It seems that even a basic but sound understanding of transformers challenges lots of hams, and even online experts that have been heard to brag of their qualifications so as to intimidate others who might question their words.

So at ARRL EFHW (hfkits.com) antenna kit transformer – revised design #1 – part 2 I estimated that at a current of 4Arms marked the onset of non-linear B-H response, ie the onset of saturation.

One online expert proposed a method that would rate this transformer at maximum 4^2*50=800W at which magnetic saturation would occur.

The referenced article estimated saturation at more like 17000W.

Some very basic transformer concepts

Let’s talk about some really basic transformer concepts.

The diagram above from Wikepedia shows a rectangular magnetic core with two windings, a primary and secondary on opposite limbs of the core.

Note the phase polarity markings (+ / -) and the direction of (conventional) alternating current.

An example for discussion

Above is an example power transformer for discussion, 240V 50Hz 100VA, n=Vs/Vp=1/20, rated primary current 0.416A, core mass around 800g, estimated core loss at 1W/kg is 0.8W, for a simple explanation, leakage is assumed zero and rated load is assumed purely resistive. Note that an AC power transformer is typically rated for primary voltage, frequency and VA, and that they are operated into the low end of BH saturation, a compromise between weight, dissipation and efficiency (and cost of course).

Let’s assume that it is a good design, and for a first analysis, let’s ignore flux leakage, ie flux due to current in one winding that does not induce voltage in the other winding (does not ‘cut’ the other winding). Let’s analyse it with no load and rated load.

No load

With load load, assume zero current flows in the secondary.

The primary winding acts like an iron cored inductor, when voltage is applied, current flows. The current produces magnetic flux and a voltage induced in the primary winding which by Lenz’s law opposes the voltage that created the current.

The current that flows with no-load is known as the magnetising current, it establishes magnetic flux in the core. In a good design, the magnetising current is small wrt rated current. We can extend that and talk of the corresponding magnetising impedance and the magnetising admittance.

For the example 50Hz transformer shown, the magnitude of magnetising current is 12mA, 2.9% of rated current. Magnetising current is a component of primary current in a loaded transformer.

You might at first think that the magnetising impedance Zm is purely inductive, but that would make it lossless and nothing is lossless. In the example case, the phase of Zm is 74°, Zm=5556+j19213Ω.

Magnetising force is given by \(mmf=n(I_p +0.012 \angle -74°  – \frac{I_s}{n})=n 0.012 \angle -74° \text{At/m}\).

Rated load

So, when rated current flows in the secondary, it induces a voltage component in the primary winding that opposes the voltage induced in the primary by the magnetising current component, so more primary current flows… rated primary current in a zero leakage scenario.

Print the diagram and annotate it with a pencil, and work through Lenz’s law and the direction of current. Sure, I could have done it, but you will learn more by working through the solutions, you will remember it better, and get confidence in your growing analytical capability.

So, the primary current under rated load is the load current divided by the turns ratio, plus the magnetising current. In this example, Ip=0.416-0.012∠-74°.

Why does the core not saturate?

You need to calculate the net magnetising force by adding the primary and secondary magnetising force components. In this case, recalling that Is=Ip/n for our scenario, \(mmf=n(I_p +0.012 \angle -74°  – \frac{I_s}{n})=n 0.012 \angle -74° \text{At/m}\), the same as the no-load magnetising force.

Note that when leakage inductance and winding resistance are factored in, loaded magnetising force is usually a little less than no-load.

If load current does not cause saturation, what does?

Two common causes:

• operating at lower that rated frequency; and
• operating at higher than rated voltage.

Because the B-H response is non-linear, a small increase in primary voltage creates a disproportionate increase in core less (due to higher flux density).

But my guitar amp can be driven to transformer saturation!

Sure, it is being driven to higher voltage and or lower frequencies than the design point, both of which contribute to saturation. Changing the load impedance does not directly cause magnetic saturation.

My ferrite cored EFHW transformer is easily saturated!

Probably not. Naive users often incorrectly blame overheating of the ferrite core beyond the Curie point as magnetic saturation.

Conclusions

• Lenz’s Law is key to understanding.
• Increasing primary voltage or lowering frequency can cause magnetic saturation.
• Increasing load current alone is unlikely to cause magnetic saturation.
• Other non-linear behavior is often wrongly attributed to magnetic saturation.

This article continues on from several articles that discussed the ARRL EFHW kit transformer, apparently made by hfkits.com, and the revised design at ARRL EFHW (hfkits.com) antenna kit transformer – revised design #1 – part 1.

This article presents a saturation calculation.

You will not often see saturation calcs (for reasons that will become apparent), though you will hear uninformed discussion promoting FUD (fear, uncertainty and doubt).

Lets assume that the core is capable of maximum continuous power dissipation of 10W (limited by factors like safe enclosure temperature, human safety, Curie point etc).

Now let’s estimate the magnetising current for 10W of core dissipation with 3t primary

Starting with the expected permeability above…

Gm is the magnetising conductance, the real part of Y above, 0.00231.

We can approximate the primary voltage for 10W core dissipation as \(V=(\frac{P}{G_m})^{0.5}=(\frac{10}{0.00231})^{0.5}=65.8V\) which implies 87W in a 50Ω load.

Magnetising current can be calculated at 10W core dissipation as \(I_m=\frac{V}{|Z|}=\frac{65.8}{222}=0.296A\)

In fact, under load, the net magnetising force may be just a little below 0.296A due to the effects of leakage inductance.

Let’s estimate saturation current for a 3t primary

Lets assume that under load, magnetising force due to current in the secondary offsets most of the magnetising force due to current in the primary and that the net magnetising force is due to magnetising current.

So, let’s solve.

Above is Fair-rite’s published data for their #43 mix (do not assume it applies to pretenders). Let’s take saturation flux density to be 1500 gauss, 0.15T.

Above is a calc of the saturation current for the 3t primary (peak), 4.01 Arms.

Conclusions

The saturation current is 14 times the magnetising current at 10W core dissipation, and is unlikely to be a significant limitation for low duty cycle modes.

This EFHW transformer is loss limited rather than saturation limited for most practical applications.

If you had in mind that this transformer was suited to peak power \(14^2 \cdot 87=17000 \text{ W}\) or more, then it may be driven into saturation.

The NanoVNA can measure and display “phase”, is it useful for antenna optimisation?

Some authors pitch it as the magic metric, the thing they lacked with an ordinary SWR meter.

In a context where it seems most hams do not really have a sound understanding of complex numbers (and phase is one ‘dimension’ of a complex quantity like voltage, current, S parameters, impedance, admittance etc), lets look at it from the outside without getting into complex values (as much as possible).

The modern NanoVNA can display three phase quantities, only two are applicable to one port measurements as would commonly be done on an antenna system:

• s11 phase; and
• s11 Z phase.

