coupler, so that I can used lumped-element analysis.

I'll use Loop-analysis.  Below I 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


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

Let's rearrange loop 4 to be an expression for Vo:

Vo = Rload*(i1 - i4)

This equation for Vo can then be substituted into the loop 3 equation's Vo term.  We can then express i4 in terms of i1:

i4 = i1*(Rload - R) / (2*R*(n^2) + Rload)

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):

Va/i1 = R/(n^2) + R*(Rload - R)/(2*R*(n^2) + Rload)

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

Flux density is simply the current passing through the 1-turn primary winding of this transformer multiplied by this "reflected resistance:"

Bmax = (Imax * (Va/i1) * 10^8) / (4.44 * f * Ae) gauss

Note that 'N' (the number of turns of the current-sense transformer primary) in the original Bmax equation has been set to one (representing the current-sense transformer primary's single turn and so it does not appear in this equation..

Now that we have this equation...


How Does Core Flux-Density Vary with SWR?

As SWR increases from the ideal 1:1, a voltage standing wave and a current standing wave will form on the line.

The directional-coupler might be placed in the transmission line at a point where the voltage of the voltage standing wave is maximum, or where the current of the current standing wave is maximum.  So it is these maximum values that I use as Vmax and Imax when calculating the flux density of these two transformers.

Let's take an example.

Let's say that I'd like to design my directional-coupler so that it could run without overheating on a transmission line where the SWR is 3:1, worst-case, and 200 watts is being dissipated by the load.

To simplify calculations, I'll assume that the load is resistive and that the directional coupler is placed at the load -- this is where the voltage maximum will occur of the load resistance is greater than the transmission line's Zo, and it is where the current maximum will occur if the load resistance is less than the transmission line's Zo.

(Note:  you don't need to assume that the directional coupler is placed at the load to calculate Vmax and Imax.  There is a bit more math, but it isn't difficult. See this blog post:  Useful Transmission Line Equations))

For a given SWR, I will use two load resistances -- the resistance that gives me Vmax, and the resistance that gives me Imax.  For example, for a 3:1 SWR and a transmission line Zo of 50 ohms, these resistances are 150 ohms and 16.67 ohms, respectively.

Given the power being dissipated by the load and the load's resistance, calculating the current and voltage at the directional coupler is straightforward.  These values are then used to calculate the transformer flux densities.

The table below shows the flux densities if the transformer core were an FT-50 Mix 43 core, given load power to be 200 watts and a transformer turns-ratio of 24:1.

At 3.5 MHz the maximum flux density (to prevent overheating) should be limited to 80 Gauss.  You can see that at all SWRs the flux density of the voltage-sense transformer is well above this value.  So the Mix 43 FT-50 core is not a core I would want to use for my voltage-sense transformer.

On the other hand, the flux density of the current-sense transformer is well below the 80 Gauss limit for my range of SWRs.  So this core would be OK to use for the current-sense transformer.

(Note, too, that the current-sense transformer's flux-density only depends upon SWR, not the value of the load resistance at that SWR -- it is the same for the maximum resistance and the minimum resistance loads that have the same SWR.)

What core could I use for my voltage-sense transformer? 

I have a handful of 2643625002 cores in my junkbox.  How would they perform?  Below is a table showing their flux densities versus SWR for this core:

As you can see, the flux density is only greater than the 3.5 MHz 80 Gauss limit when the load resistance is greater than about 130 ohms (SWR of 2.6:1).  So, although it doesn't quite meet my design goal of handling a 3:1 SWR all day long, it's close enough that I would probably change my design goal to be the ability to handle a  2.6:1 SWR all day long.

Calculating the Current in the Voltage-sense Transformer's Primary:

If calculating the heating of the wires used to wind the voltage-sense transformer, we need to know the current flowing through these windings.

Let's assume that the resistance reflected by the voltage-sense transformer's one-turn secondary into its primary is much less than the inductive reactance of the primary.  In other words, the primary's current essentially flows through the resistance.

From the loop equation, above, we can derive an equation for this current:

i4 = Vload * (Rload - R) / (R*Rload*(2*(n^2) + 1))

Where 'n' is the turns ratio (e.g. 24) and R is the characteristic impedance of the transmission line (e.g. 50 ohms).

Note a couple of things about this equation:

1.  If Rload equals R (i.e. if Rload equals 50 ohms), no current flows.  I.e. no current is flowing in the 24-turn primary, and thus no current is flowing in the 1-turn secondary of the voltage sense transformer.

2.  If Rload is less than R, then i4 will be negative.  This just means that the current is flowing in the opposite direction.  

3.  The current through the transformer secondary will be (n*i4).

LTSpice Verification:

To verify my derivations, I created the following LTSpice model (with transformer turns ratio assumed to be 16:1):

The LTSpice simulation voltage results (versus Rload and referenced to Vin) are below, in dB:

I then used Excel and my hand-derived equations to calculate the same voltages (shown in the yellow columns, below):

Note that the Va of this spreadsheet is the same as "Vin-Vo" of the simulation.  If you compare the simulation results to the results based upon my derived equations, you will see that they are the same.

Additional notes on the spreadsheet based upon my derived equations:

1.  I had to derive one additional equation to calculate the results above:

Rx = Vo/i1 = (R*Rload*(1+2*(n^2))/(2*R*(n^2) + Rload)

2.  "Ra" in the spreadsheet is simply Va/i1, the formula for which appears earlier in this post.

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| is the maximum voltage on the transmission line for the maximum SWR, at max power, that I would expect to see.

Current-sense transformer flux-density:

Bmax = ((|Imax| * Rprimary)) * 10^8) / (4.44 * f * N(primary) * Ae) gauss

where Rprimary of the current sense transformer is:

Rprimary = R/(n^2) + R*(Rload - R)/(2*R*(n^2) + Rload)

and where:

• N(primary) is the turns-count of the primary winding, and equals1.
• R equals the Zo of the transmission line (e.g. 50 ohms).
• n is the transformer turns-ratio (e.g. 24).

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 (Tandem Match)

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 and my HF PA posts:

Auto Tuner, Part 5:  Directional Coupler Design

Auto Tuner, Part 6:  Notes on Match Detection

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

HF PA, Part 5: T/R Switching and Output Directional Coupler


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.