Above is the prototype transformer wound with 14t of 0.71mm ECW tapped at 2t. The mm rule gives some scale. The turns are close wound, touching on the inner diameter of the core.

Leakage inductance is the enemy of broadband performance, so it is important to minimise it.

Leakage inductance is affected by winding geometry. It is important to avoid opportunity for flux inside and around conductors that does not also ‘flow’ in the core, such flux does not ‘cut’ the other turns and is flux leakage. The winding should hug the core, so winding with wires with thick insulation, thick inflexible wires, and topologies like the Riesert cross over may compromise leakage inductance. High core permeability and high ΣA/l help to minimise wire length which helps in minimising flux leakage.

The transformer is configured as an autotransformer rather than separate primary and secondary windings to again minimise leakage inductance.

Above is a chart derived from s11 looking into the transformer primary with the 14t secondary short circuit showing the equivalent series inductance. The value will be taken as a total value of 236nH, or 118nH a side in the split model.

Note that the inductance at low frequencies is almost independent of frequency even though core permeability changes at these frequencies (see the #43 material data sheet), showing that, for the most part, leakage flux exists elsewhere than the core itself.

A simple model works quite well for predicting nominal load performance on low ratio transformers but less well for high ratio transformers, so best to proceed to measurement of the prototype with nominal load. A load was made from two small resistors in parallel having a combined DC resistance of 2480Ω, quite close to the nominal 2450Ω.

Measurement of the uncompensated transformer hinted that some 50-100pF was the likely optimum compensation capacitance. The transformer was measured with 47pF and it was under compensated, 100pF was perhaps more than needed but InsertionVSWR at the mid to high frequencies was not compromised so since there was a 100pF silver mica capacitor on hand, it was committed to the prototype.

Above is a plot of InsertionVSWR with 100pF silver mica compensation. InsertionVSWR is less than 2 from 1-30MHz. As this type of transformer goes, this is a very good result.

Note there is some contribution of the connecting wires to this response, it is just not possible to use zero length wires to connect the secondary load circuit… but then that applies to its application circuit as well.

Loss modelling ignores conductor loss. Even though the conductors are relatively small, the effective RF resistance referred to the primary side is very low and insignificant compared to core loss. Compensation capacitor loss is modelled, Q=1000 assumed for the silver mica capacitor, but his can be quite a deal lower and very significant for ceramic capacitors.

Above is a plot of the expected loss in dB, magenta on the left scale, and watts @ 50W continuous, red on the right scale. Maximum dissipation is less than 5W which should be accommodated within safe temperature rise for an unenclosed transformer.

Next step is to measure the transformer performance under power, capturing thermographs to confirm the predicted dissipation and power rating.

]]>I have also had lengthy discussions with Faraaz, VK4JJ, who is experimenting with a similar transformer.

This article describes my own design workup and measurements using a Fair-rite suppression core, 2643251002. The cores are not readily available locally, so I bought a bunch from Digi-key.

I really resist the tendency in ham radio to design around unobtainium, it is often quite misguided and always inconvenient. In this case, the motivation for these cores that use quite ordinary #43 material is the geometry of the core, they have ΣA/l=0.002995, a quite high and rivalling the better of binocular cores. High ΣA/l helps to minimise the number of turns which assists broadband performance. See Choosing a toroidal magnetic core – ID and OD for more discussion.

- EFHW;
- InsertionVSWR<2 3-22+MHz;
- nominal 49:1 transformation;
- compensated;
- autotransformer; and
- 50W average power handing.

Some key points often overlooked in published designs of EFHW transformers:

- Insufficient turns drives high core loss; and
- leakage inductance is the enemy of broadband performance, so the design tries to minimise leakage inductance.

Note that high number of turns drives high leakage inductance, so the design is to a large extent, a compromise between acceptable core loss and bandwidth.

From models, I expect that a turns ratio of 2:14 (ie 14t tapped at 2t) is likely to deliver the design criteria (with suitable compensation capacitor).

Above is a perhaps ambitious initial objective using a simple model of the transformer, dotted line is Loss and solid line is InsertionVSWR.

The first step is to measure a 2t winding alone on the core.

