There are two elements that are critical to efficient near ideal impedance transformation over a wide frequency range, low flux leakage and sufficiently high magnetising impedance. While low magnetising loss is essential for efficiency, it does not guarantee sufficiently high magnetising impedance for near ideal impedance transformation.

Magnetising impedance can be estimated using one of the following calculators, but keep in mind that there are quite wide tolerances on ferrite cores.

- Inductance of RF cored inductors and transformers
- Calculate ferrite cored inductor – rectangular cross section
- Calculate ferrite cored inductor – circular cross section
- Calculate ferrite cored inductor (from Al)
- Calculate ferrite cored inductor – ΣA/l or Σl/A
- Ferrite permeability interpolations

Magnetising impedance can be measured (eg with an analyser), but it should be measured with only the measured winding on the core. Did I mention the wide tolerance of ferrites?

You might ask the question is 3t sufficient for the primary of an EFHW transformer that delivers a 50+j0Ω load to a transmitter.

Estimating with a calculator, we get the following.

Plugging the real part of Y into Estimate core loss for ferrite cored RF transformer we obtain the following.

Measuring a core with a 3t winding using very short wires to the AA-600 coax socket, the following results were obtained.

Plugging the R,X pair into Estimate core loss for ferrite cored RF transformer we obtain the following. (You could also just enter just the R|| from this analyser value for Rpm.)

Above, the results from measurement are a little better than expected from the datasheets, I did mention that ferrites have quite wide tolerance.

Depending on your loss criteria, the 3t winding might be adequate from a loss perspective on 3.6MHz. On the other hand you might be tempted to test 4t, but there is a limit as more turns tends to compromise the higher frequency performance, especially on a large core.

A follow up article will consider the effect of magnetising impedance on impedance transformation.

]]>The article is ‘in-brief’ as technical stuff that will not interest most hams is published privately on a members-only page. This article is based on the information in the QST article alone (ie not on the private members only supplementary information).

The core has a modest price in North America, but shipping to other parts of the world may make it very expensive… IOW unobtainium to most parts of the world.

Above is the published InsertionLoss. The article states that they were half the value obtained in a back to back measurement, and it should be noted that is a compromised measurement, and secondly that InsertionLoss comprises two components, (dissipative) Loss and MismatchLoss.

The fact that InsertionLoss increases markedly at the lower frequencies is a hint that it has too few turns for those frequencies, and that InsertionVSWR and efficiency may suffer.

The transformer is intended to be used with a load such that the input impedance Zin is approximately 50+j0Ω, Gin=0.02S.

Analysis of a simple model of the transformer with a load such that input impedance is 50+j0Ω gives insight into likely core losses in that matched condition.

Let us calculate the magnetising admittance of the 4t primary at 1.8MHz. The core is a Laird 28B1540-000 ferrite toroid.

Above is a table of complex permeability for Laird 28 material. (I have seen online experts advise that this information is not available… but here it is.) The Laird #28 material is a medium μ NiZn ferrite, somewhat similar to Fair-rite #43 but different enough to not be equivalent.

Now lets model the 4t primary at 1.8MHz.

Gcore is the real part of Y, 0.000826S.

If Yin of the loaded transformer is 0.02S, we can calculate the core efficiency as 1-Gcore/Gin=1-0.000826/0.02=95.8%, core loss is 0.18dB. Note that this efficiency is in the matched condition, and could be higher or lower at other input impedances (η=1-Gcore/Gin). Ferrite materials have fairly wide tolerance, so measurements of actual cores may have some variation.

The balance of the measured InsertionLoss is likely to be MismatchLoss, but that information is not disclosed in the main article. MismatchLoss will include contributions from the shunt magnetising admittance (moderately low in this case) and effects of flux leakage.

- 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).

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An online expert questioned the analysis and later measurements, and proposed his own transformer design as evidence.

Notably, his transformer uses #61 material and a larger binocular core, a Fair-rite 2861006802 with 2t for a nominal 50Ω primary, giving loss measurements at 7MHz of 0.08dB. Note that the confidence limits of that loss measurement because of the way in which it was obtained (eg a 1% error in the 1120Ω load resistor contributes 0.043dB error to the result), but the measurements do suggest that the loss is probably very low.

Though the loss is low and Return Loss is high at 7MHz, the limits for ReturnLoss>14dB (VSWR<1.5) is 5-18MHz. With compensation, that range may be changed.

Lets apply the method laid out at PD7MAA’s BN43-202 matching transformer for an EFHW.

The best Fair-rite data I can find quickly is a chart of the impedance of a one turn winding.

