Sunday, June 10, 2018

Transmit Common-Mode Chokes (1:1 Current Baluns); Measurements and Notes

A New Note (22 December 2020):

The post below describes two ways of using a Vector Network Analyzer (VNA) to determine a Common-Mode Choke's impedance by using either an S11 measurement or an S21 measurement.

As you will see below, the S11 measurement can be quite susceptible to parasitic capacitance, and thus the S21 measurement is preferred.

But even the S21 measurement described below can be affected by parasitic shunt capacitance.  For this reason I recommend measuring Common-Mode impedance using a third method, called the "Y21 Method," which removes the parasitic shunt capacitances from the measurement.

I describe the Y21 method in a later post, which you can find here:

If you are interested in measuring Common-Mode Impedance, I would strongly recommend you take a look at it.

And now, to continue with my original post...


Dipping my big toe into the topic of Transmit Common-mode Chokes (also known as 1:1 current baluns)...

To Verify Choking Effectiveness, Do We measure Impedance, Resistance, or...?

A choke's effectiveness depends upon the system's common-mode impedance at the point where the choke is to be installed.

Is that impedance high?  Is it low?  How reactive is it?  The answers to these questions will determine how well the choke impedes common-mode feedline radiation.  Unfortunately, they can be difficult to answer.

So let's take the worst-case scenario -- I assume that the antenna system's common-mode impedance will have a reactive component to it, and that this unknown reactance will series-resonate with the reactance component of my choke's impedance.

In other words, I assume that any reactance in my choke will be cancelled by the antenna system's common-mode reactance, and that the choke's reactance, rather than impeding common-mode radiation, will actually support it.

Therefore, simply measuring the magnitude of a choke's impedance is not an adequate indicator of its choking effectiveness.  Suppose the impedance is entirely reactive?

Instead, I assume that it is the resistance component of the choke's impedance that will provide the choking action, and thus it is this resistance that I want to maximize.

(This conclusion is discussed by G3TXQ here and by K9YC here.  Also, I'm not going to get too concerned about trying to ensure that Rchoke is greater than Xchoke (Rchoke and Xchoke being the resistance and reactance components of the common-mode choke's impedance).  The effectiveness of Xchoke is going to be unknown, so I will concentrate instead on maximizing Rchoke, the choke's resistance.)

How much Resistance is Adequate?

For a 20 meter dipole G3TXQ recommends, for a choke installed at the feedpoint of the dipole, that the choke's resistance be at least 1000 ohms to keep coax shield common-mode current 30 dB less than dipole current.

On the other hand, W8JI, modeling an 80 meter dipole 80 feet above ground and placing a 5000 ohm common-mode choke between ground and the "balanced" generator driving 1000 watts into the antenna's transmission line, finds that the 5000 ohm choke resistance dissipates 123 watts -- 12.3 percent of the transmit power!

Keep in mind that the greater the value of Rchoke, the less power it will dissipate (in most cases).  This might require that you place a number of chokes in series to achieve sufficient choking resistance to minimize common-mode power dissipation within the chokes.

My next question -- if a choke's resistance is the important parameter, how do we measure it?

Measuring a Common-mode Choke's Resistance and Reactance:

We need to quantify the resistance component of a choke's overall impedance.  This measurement can be made with a Vector Network Analyzer (VNA).

But, if using a VNA, do we make a reflection measurement (S11) or a transmission measurement (S21)?

For RF impedance measurements, HP has used the S11 technique (see their 4191A RF Impedance Analyzer).

However, this analyzer has some issues when measuring the impedance of physically-large chokes (which I will discuss in more detail, below).

G3TXQ uses an S21 measurement technique in which he extracts the resistance and reactance components of a choke's impedance from a vector S21 measurement made with the choke in series between the VNA's port 1 and port 2.

His derivation of the math to convert S21 to impedance is here.  (And I've recapitulated it towards the end of this post).

But which method should I use to measure a choke's impedance?  My 4191A with its S11 technique, or my 8753C using G3TXQ's S21 technique?

Comparing Impedance measured via S11 versus S21:

Let's compare measurements of an inductor consisting of 12 tight turns of 12 AWG wire on a Mix 31 FT-240 core.

First -- the R and X measurement made on my HP 4191A:

(Note: I've added alligator clips to the test fixture to simplify connecting various chokes-under-test to it.  Open-Short-Load calibration is done at the alligator clips to cancel out their effect.  Also, I use Matlab to plot the 4191A's data captured via GPIB).