Above, the 2t winding measurement fixture. The wire is solid 0.5mm wire stripped from CAT5 LAN cable, the one end zip tied to the external threads of the SMA connector and the other end bent and inserted into the female part without damaging the connector.

Above, a plot or impedance. Note the resonance, the self resonant frequency (SRF) is 16MHz.

Above is a screenshot of a Simsmith model that will be used to develop the design.

It is initially configured to simply expose the impedance of the 2t winding using a simple model of the system as a resonator. The model estimates magnetising impedance from manufacturer’s complex permeability data and adds equivalent shunt capacitance cse as a first approximation of its first resonance. The model is calibrated by adjusting cse so that the model SRF coincides with the measurement.

In this case there is very good reconciliation between prediction and measurement, especially given the wide tolerance of ‘suppression’ ferrite components (see

Using complex permeability to design with Fair-rite suppression products).

The next step is to wind the 2:14 autotransformer winding and to make measurements of its SRF and leakage inductance to calibrate a predictive model.

]]>This article models the transformer on a nominal load, being \(Z_l=n^ 2 50 \;Ω\). Keep in mind that common applications of a 50Ω:200Ω transformer are not to 200Ω transformer loads, often antennas where the feed point impedance might vary quite widely, and performance of the transformer is quite sensitive to load impedance. The transformer is discussed here in a 50Ω:200Ω context.

Above is the prototype transformer using a 2843009902 (BN43-7051) binocular #43 ferrite core, the output terminals are shorted here, and total leakage inductance measured from one twisted connection to the other.

The prototype transformer is a 3:6 turns autotransformer with the two windings twisted bifilar.

Above is the equivalent circuit used to model the transformer. The transformer is replaced with an ideal 1:n transformer, and all secondary side values are referred to the primary side.

- Secondary side leakage inductance Lls is divided by n^2 to obtain the value primary referred leakage inductance in the circuit diagram.
- Cse is an equivalent shunt capacitance to partially model self resonance effects.
- Bm is the magnetising susceptance (calculated from other parameters).
- Gm is the magnetising conductance (calculated from other parameters).
- Llp is the primary side leakage inductance.
- Ccomp is a compensation capacitance.

A Simsmith model was built to implement the transformer model above.

- Complex core permeability is captured from a permeability data file.
- np is the number of turns on the primary.
- ratio is the turns ratio.
- cores is the number of cores in a stack.
- cse is Cse per the circuit diagram.
- Ll is the value of Llp and Lls’ (which are assumed equal).

Having measured the short circuit input inductance to be 43nH, it is distributed equally over Llp and Lls’ so Ll is entered as 22nH.

Above is a screenshot of the Simsmith model. Block D1 is used for data entry to supply some values direct and calculated to the following blocks.

Tfmr is the model of the transformer as shown in the diagram earlier.

Above is a plot of the measured total leakage inductance over 1-30MHz.

Above is a plot of calculated 1-k where k is the flux coupling factor. Again the measured leakage inductance and winding inductances show that k is not independent of frequency, and 1-k (which determines leakage inductance in a coupled inductor model) varies over more than 2:1 range in this example. The graph demonstrates that models that are based on an assumption that k and 1-k are independent of frequency are flawed.

Above is the modelled VSWR response of the compensated transformer on a nominal load. It is very good from 3.5-30MHz.

Above, drilling down on more detail, the \(Loss=10 log \frac{PowerIn}{PowerOut}\) curve is very good. Maximum loss is at about 4MHz, and at 0.06dB loss @ 7.0MHz means that 98.6% of the transformer input power power reaches the transformer output terminals, the deficit being lost mostly in heating the ferrite core.

So, in contrast to the FT240-43 2t:14t transformer:

- ΣA/l is nearly nine times that of the FT240 core, so fewer turns are required for similar core loss;
- shorter winding length helps to reduce flux leakage;
- lower leakage flux improves VSWR bandwidth;
- smaller cores can dissipate less heat;
- reducing core loss reduces the need to dissipate as much heat; and
- compensation capacitor assumes Q of silver mica, the appropriate choice for a transmitting application;

The transformer in free air can probably dissipate around 2W continuous, an at 4MHz where transformer loss is 1.35%, continuous power rating would be 148W (200Ω load, free air). Of course an enclosure is likely to reduce power rating.

