This article presents an alternative design for the transformer for a coupler for a 5W transmitter.

The above circuit is from (Grebenkemper 1987) and is an embodiment of (Sontheimer 1966). In their various forms, this family of couplers have one or sometimes two transformers with their primary in shunt with the through line. Let’s focus on transformer T2. It samples the though line RF voltage, and its magnetising impedance and transformed load appear in shunt with the through line. T2’s load is usually insignificant, but its magnetising impedance is significant and is potentially and often a cause of:

- high InsertionVSWR;
- high core loss;

In the case of couplers embedded in a transmitter, the InsertionVSWR is hidden and frustrates obtaining expected power and PA efficiency.

Let’s model the effect of the magnetising impedance of T2 on both of these parameters using an alternative design.

The alternative design is guided by the concept that small is beautiful when it comes to design of broadband ferrite transformers with nearly ideal response. Of course practical transformers are limited by power handling considerations, but low power transmitters permit quite small transformers.

The alternative design uses a Fair-rite 5943000101 (FT23-43), note that an Amidon core is NOT a substitute (see Sontheimer coupler – transformer issues). T2 has a 14t primary, and 1t or 2t secondary. Whilst a 1t secondary is preferred, if the transformer was to be used as a replacement of the 7t:1t transformer discussed in the previous article, 14t:2t is more directly compatible (eg with firmware).

It should go without saying that best directivity is obtained with symmetric transformers, ie, T1 and T2 are identical, they have almost equal leakage inductances etc, and almost equal amplitude and phase response.

A first step is to confirm that the core is unlikely to approach saturation.

Above, expected flux density is 76mT @ 1.8MHz, well below the onset on non-linear B-H response at about 200mT, and less at higher frequencies.

Above, the test inductor with 14t of 0.3mm ECW winding on the 6mm OD core for measurement.

A Simsmith model was constructed to estimate the InsertionVSWR and core loss due to T2 based on an estimate of the magnetising impedance of the primary winding. The impedance model is based on (Duffy 2015).

Above, the model for calibration of cse.

The prototype transformer was measured and its measurement s1p file merged with the Simsmith model to compare estimated with measured, and adjust cse for reconciliation of the self resonant frequency (SRF) of the model with measurement.

The model calibrates with cse=0.29pF, SRF=74.3MHz, the two sets of curves reconcile quite well validating the model and measurement.

Above is the calibrated model showing estimated InsertionVSWR and core loss.

Above, a close up of the chart. InsertionVSWR due to T2’s magnetising impedance alone is 1.07 @ 3.5MHz, and core loss is 0.18W @ 5W (0.15dB loss), both should be quite acceptable.

This is a desk study of an alternative design, it produces a candidate for testing of the important coupler parameters (like InsertionVSWR and Loss).

Thermals are best confirmed from the working prototype, expected temperature rise is towards 100° @ 0.18W average dissipation (5W transmitter @ 3.5MHz), much lower for SSB telephony. If temperature rise is unacceptable, a 21t winding us about half the dissipation and also allows a 21t:3t configuration to suit existing firmware.

Above is a thermograph of the inductor in free air showing 61.5-10.4=51.1° rise over ambient, stabilised after 3m of continuous 3.5MHz 5W carrier through signal. (The imager’s clock is wrong.)

- Duffy, O. 2015. A method for estimating the impedance of a ferrite cored toroidal inductor at RF. https://owenduffy.net/files/EstimateZFerriteToroidInductor.pdf.
- Grebenkemper, L. Jan 1987. The tandem match – an accurate directional wattmeter In QST.
- Sontheimer,C & Frederick,RE. Apr 1966. Broadband directional coupler. US Patent 3,426,298.

Let’s look at the calibrated model estimates of choke impedance and core loss, side by side.

The left hand chart is derived from Fair-rite 2843000202 data (as calibrated for SRF), and the right hand is the same but referencing the National Magnetics Groups H material permeability chart.

Though they are very similar in form, the core loss is quite different at lower frequencies, mainly a consequence of lower impedance components increasing magnetising current and its follow on effects. The solution is probably not to simply add another turn as that compromises Idcmax.

This is not to suggest that the H material is unsuitable, but it is less suitable, though it may work well enough in the application. Be aware of the source and applicable data from your magnetics supplier, and an imitator might have good product, but use their data to assess that.

]]>Let’s look at some examples.