To measure impedance via S21, I've adapted G3TXQ's technique for my HP 8753C VNA.  Note that the 8753C returns its S21 parameters not in dB format but as the actual measurement values, so G3TXQ's math simplifies a bit (i.e. we do not need to take logs).

Here's the 8753C math, in Matlab:

% Open .S2P File...   
s= '180607 8753C S21 10 ohms series with 150P Cap.s2p';
% ...and get the S-paramters:              [R10_series_150pF_Obj,Coax_Notes,Coax_State] = spar_read(path,s);

% Retrieve the S21 parameters
s21 = R10_series_150pF_Obj.S_Parameters(2,1,:);
% Calculate Magnitude and angle of S21 
abs_s21 = squeeze(abs(s21));
angle_s21 = squeeze(angle(s21));
% Use G3TXQ equations to calculate Resistance and Reactance of Device-Under-Test (DUT)
DUT_R = (100*cos(angle_s21)./abs_s21)-100; 
DUT_X = -(DUT_R + 100).*tan(angle_s21);

And here's the R and X measurement of the same inductor that I measured with my 4191A, this time measured using the S21 method with my HP 8753C:

There is a HUGE difference between the two measurements!  Which one is the correct (or rather, more correct) measurement?  And why the huge difference between the two?


The 4191A's resonant peak is significantly lower in frequency than the peak's frequency in the 8753C measurement.  Could it be that, with the 4191A, the inductor's wiring is capacitively-coupling to the metal (ground) of the 4191A's test fixture (and chassis), thus acting as an additional parallel capacitance whose effect is to lower the inductor's resonant frequency?

To test this theory I did a simple measurement to estimate the inductor winding's capacitance-to-ground by attaching one lead of the inductor to the "hot" terminal of the 4191A and leaving the other inductor lead floating in air.

Here's that measurement.  Note that the capacitance-to-ground is about 2.4 pF.

I found a 2 pF capacitor in my junkbox and connected it in parallel with the inductor connected to the S21 measurement fixture, as shown below:

Here's the new S21 Impedance measurement (with the addition of the parallel 2 pF cap):

Note how closely it matches the 4191A measurement!

I believe this experiment shows (at least to my satisfaction) that the parasitic capacitance-to-ground effects of the 4191A, when measuring devices with a large amount of conductive "surface area" (such as my 12-turns of 12 AWG wire inductor) can greatly skew the impedance measurement, and that the S21 technique is a better way to go.

OK, with that question answered, I wondered how the S21 technique worked with components of "known" values (in an attempt to verify this technique's effectiveness):

First, measuring a high-resistance (5.0K ohms) resistor:

Looks reasonable.  (But note the capacitive reactance -- but this isn't simply a series capacitance -- note that the reactance is doubling (rather than halving) every octave.  In other words, it's acting like an inductor (X increasing with frequency), except the sign is negative, rather than positive.  Why?  I suspect parasitic effects, but I don't know what they are).

Next, measuring a low-resistance (10 ohms)  resistor:

Looks reasonable.

Here are the measurements for a 2 uH inductor and a 150 pF capacitor, both measured separately:

As expected!  Note that the measured reactance values are very close to the reactance one would calculate for a 2 uH inductor and a 150 pF capacitor.

Let's look more closely at the resistance values of these two components:

Note that the capacitance's resistance is negative! (But only slightly).  I suspect this result is due to the resolution and "uncertainty" of the 8753C.

Now let's parallel the 2 uH inductor with the 150 pF cap:

Calculated resonant frequency (assuming the component values are actually what their markings say) should be 9.2 MHz.  Measured resonant frequency is 9.28 MHz.

Pretty close!

Finally, let's put 10 ohms in series with 150 pF:

Pretty close to what they should be!  (And note that the "negative resistance" has disappeared).

So the 8753C S21 technique seems to give fairly accurate measurements of component resistances and reactances (when not near 0 ohms), and it is significantly less sensitive to the "coupling-to-ground" effect of the 4191A.

So, it seems to me that the S21 technique, when used with my 8753C, is a more than satisfactory technique for measuring impedance resistance and reactance (at least up to 30 MHz).