Note that leakage inductance is sensitive to the diameter of conductors and the spacing relative to other conductors, so changing the wire conductor diameter and insulation diameter, and wire to wire spacing all roll into changes in leakage inductance. For broadband performance, the goal is least leakage inductance.

Try changing model parameters in the sample model (link below), change mix type, measure the leakage inductance for some different winding configurations and use it.

If you have heard online experts advising the #43 mix is not suitable for this type of application, and that you should use something else… try something else in the model… if you can find a binocular of this size in a more suitable material.

The model input value aol is the core geometry ΣA/l (m) and can be calculated from dimensions using Calculate ferrite cored inductor – rectangular cross section. Some datasheets give ΣA/l or ΣA/l in various units which can be inverted / scaled as necessary. Calculate ferrite cored inductor (from Al) can calculate ΣA/l (m) from Al.

The model does not give a definitive design, but it does help to explore the effects of magnetising admittance and leakage inductance on VSWR bandwidth, loss etc.

Sample Simsmith model for download: EFHW-2843009902-43-2020-3-6k.7z . (Compressed with 7zip.)

]]>This article models the transformer on a nominal load, being \(Z_l=n^ 2 50 \;Ω\). Real EFHW antennas operated at their fundamental resonance and harmonics are not that simple, so keep in mind that this level of design is but a pre-cursor to building a prototype and measurement and tuning with a real antenna.

Above is the prototype transformer measured using a nanoVNA, the measurement is of the inductance at the primary terminals with the secondary short circuited.

The prototype transformer follows the very popular design of a 2:14 turns transformer with the 2t primary twisted over the lowest 2t of the secondary, and the winding distributed in the Reisert style cross over configuration.

The winding layout used in the prototype is that recommended at 10/(15)/20/40 Mini End fed antenna kit, 100 Watt 1:49 impedance transformer .

Above is a plot of the equivalent series primary inductance of the prototype transformer with short circuit secondary calculated from s11 measured with a nanoVNA from 1-31MHz. Note that the inductance is fairly independent of frequency, rising a little at the high frequency end probably due to effects of distributed capacitance and self resonance. This suggests that leakage flux is for the most part not immersed in the ferrite core, and it provides hints as to how to minimise it.

Note that since the inductance of the primary and secondary are frequency dependent (by virtue of the ferrite characteristic), and that leakage inductance is relatively independent of frequency (see above), that the flux coupling coefficient k is frequency dependent, and making it constant is not a very good model at these frequencies.

It might appear that k is fairly independent of freq, but 1-k is not, and it is 1-k that is used to evaluate leakage inductance in the k based approach, so it delivers a poor estimate of leakage inductance when the magnetising inductance is frequency dependent (as it is likely to be with ferrite).

It can be seen above that 1-k varies over a 2:1 range in this model, which would drive a 2:1 variation in leakage inductance… when leakage inductance is almost constant (see the earlier chart).

Above is the equivalent circuit used to model the transformer. The transformer is replaced with an ideal 1:n transformer, and all secondary side values are referred to the primary side.

- Secondary side leakage inductance Lls is divided by n^2 to obtain the value primary referred leakage inductance in the circuit diagram.
- Cse is an equivalent shunt capacitance to partially model self resonance effects.
- Bm is the magnetising susceptance (calculated from other parameters).
- Gm is the magnetising conductance (calculated from other parameters).
- Llp is the primary side leakage inductance.
- Ccomp is a compensation capacitance.

A Simsmith model was built to implement the transformer model above.

- Complex core permeability is captured from a permeability data file.
- np is the number of turns on the primary.
- ratio is the turns ratio.
- cores is the number of cores in a stack.
- cse is Cse per the circuit diagram.
- Ll is the value of Llp and Lls’ (which are assumed equal).

Having measured the short circuit input inductance to be 360nH, it is distributed equally over Llp and Lls’ so Ll is entered as 180nH.

Above is a screenshot of the Simsmith model. Block D1 is used for data entry to supply some values direct and calculated to the following blocks.

Tfmr is the model of the transformer as shown in the diagram earlier.

Above is the modelled VSWR response of the compensated transformer on a nominal load. It is not brilliant, but might be acceptable to many users.