A poster advising on how to measure inductance using a NanoVNA posted a .s1p file of his measurements of a SM inductor of nominally 4.7µH from 1-5MHz and discussed the use of phase in determining the inductance.

Above is a plot of the data in the VNWA PC client. Four values are plotted:

- s11 magnitude;
- s11 phase (°);
- Z magnitude; and
- Z phase (°).

An explanatory note, the y-axis zero for the magnitude plots is the bottom of the chart, for the phase plots it is at the fifth division and the scale is 50°/div.

Since this article is about phase, lets look at the phase plots.

For an ideal inductor, the phase of Z would be 90° independent of frequency. For a good inductor, it will be close to 90° independent of frequency.

In this example, the phase of Z (red) is very close to 90° above about 250kHz, quite as expected for the SM inductor over the measured frequency range (a tribute to the measurer).

Now look at the phase of s11 (orange) over a wide range, it is clearly not the same as the phase of Z.

This example is measurement of a lossy ferrite inductor from 1-30MHz.

Above is a plot of the data in the VNWA PC client. Four values are plotted:

- s11 magnitude;
- s11 phase (°);
- Z magnitude; and
- Z phase (°).

An explanatory note, the y-axis zero for the magnitude plots is the bottom of the chart, for the phase plots it is at the fifth division and the scale is 50°/div.

Since this article is about phase, lets look at the phase plots.

For an ideal inductor, the phase of Z would be 90° independent of frequency. For a good inductor, it will be close to 90° independent of frequency. For a lossy inductor, it will vary from 0-90° depending on the ratio of X/R at the frequency.

In this example, the phase of Z (red) varies from about 87° to 23° over 1-30MHz, quite as might be expected from the datasheets.

Now look at the phase of s11 (orange) over a wide range, it is clearly not the same as the phase of Z.

This example is measurement of a nominally 50+j0Ω load.

Above is a plot of the data in the VNWA PC client. Four values are plotted:

- s11 magnitude;
- s11 phase (°);
- Z magnitude; and
- Z phase (°).

An explanatory note, the y-axis zero for the magnitude plots is the bottom of the chart, for the phase plots it is at the fifth division and the scale is 50°/div.

Since this article is about phase, lets look at the phase plots.

For an ideal 50+j0Ω, the phase of Z will be very small.

In this example, the phase of Z (red) varies and is <0.1° over 1-30MHz, quite as might be expected.

Now look at the phase of s11 (orange) over a wide range, it is clearly not the same as the phase of Z.

Some argue that phase of s11 is a good indicator of resonance or a non-reactive load. Whilst it is true that a non-reactive load has s11 phase of 0° OR 180° (depending on its magnitude) , the reality is that measurement noise makes this a pretty impractical metric for that purpose.

It depends on the use.

Very often, users are interested in the phase of Z and mistakenly / unknowingly use phase of s11.

Here is the catch, VNAs and PC clients may not offer a plot of phase of Z, in the plots give above, a custom function was written to plot phase of Z.

]]>A magnetic core increases the flux Φ due to a current flowing in the inductor, and since \(L \propto \phi\), the magnetic core increases inductance.

Magnetic core materials are not usually linear, they exhibit saturation and hysteresis (which brings core loss), and changing magnetic field induces eddy currents in the material which also brings core loss.

The B-H curve relates flux density to magnetising force, and as mentioned, the underlying material is non-linear and exhibits saturation (where at some point, B increases very little for increased H).

Above is a generic BH curve for magnetic core material. It shows saturation and hysteresis. Note that saturation (Bs) is total saturation of the core, but saturation begins at half that flux density in this case.

Permeability is mathematically given by \(\mu = \frac{B}{H}\), and \(\mu = \mu_0 \mu_r = 4 \pi \text{e-7} \mu_r\), where µ0 is the permeability of free space and µr is the relative permeability.

Techniques to work around some of these characteristics include alloying of the material, lamination, powdering / sintering, and air gapped magnetic paths.

Ferrite cores offer more extreme examples and are in common use, so will be discussed as a vehicle for exploring the issues.

Ferrite is a ceramic like material that exhibits ferromagnetism, it may have hight to extremely high resistivity (differently to iron, laminated steel, and powdered iron), has relatively permeability from single digit numbers to thousands, and exhibits low to very high core loss.

You might think all this makes it pretty unattractive, and lots of pundits claim so, but it is an option that can be very effective in many of applications.

