This article compares the results for the FT240-43 example at 3.5MHz with calculation using tools on this web site.

Above is a very simple approximation of an ideal 1:1 transformer where the effects of flux leakage and conductor loss are ignored. A 1:n transformer can be modelled the same way, as if flux leakage and conductor loss are ignored, the now ideally transformed secondary load becomes 50Ω.

First step is to find the complex permeability of the core material.

Next, calculate the RF impedance and admittance at 3.5MHz of a 4t winding.

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

We can calculate core magnetising loss as \(Loss_m=10 log \left(1 + \frac{Gm }{0.02}\right)=10 log \left(1 + \frac{0.00168}{0.02}\right)=0.35 \; dB\).

We can calculate InsertionVSWR using Ym+0.02 S as the load admittance.

Above, InsertsionVSWR is 1.21.

Measured | Predicted | |

Loss (dB) | 0.32 | 0.35 |

InsertionVSWR | 1.15 | 1.21 |

Given the tolerance of ferrite cores, the reconciliation is very good.

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I note that recently, the published table of #43 permeability changed subtly but significantly. Does this table apply to historical product, or does it only apply to new product, ie was there an actual change in the mix, or what it the result of better measurement of characteristics?

Hi Owen, It is a little bit of both. We have acquired new more accurate test and measurement equipment since the previous curves have been published. Additionally due to availability of raw materials over time; there has been some drift relative performance in some of our materials.

The actual compositions of the materials are the same but, in some cases the composition/concentrations of some of the impurities or differences in some of the processing parameters of the various oxides our materials are comprised of has seen changes.

We use average recent lots for producing curves for our catalog and website.

It is a little difficult to separate out exactly what change can be attributed to the measurement equipment versus what is due to the material. What I can say is that; In general, the shift due to equipment changes were most pronounced at higher test frequencies.

Thanks, Michael Arasim Product Manager for Power and Inductive Applications.

I have published various tools and articles based on the pre 2020 published data, and measurement results reconciled quite well with that data, taking into account the tolerance on the products.

I have added the #43 data I downloaded in 2020 (unfortunately the files do not contain a date), and included the data in Ferrite permeability interpolations as shown in the above screen shot.

]]>The discussion of an example worked up the loss components at 3.5MHz of the example configuration, a 4t winding on a FT240-43.

This article demonstrates that a graph of the loss components from the saved .s2p is possible.

Let’s review some meanings of terms (in the 50Ω matched VNA context):

- \(Loss=\frac{PowerIn}{PowerOut}=\frac{P_1}{P_2}\);
- \(s11=\frac{V_{1r}}{V_{1i}}\);
- \(MismatchLoss=\frac{P_{1i}}{P_{1}}=\frac1{(1-|s11|^2)}\);
- \(s21=\frac{V_{2i}}{V_{1i}}\); and
- \(InsertionLoss=\frac{P_{1}}{P_{2}} \approx \frac{P_{1i}}{P_{2i}} =\frac1{|s21|^2}\).

It is assumed that Zin of VNA Port 2 is 50+j0Ω, and that therefore P2r=0. Error in Zin of VNA Port 2 flows into the results. A 10dB attenuator is fitted to Port 2 prior to calibration to improve accuracy of Zin.

With the quantities expressed in dB, we can derive that \(Loss=-|s21|-MismatchLoss\).

The graph gives a wider perspective of the contribution of Mismatch Loss and (Transmission) Loss to Insertion Loss.

Loss is core and copper loss, mostly core loss in this case.

Try 2, 3, 4, 5 turns on a FT240-43, what does that say about all the articles on the ‘net using 2t primaries?

Try FT240-61, how many turns are sufficient, how does the Loss compare? Before you jump to the conclusion that #61 is superior, you need to measure the broad band performance which may be impacted adversely by the length of the windings… a story for another article.

]]>Above is a very simple approximation of an ideal 1:1 transformer where the effects of flux leakage and conductor loss are ignored. A 1:n transformer can be modelled the same way, as if flux leakage and conductor loss are ignored, the now ideally transformed secondary load becomes 50Ω.