Re-measurement of crossover winding versus normal winding techniques:

In my previous blog post I compared two different winding techniques" the "normal" method of winding a toroidal coil versus the "crossover" technique.  And I concluded that there was little difference between the two.

But these measurements were made with my HP 4191A, and, as I've shown above, this instrument can have significant accuracy issues with components having large conductive surface areas (due to capacitive coupling to ground).

So I thought it worthwhile to repeat my measurements, but this time using the 8753C S21 technique.

First, a recapitulation of the four test windings on an FT-240, Mix 43 core:

And here were my HP 4191A S11-derived measurements:

Repeating the same measurements on my 8753C using S21, the resonant peaks are far to the right, making it difficult to draw any conclusions.

So let's repeat the measurements, but I will use Mix 31 in lieu of Mix 43.  Below are the results:

Again, as before, I do not see a significant difference in winding techniques.

Transmit Balun Measurements:

Now let's measure some transmitting common-mode chokes!

These chokes are all 1:1 current baluns.

I'm going to make these baluns by winding RG8-diameter coax through FT-240 cores of various mixes.  I will limit the maximum number of turns to five, because (from experience) this is the maximum number I can wind through an FT-240 if the coax has an N-connector on its end.

And, I will only plot the choke's resistance (and ignore its reactance), per my discussion of the importance of resistance over reactance earlier in this post.

Five turns through four Mix 31 FT-240 Cores:

Let's look at some chokes made by winding 5 turns of RG-213 through four FT-240 Mix 31 cores.  I'm going to make one measurement with the turns wound close together, and then I will move turns apart and repeat the measurements, as shown below:

Here are the measurements (I've also included a measurement of 5 turns through 3 cores, instead of 4 cores):

You will note that some of the measurements are annotated in the legend box with the term, "with wires".  I noticed when I first was making these measurements that the coax quickly bent away from the coil, and I thought it more representative of how the coax actually ran up to my loop antenna to have the ends stick out straight, rather than bend at right angles.

This change should be reflected in slightly lower measured resonant frequencies due to the increased capacitive coupling of the ends of the coax to the coiled turns, as shown below:

And here are two examples, with the wires attached and less coax "bending":

Four turns on four Mix 31 FT-240 cores (as well as three turns through a Mix 31 clamp):

Next, some more Mix 31 four-core measurements

Various turns through various numbers of Mix 43 FT-240 Cores:

And some Mix 43 measurements:

(Note the difference between the solid-orange and the solid-blue curves.  Is the difference in the peak-frequencies due to winding diameter?  Or in differences in construction?  E.g. are the turns of one choke "tighter" (more close together) than the turns of the other?)

Three turns through three Mix 61 FT-240 cores:

Note the significant differences in the peak frequencies between the three different times I tried to wind three 7-inch loops!

Below is a photo of my third "seven-inch loop attempt", which uses the same windings as my second attempt, but with tie-wraps added at 9 and 3 o'clock to bring the turns in closer together.

Comments on the Measurementss:

To maximize choking action on 80 meters, keep the five turns close together.

As my measurements of the chokes made with Mix 61 cores show, inter-winding capacitance is very important!  Therefore, coax size (and inter-winding spacing) will matter.  Do not assume that you will get the same results if you wind your chokes with RG58 in lieu of the bulkier RG-8.

The three promising baluns for my 80 meter loop:

From the data above, I will add the following three baluns to my 80 meter loop's feedline (with choking designed specifically for 80, 40, and 20 meters).  They will be placed with the 20-meter choke closest to the feedpoint, followed by the 40-meter choke and then the 80-meter choke.

Choke Placement on the Feedline Coax:

As I mentioned above, my three chokes will be placed with the 20-meter choke closest to the feedpoint, followed by the 40-meter choke and then the 80-meter choke.

How far apart should one choke be placed from the other?  I don't have a good answer to that -- you do not want the magnetic-field from the turns of one choke to interact with the turns of another choke.  But, on the other hand, most of the magnetic field for each choke should be concentrated in its ferrite cores, not in the air.

I'm going to try to separate mine by a couple of feet between each choke.  Hopefully that will be sufficient.

And I will note, there are some places you do not want to place a choke!

For example, do not put a choke anywhere near 1/4 wavelength away from the feedpoint (or at odd-integer multiples of 1/4 lambda) -- you will make the coax between the choke and the feedpoint an efficient radiator, as the EZNEC plot (below) of currents (simulating a 20 meter dipole with a choke installed 18 feet away from the feedpoint) shows:

Note that this coax-radiation drops the power of the antenna pattern's main lobe (per the elevation plot) by at least 2.8 dB!

