In the case of the Sontheimer coupler the winding with the higher number of turns appears in shunt with the nominal 50Ω load, and its effect on InsertionVSWR and the core loss can be predicted reasonably well and confirmed by measurements as in the referenced article.

In that instance, a 7t winding in shunt with the nominal 50Ω load causes excessive core heating, a 3t winding will be worse, and 2t worse again.

The case of an EFHW transformer is somewhat similar, the difference is now that the winding with less turns in approximately in shunt with the nominal 50Ω primary referred load. The same Simsmith model can be used to predict likely InsertionVSWR due to primary magnetising admittance, and the core loss.

Let’s try the 3t case first, with the experience of the referenced article we can expect it will have insufficient turns for good performance.

Above is the Simsmith model of a Fair-rite 5943000201 core (equivalent dimensions to FT37-43) with a 3t winding. Note this does not apply to Amidon #43 as their material is significantly different in characteristic.

So, the model scenario is of a 50Ω source that would deliver 5W into a matched load. With the magnetising admittance appearing in shunt, at 3.5MHz core loss is around 1.2W and power delivered to the transformed load is 3.4W. InsertionVSWR due to the magnetising admittance is 1.8.

So, will it work?

Well, anything ‘works’, it depends on your meaning of ‘works’.

There are two main questions:

- is the loss of power to the antenna a concern; and
- will the associated heat be an issue?

It seems that QRP aficionados are less concerned about loss of power to the antenna, QRP^2 if you like.

It is a personal matter.

Above, the predicted temperature rise for the core in free air of just 0.1W of dissipation is 22°, so on a hot day at 40° ambient, the core would reach 62° which is just enough to burn skin.

The good news is that the average power of uncompressed SSB telephony is around 5% of PEP, so this core scenario should withstand that mode at up to 2W PEP, less for compressed speech.

If your choice was FT8, the core would get dangerously hot, and even reach the Curie temperature (where the core loses its magnetic properties) in time.

If the transformer is enclosed, its power rating is reduced.

I have had enquiries from hams about similar transformers on small #43 cores, FT82-43 matching transformer for an EFHW discusses one of the popular designs. It seems everyone is a designer, and adapting an often poor design by changing the core size is seen and an obvious path to success.

The promotion of these ‘designs’ speaks to the credibility of the ‘designer’.

There are several designs on this web site, start with Small efficient matching transformer for an EFHW.

Search the net for published designs that include exposition of the magnetic design, prediction of core loss, measurement of a prototype, and thermographs to validate the design performance.

This 3t primary on a Fair-rite 5943000201 (FT37-43) is not a good choice for 80m, it has high InsertionVSWR and high core loss.

- RF transformer design with ferrite cores – initial steps
- 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.

Above is the graph scaled R/ω and jX/ω, and an untitled X axis, though it would appear to be frequency in Hz (scaled by the M multiplier).

I had difficulty reconciling the Y values plotted for R/ω and jX/ω with the displayed R,jX values.

David F4HTQ offers the following explanation online.

I add some explanations.

I asked Rune if he could add this graphic because it is very useful.

It display curves that have exactly the same shape as the complex permittivity curves (μ’r and μ”r) of the ferrite datasheets.The values do not match those of the constructor curve ( to have the right value the software might know the exact geometry of the inductor) , but the shape is absolutely identical.

This allow to easy identify unknown ferrite core, and to better understand how to use it in a RF device.

He says permittivity… but he is talking about permeability.

The quote seems to say the Y axis scale is worthless?

In any event, the underlying R,X data only follows µ at frequencies well below the self resonant frequency (SRF) of the inductor.

I have a small ferrite cored inductor, 4t on a FB43-2402, which I will measure s11 from 1-10MHz. A marker was set at nominally 1.5MHz, it is actually 1.54MHz due to the scan set. Normallys one would use the least number of turns for good measurement, the 4t inductor happened to be at hand and suits this study.

Firstly, lets estimate the permeability of the core material.

Note that ferrite has a wide tolerance range, and is temperature sensitive.

Now lets estimate Z # 1.54MHz, ignoring the effects of self resonance, ie Cs=0.

So, we estimate Z=33+j252.

Looking at the first marker report at left, Z=31.4+j198. That is in the ballpark of estimate value of Z=32+j245, so the measurement looks valid.