Above, drilling down on more detail, the \(Loss=10 log \frac{PowerIn}{PowerOut}\) curve is troubling. 1dB loss @ 7.0MHz means that only 74% of the transformer input power power reaches the transformer output terminals, the deficit being lost mostly in heating the ferrite core. Of course you transmitter might not develop its rated power into the load that it sees, there could be a further reduction in power output.

So despite its popularity, this is an appalling design. It has high loss due to insufficient turns, and high leakage inductance due to winding layout and high turns. Acceptable designs are a compromise between bandwidth and loss for a give core, and small is beautiful from the transmission parameters, but not for power handling.

Try changing model parameters in the sample model (link below), change mix type, measure the leakage inductance for some different winding configurations and use it.

The model input value aol is the core geometry ΣA/l (m) and can be calculated from dimensions using Calculate ferrite cored inductor – rectangular cross section. Some datasheets give ΣA/l or ΣA/l in various units which can be inverted / scaled as necessary. Calculate ferrite cored inductor (from Al) can calculate ΣA/l (m) from Al.

The model does not give a definitive design, but it does help to explore the effects of magnetising admittance and leakage inductance on VSWR bandwidth, loss etc.

Sample Simsmith model for download: EFHW-5943003801-43-2020-2-14xk.7z . (Compressed with 7zip.)

]]>This article models the transformer on a nominal load, being \(Z_l=n^ 2 50 \;Ω\). Real EFHW antennas operated at their fundamental resonance and harmonics are not that simple, so keep in mind that this level of design is but a pre-cursor to building a prototype and measurement and tuning with a real antenna.

The prototype transformer follows the very popular design of a 2:16 turns transformer with the 2t primary twisted over the lowest 2t of the secondary, and the winding distributed in the Reisert style cross over configuration.

Above is a plot of the equivalent series impedance of the prototype transformer with short circuit secondary calculated from s11 measured with a nanoVNA from 1-31MHz. Note that it is almost entirely reactive, and the reactance is almost proportional to frequency suggesting close to a constant inductance.

Above is a plot of the equivalent series primary inductance of the prototype transformer with short circuit secondary calculated from s11 measured with a nanoVNA from 1-31MHz. Note that the inductance is fairly independent of frequency, rising a little at the high frequency end probably due to effects of distributed capacitance and self resonance. This suggests that leakage flux is for the most part not immersed in the ferrite core, and it provides hints as to how to minimise it.

Note that since the inductance of the primary and secondary are frequency dependent (by virtue of the ferrite characteristic), and that leakage inductance is relatively independent of frequency (see above), that the flux coupling coefficient k is frequency dependent, and making it constant is not a very good model at these frequencies.

Above is the prototype transformer measured using a LCR meter, the measurement 335nH @ 100kHz is of the inductance at the primary terminals with the secondary short circuited.

Above is the equivalent circuit used to model the transformer. The transformer is replaced with an ideal 1:n transformer, and all secondary side values are referred to the primary side, eg \(L_{ls}^\prime=\frac{L_{ls}}{n^2}\).

- Secondary side leakage inductance Lls is divided by n^2 to obtain the value primary referred leakage inductance in the circuit diagram.
- Cse is an equivalent shunt capacitance to partially model self resonance effects.
- Bm is the magnetising susceptance (calculated from other parameters).
- Gm is the magnetising conductance (calculated from other parameters).
- Llp is the primary side leakage inductance.
- Ccomp is a compensation capacitance.

A Simsmith model was built to implement the transformer model above.

- Complex core permeability is captured from a permeability data file.
- np is the number of turns on the primary.
- ratio is the turns ratio.
- cores is the number of cores in a stack.
- cse is Cse per the circuit diagram.
- Ll is the value of Llp and Lls’ (which are assumed equal).

Having measured the short circuit input inductance to be 350nH, it is distributed equally over Llp and Lls’ so Ll is entered as 175nH.

Above is a screenshot of the Simsmith model. Block D1 is used for data entry to supply some values direct and calculated to the following blocks.

Tfmr is the model of the transformer as shown in the diagram earlier.

Above is the modelled VSWR response of the compensated transformer on a nominal load. It is not brilliant, but might be acceptable to many users.