If you have a half-wavelength of coax between transmitter and, say, a dipole antenna feedpoint, it must be choked otherwise it will radiate, as shown below (wire 5 simulates the outer-surface of the coax shield connecting between one side of the antenna feedpoint and ground).

But in this case (and all cases where the feedline length is integer lengths of 1/2 lambda), you can put the choke at either the feedpoint end of the coax or at the transmitter's end.  The latter placement is simulated, below (with wire 5 now disconnected from ground but still connected at the antenna's feedpoint).

In the above, example, though, I would recommend that the choke never the less be placed at the antenna's feedpoint, to cover those cases where the length of the coax is not 1/2 lambda.

Sidebar:  A Recapitulation (but without logarithms) of G3TXQ's Derivation of Impedance from S21 Magnitude and Phase:

(Note that because the HP 8753C S21 data in the Matlab .S2P file is not in dB, I will derive these equations without logs).

Given the diagram above:

At calibration with the Device-Under-Test (i.e. DUT) replaced with a short:

Ical = V/100

(Note: after calibration Ical is a real value, not complex.)

With the DUT (whose impedance Zdut = Rdut + jXdut) inserted into the text fixture, let us define the the total series resistance in the diagram, above:

r = Rdut + 100

And let us define the magnitude of the current flowing in the loop:

|Idut| = V/|Ztotal| = sqrt(r^2 + Xdut^2)

Given that S21 is the ratio of the current (with DUT inserted) to Ical, then:

S21 Magnitude (|S21|) = |Idut|/Ical = 100/sqrt(r^2 + Xdut^2)


S21 Angle ( θ ) = -arctan(Xdut/r)

Solving the latter for Xdut:

Xdut = -r*tan(θ)

and this equation for Xdut substituting into the |S21|:

|S21| = 100/sqrt(r^2 + r^2*(tan(θ))^2)

Now, given that 1/(sqrt(1 + (tan(θ))^2) = cos(θ), we then can rewrite |S21| as:

 |S21| = 100*cos(θ)/r

Solving for r:

r = 100*cos(θ)/|S21|

and thus

Rdut = r-100 = 100*cos(θ)/|S21| - 100


Xdut = -r*tan(θ) = -(Rdut + 100)*tan(θ)

A Simpler Derivation of series-Z calculated from S21:

The above recapitulation of G3TXQ's method of finding Z of a device inserted in series between a VNA's 2 ports is straightforward, but, in my opinion, unnecessarily complex.

Below is a simpler approach.  And again, let's use the same diagram:

First, a given:  S21 = 2*VB / V  (see page 17 of UCSB S-parameter Notes)

where "VB" is the voltage measured at the VNA's Port B, and "V" is the driving source at the left of the diagram, above.  (This definition assumes both Port A's source R and Port B's load R equal Zo, which in this case is 50 ohms).

Next, from inspection of the circuit, above (i.e. a simple voltage divider), we can write the following equation for VB:

VB = V*50/(100+Zdut)

Substituting this equation for VB into the equation for S21 and solving for Zdut, we get:

Zdut = (100/S21) - 100

or, expressed in terms of Zo...

Zdut = Zo*((2/S21) - 2)

And to get resistance and reactance, simply find the real and imaginary parts of Zdut:

Rdut = real(Zdut)
Xdut = imag(Zdut)

That's it!

Useful Choke and Balun Links:

These are links I have found useful.  I've added notes, where appropriate.

Common-Mode Choke Basics (as applied in switching power supplies):

·         Note model of Common Mode Noise Current and Path.

Balun Basics:

Ruthroff, C.L.: Some Broad-Band Transformers. Proceedings of the IRE, Vol 47, No. 8, August 1959, pp. 1337-1342.
·         The classic article on broad-band transformers.

Good primer.

Good discussion of balun basics as well as the W2DU bead balun.

W2FMI (Sevick), “More on the 1:1 Balun”:
Good review of 1:1 balun articles.

DX Engineering (W8JI?): 
·         Discussion of the basics.
·         Myth:  equal conductor currents (if measuring only current magnitude) mean a feedline is balanced, because does not include phase info. 
·         Better to measure current sum by placing a current-sensing clamp over both conductors.
·         Check at two points 1/8 lambda apart.
·         Common rule of thumb of 10-to-one CM impedance to DM impedance is not always correct.