Let’s look at a sweep of a real antenna system from the connector that would attach to the transmitter (this is the reference plane), plotting the two phase quantities s11 phase and s11 Z phase, and SWR (VSWR) and a Smith chart presentation of the s11 measurement.

Above is the measurement of the antenna system.

Like most simple antenna systems (this is a dipole, feedline, ATU), the most appropriate optimisation target is SWR, and minimum SWR well above 7.1MHz.

The SWR is 2.568 at the desired frequency, it is poor.

Do either or both of the phase plots give useful information on the problem, and leads to fix it?

s11 phase

s11 phase is -179.62° at the desired frequency (the marker).

Some authors insist optimal s11 phase is zero, some with a little more (and only a little more) knowledgeable insist it should be either 0° or 180°, take your pick. In fact the latter criteria essentially means the load impedance is purely resistive… but let’s deal with that under the more direct measurement s11 phase of Z.

Phase of -179.62° is approximately -180°=180°.

This metric is not very useful in this case.

s11 phase of Z

s11 phase of Z is -0.4°, approximately zero, which means the load impedance is almost purely resistive.

Of itself, s11 phase of Z does not identify the shortcoming.

So, what is the shortcoming?

If SWR is the optimisation target as proposed for this type of antenna, the SWR is poor, and the minimum is at a significantly higher frequency.

The SWR plot is revealing.

For more information, the value of Z is reported for the Smith chart marker as 19.47-j0.140Ω.

The reason that SWR is not 1.0 is that the feed point impedance is not exactly 50+j0Ω, and the main reason is that the real component is quite low at 19.47 and less importantly there is some very small reactance.

So, this provides information that to improve the match, the real component needs to increase significantly, and some minor trimming of the imaginary component.

Let’s make some matching adjustments

The sweep above is after some adjustment seeking to optimise the match.

Overall, the SWR plot shows that SWR is now fairly good at 7.1MHz, the Smith chart shows the marker just left of the prime centre so R is a little low and X is close to zero, the marker detail shows that Z is 45.57-j0.426Ω, so a little more information than the SWR curve, and with more resolution than reading the Smith chart graphically, R is a little low, X is close to zero. This is good information to guide the next matching steps if one wanted to refine the match.

The phase plots are of almost no value.

Conclusions

• Neither of the available s11 derived phase plots are of much use for this matching task.
• The SWR plot gives the best high level indication of the match.
• Knowledge of R and X components of Z can be helpful in understanding more detail of the match and guiding matching adjustments.
• This article has not explained the Smith chart in detail, it requires an understanding of complex quantities, so outside the scope and prerequisite knowledge set out for this article. In fact the Smith chart provides insight well beyond any and all of the other plots.

Jeff, K6JCA, kindly sent me a paper, (Dunsmore 1991) which gives design details for a variation of the common resistive Return Loss Bridge design.

This article expands on the discussion at Return Loss Bridge – some important details, exploring Dunsmore’s design.

Dunsmore’s design

Above is Figure 3a from (Dunsmore 1991).

Exploration

The Dunsmore’s circuit has been rearranged to be similar to that used in my earlier articles.

Above is the rearranged schematic for discussion. It is similar to that used in the earlier article, but three components are renamed, R1, R2 and R3.

To analyse the circuit, we can use the mesh currents method. Mesh currents i1, i2 and i3 are annotated on the schematic.

The mesh equations are easy to write:

This is a system of 3 linear simultaneous equations in three unknowns. In matrix notation:

Lets solve it in Python (since we are going to solve for some different input values) for Vs=1V.

First pass: zs, zref, zd are 50Ω, and zu is short circuit (1e-300 to avoid division by zero) for calibration and 25Ω for measurement. This example uses Dunmore’s 16dB coupling factor to derive the values for r1 and r3.

from scipy.optimize import minimize
import numpy as np
import cmath
import math

zs=50
zref=50
zd=50
c=0.5
c=10**(-16/20)
r3=zref/c-zref
r1=50
r2=zref**2/r3

print(c,r1,r2,r3)

#equations of mesh currents
#vs=(zs+r1+r3)⋅i1−r1⋅i2−r3⋅i3
#0=−r1⋅i1+(r1+r2+zd)⋅i2−zd⋅i3
#0=−r3⋅i1−zd⋅i2+(r3+zd+zu)⋅i3

#s/c cal
zu=1e-300
#solve mesh equations
A=[[zs+r1+r3,-r1,-r3],[-r1,r1+r2+zd,-zd],[-r3,-zd,r3+zd+zu]]
b=[1,0,0]
#print(A)
#print(b)
res=np.linalg.inv(A).dot(b)
#print(res)
vcal=(res[1]-res[2])*zd
#print(vcal)

#check oc
zu=1e300
#solve mesh equations
A=[[zs+r1+r3,-r1,-r3],[-r1,r1+r2+zd,-zd],[-r3,-zd,r3+zd+zu]]
b=[1,0,0]
#print(A)
#print(b)
res=np.linalg.inv(A).dot(b)
#print(res)
vm=(res[1]-res[2])*zd
#print(vm)
rl=-20*math.log10(abs(vm)/abs(vcal))
print('ReturnLoss (dB) {:0.2f}'.format(rl))

#check 25
zu=25
#solve mesh equations
A=[[zs+r1+r3,-r1,-r3],[-r1,r1+r2+zd,-zd],[-r3,-zd,r3+zd+zu]]
b=[1,0,0]
#print(A)
#print(b)
res=np.linalg.inv(A).dot(b)
print(res)
vm=(res[1]-res[2])*zd
#print(vm)
rl=-20*math.log10(abs(vm)/abs(vcal))
print('Zu: {:.2f}, ReturnLoss: (dB) {:.2f}'.format(zu,rl))

This gives calculated rl=9.54dB which is correct. Also ReturnLoss for OC reconciles with the SC calibration.

This result is sensitive to the value of Zs and Zd, changing them alters the result. It may be that there are other combinations of Zs, Zd, r1, r2, r3 that give correct results in all cases.

A common manifestation of design failure is that ReturnLoss of a short circuit termination will be significantly different to an open circuit termination, and the difference may be frequency dependent when combined with imperfections in the bridge.

RF Directional Bridge: Operation versus Source and Detector Impedances

Jeff, K6JCA, published a very interesting article RF Directional Bridge: Operation versus Source and Detector Impedances that is relevant to the wider understanding of Return Loss Bridges.

Conclusions

Dunsmore gives an alternative design, though his formulas still depended on Zs=Zd=r1=Zref.

References

Dunsmore, J. Nov 1991. Simple SMT bridge circuit mimics ultra broadband coupler In RF Design, November 1991: 105-108.

This article continues on from several articles that discussed the ARRL EFHW kit transformer, apparently made by hfkits.com.