Sometimes the distinction is made by some ‘experts’ that ferrite is not suitable for “resonant applications”, or not suitable for “transformers”, that it is a “suppression only product”, but better to set those notions aside and approach ferrite with an open mind discovering and exploiting the characteristics are suited to different applications.

A good model to account for ferrite core loss is to use a complex quantity for permeability, and the imaginary component gives rise to an equivalent resistance in series with the reactance. The permeability components are also often designated µ’ and µ”, \(\mu_r=\mu^{\prime}-\jmath \mu^{\prime \prime}\).

For a deeper discussion of estimating the impedance of a ferrite cored inductor, see A method for estimating the impedance of a ferrite cored toroidal inductor at RF.

Permeability is temperature sensitive. Further, ferrites exhibit Curie effect where at sufficiently high temperature, the lose their magnetic properties.

Ferrite permeability has wide manufacturing tolerance.

Ferrite permittivity is different to those of a vacuum or air which influences electric field distribution.

Ferrite cored inductors exhibit all of the effects discussed with the air cored solenoid, and additionally the frequency and temperature dependence of complex permeability, and these combinations and influences vary widely for different inductors.

So again, it is wise to think of a ferrite cored inductor as a resonator in the general sense, and apply simpler models where they are adequate approximations for the application at hand.

Even more so than in the case of the air core solenoid discussed earlier, the specification of a ferrite cored inductor as simply an inductance value is pretty naive if it is intended to be used at frequencies where µ’ varies with frequency, or µ” is significant.

It is commonly the case that ferrite cored inductors used in continuous wave (CW) applications above 1MHz will overheat (even to the Curie point) long before flux density reaches saturation, and so saturation tends to be ignored in design calcs.

This assumption needs to be tested for each application, and is less likely to apply for pulse or other low duty cycle applications. More importantly, if there is a non-zero DC component of current / magnetising force, saturation may occur.

Applications such as RF chokes, common mode chokes on power conductors etc require consideration of saturation.

Be aware that measurement of the inductor with most instruments does not approach saturation, and it takes more complicated test setups to safely measure impedance with a DC current bias applied.

There is a plethora of ‘inductance calculators’ that purport to give valid results, not just for air cored solenoids, but for magnetic cored inductors including powdered iron and ferrite in various shapes and materials.

I will simply say, use with caution, it is my experience that most are not to be trusted. The ‘net being what it is, the fact that two calculators might agree is not evidence they are valid, they may simply have common heritage.

It is often the case that popular tools are popular because they are popular, and they need not be valid to be ‘liked’.

There is no substitute for understanding basic Electricity & Magnetism.

Magnetic cores add another level to the complexity of inductor design and measurement.

Common materials used for RF inductors are powdered iron and ferrite in various formulation, none are ideal materials, but informed design work can produce components well suited to application.

]]>This article discusses a common issue with the design of the RF choke providing DC to the Class-E stage.

Above is a circuit above is from (Sokal 2001) which explains the amplifier and gives guidance on selection of components. One key recommendation is that the usual choice of XL1 being 30 or more times the unadjusted value of XC1.

This spells out that L1’s role is essentially an RF choke, it is intended to pass DC but to largely prevent RF current, it needs a high impedance at RF, and low DC resistance.

Let’s estimate a design value for the choke’s DC current for a PA specified for 5W out at 13.8Vdc.

The saturation voltage of a HexFET is very low, so taking it as zero, and assuming say 85% efficiency, the DC current through L1 would be \(I_{dc}=\frac{5}{13.8 \cdot 0.85}=0.43A\). Given tolerances on ferrite, it would be wise to design for Idc=0.6A.

One popular design ((tr)uSDX) specifies L1 equivalent to be “22t on FT37-43 – abt 200µH”.

Before even considering whether 200µH is correct or adequate characterisation, lets focus on the DC current, #43 material characteristics, core dimensions and B-H curve.

The inner radius of the FT37-43 is 2.375mm. Let’s calculate the magnetising force at the inner part of the core: \(H=\frac{I N}{l}=\frac{0.6 \cdot 22}{2 \pi 0.002375}=884 \text{A/m}\). Since the BH curve is scaled in cgs units, \(H=884 \text{A/m} =11.1 \text{Oe}\).

The calculated magnetising force is off the scale to the right, the core is fully saturated and some.