Scaling from this graph, Xs is close of 35Ω at 7MHz, so lets used that to derive some basic parameters for the core.

Firstly, lets find the permeability of #61 at 7MHz.

Freq (MHz) |
µ’ |
µ” |

7.000e+0 |
1.214e+2 | 1.159e+0 |

Using that in a calculator to iteratively find the value of ΣA/l that gives Xs=35Ω at 7MHz, we obtain ΣA/l=0.0054m, this captures the magnetic path geometry of the binocular core.

Let us now use that core characteristic to calculate the magnetising admittance of the 2t primary winding.

Gcore is the real part of Y, 0.0000421S.

If Yin of the loaded transformer is 0.02S, we can calculate the core efficiency as 1-Gcore/Gin=1-0.0000421/0.02=99.8%, core loss is 0.009dB.

The total loss of this type of transformer will be dominated by the core loss.

The posted measured results, though having wide confidence limits, fall quite in line with a theoretical prediction using the method laid out at PD7MAA’s BN43-202 matching transformer for an EFHW. The measurements are evidence that the design method works.

The superior efficiency of the tested transformer is due to a better magnetic design, better than the PD7MAA design.

- PD7MAA EFHW antenna for 40-10m qrp
- 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).

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I have been offered input VSWR curves for such a configuration, and they are impressive… but VSWR curves do not address the question of loss / efficiency.

Note that building loss into antenna system components is a legitimate and common method of taming VSWR excursions, eg TTFD, CHA250, many EFHW transformers, but in some applications, users may prioritise radiated power over VSWR.

Objectives are:

- used with a load such that the input impedance Zin is approximately 50+j0Ω, Gin=0.02S;
- broadband operation from 3.5-30MHz;
- VSWR < 2 with nominal 3200Ω load; and
- transformer efficiency > 90% at 3.6MHz.

The following describes such a transformer using a Fair-rite 2643625002 core (16.25×7.29×14.3mm #43).

I mentioned in the reference article that the metric ΣA/l captures the geometry, the larger it is, the fewer turns for same inductance / impedance. ΣA/l for the chosen core is 3.5 times that of a FT82-43 yet it is only 1.6 times the mass.

The transformer is wound as an autotransformer, 3+21 turns, ie 1:8 turns ratio.

Firstly, lets estimate at 3.6MHz minimum number of turns to ensure that magnetising conductance is less than about 0.002S (for better than 90% core efficiency).

Above, 3t on the primary delivers Gcore<0.002S.

Above is a sweep of the uncompensated prototype with a 3220+50Ω load.

Let work through a loss analysis.

Because of the division of power between the 3220Ω resistor and VNA input, there is effectively an attenuator of -10*log(50/(50+3220))=18.16dB, so |S21| has a component due to this division. Lets call this element the LoadAttenuator.

Zin=46.52+j6.72Ω. From that we can find Mismatch Loss.

MismatchLoss is 0.03dB.

Loss (to mean PowerIn/PowerOut) can be calculated in dB as -|S21|-LoadAttenuator-MismatchLoss=–18.64-18.16-0.03=0.450dB, or an efficiency of 10^(-0.45/10)=90.2%.

Note that there is some uncertainty in the measurements, but we can be confident that the loss is no where near the figure estimated for the FT82-43 design.

A 100pF silvered mica was connected in shunt with the transformer primary. This is not an optimal value, benefit may be obtained by exploring small changes to that value.

Above is a sweep of the roughly compensated transformer. The capacitor makes very little difference to the low frequency behavior, but it reduces the input VSWR significantly at the high end. VSWR<1.8 over all of HF.

This transformer has more surface area than a FT82-43 based one, so it has higher capacity to dissipate heat, and it is more efficient, so it will have higher power capacity than the FT82-43 based one.

Above is a thermograph of the transformer at 20W input at 3.6MHz. Ambient temperature is 20°, and the core temperature increased by 5° over 120s @ 20W continuous input. That does not seem inconsistent with the expectation calculated above of about 10% core loss at 3.6MHz.

The tests here were using a dummy load on the transformer, and that did allow confirmation of the design and expected loss at 3.6MHz.

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.

Well, in ham radio, everything works. But systems that work better increase the prospects of contacts.

- 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).

Like almost all such ‘designs’, they are published without supporting measurements or simulations.

The transformer is intended to be used with a load such that the input impedance Zin is approximately 50+j0Ω, Gin=0.02S.

Analysis of a simple model of the transformer with a load such that input impedance is 50+j0Ω gives insight into likely core losses.

Let us calculate the magnetising admittance of the 3t primary at 3.6MHz. The core is a FT82-43 ferrite toroid.