This simple equivalent circuit does contain the elements that are most important to low frequency performance, the inductor and resistor represent the magnetising impedance as a parallel equivalent circuit of the magnetising inductance and core loss.

Let’s simulate that circuit.

Above is a simulation of the |s11| and |s21| we would expect to measure for our simplified transformer.

The design object is:

- low |s11|; and
- high |s21|.

In this case, at the low frequency end, |s11| increases, and |s21|decreases, both due to the combined effects of the winding inductance and core loss.

Now ferrite cores yield a frequency dependent inductance and core loss. There are many articles on this website explaining how to design with ferrite cores using the published core characteristics, but this article is about using a nanoVNA to validate such a design, or even to find a combination by cut and try.

Most failed published ham designs failed to provide sufficient magnetising impedance to deliver adequate low frequency performance.

Let’s measure a couple of examples by winding the primary winding alone, and measuring it in shunt with a through connection from Port 1 to Port 2 (nanoVNA CH0 and CH1). As always, the fixture is very important.

Let’s try a FT240-43 core with a 2t winding connected in shunt with a through connection from VNA Port 1 to Port 2.

Above is a plot of |S21|, and recall that InsertionLoss=-|s21|.

Without capturing the effects of a secondary winding and flux leakage, the primary winding is not at all suitable for a low InsertionVSWR broadband transformer, it has low magnetising impedance, a result of the combination of ferrite characteristic and the number of turns.

Let’s try a FT240-43 core with a 4t winding connected in shunt with a through connection from VNA Port 1 to Port 2.

Above is a plot of |S21|, a huge improvement on the 2t case.

Above is a plot of |s11| which tells us there is some mismatch at the lowest frequencies, but mismatch is unlikely to make a large contribution to the InsertionLoss in this case.

Above is a plot of InsertionVSWR, another presentation of the |s11| measurement.

We can calculate the Loss from the s11 and s21 measurements recorded in the saved .s2p file at 3.5MHz.

As suggested, the main contribution to InsertionLoss is Loss (conversion of RF energy to heat) in the core material and winding, mainly the core material in this case.

Increasing turns increases magnetising impedance and reduces losses, but more turns means longer conductors which compromises high frequency performance, so for a broadband transformer, you need sufficient turns for low frequency response, but no more.

Stacking cores increases magnetising impedance, reduces losses and increases surface are, but longer turns means longer conductors which compromises high frequency performance, so for a broadband transformer, you need sufficient turns for low frequency response, but no more.

Based on measurement results, you may choose different turns and / or different core material.

This measurement does not capture all the important influences on transformer performance, but it does provide a very useful first step for selection / screening / validation test to select a ferrite core material and sufficient primary turns for a low InsertionVSWR 50Ω broadband RF transformer.

If a DUT does not pass muster, it is going to be worse when built with a secondary winding and load.

]]>This article provides more detail on the practical test case.

A small ferrite cored test inductor was measured with a ‘bare’ nanoVNA SOLT calibrated, firstly using s11 reflection.

Above is the R,X,|Z| plot from the s11 reflection measurement of the unknown Zu.

The series calibration used in this case was in fact the same one used for SOLT calibration, and as my might expect, the s11 reflection impedance is essentially 50+j0Ω with measurement noise superimposed. This might not be the case in all fixtures.

Above is the R,X plot from the s11 reflection measurement of Port 2.

Above is the R,X plot of Port 1 calculated as detailed at Improving ‘s21 series-through’ measurement of high impedances.

Above is the test fixture with the test inductor in the series-through configuration.

Above is the R,X plot of the unknown test inductor Zu calculated as detailed at Improving ‘s21 series-through’ measurement of high impedances.

The analysis performed above was done in iPython, the following is a Python export of the script.