This article presents a redesign of the transformer to address many of the issues that give rise to poor performance, and bench measurement of the prototype. Keep in mind that the end objective is an antenna SYSTEM and this is but a component of the system, a first step in understanding the system, particularly losses.

This is simply an experimental prototype, it is not presented as an optimal design, but rather an indication of what might be achieved if one approaches the problem with an open mind instead of simply copying a popular design.

• This prototype uses a Fair-rite 5943003801 core, equivalent in size to a FT240-43, it does not use NMG-H material (“Amidon 43”).
• A 3t primary is used, barely sufficient but a substantial improvement on the original 2t primary.
• The winding configuration is a primary of 3t and secondary of 21t, the secondary is close wound with 0.7mm ECW and the primary is wound using CAT5 cable wire wound over the middle of the secondary winding. This is done for two main reasons:
  • to permit some level of isolation of the ‘radiator’ and ‘counterpoise’ from the coax feed line with a view to reducing common mode current (most effective at the lower frequencies); and
  • to reduce leakage inductance with a view to improving broadband InsertionVSWR.

Leakage inductance

Leakage inductance is the enemy of broadband performance.

Total leakage inductance was assessed by measuring input impedance at 5MHz with the secondary shorted. Total leakage inductance is about 190nH (this slightly overestimates  leakage inductance due to resonance effects). This is two thirds that of the hfkits.com / ARRL original winding layout.

Through measurement with nominal load

The transformer is loaded with the nominal load comprising several 1% SMD resistors and VNA Port 2.

Above is a SimNEC design model, calibrated against measured input impedance with the nominal load.

Above is a pic of the measurement setup, a NanoVNA-H4 is to the right of the equipment shown. Note that the NanoVNA does not correct Port 2 mismatch error.

The transformers are described at Conversion of NOELEC style balun board to 1:1. The setup was SOLIT calibrated, the reference plane was the output side of the grey terminal blocks.

Above is a close up view of the prototype transformer with compensation capacitor. The loads are two 1% 2400Ω in parallel making 1200Ω, one in each secondary leg.

Measurement of the transformer was saved as a .s2p file.

Above is a SimNEC model importing the s2p file and adjusting for the voltage division of the 2400Ω and Port 2 impedance (assumed 50Ω, but uncorrected). The indicated Loss is just a little higher than predicted by the model, keep in mind that ferrites have quite wide tolerances.

Above is a plot of measured ReturnLoss and InsertionVSWR from the .s2p file.

Above is a plot of measured InsertionLoss from the .s2p file, and its components Loss and MismatchLoss. See Measurement of various loss quantities with a VNA for discussion of loss terms.

Common mode impedance

Mention was made that the common mode impedance may help to reduce common mode current at lower frequencies. Zcm will look like a small capacitance below 10MHz, of the order of 10pF for the layout shown. For that reason, I would not deploy an inductive common mode choke near to the transformer, put it at the other end of the feed line.

Conclusions

This is simply an experimental prototype, it is not presented as an optimal design, but rather an indication of what might be achieved if one approaches the problem with an open mind instead of simply copying a popular design.

The revised transformer has substantially better performance than the original: ARRL EFHW (hfkits.com) antenna kit transformer – measurement .

The modifications are mainly about:

• sufficient magnetising impedance; and
• reducing leakage inductance.

Loss remains a little higher than I would, an opportunity for the reader to find further improvement, a learning opportunity!

Jaycar’s LO1238 ferrite toroid is readily available in Australia at low cost and quite suits some HF RF projects.

The published data is near to useless, so a long time ago I measured some samples and created a table of complex permeability of the L15 material which I have used in many models over that time. It did concern me that measured µi was about 25% higher than spec, which is the limit of stated tolerance. Keep in mind that this is Chinese product with scant data published.

I have measured some samples purchased recently, and µi is closer to the specified 1000, so I intend using this new data in future projects.

Above is complex permeability calculated from s11 measurement of a single turn on the LO1238.

Downloads

L15.7z

Derivation of the expression for the unknown impedance in an s21 series through measurement arrives at the following expression:

\(Zu=(Zs+Zl)(\frac{1}{s_{21}}-1)\).

The diagram above is from (Agilent 2009) and illustrates the configuration of a series-through impedance measurement.

It is commonly assumed that Zs+Zl=100Ω, as is done in (Agilent 2001). That might be a reasonable assumption if the VNA correction scheme corrects source and load mismatch, but let’s consider the NanoVNA-H running NanoVNA-D v1.2.29 (Apr 2024) firmware (the current release).

It is good practice to validate a measurement system by measuring a known component. Let’s measure a 200Ω 1% resistor that measures 200.23Ω at DC (it is actually 2 x 0806 100Ω 1% resistors in series.

Above is the test setup, the SDR-kits test board fixture was SOLIT calibrated 1-31MHz using the parts shown at centre of the pic. The fixture is shown with a 200Ω 1% DUT that measures 200.23Ω at DC.

NanoVNA-H v3.3 with NanoVNA-D v1.2.29

Above is a screenshot from the NanoVNA-H measuring the SMD resistor that measures 200.23Ω at DC. The red and green traces use the internal feature to transform an S21 measurement into series thru impedance. The measured value of 204.9-j2.042Ω is significantly different to the expected 200.23Ω.

Above is a plot of calculated DUT assuming the Zs+Zl=100Ω (as does the screenshot above).  The measured values are significantly different to the expected 200.23Ω.

NanoVNA-D v1.2.29 does not correct source and load mismatch, and that is probably the main cause of the apparent error.

Lets turn the measurement around, if we measure some known impedance Zk, we can calculate Zs+Zl:

\(Zs+Zl=\frac{Zu}{(\frac{1}{s_{21}}-1)}\).

#import the network
nws21k=rf.Network(name+'-S21k.s2p')
zu=(50+50)*(1/nws21k.s[:,1,0]-1)
zk=200.23 #precision measurement of DUT resistor at DC
#calculate zs+zl using zu=(zs+zl)(1/s21-1) and zu=zk
zs_zl=zk/(1/nws21k.s[:,1,0]-1)

Above is a snippet of code in Jupyter to import the network and calculate the value of zs_zl (Zs+Zl) inferred by the s21 measurement of the DUT.

Above is a plotting of calculated zs_zl impedance components, real and imaginary, or R,X.

NanoVNA-H4 v4.3 with NanoVNA-D v1.2.40 using ERC

NanoVNA-D v1.2.40 (released 07/09/2024) includes optional Enhanced Response Correction which corrects source mismatch. The feature is turned on for these measurements.

The instrument was calibrated with a LOAD that measured 50.17Ω at DC, this was specified in the calibration.

The NanoVNA-H4 v4.3 has better Port 2 input impedance than the NanoVNA-H used in the previous tests.

All of these measures improve the accuracy of s21 series through measurement of Z.