At saturation, the inductor will not have the RF impedance needed for the application.

A prudent limit for H considering some small temperature rise in operation might be 0.6Oe or 50A/m.

A better approach is to design for Bmax, and a value of 2000gauss or 0.2T would be appropriate.

Above is a calculation of maximum current for Bmax=0.2T, Imax falls a long way short of the 0.6A design value.

So, the single FT37-43 inductor seems quite flawed, it is unlikely to withstand sufficient DC component of current to deliver most of its design RF impedance.

Mosaic published a LTSPICE simulation of a Class-E PA design for 3.5MHz. It appears to be a detailed and valid simulation, so let’s use it for a quick path to explaining the role of the RFC. The simulation provides a convenient way of showing currents that can be very hard to measure in practice.

Above is an extract of the simulation schematic. Note L4, the RFC feeding DC to the output stage. It is 80µ and has |Z| of 1800Ω, a relatively high impedance.

The above figure from (Sokal 2001) gives advice on the expected current and voltage at the low order Class-E switch.

Above is a plot from the simulation showing:

- green: drain voltage;
- cyan: gate voltage;
- magenta: RF output voltage;
- blue: drain current; and
- red: RFC current.

Note that the RFC current is almost steady DC, there is in fact 0.06App ripple, but if the RFC is effective as such, then the ripple current will be very small.

Note that there are designs published (eg QCX) where the DC feed inductor is not a high impedance itself, it is not of itself an effective RFC, and the current will not be as shown above for an effective RFC.

So, to design an effective RFC, we must design a component of:

- sufficiently high impedance;
- sufficiently low RF loss; and
- sufficiently low DC resistance.

Note that that all these things need to be delivered with commercial tolerance components, it is not proven by apparent operation of a single prototype alone.

Let’s look at an alternative to the “22t on FT37-43 – abt 200µH” RFC discussed earlier that might suit (if it fits).

This section applies to Fair-rite 2843000202 cores, and measurements were of Fair-rite product. These cores are often sold as BN43-202, but be aware that other manufacturers pass off cores as #43 material thought they may not be exactly comparable. For example, Amidon’s current #43 datasheet appears to be National Magnetics Group H material.

Firstly, since the application has a significant DC component of current, we need to check for magnetic saturation.

You may see in datasheets, Bsat specified as 300mT, but for #43, that is the absolute maximum B in full saturation at 25°, it is lower at higher temperatures, and the maximum usable (meaning most of the design impedance is still available) B is more like 200mT (look at the B_H curves, and allow some margin for tolerances).

At a fundamental level, we can write \(L=n\frac{d \phi}{di}\). If the operating point due to DC bias is so far up the BH curve that flux hardly changes with change in current, the inductance has been reduced greatly.

Iteratively, it was found that the most turns for acceptable Imax was four.

Next question is that sufficient at RF?

A reminder that the driving waveform is not sinusoidal.

The above voltage waveform from (Sokal 2001) shows the expected waveform from a properly tuned low-order Class-E amplifier, there will be a substantial fundamental component and some less harmonic components. A simple analysis considers only the fundamental component, but it gives good guidance on design of an inductor, and should be suitable if it presents a high impedance at the harmonics.

Above is a Simsmith model for the inductor with an informed guess for Cse. If the design was inspired by (Sokal 2001), it also includes calculated corrections for C1 and C2… which are quite small due to the high equivalent shunt inductance of the inductor.

Core loss is calculated for 20Vpk applied, and it is very small, around 130mW.

Note that the calculated temperature rise is for average power dissipation of 130mW and there is considerable thermal latency in ferrite cores, it may take many minutes to approach that temperature rise.

Ok, looks good… let’s make and measure one.

Above, measured R,X for the prototype look good… let’s compare them to the estimate and calibrate the model.

Above is the model extended to compare the model estimate with measured impedance. It is quite good (within the tolerances of ferrite materials and the model), and it turned out the experienced guess at Cse was right on the money.

Above is a thermograph of the test in free air of the RFC with the same RF voltage at 3.5MHz as that impressed in the real transmitter application. Temperature stabilised after 2m, temperature rise is 22.5-12.2=10.3°, a little lower than expected but within tolerance.

The next, and necessary step, is to try a prototype in the application circuit… though I do not have the transmitter and will not be doing that.