Marki Microwave: Balun Basics Primer

Introductory PPT presentation:
·        States that a 4:1 Guanella Current Balun should be made with 2 cores, but note that his 4:1 Ganella Current Balun schematics (pages 18 and 19) show the polarity dots of the two 1:1 transformers on different ends of each transformer, which is what they would need to be IF WOUND ON A SINGLE CORE.  (If wound on 2 cores, don’t really care which end the two dots are on, as long as each dot within a transformer is on  the same end as its other dot).
·        Also describes the Trask single-core 4:1 Current Balun.

Transmission-Line Transformers:

·         Good App Note on “Design of HFWideband Power Transformers.
·         Section 6.3 describes a 1:4 Impedance Transformer (note that it is 1:4, NOT 4:1).  But wound on a single core, not two cores.
·         Note that Fig. 24 shows a 1:4 transformer (12.5 to 50 ohm) with an additional 1:1 common-mode choke.

·         An informative power-point presentation of Transmission-Line Transformers, including overview as well as PA applications.

·         Tutorial on Transmission Line Transformers.

Measuring an Antenna System's Common-Mode Impedance:

W8JI, Links for Common-Mode Impdedance:
· (Note: short feed line terminals and feed the antenna system as a single wire antenna)
· (part of 4:1 balun discussion)

W9CF “Putting a Balun and a Tuner Together:
·         Modeling a 2-wire transmission line as a 3-terminal device and measuring the three values of that model (Zd, Zc, and Zu).
·         Note analysis of the effectiveness of a balun!

Common-Mode Choke and Balun Implementations:

1.       Excellent power-point presentation.
2.       Resistance always reduces current.  Can get current increases via series resonances.
3.       W2DU choke uses #73 mix beads (#73 a very good choice).
4.       Prefers W2DU bead-choke (mix 73) to W0IYH bead-choke (mix 43).
5.       Choking Z: If well matched and balanced, 5K ohms is plenty, but if severe imbalance, 10K ohms may not be enough (chokes on Windom antennas notorious for failing).  See ppt pages 101 and 102.
6.       Heating of chokes wound with THHN wire: at 1.5 KW they barely got warm:  but no heating of the core itself, all dissipation was in the wire. (Note, this app had no common-mode current).
7.       14 turns of #14 THHN on #31 toroid.  Add second choke for higher power.
8.       Note that DX Engineering 1:4 baluns look like they use two xfrmrs, each made with 6 of the smaller cores and 4 turns each.  But measurements show Z about 1.5K at 4 MHz and 600 ohms at 7 MHz.

·         (Essentially a shorter version of the above power-point presentation).

·         K9YC’s excellent paper on understanding and solving RFI problems.  Note that chapter 6 (page 24) describes baluns and has LOTS of info.  Chapter 7 (page 36) is his Choke Cookbook (manly 1:1 chokes, but good info – e.g. 80/40 meters use 6 turns coax through 5 2.4” #31 cores as well as construction info.

W1HIS Chokes: 
·         |Z| should be t least 1K ohms for noise/RFI issues.
·         Prefers W0IYH choke to W2DU choke (page 12).
·         In an antenna application, you could need much more than one 1000 ohm choke (page 13).
·         Discusses how to identify ferrites at swapmeets.
·         Uses “4:1” turn (secondary to primary step-up of 2).  Page 29
·         His 1:4 current balun uses 6 (FT-240?) toroids, mix 61, with 7 turns through each.  Note that the cores are not stacked 3 high, rather, each transformer in the balun consists of 3 toroids in series.

G3TXQ’s page on chokes:
·         43 or 31 mix for 80 meters?.  17T on 1x or 2xFT240-43 best?
·         Best CM Impedance:  > 8K. (>4K light green, >2k yellow)
·         Want Rs>|XS| -- don’t want reactive resonance.
·         Significant core heating can result if resistance is not sufficient to reduce CM current.
·         Measures Choke impedance via S21

·         Discusses Guenella and Ruthroff 1:4 baluns (called 4:1 here).
·         Introduces his design for an improved 1:4 current balun.
·         Note this design is on a single core, and my simple SPICE simulations do not show it to be better than a 2-transformer implementation.