Above is a screenshot from the NanoVNA-H4 measuring the SMD resistor that measures 200.23Ω at DC. The red and green traces use the internal feature to transform an S21 measurement into series thru impedance. The measured value of 199+j0.594Ω is different to the expected 200.23Ω, about 0.5%.

Above is a plot of calculated DUT assuming the Zs+Zl=100Ω (as does the screenshot above).  The measured values are significantly different to the expected 200.23Ω.

NanoVNA-D v1.2.40 does not correct load mismatch, and that is probably the main cause of the apparent error.

Lets turn the measurement around, if we measure some known impedance Zk, we can calculate Zs+Zl:

\(Zs+Zl=\frac{Zu}{(\frac{1}{s_{21}}-1)}\).

#import the network
nws21k=rf.Network(name+'-S21k.s2p')
zu=(50+50)*(1/nws21k.s[:,1,0]-1)
zk=200.23 #precision measurement of DUT resistor at DC
#calculate zs+zl using zu=(zs+zl)(1/s21-1) and zu=zk
zs_zl=zk/(1/nws21k.s[:,1,0]-1)

Above is a snippet of code in Jupyter to import the network and calculate the value of zs_zl (Zs+Zl) inferred by the s21 measurement of the DUT.

Above is a plotting of calculated zs_zl impedance components, real and imaginary, or R,X. Zs+Zl is considerably closer to ideal (100+j0Ω) than the previous case.

Conclusions

• The analysis here is specific to a NanoVNA-H v3.3 running NanoVNA-D v1.2.29 and NanoVNA-H4 v4.3 running NanoVNA-D v1.2.40 configured as described.
• The formula often given and implemented for transformation of a series through s21 measurement to impedance depends on an assumption that source and load ports have zero mismatch error (eg that any mismatch is corrected).
• NanoVNA-D v1.2.29 does not correct source and load mismatch, the results are mediocre.
• NanoVNA-D v1.2.40 combined with the improved NanoVNA-H4 v4.3 hardware does not correct load mismatch, but other improvements improve the accuracy.

References

• Agilent. Feb 2009. Impedance Measurement 5989-9887EN.
• Agilent. Jul 2001. Advanced impedance measurement capability of the RF I-V method compared to the network analysis method 5988-0728EN.

Two previous articles were desk studies of the the ARRL EFHW kit transformer, apparently made by hfkits.com:

• Thoughts on the ARRL EFHW antenna kit transformer; and
• Thoughts on the ARRL EFHW antenna kit transformer – improvements?

This article documents a build and bench measurement of the component transformer’s performance, but keep in mind that the end objective is an antenna SYSTEM and this is but a component of the system, a first step in understanding the system, particularly losses.

The prototype

Albert, KK7XO, purchased one of these kits from ARRL about 2021, and not satisfied with its performance, set about making some bench measurement of the transformer component.

Above is Albert’s build of the transformer.

The kit parts list is as follows:

Magnetics

The first point to note is that Amidon’s 43 product of recent years has published characteristics that are a copy of National Magnetics Group H material (though they changed the letterhead).

Above is a side by side comparison of the NMG H material datasheet and Amidon 43, the Amidon appears to be a Photoshop treatment of the NMG.

Fair-rite have been a long term manufacturer of a material they designate 43 (which has changed over time), and it has been resold as such by many sellers. NMG H material is somewhat similar, but to imply it is equivalent to Fair-rite 43 might be a reach.

Overall design

I might note at the offset that the design is not original, there are countless articles on the net describing a 2:14 turn on a ‘FT240-43’ transformer for an EFHW using exactly the same winding layout.  As discussed in other articles on this site, 2t is insufficient for operation down to 3.5MHz using Fair-rite #43, even worse for the National Magnetics H material published as Amidon 43.

Measurement

Measurements were made with a NanoVNA-H4. It is not a laboratory grade instrument, but well capable of qualifying this design and build.

Albert performed a two port measurement using the setup above. This places a nominal load on the transformer, and the voltage division of the series resistor and Port 2 input impedance is used to correct the measured s21 figure.

So let’s take the measurements and calibrate a SimNEC model of the transformer.

Above is the model using Fair-rite #43 material calibrated to measured leakage inductance and InsertionVSWR. The reconciliation of model (magenta) with measurement (green) is good. The equivalent Fair-rite part is 5943003801.

Let’s compare measurement to the same model but with Amidon 43 material (NMG-H):

Reconciliation is very poor at lower frequencies, the measured transformer appears to have significantly higher permeability.

Measure core complex permeability

Let’s take a diversion for a moment and put this question to bed.

Above is complex permeability calculated from measurement of a 1t winding on the core used above compared with the Fair-rite #43 2020 published data. Keep in mind that tolerances on ferrite are relatively wide, these two reconcile very very well, there is no doubt in my mind that the material is Fair-rite #43. This questions the seller’s specified parts list.

A similar plot against NMG-H shows a stark difference.

InsertionVSWR and ReturnLoss

So let’s return to the saved .s2p file from the two port measurement.

Above is a plot of bench measurement of ReturnLoss and (related) InsertionVSWR for the transformer with a nominal load.

The InsertionVSWR might look acceptable, but using InsertionVSWR as a single metric is a very limited view.

Above is a plot of InsertionLoss (-|s21|dB), and its components MismatchLoss and (Transmission) Loss. See Measurement of various loss quantities with a VNA for discussion of Loss terms.

Let’s dismiss performance below 7MHz, the plot shows it has insufficient turns.

Importantly, (Transmission) Loss is around 1dB at 7MHz, so 20% of the input power is converted to heat in the core and winding, mostly in the core. If you look back to the first SimNEC screenshot, the model predicts just under 1dB Loss, so measurement reconciles well with the prediction model.

Experience and measurement of Loss and thermographs informs that a transformer of this type in this type of enclosure is not capable of more than about 10W of continuous dissipation in typical deployments, less if it is in direct sun in a hot climate. That means the transformer is not likely to withstand more than about 50W average input power without damage or performance degradation.

Now that might be quite acceptable to some users, gauging by the number of web articles and Youtube videos recommending this, a lot of users apparently.

Credit

Credit to Albert for his interest in understanding these things, careful measurement of the prototype, and preparedness to dismantle the prototype for science.

Summary

So let’s end this article with the results of desk study and measurement:

• though the kit specifications state the core is Amidon #43, the sample kit is almost certain to contain a Fair-rite 43 core which is significantly different;
• two port measurement of the sample of one with nominal load showed around 1dB of Loss at 7MHz;
• MismatchLoss grows rapidly as frequency is reduced below 7MHz suggesting it has insufficient magnetising impedance, a result of insufficient turns;
• the winding configuration is not optimised for leakage inductance;
• popularity is not a good indicator of performance.

Where to from here?

These problems beg a redesign and measure of the transformer… more to follow.