There seems a consensus in the design of the RFC for several projects, but none discuss the magnetic design and in particular saturation calcs. WB2CBA’s ADX specifies 20t on FT37-43. Dan Tayloe N7VE has a design using different number of turns on different bands, 12.5t on FT37-43 for 2.25W on 40m (Q: how do you wind 0.5t on a toroidal inductor?), so lower turns and lower current.

This article is not about a somewhat similar design that uses 24t at 80m on a (powdered iron) T37-2 (eg QCX). It has very high Imax (much higher than needed), but it has much lower RF impedance than the options discussed above, and that may be an issue.

- Sokal, N. Jan 2001. Class-E RF power amplifiers In QEX Jan 2001.
- BN43-202-4t-c.7z

]]>

That sort of thinking betrays a lack of understanding of how a transformer works.

If you take a good 50/60Hz 1:1 power transformer, assume no losses, no flux leakage, and ignoring distributed capacitance, you might ask:

- what is the primary current with nominal voltage applied to the primary and the secondary open circuit;
- what is the primary current with nominal voltage applied to the primary and rated current flowing in the secondary circuit (resistive load);
- what is the core flux density in case 2 relative to case 1, much higher, about the same, much lower.

The following expression gives the voltage present on a winding \(E=4.44fNAB_m\).

where:

E Applied rms voltage

f frequency [Hz]

N turns on the winding where the voltage E is applied

A Magnetic circuit cross-sectional-area enclosed by the winding [m^2]

B_{m} the maximum flux density [Tesla]

We can rearrange it to make B_{m} the subject \(B_m=\frac{E}{4.44 f N A}\).

Readers will observe for that simple model of a transformer, flux density is not a function of load current.

If leakage inductance is taken into account, the applied voltage E is reduced due to the leakage inductance and in fact flux density under load (in a transformer with low leakage inductance) is likely to be slightly less than at no load.

That is not to say transformer losses might not increase significantly with load current, but that it is not due to increased magnetic flux.

Ferrite cored transformers inherit much of this model, though flux leakage may be significantly higher, RF conductor resistance is higher that at 50/60Hz, core materials may be quite lossy at some frequencies (but that does not prevent them being used for a transformer with efficiency acceptable for an application).

Read widely, and think… there is some pretty wooly stuff elaborated on social media.

]]>Equally selfevidently we don’t want ANY real part of the reactance in a transformer and, for a practical transformer, we want the self inductance on each side (primary and secondary) to be at least j10*R(Load or Source) and the coupling to be as close to 100% from primary to secondary. It is the real part that heats up transformers a LOT and, since ALL of the current is seen by the ferrite in a transformer, not just the part that got reflected back on the outside of the coax in a choke, losses are abos-posilutely-undubiously NOT desired and the u”

_{R}needs to be as close to zero as we can get at the designed frequency for minimum loss and minimum power dissipation.

Setting aside the hyperbole and the wooly thinking, let’s drill down on u”

needs to be as close to zero as we can get at the designed frequency for minimum loss and minimum power dissipation._{R}

It is a pretty general statement without really specific quantities, needs to be as close to zero as we can get

and minimum loss and minimum power dissipation

does not give useful guidance of acceptable values of µ”, and may even impart the impression that the following chart is for material that is not suitable above perhaps 200kHz, if that.

Above, µ” is greater than 10 above about 200kHz, greater than 100 from about 2 to 100MHz. Is this what the quote condemns?

Let’s pull up a previous design.

A prototype small 4:1 broadband RF transformer using medium µ ferrite core for receiving use gave details of the design of a small transformer which used the above #43 material.

Above is the prototype transformer in a test jig which has been SOL calibrated. One end of the winding has a 150Ω resistor (measured as 149Ω) in series to ground, the other connects to the output connector centre conductor. This arrangement puts a 200 load on the transformer, and there will be 6dB of loss from input to output due to the divider action of the 150Ω resistor and 50Ω instrument input.

Above is a scan of S11 and derived input impedance and VSWR. As expected from the earlier work, Insertion VSWR is below about 1.2 from 1.8MHz up, and looking at the Smith chart, you can see that the departure from an ideal transformation is a little +ve reactance and slightly lower resistance which is the expected effect of the shunt magnetising impedance.

The S21dB (black) plot is |S21| in dB, the gain through the device, and bear in mind that due to the resistor load division, -6dB represents zero loss. So, the InsertionLoss is tenths of a dB, demonstrating that efficient transformers can be made from lossy ferrites.