Gcore is the real part of Y, 0.00691S.

If Yin of the loaded transformer is 0.02S, we can calculate the core efficiency as 1-Gcore/Gin=1-0.00691/0.02=65.4%, core loss is 1.87dB.

Now as to whether 65% efficiency is acceptable is a question for the user. This is intended for QRP use, so 5W SSB telephony input is not like to damage it, and you could think that an inefficient antenna system doubles the benefit of QRP, QRP^2 if you like.

If an efficiency target for the transformer is set at say 90% or better, it takes an 6t primary to achieve that on 3.6MHz.

Gcore is the real part of Y, 0.00173S.

If Yin of the loaded transformer is 0.02S, we can calculate the core efficiency as 1-Gcore/Gin=1-0.00173/0.02=91.3%, core loss is 0.39dB.

Of course twice the turns are needed on the other winding, leakage inductance is increased, and the compensation capacitor will need review. It is likely that even with compensation, less VSWR bandwidth is achieved.

Another option is to stack two or three of the cores, or use a different core better suited to the application.

It seems 2t or 3t primary windings on #43 cores are very common which might suggest ‘designers’ have simply changed the core dimensions.

The geometry of the core varies from size to size, so just as inductance and impedance are very dependent on magnetic properties of the material (complex permeability), the also depend on cross section area and path length. The calculator shots above show a metric, ΣA/l, which captures the geometry, the larger it is, the fewer turns for same inductance / impedance. ΣA/l for and FT240-43 core is 0.00106 whereas for the FT82-43 core discussed in this article, it is less than half of that at 0.000468 and so drives a need for more turns.

Note that the next size core up or down in a series could have greater or lesser ΣA/l (m), there is no general rule that going to a smaller or a larger core requires more or less turns. Some datasheets show ΣA/l or the inverse Σl/A .

Well, in ham radio, everything works. But systems that work better increases the prospects of contacts.

- 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

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Like almost all such ‘designs’, they are published without supporting measurements or simulations.

The transformer is intended to be used with a load such that the input impedance Zin is approximately 50+j0Ω, Gin=0.02S.

Analysis of a simple model of the transformer with a load such that input impedance is 50+j0Ω gives insight into likely core losses.

Let us calculate the magnetising admittance of the 2t primary at 7MHz. The core is a stack of 3 x FT240-52 ferrite toroids.

Gcore is the real part of Y, 0.00104S.

If Yin of the loaded transformer is 0.02S, we can calculate the core efficiency as 1-Gcore/Gin=1-0.00104/0.02=94.8%, core loss is 0.23dB.

Whilst this might be smaller than many similar designs, it must be considered in the context of the 500W CW rating.

The average power of A1 Morse code modulation is about 44%, so the average power of a 500W CW transmitter is about 220W, and 5% dissipation is 11W average which is probably within the capability of the cores, especially with ventilation as shown.

Losses in the matching transformer (in this case 25W of the 500W CW input) are only part of the total system loss, and overall system efficiency will be lower than estimated here for the transformer alone.

The core loss for this configuration is worse at 14MHz, nearly double and solution is left as an exercise for the reader.

The magnetising impedance is so low on the lower bands that near ideal impedance transformation is unlikely, it may nevertheless perform adequately in some specific EFHW scenarios.

In summary, it uses three relatively expensive cores, has poor magnetising impedance on the lower bands compromising near ideal impedance transformation, and moderate core loss in mid HF, but on the positive side, large surface area to assist is dissipating heat.

- 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

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A recent article by PD7MAA describes such a transformer using a BN43-202 balun core for up to 20W PEP from 7-29MHz.

Above is PD7MAA’s graphic for his transformer. It is a little confusing as an 11t wind will start and finish with ends as the blue wind, so the red winding must have and odd number of half turns which suggests the windings are actually 1t and 5.5t (pity he did not show a picture of the real transformer). Let’s proceed under that assumption.

Let us calculate the magnetising admittance of the 1t primary at 7MHz.

Gcore is the real part of Y, 0.0115S.

If Yin of the loaded transformer is 0.02S, we can calculate the core efficiency as 1-Gcore/Gin=1-0.0115/0.02=43%, core loss is 3.7dB.

The average power of uncompressed SSB telephony is about 3%, so the average power of a 20W PEP SSB transmitter is about 600mW, and 57% dissipation is only 340mW which the core should safely withstand.

Losses in the matching transformer are only part of the total system loss, and overall system efficiency will be lower than estimated here for the transformer alone.