Derivation of the expression for the unknown impedance in an s21 series through measurement arrives at the following expression:

\(Zu=(Zs+Zl)(\frac{1}{s_{21}}-1)\).

The diagram above is from (Agilent 2009) and illustrates the configuration of a series-through impedance measurement.

It is commonly assumed that Zs+Zl=100Ω, as is done in (Agilent 2001). That might be a reasonable assumption if the VNA correction scheme corrects source and load mismatch, but let’s consider the NanoVNA-H running NanoVNA-D v1.2.29 (Apr 2024) firmware (the current release).

It is good practice to validate a measurement system by measuring a known component. Let’s measure a 200Ω 1% resistor that measures 200.23Ω at DC (it is actually 2 x 0806 100Ω 1% resistors in series.

Above is the test setup, the SDR-kits test board fixture was SOLIT calibrated 1-31MHz using the parts shown at centre of the pic. The fixture is shown with a 200Ω 1% DUT that measures 200.23Ω at DC.

Above is a screenshot from the NanoVNA-H measuring the SMD resistor that measures 200.23Ω at DC. The red and green traces use the internal feature to transform an S21 measurement into series thru impedance. The measured value of 204.9-j2.042Ω is significantly different to the expected 200.23Ω.

Above is a plot of calculated DUT assuming the Zs+Zl=100Ω (as does the screenshot above).  The measured values are significantly different to the expected 200.23Ω.

NanoVNA-D v1.2.29 does not correct source and load mismatch, and that is probably the main cause of the apparent error.

Lets turn the measurement around, if we measure some known impedance Zk, we can calculate Zs+Zl:

\(Zs+Zl=\frac{Zu}{(\frac{1}{s_{21}}-1)}\).

#import the network
nws21k=rf.Network(name+'-S21k.s2p')
zu=(50+50)*(1/nws21k.s[:,1,0]-1)
zk=200.23 #precision measurement of DUT resistor at DC
#calculate zs+zl using zu=(zs+zl)(1/s21-1) and zu=zk
zs_zl=zk/(1/nws21k.s[:,1,0]-1)

Above is a snippet of code in Jupyter to import the network and calculate the value of zs_zl (Zs+Zl) inferred by the s21 measurement of the DUT.

Above is a plotting of calculated zs_zl impedance components, real and imaginary, or R,X.

Conclusions

• The analysis here is specific to a NanoVNA-H v3.3 running NanoVNA-D v1.2.29.
• The formula often given and implemented for transformation of a series through s21 measurement to impedance depends on an assumption that source and load ports have zero mismatch error (eg that any mismatch is corrected).
• NanoVNA-D v1.2.29 does not correct source and load mismatch.

References

• Agilent. Feb 2009. Impedance Measurement 5989-9887EN.
• Agilent. Jul 2001. Advanced impedance measurement capability of the RF I-V method compared to the network analysis method 5988-0728EN.

This is a follow up to Thoughts on the ARRL EFHW antenna kit transformer.

The first point to note is that Amidon’s 43 product of recent years is specified identically to National Magnetics Group H material. It is significantly different to Fair-rite’s 43 mix.

Though the parts list specifies an Amidon #43 core, I note that W1VT posted recently:

The ARRL kits don’t use Amidon parts as specified in the Parts List.

That was done as a “service” to those who wanted to know where to get parts for building their own without buying the kit.

The parts are sourced by a European company and shipped as kits to ARRL HQ, which acts as the distributor.

ARRLHQ publishes a Youtube video which shows a label by hfkits.com, and their website also lists Amidon FT240-43. hfkits.com may use a ‘genuine’ Amidon FT240-43 in their kits… this article applies to ‘modern’ Amidon FT240-43.

Trusting hfkits.com and ARRL, lets take the core as a ‘modern’ Amidon FT240-43 (equivalent to NMG-H).

Estimate the power dissipated in the core magnetised to 50V applied

Let’s make a first estimate of the power dissipated in the core with 50V impressed on the nominal 50Ω input winding alone (ie no secondary winding on the core), equivalent to 50W in 50Ω.

We  will calculate the magnetising admittance Gm+jBm, and the power dissipated is given by \(P=V^2G_m\).

Amidon 43 / National Magnetics H case

The first point to note is that Amidon’s 43 product of recent years is sourced from National Magnetics Group, and is their H material. It is not a good equivalent to Fair-rite’s 43 mix.

Let’s make a first estimate of core loss at 3.5MHz.

We can estimate the complex permeability which is needed for the next calculation.

The real part of Y is the magnetising conductance Gm (the inverse of the equivalent parallel resistance).

\(P_{core}=V^2G_m=50^2 \cdot 0.00950=23.8 \text{ W} \).

Amidon 43 / National Magnetics H case with 4t primary

Let’s recalculate with a 4t primary.

The real part of Y is the magnetising conductance Gm (the inverse of the equivalent parallel resistance).

\(P_{core}=V^2G_m=50^2 \cdot 0.00232=5.8 \text{ W} \).

To me, 5.8W core heating due to 50W RF input is a lot more acceptable that the 23.8W with a 2t primary. It is not stunning by an means but borderline acceptable. This configuration might stand 100W FT-8 without overheating (depending on the enclosure, environment etc).

So, can you use too many turns?

Yes, increasing the turns will increase leakage inductance which is a very important, if not most important, constraint on high end Insertion VSWR.

Try it and measure it.

Oh, but I only want to use 40m and up

You can follow the same process and estimate the minimum primary turns that suits your own acceptable core loss criteria.

Background

From time to time, ham radio operators may question whether a section of installed and used coax is still good or significantly below spec and needs replacement.

A very common defect in coax installed outside is ingress of water. The earliest symptoms of water ingress are the result of corrosion of braid and possibly centre conductor, increasing conductor loss and therefore matched line loss (MLL). Any test for this must expose increased MLL to be effective.

Introduction

This article describes a simple but effective test of MLL for coax of known Z0 and length using a suitable one port antenna analyser (or VNA such as the NanoVNA), the nominal Z0 is sufficient to demonstrate cable is good.

The test involves measuring the resistance looking into a resonant length of coax with either an open circuit or short circuit termination.

The concept is to measure MLL and compare it to specification. Defects that increase MLL are not usually narrowband, but will be evidenced over a very wide range of frequencies so measurement at the exact operating frequency is not necessary.

Analyser requirements

Frequency

The analyser will need to cover a suitable frequency range for measurement. For cables for use on HF, I would advise measurement above 10MHz as actual Z0 is closer to nominal Z0. For higher frequencies, choose a range near to the operating frequency.

The analyser needs to be able to measure R and X reasonably accurately at a low impedance or high impedance resonance of the line section with either SC or OC termination.

Access

Access is needed to one end for the analyser, and at the other end for a SC or OC termination (the cable has to be disconnected from the antenna). It is not necessary to connect the ‘far end’ back to the analyser as you would for a two port transmission test.