We can extract from the s11 and s12 figures and the 149Ω series resistor, the InsertionLoss and its components TransmissionLoss (or simply Loss) and MismatchLoss.

It is the TransmissionLoss that heats the core and wire, MismatchLoss does not cause heat in the transformer (and does not necessarily cause heat anywhere else).

In this case, the TransmissionLoss varies from 0.1dB to 0.25dB, by most measures a reasonably efficient transformer.

Let’s look at a Simsmith model of the transformer based on published material characteristics and calibrated to the measured response.

Referring to the manufacturer’s chart given earlier, µ” is greater than 200 from 2 to 30MHz, yet the TransmissionLoss is quite low.

The Rm value plotted in blue against the right hand Y axis is the series equivalent resistance component of the of the magnetising impedance due to µ”, it is what the quote refers to with we don’t want ANY real part of the reactance in a transformer.

It is near zero at the left but goes up to quite large values on the right, yet the TransmissionLoss is quite low.

It is easy to wave hands around and say core materials must be nearly lossless and to talk about achieving minimum loss, but the challenge for designers of practical devices is to find a compromise set of many competing effects / parameters that provides a design that is acceptable on many criteria for a particular application.

Read widely, and think… there is some pretty wooly stuff elaborated on social media.

]]>Above is a diagram from the manual. It is a pair of cores stacked, each core is wrapped in self adhesive glass fibre tape.

Straight away, the glass fibre tape is an important fact. Hams often use glass fibre tape on any and everything, but for a high power application like this, a conductive core (eg powdered iron) requires additional insulation, the paint is not really sufficient. Powdered iron cores usually have radiused corners to reduce the risk of punching through insulation, whereas ferrite cores are more commonly chamfered with relatively sharp edges.

Above, the schematic shows the balun to be a Ruthroff 4:1 balun, a voltage balun.

Pics of the balun in the ATU show quite long connecting wires, so any attempt to measure the component will be frustrated by impedance transformation due to the transmission line effects, more so at higher frequencies.

The poster had made measurements with a Rixexpert analyser though a quite short coax lead and pigtails, perhaps 80mm in all. the test equipment was connected between J4 and ground, and a 200Ω resistor of unknown quality connected from J4 to J5. Again, there will impedance transformation due to the transmission line effects, more so at higher frequencies.

The good thing is his measurements were from 25kHz to 30MHz.

Above, the InsertionVSWR looks pretty awful, but remember that this is for use with an ATU, and non-ideal impedance transformation is corrected by the ATU adjustment.

Above, the same information presented as Return Loss.

Above is a plot of admittance. Note that over much of the plot, G is almost constant, and at the very low frequency end, B varies almost inversely proportional to frequency. The curves hint that at the lowest frequencies, the admittance is dominated by B, B is -ve, and it varies almost inversely proportional to frequency… so we are looking at an admittance of some fairly constant G and (in parallel with) some inductive susceptance inversely proportional to frequency… the latter hinting a constant inductance.

Above is the Smith chart presentation of the same data. Note that the curve almost exactly follows a circle of constant G from 25kHz (LHS) to the marker at 400kHz. The combination looks like about 51Ω resistance in parallel with 4.7µH of inductance… the magnetising inductance looking into one winding of the transformer.

One contribution to the non-ideal impedance transformation is inductance, worse with lower permeability cores, and worse with long windings, both are features of this transformer.

The conductance figure is attributable to the transformed 200Ω load, so there is not strong evidence of substantial loss, further hint of a powdered iron core.

Let’s look at the inductance of the common T200-2 core with 14t as an example.

A stack of two cores will simply double this at the frequency being discussed, so a pair of T200-2 cores with 14t should have an inductance @ 400kHz of around 4.8µH. It is quite likely that the transformer uses these cores or ones with similar characteristic.

So, it is a 4:1 voltage balun integral to a high power ATU, it has quite poor InsertionVSWR but that is relatively unimportant in an ATU, it appears to use a powedered iron core which has relatively low losses. The matter of whether a voltage balun is a good choice for HF wire antennas with two wire line feed is another question.

The Smith chart is so informative!

]]>The test uses a small test inductor, 6t on a BN43-202 binocular core and a small test board, everything designed to minimum parasitics. This inductor has quite similar common mode impedance to good antenna common mode chokes.