Above, drilling down on more detail, the \(Loss=10 log \frac{PowerIn}{PowerOut}\) curve is troubling. 1dB loss @ 3.5MHz means that only 74% of the transformer input power power reaches the transformer output terminals, the deficit being lost mostly in heating the ferrite core. Of course you transmitter might not develop its rated power into the load that it sees, there could be a further reduction in power output.

So despite its popularity, this is an appalling design. It has high loss due to insufficient turns, and high leakage inductance due to winding layout and high turns. Acceptable designs are a compromise between bandwidth and loss for a give core, and small is beautiful from the transmission parameters, but not for power handling.

Try changing model parameters in the sample model (link below), change mix type, measure the leakage inductance for some different winding configurations and use it.

The model input value aol is the core geometry ΣA/l (m) and can be calculated from dimensions using Calculate ferrite cored inductor – rectangular cross section. Some datasheets give ΣA/l or ΣA/l in various units which can be inverted / scaled as necessary. Calculate ferrite cored inductor (from Al) can calculate ΣA/l (m) from Al.

The model does not give a definitive design, but it does help to explore the effects of magnetising admittance and leakage inductance on VSWR bandwidth, loss etc.

Sample Simsmith model for download: EFHW-5943003801-43-2020-2-16x.7z . (Compressed with 7zip.)

]]>Above is the transformer with 100pF compensation capacitor across the input, and two resistors to make up a 3300Ω load in combination with the VNA port.

The transformer is an autotransformer of 16t tapped at 2t, so the nominal turns ratio is 1:8, or impedance 1:64.

Looking into the 2t tap with the transformer loaded with 3250+50=3300Ω, Luis measured s11, s21 parameters from 1-45MHz with a nanoVNA. (Nominally this should be a 3200Ω load, but the error is very small, VSWR equivalent of 1.03.)

Above is my analysis of this data for InsertionVSWR and ReturnLoss. It gets a bit shabby above 15MHz.

Above is a plot of InsertionLoss and Loss (or TransmissionLoss). See On Insertion Loss for explanation of the terms.

Compensation of the transformer with 100pF in shunt with the input improves the broadband response.

Above is InsertionVSWR and ReturnLoss. It is not too bad over all of HF (3-30MHz).

Above is a plot of InsertionLoss and Loss (or TransmissionLoss). See On Insertion Loss for explanation of the terms.

Almost all of the Loss is in core loss, and it gives us a good indicator of core heating. The worst case is at 30MHz where the Loss is 0.7dB, so about 15% of input power is converted to heat.

The tests here were using a dummy load on the transformer, and that did allow confirmation of the design.

Real end fed antennas operated harmonically do not present a constant impedance, not even in harmonically related bands. Note that the resonances do not necessarily line up harmonically, there is commonly some enharmonic effect.

Being a more efficient design that some, it might result is a wider VSWR excursion that those others as transformer loss can serve to mask the variations in the radiator itself.

Hats off to Luis, CT2FZI for his work in building and measuring the transformer, and to have the flexibility to consider more than FTxxx-43 cores.

]]>Pictured is a dual UnUn. I made this for experimenting. It’s both a 49 and 64 to 1 UnUn.

The 49 to 1 tap uses the SS eye bolt for the feed through electrical connection and the SS machine screw on the top is the 64 to 1 connection. If I want to use the 49 to 1 ratio, there’s a jumper on the eye bolt that connects to the top machine screw where the antenna wire is attached. The jumper shorts out the last two turns of the UnUn. Disconnect the jumper from the top connection and now you have a 64 to 1 ratio.

The advice to short the section of the winding (the white wire in the pic) is really bad advice.

Tapping an air cored solenoid can be effective and with low loss… can be… but not unconditionally. Tapping a ferrite cored inductor almost always has quite high Insertion Loss, it is akin to shorted turns in an iron cored transformer, … so if you try it, measure it to see if the outcome is acceptable.

Keep in mind that flux leakage degrades broadband performance, so conductors wound loosely around the core (as in the pic) and wide spaced single layer windings (as in the core) tend to have higher flux leakage and poorer broadband performance. Measure what you make to verify that it did what you think.

An S parameter file from a two port sweep over HF would be informative.

I offer this analysis without measurement evidence to prove the case, but sometimes an understanding of basic circuit analysis allows one to avoid wasting time on poor designs.