Connectors

If you use connectors with a loose coupling sleeve (UHF, SMA etc), do not use a loose male connector as OC, connect it to a F-F adapter so nothing is loose.

To get accurate results, all connectors must be secure, clean and properly tightened.

Got all that under control? Let’s measure…

A practical example

The DUT is 10m of quite old budget RG58A/U fitted with crimp BNC connectors. A sample of this cable has previously been measured for braid coverage, it is just 78% so we might expect it to be a little poorer than Belden 8259.

Above, the top is a sample of the cable under test, and lower is Belden 8259. One can see the poorer braid coverage of the test cable… but does that alone condemn it?

The analyser is an AA-600 which uses an N(F) connector so a N(M)-BNC(F) adapter is used. The AA-600 uses a 16bit ADC, so it gives very good accuracy of extreme impedances (which is the case for this test).

Taking my own advice to measure above 10MHz, the third low impedance resonance of the cable section with OC termination is about 15MHz… let’s measure that.

Using the AA-600’s measure All facility and scrolling frequency up and down with the arrow keys until X passes through zero, we get the above measurements. I have taken a screen shot for the article, but no USB connection is needed for a practical measurement, just write down the frequency and R when X=0.

Now using Calculate transmission line Matched Line Loss from Rin of o/c or s/c resonant section we will calculate MLL (see Measuring matched line loss).

So, now we know the frequency of measurement and MLL, we need to find the specification MLL at that frequency using a GOOD line loss calculator.

Specification MLL is 0.06dB/m, we measured 0.07dB/m, it is a little higher than spec, probably a result of the budget construction, and no reason to condemn it, it is probably as good as the day it was made.

Can you use a NanoVNA?

Yes, you can any instrument that can measure R and X at resonance. I have demonstrated the technique using a noise bridge, an antenna impedance bridge (GR1606B), and a NanoVNA.

Conclusions

The technique and formulas used gives a practical simple but effective method of measuring matched line loss using a one port analyser (or any instrument that can measure R and X at resonance).

This article explains some very useful equivalences of very short very mismatched transmission lines. They can be very useful in:

• understanding / explaining /anticipating some measurement errors; and
• applying port extension corrections to VNA measurements where the fixture can be reasonably be approximated as a uniform transmission line.

Port extension commonly applies a measurement correction assuming a section of lossless 50Ω transmission line specified by the resulting propagation time, e-delay, but as explained below, can be used to correct other approximately lossless uniform transmission line sections of other characteristic impedance.

In the following, Z0 is the characteristic impedance of the transmission line, and Zl is the load at the end of that transmission line.

The line section can usually be considered lossless because of its very short length, but whilst its loss might be insignificant, the phase change along the line may be significant and is considered in the following analysis.

The expression for Zin of a lossless transmission line terminated in Zl is:

where:

• Z0 is the characteristic impedance of the line;
• β is the phase velocity of the wave on the transmission line, the imaginary part of the complex wave propagation constant γ (the real part α is zero by virtual of it being assumed lossless);
• l is the length of the line section.

βl is the electrical length or phase length in radians.

β can be calculated:

\(\beta= \frac{2 \pi f}{c_0 v_f}\) which allows the equivalent shunt capacitance to be calculated (an exercise for the reader).

• where f is the frequency;
• c0 is the speed of an EM wave in a vacuum, 299792458m/s; and
• vf is the applicable velocity factor.

The transmission line has a propagation time t:

\(t=\frac{l}{c_0 v_f}\) or rearranged \(l=t c_0 v_f\), or \(t=\frac{3.336}{v_f} \text{ps/mm}\).

Two cases are discussed:

• Zl>>Zo; and
• Zl<<Zo.

Zl>>Zo

Recalling that the expression for Zin of a lossless transmission line terminated in Zl is:

Inverting both sides:

For \(Z_l \gg Z_0 \text{, } Z_0 tan(\beta l) \ll Z_l\) and can be ignored.

Where \(\beta l \lt 0.1\) applies:

For short electrical length, \(\beta l<0.1\), \(tan(\beta l) \approx \beta l\):

So, the effect of this transmission line is to add a small +ve (capacitive) shunt susceptance to Yl.

\(\frac{\beta l}{Z_0}\) is interesting:

Where \(\beta l\frac{50}{Z_0} \lt 0.1\) also applies:

At a given frequency, we can say that \(\frac{\beta l}{Z_0} \propto \frac{t}{Z_0}\) and that \(t_{50} \equiv t_{Z_0}\frac{50}{Z_0}\) and since \(l \propto t \text{, } l_{50} \equiv l_{Z_0}\frac{50}{Z_0}\). These equivalences allow transforming a length or propagation time at actual Z0 into an equivalent length or time at Z0=50. For example if the propagation time of a fixture of 6mm length of air spaced line with Z0=200Ω was 20ps, an equivalent one way e-delay at Z0=50Ω (the instrument reference) of \(t_{50} =20\frac{50}{200}=5 \text{ ps}\) would approximately correct the transmission line effects of the 20ps of 200Ω line. Note that for an s11 correction, the two way e-delay is needed, 10ps in this example.

An important thing to remember is that port extension using e-delay assumes a lossless port extension using transmission line where phase length is proportional to f.

Above is a SimNEC simulation of Zl=10000+j0Ω with 0.025rad 200Ω lossless line backed out by -0.1rad of lossless 50Ω line (comparable to e-delay). The green reversal path lies almost exactly over the magenta path of the original transmission line transformation.

Summary for Zl>>Zo

Zl<<Zo

Recalling that the expression for Zin of a lossless transmission line terminated in Zl is:

For \(Z_l \ll Z_0 \text{, } Z_l tan(\beta l) \ll Z_0\) and can be ignored.

Where \(\beta l \lt 0.1\) applies:

For short electrical length, \(\beta l<0.1\), \(tan(\beta l) \approx \beta l\):

So, the effect of this transmission line is to add a small +ve (inductive) series reactance to Zl.

\(\beta l Z_0\) is interesting:

Where \(\beta l\frac{Z_0}{50} \lt 0.1\) also applies:

At a given frequency, we can say that \(\beta l Z_0 \propto t Z_0\) and that \(t_{50} \equiv t_{Z_0}\frac{Z_0}{50}\) and since \(l \propto t \text{, } l_{50} \equiv l_{Z_0}\frac{Z_0}{50}\). These equivalences allow transforming a length or propagation time at actual Z0 into an equivalent length or time at Z0=50. For example if the propagation time of a fixture of 6mm length of air spaced line with Z0=200Ω was 20ps, an equivalent one way e-delay at Z0=50Ω (the instrument reference) of \(t_{50} =20\frac{200}{50}=80 \text{ ps}\) would approximately correct the transmission line effects of the 20ps of 200Ω line. Note that for an s11 correction, the two way e-delay is needed, 160ps in this example.