Above is the SDR-KITS VNWA testboard.

The nanoVNA-H4 v4.3 was calibrated using the test board and its associated OSL components. The test board is used without any additional attenuators, it is directly connected to the nanoVNA-H using 300mm RG400 fly leads.

Above, the test inductor mounted in the s11 shunt measurement position.

Everything is highly symmetric, so measurements of s11 and s21 are copied to s22 and s12 to avoid the measurement noise of actually reversing the test jig and a second calibration set. This would not be appropriate for test fixtures that used floating clip leads.

The VNA does not support 12 term correction, so errors in Zin of Port 2 are uncorrected, and flow into the results for the methods using s21.

Above is a plot of ReturnLoss of Port 2 over the measurement range.

Above is a plot of the components of Zin of Port 2 measured in through configuration of the test board. It is good, but not perfect. Errors flow into methods that depend on s21 through measurements.

Connecting the DUT in shunt to Port 1 allows measurement of s11, and from that, calculation of impedance. It is straightforward, and most VNAs and PC client software can display the R and X components of impedance directly.

(Agilent 2001) discusses the use of this method on extreme impedances. Note that observed measurement noise is an indicator of whether the method suits the application at hand.

Above is the impedance plot directly from the PC client software (NanoVNA-App). Notwithstanding the advice that you just cannot measure such high impedances with a VNA in reflection mode, much less a $100 jobbie, the curves show very little measurement noise and behave much as would be predicted from knowledge of core, material and winding.

Above is a chart comparing s11 measurement with estimated (calibrated to SRF). Bearing in mind the wide tolerance of ferrite material, the measurement reconciles well with the estimate.

Connection of the DUT between ports 1 and 2 allows measurement of higher impedances with relatively lower measurement noise, and might seem to be the preferred method.

An important issue that is not usually given by supporters is that the calculation of Z from s21 depends on an assumed value for Port 2 input impedance, and error in this flows into s21 unless 12 term correction has been used in the measurement. The VNA used for this experiment does not support 12 term correction, and the measured Port 2 input impedance was given earlier..

The transformation of s21 to series impedance is not native to lots of VNAs and PC clients. Calculation of impedance from s21 is not difficult, but it appears from my correspondents that it is commonly messed up.

The s21 pi method calls for connecting the DUT between Port 1 and Port 2 of the VNA, and taking a full s parameter scan of the DUT as in the s21 series method. The s parameters are converted to y parameters and then a pi equivalent circuit.

This method is also known as the Y21 method of measuring common mode choke impedance.

Above, calculation of a pi equivalent circuit from y parameters is trivial.

The assertion by its supporters is that naturally, the shunt elements (y11+y12 and y22+y21) are parasitics due to the test fixture and separating them from the series element -y12 gives the ‘true’ admittance of the common mode choke, or -1/y12 for the choke impedance. (For a reciprocal device such as this, y21=y12, perhaps the reason for the “y21 method” nomenclature.)

It is not obvious to me that allocation of all of the shunt elements to fixture parasitics is valid.

Above is a plot of the components of common mode impedance.

Above is a plot of the admittance of the shunt elements of the pi equivalent network. If those elements were mainly a fixed equivalent capacitance, we would expect the susceptance curves to dominate, to be +ve, and close to a straight line with a slope that implied the equivalent capacitance. They are not so.

Above is a plot of the equivalent shunt capacitance of the shunt elements of the pi equivalent network. They do not look like a constant capacitance, in fact they are a -ve capacitance at all frequencies… so the explanation of parasitic capacitance does not fit the measurements.

The above chart compares the three methods. Both s21 methods are almost coincident in this case… but they are different.

A comparison of a different measurement scenario might give quite different results.

There is great benefit in direct reading of R and X, benefit that should not be overlooked.

Uncertainty in Zin of Port 2 rolls up into uncertainty of s21 series impedance measurement.

An attenuator may be used to better control Port 2 Zin, but at the expense of noise performance. In this case, the VSWR of Port 2 is no worse than affordable attenuators.

There was little difference in the s21 based methods here.

The s21 pi method did not reveal a convincing parasitic equivalent capacitance.

Common mode chokes with high impedance are very sensitive to stray capacitance and distributed inductance of connections, and the measurer must consider the design of a test fixture appropriate to the deployment.