G3TXQ’s page on baluns:
·         Analysis and discussion of some common balun topologies.
·         Analysis and discussion of the Trask balun.

·         Discussion why 4:1 balun should be on 2 cores, not 1.
·         Towards the end he seems to state that the Trask 4:1 xfrmr is NOT operating in TEM  mode.
·         Note 80 meter dipole example in which 5000 ohms common-mode impedance would not be enough at 1KW operation.
·         Another stressful load:  80 meter doublet on 40 meters.
·         Myth that the only requirement for a balanced transmission line not to radiate is to have equal and opposite currents at one point in the system.  Instead, for a balanced transmission line not to radiate, “line currents have to be exactly equal and opposite and voltage with respect to the outside world has to be exactly equal and opposite at any and all points along the system.”  
·         The 2-core current balun can have uneven heating in the two cores.
·         To minimize balun heating:  CM impedance should be as high as possible on the balun output, and DM impedance should be as low as possible across the balun output.
·         Most antenna systems are probably 1000 ohms or less common mode impedance.

W8JI, Core material section:
·         Note his recommendation of 65, 61, or (in extreme cases) 43 mix.

W8JI,  parallel winding vs. split winding (this is what I call "crossover-winding") of toroids:
·         Makes case that split winding not as good as parallel winding for CM impedance.
·         (Note: Joe Reiser, W1JR, describes the “split” or “crossover” technique of winding a 1:1 balun in “Simple and Efficient Broadband Balun”, Ham Radio, Sept 1978)

·         Interesting discussion of 4:1 Baluns.

·         Discussion of 1:4 baluns.

·         Describes some his 1:1 common-mode chokes

VK6YSF, building a 1:4 balun:
·         Describes construction of a 1:4 Guanella Current Balun.
·         Note that he uses 2 cores.

·         1:1 CM balun (12 turns on T240-K core) had loss above 10 MHz.  Impedance increases from 50 ohms.
·         If high impedance loads (and thus high voltages), coax outer insulator may not have sufficient voltage isolation – arcing.
·         With 100 watts uses eight turns of 100 ohm line on 5xFT240-31 cores.
·         Should have at least 1K ohm common mode rejection.
·         For a 4:1 Current Balun use type 61 or type K material, otherwise can have “a   noticeable shunt resistance which restricts the maximum impedance transformation   ratio that can be obtained.”
·         Measure loss using 2 baluns (preferred) or using ATU to tune to max power.
·         Discusses how to identify ferrites at swapmeets.

Biggins, Paul, Widband Balun Design with Ferrite Cores, (Cal Poly Senior Project).

·         A good collection of Balun posts! 

·         Comments on the Trask balun topology.

Taranovich, Steve, EDN Series reviewing excerpts from the the fifth edition of Sevick’s Transmission Line Transformers  (2014, Authored by Raymond Mack and Jerry Sevick).
·         Discusses Ruthroff and Guenella 1:4 topologies.
·         Note the recommendation that the Guenella topology use 2 cores if Rload is center-tapped to ground.
·         Taranovich builds a Guenella and a Ruthroff 1:4 baluns and tests them.
·         Note that 9 bifilar turns on the Mix 61 FT-240 core.  And only one core (not two) is used.

·         Quanella 4:1 Balun, analysis and construction.

Palomar Engineers: Ferrite Mix Selection.
·         Good discussion of the best applications for various ferrite mixes.  Note:
·         Mix 31 excellent for CM suppression between 1-10 MHz, then about the same as 43 to 250 MHz.
·         Mix 31 not recommended for multi-ratio impedance transformers.
·         Mix 43 excellent for CM chokes from 2-300 MHz.
·         Mix 61 will withstand high power on multi-ratio impedance transformers

Measuring Balun Performance:

Skelton, Ron, W6WO: Measuring HF Balun PerformanceQEX, Nov/Dec. 2010
·         Interesting article on measuring balun CMRR.

W8JI, Balun Testing:
·         Note poor balance for a single-core 4:1 current balun.

VK2OMD (Owen Duffy):
·         Comments on |S21| measurements of common-mode chokes (he does not agree with this technique).

·         Use S21 amplitude and phase to derive the complex impedance of a two-terminal device.  See applied here:

My Balun (and 80-Meter Loop) posts:

Standard Caveat:

I might have made a mistake in my 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 design and any associated 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.

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