Now lets focus on the graph of R/ω and jX/ω above. I have scaled values by eye and tabulated them along with the reported f,R,X values.

I have also calculated R/ω and jX/ω from the measurement data, and the ratio of the displayed values with the calculated values, and the ratio is consistently around 6.3e1, probably actually 2πe1 if the plotted values were captured more accurately.

So, it does not look like the plotted values are actually R/ω and jX/ω at all, but the result of some untidy mathematics and a failure to test the solution, possibly all a result of the value of the underlying concept.

Recalling that the apparent inductance of a toroidal inductor of medium to high permeability well below SRF is \(L=N^2 \mu \sum \frac{A}{l}\) where l is the path length \(l=2 \pi r\), N is the number of turns and A is the cross section area, and noting for µ is a complex value for ferrite, so jωL has a real component which models core loss.

The quantity \(\sum \frac{A}{l}\) captures the core geometry, and it or transforms are often given in datasheets, eg Fair-rite often gives the inverse in /cm, \(\sum \frac{l}{a}\).

So, we can say that \(Z=R+\jmath X=\jmath \omega N^2 (\mu^{\prime}-\jmath \mu^{\prime\prime}) \sum \frac{A}{l}\) and therefore \(\frac{R+\jmath X}{\omega}=\jmath N^2 (\mu^{\prime}-\jmath \mu^{\prime\prime}) \sum \frac{A}{l}\) and rearranging that, \(\mu^{\prime}+\jmath \mu^{\prime\prime}=\frac{X+\jmath R}{\omega} \frac{1}{N^2 \sum \frac{A}{l}}\) .

We can factor permeability of free space out so that we see relative permeability: \(\mu_r^{\prime}=\frac{X}{\omega} \frac{1}{\mu_0N^2 \sum \frac{A}{l}}\) and \(\mu_r^{\prime\prime}=\frac{R}{\omega} \frac{1}{\mu_0N^2 \sum \frac{A}{l}}\).

So as the quote states, the shape of the R/ω and jX/ω does follow that of relative permeability, but only well below SRF, and the constant of proportionality is \(\frac{1}{\mu_0 N^2 \sum \frac{A}{l}}\)

Is it a magic view (even if implemented accurately)? You decide.

]]>A quick Google search did not turn up any published design rationale or measurement data for the AT-100 coupler design.

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, and another which is in series with the through line to sense current. To achieve good Directivity, these transformers must be symmetric, nearly ideal, and they must be independent, ie no significant coupling between the transformers by magnetic or electric fields.

The AT-100 uses a Sontheimer coupler, they are very popular with ham users for perceived better performance, notably better Directivity over a wide frequency range.

Above is an extract from the schematic of the AT-100 ATU. Note T1 and T2. The designations might imply they are two separate transformers with negligible coupling (as per Sontheimer), the diagram does not show that they have a magnetic core, nor more importantly, that they share magnetic paths to some extent.

Above, the pic shows T1 and T2 as a binocular core, and the winding configuration can be seen as a single pass of one conductor through an aperture hole, and a bunch of turns on the ‘outer limb’ to form T1 and similarly for T2.

The article Thoughts on binocular ferrite core inductors at radio frequencies discusses the behavior of binocular cores from a different but related perspective.

Let’s look at the magnetic flux due to current in a wire passing through one aperture hole.

Above is an end view of a binocular core, the blue + is the tail of a current flowing into the page. Oersted’s rule tells use that current flowing into the page like that will produce magnetic flux surrounding the current.

At radio frequencies, current distribution is such that most of the current flows in a very thin layer at the surface (skin effect), and as a result there is very little flux interior to the conductor.

Think of the flux as closed loops. In the space between the outer surface of the conductor and the inner surface of the core aperture, there will be flux loops that are approximately circular.

In the space between the inner surface of the core aperture and say half wave to the outside of the core, there will be flux loops that are approximately circular, but much higher density due to the high relative permeability of the ferrite material.

As we approach the outer edge of the core, some of those flux loops are around both core apertures, some of the flux in the left hand limb fringes into the right hand limb.

Outside the core, there is also some flux, out to infinity, but it is much weaker now due to low permeability and greater flux path length.