An important thing to remember is that port extension using e-delay assumes a lossless port extension using transmission line where phase length is proportional to f.

Above is a SimNEC simulation of Zl=1+j0Ω with 0.1rad lossless 200Ω line backed out by -0.025rad of lossless 50Ω line (comparable to e-delay). The green reversal path lies almost exactly over the magenta path of the original transmission line transformation.

Summary for Zl<<Zo

I have found you can never have enough of these things. It is very convenient to leave some measurement projects set up while work continues on some parallel projects.

Above is the kit as supplied (~$8 on Aliexpress). Note that it does not contain any male turned pin header… more on that later.

I was critical of Alixpress four years ago when I purchased the last one, but things have improved greatly in that time, to the point they are often faster delivery that eBay’s “Australian stock” sellers.

Carefully break off 7 x 7 way pieces of the turned pin female header and ‘dry’ fit them to the board, Now get two more pieces of header that are at least 7 way, and plug them at right angles to the ones already placed to set the spacing. Now solder them to the board (hint: liquid flux makes this job easier.) The ‘donuts’ are quite small, use a tip that gives contact to the donut so that heat is applied to the pin and the ‘tube’ for a good solder joint.

Test the coax connectors on a good male connector to be sure they are not defective… quality of these is poor. I tighten them to 0.8Nm to seat and form the female connector. If the threads bind, chuck them now rather than after you have soldered them in place.

I installed only the mid connectors on this board, I have another which has all the connectors and I have never used the other four. Carefully position and solder the connectors, again liquid flux helps.

The clear plinth does not come in the kit, it is my addition.

Above, the plinths designed in Freecad were cut out of 3mm clear PVC.

They are cut using a single flute 2mm carbide cutter.

You could easily make them with hand tools and a drill. M2.5×6 nylon screws are used to attach the plinth to the hex spacers (supplied), giving the assembly four non-scratch feet.

Now the kit is incomplete. You are going to need some parts you see above built on male turned pin header strip. The kit does contain some 49.9Ω resistors you can use for a LOAD, you will also need an OPEN (centre left) and a SHORT (lower right). Others are for connecting the sections of the test board and THROUGH calibration.

The SMA connector at left is another test fixture which uses the same calibration parts. It can be used directly on the NanoVNA or at the end of a convenient length of cable. I originally made it to use on a Rigexpert AA600, either on a N(M)-SMA(F) adapter, or a short N(M)-SMA(F) cable.

It is bit hard to see the connections on the board when it is populated, so I have made the graphic above, printed it and laminated it for handy reference.

Note that the connections are a little different to the SDR-Kits jig (that they probably copied), in particular C2, C6, E2 and E6 are each not connected to anything.

I have some custom made 300mm long RG400 cables that I use with these, labelled for calibration purposes. IIRC they cost about $30 per pair from RFSupplier.com.

Following on from Common mode choke measurement – length matters #2 which demonstrated that the following test fixture gave invalid results due to the 20mm length of resistor pigtails, the connection length in general terms

Above is a pic of my experimental setup. The resistor on Port 1 is a 10k 1% metal film resistor. The NanoVNA has been SOL calibrated at the Port 1 jack.

These plots should be horizontal lines at R=10k and X=0. They can be improved by shortening the connections, try the experiment yourself, you will learn more than if I show you the results.

If you do the experiment, you will be better placed to critically appraise the test configurations that abound online, especially on Youtube, and form a view as to whether the results are credible.

A thought exercise: do you think the fixture for this choke is good?

Following on from Common mode choke measurement – length matters…

Lots of people have reported experiments to show gross failure of s11 reflection measurement of high impedances such as those encountered measuring common mode chokes.

Above is a chart of a “10k resistor with leads” from (G4AKE 2020), the curve of interest is the s11 curve which he describes as unsuitable. He did not publish enough information to critique his measurement… so I will conduct a similar experiment.

My experiment

Above is a pic of my experimental setup. The resistor on Port 1 is a 10k 1% metal film resistor. The NanoVNA has been SOL calibrated at the Port 1 jack.

Above is a screenshot of the measurement. Quite similar result to that shown in (G4AKE 2020). (Note R, X, s11 phase at the marker.)

Note the s11 phase plot, we see the phase of s11 falling at a uniform rate from 1 to 101MHz.

What should we expect?

s11 for a 10k+j0 DUT should be 0.990050+j0 or 0.990050∠0.000° independent of frequency.

But we see this linearly decreasing phase. It is a big hint, transmission lines do this sort of thing.

So what if we attempt an approximate correction using e-delay.

e-delay of 54.5ps flattens the phase response, and the R and X values are closer to ideal, not perfect, but much closer than the uncompensated plot earlier.

A SimNEC simulation

Above is a SimNEC simulation of my experiment.

Conclusions

• An experiment to duplicate G4AKE’s measurement achieves similar response.
• Drilling down on the detail of the experiment response hints that the resistor pigtails contribute transmission line effect that are the main cause of the poor response.
• An approximate compensation of the transmission line effects gives an impedance measurement that is much better than G4AKE’s recommended s21 series through measurement.
• G4AKE condemns s11 impedance measurement as unsuitable, but there is good reason his fixture was the main reason for poor results… length matters.
• Read widely, question everything.

References

Mar 2020. G4AKE. Measuring high and low impedance at RF.

There must be thousands of Youtube videos of “how to measure a common mode choke” to give a picture of some sort of the test configuration… though most lack important detail… and detail IS important in this case. Likewise there are lots of web pages on the same subject, and some have pics of the test configuration, again mostly lacking important detail.

For the most part, these show test configurations or ‘fixtures’ that might be appropriate for audio frequencies, but are unsuitable at radio frequencies, even at HF.

Connecting wires at radio frequencies are rarely ideal, the introduce some impedance transformation that may or may not be significant to the measurement project at hand. Such connections can be thought of as transmission lines, often mismatched so they have standing waves (meaning the impedance of the load appears to vary along the line.

Let’s take the DUT in my recent article Baluns: you can learn by doing! as an example for discussion.

Let’s take the saved s1p file from a S11 reflection impedance measurement as the example.

Above is a plot of the common mode impedance of the choke, solid line is |Z|, dashed line is R, dotted line is X. This was measured with connecting wires <10mm, see the original article.

Now lets transform that to what we would see if just 100mm of 300Ω lossless line was used to connect the VNA to the balun.

The green curves are what would now be measured by the VNA. Observe the shift in the self resonant frequency (where X passes through zero), observe the shift in frequency of maximum |Z|, and the change in maximum |Z|.

These curves are like they were from different baluns.

Above is an example from a tutorial Youtube video by Fair-rite (a ferrite core manufacturer). Can you work out and draw a schematic of the test fixture? IIRC, the fixture itself was not calibrated, it cannot be because some of it is coax with the shield at one end disconnected.