Low efficiency is not uncommon in QRP systems as even when more than half the transmitter power is lost in the core, the core survives by virtue of the low input power whereas a 1kW transmitter would destroy the core in short time.

Some hams might question the wisdom of “chucking away” transmitter power when you don’t have very much to start with, but the logic might be that the purest form of QRP is to start with a low power transmitter and then waste more than necessary in an inefficient antenna system.

PD7MAA does give a measurement of Zin with a 3k3Ω resistor load as VSWR=1.48 and |Z|=60.3Ω.

Above is a calculation of Gin and |Bin|, and if the 3k3Ω load (G=0.000303S) is transformed approximately by the turns ratio (though that is a gross approximation), it would present G approximately 0.0092S in parallel with the magnetising admittance estimated at 0.0115+j0.0165S above for total Yin of 0.021+j0.016S which is in the ball park of the admittance implied by PD7MAA’s measurements (given the tolerances of ferrite and the approximations mentioned). PD7MAA’s measurements support the assumption of a 1t primary and 5.5t secondary.

- PD7MAA EFHW antenna for 40-10m qrp
- 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

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If the transformer is simply used without an ATU between it and the radio, and we assume that the antenna system is adjusted to present low VSWR(50) to the radio, a simple approximation involves calculating the magnetising admittance of the 3t 50Ω winding, and calculating the portion of total input power that is dissipated in that admittance.

Using the calculator at Calculate ferrite cored inductor, the admittance (G+jB) of the 3t winding is 0.00177-j0.00204S. (The impedance of a sample wind could be measured with a suitable analyser and converted to admittance.)

For a 50Ω match, the total conductance G is 1/50=0.02, and the percentage power lost in the magnetising admittance is Gcore/Gtotal*100=0.00177/0.02*100=8.9%.

Now cores of this type heat up relatively slowly, but long term the temperature rise is around 10K/W/m^2 in free air. (This is a very conservative figure based on design methods for heatsinks, up to five times this dissipation might be achieved by radiation from high emissivity surfaces such as many ferrites.) The surface area of the core is about 0.0035m^2, and for temperature rise of say 70K, maximum long term average power dissipation is around 2.5W, less if enclosed.

Pulling all this together, maximum continuous average RF power is around 2.5/0.125=20W, again less if enclosed.

Heat wise, this transformer might work ok at up to 600W PEP SSB telephony (no speech processing) because of the very low average/PEP ratio of SSB telephony, but it would probably fail quite quickly tuning up with a 600W carrier. (It might not withstand the voltage associated with 600W PEP.

Because of the risk of explosion of the ferrite (a ceramic material with risk of dangerous flying shards) under rapid temperature rise, such a device should be enclosed if used with a transmitter over a hundred watts… exacerbating the heating problem.

For this application, I would be considering a higher Q core material (which could be ferrite or powdered iron), and then a size appropriate to the power requirement. Note though that reducing the core loss will reduce the bandwidth, so system tuning becomes sharper.

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A correspondent having read End fed matching – design review raised a similar design by PA3HHO which uses a#43 ferrite toroid as part of an end-fed matcher, see Multi band end-fed (English).

Further to End fed matching – PA3HHO design review , this article has a solution to PA3HHO’s “original 40/20/10 version” using FT140-43 with a 2t primary.

If the transformer is simply used without an ATU between it and the radio, and we assume that the antenna system is adjusted to present low VSWR(50) to the radio, a simple approximation involves calculating the magnetising admittance of the 2t 50Ω winding, and calculating the portion of total input power that is dissipated in that admittance.

Using the calculator at Calculate ferrite cored inductor, the admittance (G+jB) of the 2t winding is 0.00715-j0.0104S. (The impedance of a sample wind could be measured with a suitable analyser and converted to admittance.)

For a 50Ω match, the total conductance G is 1/50=0.02, and the percentage power lost in the magnetising admittance is Gcore/Gtotal*100=0.0071/0.02*100=36%.

That gives a core efficiency of 64% or core loss of 2dB.

The 2t primary on FT240-43 is better…

Using the calculator at Calculate ferrite cored inductor, the admittance (G+jB) of the 2t winding is 0.00573-j0.00834S. (The impedance of a sample wind could be measured with a suitable analyser and converted to admittance.)

For a 50Ω match, the total conductance G is 1/50=0.02, and the percentage power lost in the magnetising admittance is Gcore/Gtotal*100=0.00573/0.02*100=29%.

That gives a core efficiency of 71% or core loss of 1.5dB.

PA3HHO Multi band end-fed (English)

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This article walks through practical measurement of a ferrite toroidal inductor using an antenna analyser.