For a medium permeability core such as #43 material, almost all of the flux is within the core, sufficiently so that we tend to ignore the other flux with little loss in accuracy.

But let’s return to the flux loops that go right around the outer thickness of the core, that flux cuts a conductor placed in the right hand aperture.

So, current in at AT-100 T2 1t primary causes some flux to cut the T1 10t primary winding and induce a voltage in it proportional to the through current. This cross coupling of the transformers due to the partially shared magnetic circuit compromises coupler Directivity.

The configuration used in the AT-100 Sontheimer directional coupler creates some magnetic coupling between T1 and T2 due to the geometry of the single binocular core used, compromising coupler directivity.

Since its Directivity is compromised, it begs the question, why is a Sontheimer coupler used over simpler circuits.

In any event, this coupler is principally to guide the auto tuner which appears to have only 256 steps of L and C and one doubts that it has the fineness of control to achieve extremely low VSWR which would require the use of a coupler with high Directivity. A useful metric is the frequency normalised number of steps available in each adjustable component, \(steps_n=\frac{steps}{\frac{f_{max}}{f_{min}}}=\frac{256}{\frac{54}{1.8}}=8.5\), it is a very small number which suggests quite coarse adjustment steps at the ends of its range.

All in all, a questionable design and to my mind, one that is not a good model to copy.

Using two separate small suppression sleeves such as the Fair-rite 2643250402 with a little spacing might well produce a higher Directivity coupler.

Another example that questions apparent ham preference for a Sontheimer coupler.

- Grebenkemper, J. 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.

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 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 5943000201 (FT37-43), 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 21mT @ 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 9.3mm 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=0.415pF, SRF=41MHz, the two sets of curves reconcile quite well validating the model and measurement.

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

InsertionVSWR due to T2’s magnetising impedance alone is 1.03 @ 3.5MHz, and core loss is 0.08W @ 5W (0.07dB 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 20° @ 0.08W average dissipation (5W transmitter @ 3.5MHz), much lower for SSB telephony.

Above is a thermograph of the inductor in free air showing 25.9-12.5=13.4° rise over ambient, stabilised after 3m of continuous 3.5MHz 5W carrier through signal.

- 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.

End Fed Half Wave matching transformer – 80-20m described a EFHW transformer design with taps for nominal 1:36, 49, and 64 impedance ratios.

Keep in mind that this is a desk design of a transformer to come close to ideal broadband performance on a nominal 2400Ω load with low loss. Real antennas don’t offer an idealised load, but this is the first step in designing and applying a practical transformer.

The transformer comprises a 32t of 0.65mm enamelled copper winding on a Fair-rite 5943003801 core (FT240-43) ferrite core (the information is not applicable to an Amidon core), to be used as an autotransformer to step down a EFHW load impedance to around 50Ω. The winding layout is unconventional, most articles describing a similar transformer seem to have their root in a single flawed design, and they are usually published without meaningful credible measurement.

This article presents a model of the transformer using the 1:49 taps, and measurements used to calibrate the model.

See also On ferrite cored RF broadband transformers and leakage inductance.

Above is a pic of the prototype being measured with a 2400+j0Ω load in a 4t:28t connection. Sweeps of the transformer with OC and SC terminations were also made, and all three used to calibrate the Simsmith model.

Above is the calibrated Simsmith model with 65pF compensation capacitor added. The blue curves are the uncompensated VSWR and losses, and the magenta are with compensation. Note that the compensation capacitor is a high quality capacitor, eg silvered mica.

Optimal compensation capacitance on a real antenna may be a little different, pre prepared to measure and trim.

So, InsertionVSWR on a nominal 2400Ω load is less than 2.3 from 80m to 20m.

Core loss is highest at 20m, 0.36dB, which equates to 8.6W of core loss at 100W input.

Above, worst expected core temperature rise in free air is at 14MHz, about 51°.

Above, at 3.6MHz, expected core temperature rise in free air is a little lower at about 37°.

The core measured showed 35° in free air @ 100W through @ 3.6MHz suggesting it is probably best rated for no more than 300W continuous, perhaps less depending on the enclosure. These results are consistent with the measured impedance of the prototype, but it is wiser to use the model prediction of expected average characteristic of the cores.

]]>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 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 / trusdx) 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

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