A clear pic of the detail of all connections and how / where the fixture is calibrated is essential to interpreting any measurements.

A guide I often give people is this:

if you reduce the length of connections and measure a significant difference, then:

• they were too long; and
• they may still be too long.

Iterate until you cannot measure a significant difference.

Make some measurements with different fixture configurations, analyse the results and learn more about fixtures that you will glean from this or probably any written article or Youtube video.

If you believe s21 series through measurement technique or some other technique magically corrects poor fixtures, measure them and critically analyse the results… are they magic?

Read widely, question everything.

The article Baluns: you can learn by doing! presented measurements of a Guanella 1:1 Balun, a common mode choke.

Above is the prototype balun being a Fair-rite 5943003801 (FT240-43) wound with 11t of solid core twisted pair stripped from a CAT5 solid core LAN cable and wound in Reisert cross over style. Note that Amidon #43 (National Magnetics Groups H material) is significantly different to Fair-rite #43.

Above is a plot of the R and X components of Zcm taken from the .s1p file saved during measurement.

Note that it exhibits a self resonance at 13.5MHz. It is not simply an inductor, it is a resonator with an observed self resonant frequency (SRF) of 13.5MHz.

We can predict the impedance of the choke at low frequencies using the complex permeability, core parameters and turns, but as frequency approaches SRF some adjustment needs to be made to accommodate the self resonance. A simple adjustment is to shunt the simple inductor with some value of equivalent capacitance Cse that would cause the simple inductor to resonate at the observed SRF. This measure improves prediction of impedance up to SRF and a little higher depending on your accuracy requirement.

Above is a prediction based on µ, turns etc at the SRF. You will note that the calculated impedance is not consistent with resonance, but it can be used to calculate the Cse that would resonate it.

See also Ferrite permeability interpolations.

You could discover that Cse by inputting a value for Cse and iteratively increasing / decreasing it to find the value where the X part of Z passes through zero… but it is very tedious. You could use a hand calculator to find the value of C with B=0.0001116S.

Or you could use Solve Cse for self resonant inductor.

Above is calculation directly of Cse from calculated R, X at SRF and SRF. Note that you cannot measure R, X of the simple inductor as a practical inductor has the inseparable effects of self resonance, it comes as a bundle.

You could then return to the first calculator and enter the calculated value for Cse and so calculate the expected choke impedance etc.

Above, calculation of the choke using the calculated Cse. The calculated values to not reconcile exactly with measurement, measurement has errors, but worse, ferrites have quite wide tolerances.

Read widely, question everything.

NanoVNA examination of stacked ferrite cores of different mixes studied an example stacked core scenario, presenting measurements of a stack of BN43-202 and BN73-202, 5t wound through both.

The article stated:

They are somewhat similar (but only somewhat) to two series chokes with the same number of turns, so you might expect overlap of the responses.

Above is the measurement of the stacked configuration.

This article compares the stack with measurement of two series chokes of the same number of turns.

Above is the measurement of the series configuration.

They appear quite similar up to perhaps 5MHz or so, lets compare R, X on the same graph.

The magenta and cyan traces are R and X for the stacked configuration.

Below self resonance peaks, they are very similar which hints that the stacked cores are approximately the sum of responses of each of the cores at lowish frequencies, but as resonance comes into play, they differ.

It is clear that the stacked cores do not behave as equivalent chokes in parallel, that proposition is very wooly thinking.

This is measurement of a specific scenario, and the results cannot simply be extended to other scenarios. I have not (yet) been able to create a simple model of the stacked cores above self resonance, I would suggest that for reliability, any proposed configurations be measured. Whilst in this case, the series configuration appears to have a better impedance characteristic, that might not apply to other geometries and mixes.

Build, measure, learn.

Read widely, question everything.

This article reports and analyses a user experiment measuring current in a problem antenna system two wire transmission line.

A common objective with two wire RF transmission lines is current balance, which means at any point along the transmission line, the current in one wire is exactly equal in magnitude and opposite in phase of that in the other wire.

Note that common mode current on feed lines is almost always a standing wave, and differential mode current on two wire feed lines is often a standing wave. Measurements at a single point might not give a complete picture, especially if taken near a minimum for either component.

MFJ-854

The correspondent had measured feed line currents using a MFJ-854.

Above is the MFJ-854. It is a calibrated clamp RF ammeter. The manual does not describe or even mention its application for measuring common mode current.

So, my correspondent had measured the current in each wire of a two wire transmission line, recording 1.50 and 1.51A. He formed the view that since the currents were almost equal, the line was well balanced.

I have not used one of these, I rely on my correspondents guided measurements. (I have used the instrument described at Measuring common mode current extensively.)

MFJ-835

This is the instrument that MFJ sell for showing transmission line balance. One often sees recommendations by owners on social media, it is quite popular.

If the needles cross within the vertical BalancedBarTM the balance is within 10%. If not, you know which line is unbalanced and by how much.

Note the quote uses current like it is a DC current, not an AC current with magnitude and phase.

So, in the scenario mentioned earlier, the needles would deflect to 50% and 50.3% on the 3A scale, the needles would cross right in the middle of the BalancedBarTM, excellent.

… or is it?

One more measurement with the MFJ-854

I asked the chap to not only measure the (magnitude) of the current in each wire, but to pinch the wires together and close the clamp around both and measure the current. The remeasured currents were of 1.50 and 1.51A in each of the two wires, the current in both wires bundled together was 1.2A.

What does this mean?

With a bit of high school maths using the Law of Cosines, we can resolve the three measured currents into common mode and differential mode components.

Above is the result, the current in each wire comprises a differential component of 1.38A and a common mode component of 0.6A. The common mode components in each wire are additive, so the total common mode current on the feed line is 1.2A.

Above is a phasor diagram of I1, I2 and I12, and the components Ic and Id.

Note in this diagram that whilst the magnitude of i1 and i2 are similar, they are not 180° out of phase and that gives rise to the relatively large sum I12 (the total common mode component of I1 and I2).

This is a severe imbalance, sufficient to indicate a significant problem and to prompt a physical and electrical check of the antenna and feed line conductors and insulators.

Repairs were made and the measured result was quite good.

Above are the measurements and calcs.

Above is the phasor diagram… a bit harder to read as there is very little common mode current.

By contrast with the previous case I1 and I2 are almost 180° out of phase and the sum of them, I12 has very small magnitude.

Conclusions

The MFJ-854 can be used effectively for measuring current balance.

Understanding the relative common mode and differential components hinted there was something very wrong in the antenna system.

Forget the MFJ-835 for proving balance. If the needles do not cross in the BalancedBarTM it indicates unbalanced amplitudes. If they do cross in the BalancedBarTM it indicates approximately balanced amplitude, but does not prove the phase relationship is approximately opposite and as shown in this example, is a quite erroneous result.