To be relevant practically, lets use an example from N4SPP’s end fed wire antenna on 3.6MHz. His coupling transformer uses a two turn winding on an FT240-43 core for the nominal 50Ω connection to the antenna system.

We could calculate the impedance of this winding using one of the plethora of online and desktop inductance calculators, but lets first fetch the data from the manufacturer.

A simple statistic that is widely used is Al, and above, Fair-rite gives Al=1075nH +/-20%. Note that although they give a tolerance of +/-20%, it is not uncommon that manufactured product has greater error, they may have optimistically quoted the standard deviation and it is easy to fall outside that (37% chance).

What might we expect. Let’s check that the likely measurements will be in-range of the type of instrument that we will use.

toroids.info is a very popular online calculator for toroidal ferrite cored inductors.

Above is the toroids.info results based on Al=1075nH (they often have incorrect Al values).

They give Z (ohms) as the scalar value 97.3Ω, and the naive would assume that Z=0+j97.3Ω, but is the R component really zero?

Above is the popular Mini ring core calculator v1.2.

First thing to note is that it gives Al as 930nH (for 1t), which is 13.5% lower than manufacturer Fair-rite gives for their cores, so we expect incorrect results.

This gives Xl=84.144Ω with no mention of R.

Above, the calculator results include Z and Y, both as complex numbers that include the effect of core loss.

Let’s make a measurement of impedance of a core with two turn winding.

Above is a plot using an AIMuhf. Due to experience that has undermined confidence in this system, I have checked the results on a Rigexpert AA-600 and they are acceptably close.

The figures on the right hand side give the spot values at 3.6MHz.

Dismiss the calculators that do not give complex impedance with non-zero R, they are toys that are not suited to RF ferrite cored inductors.

The sample core has measured Z that is significantly higher than predicted by the owenduffy.net calculator, measured R is 45.3 vs 27.4 (+65%), and measured X is 79.7 vs 57.4 (+39%). This difference highlights an important dimension of designing with ferrites.

Can we rewrite the books about FT240-43 based on measurement of a single sample? Certainly not, that cannot possibly capture the nature of a product that is actually highly variable.

There are a host of methods of characterising an inductor, and various forms of LCR meter are very popular with hams. The problem is that they usually make measurements at a low and often unknown frequency which is of little use in characterising a ferrite cored inductor at RF.

Q meters, RF impedance and admittance bridges may provide valid measurements, but the modern two port VNA or quality antenna analyser can provide sufficient accuracy for a wide range of practical inductors.

The design process often calls for choosing some key parameters of a component, and in the case of N4SPP’s transformer, one of the rules of thumb (RoT) is that number of turns is chosen so that Xl is greater than 50Ω. Such a RoT will become evident as ROT as we go on.

Assuming that the loaded transformer input impedance= 50+j0Ω, characterising the transformer two turn primary as a complex impedance or admittance allows us to calculate the percentage of power lost in magnetising the core.

We have two sets of data, one representing average core characteristics from the manufacturer and one from measurement of a single core. Lets calculate the percentage core loss of both.

Just to recap, if the input Zin of the loaded transformer with VSWR=1 is 50+j0Ω, input Yin=1/Zin=0.02S, Gin=0.02S.

Measurement of the actual core gives insight into the performance of a single core, the DUT.

From the AIMuhf results, Gcore=0.005388S.

CoreLoss=Gcore/Gin=0.005388/0.02=0.269=26.9%.

Note that these figures apply to the actual core measured. The best estimate of N4SPP’s core, or any you might buy is the manufacturer’s data (calculated below).

Some low end analysers give only VSWR(50) and |Z|. We can simulate use of those by selecting just those values from the AIMuhF report:

- VSWR=4.597; and
- |Z|=Zmag=91.736.

Using the calculator Find R,|X| from VSWR,|Z|,Ro, we can find R and |X|, and more conveniently, G and |B|.

Above, the value of G at 0.00539 agrees with that directly from the AIMuhf report.

A significant issue is that the precision and accuracy of many of the low end devices is poorer, and the limits of Z that provide acceptable accuracy are narrower. However, this example was chosen to be in range of most analysers, you can try a similar measurement to reinforce what you have read.

The manufacturer’s data should be based around measurement of a number of cores, and reduction of those measurements to an average value and some variance or tolerance. This data is more relevant to design for replication.

From the owenduffy.net calculator above, Ycore=0.00677-j0.0142S, Gcore=0.00677S.

CoreLoss=Gcore/Gin=0.00677/0.02=0.339=33.9%.

Watch the blog for continuing postings in the series Exploiting your antenna analyser. See also Exploiting your antenna analyser – contents.

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