]]>Above is the model topology. D1 is a daemon block which essentially, calculates key values for the other blocks based on exposed parameters and the named ferrite material complex permeability data file. The prototype used a Fair-rite 2643625002 (#43) core.

D1 code:

//Misc //Updates Tfmr, CoreLoss, and Cse. $data=file[]; // core mu aol; np; ratio; cse; cores; k; $u1=$data.R; $u2=$data.X; //u1=$u1; //u2=$u2; $ns=np*ratio; Ym=(2*Pi*G.MHz*1e6*(4*Pi*1e-7*$u1*aol*cores*1e9)*np^2*1e-9*(1j+$u2/$u1))^-1; Cse.F=cse; CoreLoss.ohms=1/Ym.R; $l1=1/(2*Pi*G.MHz*1e6*-Ym.I); $l2=$l1*(($ns-np)/np)^2; $lm=k*($l1*$l2)^0.5; Tfmr.L1_=$l1+$lm; Tfmr.L2_=$l2+$lm; Tfmr.L3_=-$lm;

L is the load block.

Cse models the self resonance of the transformer at lowish frequencies, Cse is an equivalent shunt capacitance.

Tfmr is a coupled coils model (above) of a (nearly) lossless autotransformer (core loss will come later). Wire loss is usually insignificant in these type of transformers and is ignored. (It seems that Simsmith will not simulate a pure inductance, in this cases the loss of the ‘lossless transformer is of the order of 5e-6dB, so satisfactory.) It is possible to simplify declaration of this component by using Simsmith’s coupled coils in a RUSE block, I have chosen to take full control and solve the mutual inductance effects explicitly.

(The model assumes that k is independent of frequency which is not strictly correct, but for medium to high µ cores, measurement suggests it is a fairly good assumption.)

Coreloss brings the ferrite core loss to book.

Ccomp models a compensation capacitor used to improve broadband InsertionVSWR.

The G block provides the source and plot definitions.

Plots code:

//Plots //check lossless Tfmr behavior //tfmrloss_dB=10*Log10(Tfmr.P/(Tfmr.P-Tfmr.p)); //Plot("TfmrLoss",tfmrloss_dB,"LossdB",y1); coreloss_dB=10*Log10(CoreLoss.P/(CoreLoss.P-CoreLoss.p)); loss_dB=10*Log10(G.P/L.P); mismatchloss_dB=10*Log10(1/G.P); Plot("CoreLoss",coreloss_dB,"LossdB",y1); Plot("Loss",loss_dB,"LossdB",y1);;

The model was calibrated to measurement of the prototype, and the fit is quite good given tolerances on components.

The model allows convenient interactive sensitivity analysis where parameters can be dialed up and down with the mouse wheel and the response changes observed.

- FT82-43 matching transformer for an EFHW
- Find |Z|,R,|X| from VSWR,|Z|,R,Ro
- A new impedance calculator for RF inductors on ferrite cores
- Calculate ferrite cored inductor (from Al)
- Calculate VSWR and Return Loss from Zload (or Yload) and Zo
- Duffy, O. 2015. A method for estimating the impedance of a ferrite cored toroidal inductor at RF. https://owenduffy.net/files/EstimateZFerriteToroidInductor.pdf.
- ———. 2006. A method for estimating the impedance of a ferrite cored toroidal inductor at RF. VK1OD.net (offline).

Having selected a candidate core, the main questions need to be answered:

- how many turns are sufficient for acceptable InsertionVSWR at low frequencies and core loss; and
- what value of shunt capacitance best compensates the effect of leakage inductance at high frequencies?

Lets look at a simplified equivalent circuit of such a transformer, and all components are referred to the 50Ω input side of the transformer.

Above is a simplified model that will illustrate the issues. For simplicity, the model is somewhat idealised in that the components are lossless.

- L1 represents the leakage inductance;
- L2 represents the magnetising inductance; and
- C1 is a compensation capacitor.

Since the magnetising inductance is assumed lossless, this article will not address design for core loss.

So, it is obvious that the InsertionVSWR curve is pretty poor at both high and low end.

Let’s look at a Smith chart presentation of the same information, it is so much more revealing.

Above is the Smith chart plot. Remember that the points go clockwise on the arc with increasing frequency, and that InsertionVSWR is a function of the distance from the centre to the point on the locus… we want to minimise that distance. Remember also that the circles that are tangential to the left had edge are conductance circles, they are the locus of constant G.