In the case of common mode chokes with high impedance one is less interested in the peak impedance values, or even impedance at some specific frequency (since it may be very sensitive to stray capacitance), but rather the range of frequencies where common mode impedance exceeds some criteria.

- Agilent. Jul 2001. Advanced impedance measurement capability of the RF I-V method compared to the network analysis method 5988-0728EN.
- The Y21 Method of Measuring Common-Mode Impedance

These are used in many things, including medium to high power applications such as EFHW matching transformers.

Leakage inductance is the equivalent series inductance due to flux that cuts one winding and not the other, and vice versa. For most simple transformers, the total primary referred leakage inductance is twice the primary leakage inductance. Since the leakage inductance appears in series with the signal path, it causes degradation of nominal impedance transformation, the very simplest approximation of the frequency response is that of a LR circuit.

Above is a Simsmith model of a 1µH total leakage inductance in series with a 50+j0Ω load, the InsertionVSWR is greater than 1.5 above 3MHz.

Is this a common scenario?

Above are measurements of equivalent total leakage inductance of two transformers, both 4t:32t on a 5943003801 (FT240-43) EFHW transformer. There are lots of designs with a 2t primary, but they have high InsertionVSWR at 3.5MHz, 4t is minimum for good InsertionVSWR at 3.5MHz. The difference between the two transformers is that:

- the purple trace (higher leakage inductance) is the common ham EFHW transformer with the primary bifilar twisted over the low turns of the secondary, and the secondary spread out and crossed over Reisert style;
- the green trace is a close wound autotransformer using only a quarter of the core perimeter.

Note the the common winding topology has about 20% higher leakage inductance.

Flux coupling factor k is a quantity that is often used to quantify the fundamental problem. For good RF transformers using medium to high µ cores, k is very close to 1, but we are more interested in 1-k, the portion of flux that leaks as that is what drives leakage inductance.

Some authors assume that since k may be around 0.99, that they can assume it is 1… but in so doing, they are assuming leakage inductance is zero… a gross modelling error.

Some authors assume k is independent of frequency. Whilst it may be for air cored transformers, or magnetic cored transformers were µ is independent of frequency, hams often (mostly) use ferrite cores at a frequency where µ is not independent of frequency, and a consequence is that k is not independent of frequency… another modelling error.

Here is another view of the same measurements above.

Above is a plot of Q (Xl/R) for the same case. Note that Q is around 30 for the autotransformer, and around 60 for the other transformer. Importantly, the characteristic of the #43 ferrite core at 5MHz is that Qcore is 1.2, so this inductance is not hosted by the ferrite core, it is mainly hosted in the air around the conductors. Looking at the green trace, leakage inductance does not vary greatly with frequency, and can be approximated reasonably as a frequency independent inductance, certainly a much better model than frequency independent k (as is popular).

I asked is this a common scenario?

Well, the better transformer has leakage inductance a bit lower than in the Simsmith model, the more common configuration is 14% higher… so the answer is yes, this is a common scenario.

Now it would be incomplete to ignore the fact that, leakage inductance may be partially compensated by a shunt capacitor.

Above, the optimally compensated circuit. In this case, the improvement is tiny.

So, what are the things that exacerbate leakage inductance?

- high number of turns;
- high turns ratios;
- small diameter wires;
- thick wire insulation;
- space for flux between the conductor and the core;
- spreading the winding out around the core;
- splitting the winding into parts;
- the Reisert cross over connection;
- wrapping the core with insulation;
- long wires (a result of core geometry including stacking cores); and
- low µ cores which drive higher turns.

How many designs have you seen that have some or most of these features?

Above is measurement of the inductance looking into the primary with the secondary short circuited, this is the total leakage inductance. This is an L/C meter that cost about $25 incl post on Aliexpress a few years ago. It takes great care to zero the meter with the leads in exactly the same configuration as the measurement, I use a coin as the short circuit for instrument zero. Not a 1% measurement by any means, but you can see that leakage inductance is hopelessly high for a broadband 50Ω:3200Ω for HF.

Total primary referred leakage reactance of 20% of the nominal impedance causes an InsertionVSWR of about 1.2.

For a nominal 50Ω transformer to 30MHz, I would like total leakage reactance less than 10Ω @ 30MHz, so leakage inductance (referred to 50Ω side) would be 50nH or less.

That is very hard to achieve on a medium sized high ratio transformer, much less a stack of three FT240 cores.

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