Now lets analyse the response.

Note that from 1 to 3MHz, the shape of the response tends to a circle tangential to the left hand edge, it a constant G circle. So, G is constant but susceptance B is frequency dependent and -ve. This the the response of a constant resistance R in parallel with a constant inductance (\(B=\frac {-1} {2 \pi f L}\), \(Y= G + \jmath B = \frac 1 R – \frac {\jmath} {2 \pi f L}\)). A part of that susceptance (shunt inductance) is due to the magnetising inductance L2 which contributes to the poor Insertion VSWR at low frequencies.

Note that from 12 to 15Hz, the shape of the response tends to a circle tangential to the left hand edge, it a constant G circle. So, G is constant but susceptance B is frequency dependent and +ve. This the the response of a constant resistance R in parallel with a constant capacitance (\(B=2 \pi f C\), \(Y= G + \jmath B = \frac 1 R + \jmath 2 \pi f C\)). A part of that susceptance (shunt capacitance) is due to the compensation capacitor C1 which contributes to the poor Insertion VSWR at high frequencies.

Lets adjust L2 and C1 for a better InsertionVSWR response.

Above is the response with L2=12µH and C1=80pF. Note that the distance to the centre is improved (and therefore InsertionVSWR is improved). The kink in the response is common, that is typically the mid region where InsertionVSWR is minimum.

It is still not a good response, the InsertionVSWR at the high end is too high, and compensation with C1 does not adequately address the leakage inductance. So, as a candidate design, this one has too much leakage inductance which might be addressed by improving winding geometry and increasing core permeability.

As mentioned, real tranformers using ferrite cores have permeability that is complex (ie includes loss) and dependent on frequency (ie inductance is constant).

Above, the magenta curve is measurement of a real transformer from 1-11MHz with nominal resistance load and three compensation options:

- cyan: 0pF, too little compensation;
- magenta: 80pF, optimal compensation; and
- blue: 250pF, to much compensation.

It should be no surprise that 80pF is close to optimal. Susceptance B at the cyan X is -0.00575S, and broadly, we want to cancel that with the compensation capacitor so we come so \(C=\frac{B}{2 \pi f}=\frac{0.00575}{2 \pi 11e6}=83pF\).

With optimal compensation (80pF in this case) The insertionVSWR at 3MHz is 1.8, probably acceptable for this type of transformer but it is still quite high (4.3) at 11MHz, which hints that leakage inductance needs to be addressed by improving winding geometry and possibly increasing permeability.

Keep in mind that measurements with a nominal resistive load are a guide, measurements with the real antenna wire are very important.

]]>This article builds an NEC model for an EFHW antenna at 3.6MHz incorporating a realistic model of the above transformer.

NEC provides for a NT card characterising a two port network using Y parameters.

The Y parameter model is based on measured input impedance with port 2 open circuit, and short circuit, and the observed turns ratio.

Impedance was measured with the transformer at 3.6MHz using an AA-600.

Above, the calculated Y parameter model including a prototype NT card. This model captures the various loss components of the transformer, mainly magnetising loss, at 3.6MHz.

Note that the Y parameter model is frequency specific.

Above is a graphic showing the geometry of the NEC model. Essentially the feed point has about λ/10 ‘counterpoise’ at a height of 0.3m to the left of the feed point, and a wire slopping upwards at about 45° for the main antenna conductor.

Although the NT card is frequency specific to 3.6MHz, we get a fair idea of the VSWR response over a narrow frequency range. The minimum VSWR of 1.06 at 3.6MHz is correct.

Above is a summary of the NEC model. Network loss captures the loss in the transformer, and at 100W input the transformer loss is 9W. This is much better transformer efficiency than almost all of the published designs that I have reviewed.

Radiation efficiency is 39%, a combination of conductor loss, transformer loss and ground loss, mostly ground loss.

Above is the pattern, highest gain is at towards the zenith as can be expected of a low antenna. Maximum gain is about 1dBi.

Above, at lower elevation (30°) the pattern shows a little skew due to the sloping radiator.

If this looks like an improvised antenna that performs well, keep in mind that the top of the sloping wire is at 28m height.