#!/usr/bin/env python # coding: utf-8 # In[379]: #Copyright: Owen Duffy 2021. All rights reserved. import math import cmath import skrf as rf import os import numpy as np import matplotlib import matplotlib.pyplot as plt #allow to run in iPython try: %matplotlib inline except: pass # In[380]: #parameters name='TestInductor' # In[381]: #populate the networks nws11c=rf.Network(name+'-S11c.s1p') nws11l=rf.Network(name+'-S11l.s1p') nws11u=rf.Network(name+'-S11u.s1p') nws21c=rf.Network(name+'-S21c.s2p') nws21u=rf.Network(name+'-S21u.s2p') rf.stylely({"savefig.dpi":100,"figure.figsize":(10.24,7.68)}) #common zplot framework def zplot(f,z): global fig global ax1 global ax2 plt.figure(figsize=(10.24,7.68),dpi=100) fig,ax1=plt.subplots() color='r' l1,=ax1.plot(f/1e6,z.real,color=color,label='R') ax1.set_xlabel('Frequency (MHz)') ax1.set_ylabel('R ($\Omega$)',color=color) color='b' ax2=ax1.twinx() #instantiate a second axes that shares the same x-axis l2,=ax2.plot(f/1e6,z.imag,color=color,label='X') ax2.set_ylabel('X ($\Omega$)',color=color) lines=[l1,l2] ax1.legend(lines,[l.get_label() for l in lines]) # In[382]: zc=nws11c.z[:,0,0] #print(zc) zplot(nws21c.f,zc) ax1.set_ylim(48,52) ax2.set_ylim(-1,1) ax1.set_title('Zc - {}'.format(name)) fig.tight_layout() filename='{}-Zc.png'.format(name) print(filename) plt.savefig(filename) zc=50 #override since part used for SOLT cal # In[383]: zl=nws11l.z[:,0,0] #z from s11 slice #print(zl) zplot(nws21c.f,zl) ax1.set_ylim(48,52) ax2.set_ylim(-1,1) ax1.set_title('Zl - {}'.format(name)) fig.tight_layout() filename='{}-Zl.png'.format(name) print(filename) plt.savefig(filename) # In[384]: #print(nws21c.s[:,1,0]) #s21 slice zs=(zl-nws21c.s[:,1,0]*(zc+zl))/(nws21c.s[:,1,0]-1) #print(zs) zplot(nws21c.f,zs) ax1.set_ylim(48,52) ax2.set_ylim(-0.5,0.5) ax1.set_title('Zs - {}'.format(name)) fig.tight_layout() filename='{}-Zc.png'.format(name) print(filename) plt.savefig(filename) # In[385]: #zu=(zs+zl)/s21-(zs+zl) zu=((zs+zl)/nws21u.s[:,1,0]-(zs+zl)) #print(zu) print(zu[68]) print(zu[-1]) zplot(nws21c.f,zu) ax1.set_ylim(-5000,5000) ax2.set_ylim(-5000,5000) ax1.set_title('Zu - {}'.format(name)) fig.tight_layout() filename='{}-Zu.png'.format(name) print(filename) plt.savefig(filename) # In[386]: #zu=(zs+zl)/s21-(zs+zl) zu=((50+50)/nws21u.s[:,1,0]-(50+50)) #print(zu) print(zu[68]) print(zu[-1]) zplot(nws21c.f,zu) ax1.set_ylim(-5000,5000) ax2.set_ylim(-5000,5000) ax1.set_title('Zu- - {}'.format(name)) fig.tight_layout() filename='{}-Zu-.png'.format(name) print(filename) plt.savefig(filename) # In[ ]:

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

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

Agilent gives the expression for s21 as \(S_{21}=\frac1{1+\frac{Z_x}{2 Z_0}}=\frac{100}{Z_x+100}\) (taking Zo=50).

We can rearrange that making Zx the subject \(Z_x=\frac{100}{S21}-100\) and expanding the values 100 to their components \(Z_x=\frac{Z_s+Z_l}{S21}-\left(Z_s+Z_l\right)\) where Zs and Zl are the equivalent Thevenin impedances of the source and load.

That begs the question whether we might compensate for small errors in Zs and Zl by measuring their values and substituting into the calculation.

Measuring Zl is simple enough, after SOLT calibration of the VNA, an s11 reflection measurement of Port 2 gives a value from which Zl can be calculated: \(Z_l=50 \frac{1+S11}{1-S11}\).

Zs cannot be measured directly, but can be found indirectly by measuring s21 of a known series impedance… so the first step is to measure Z of this calibration part, let’s call it Zc. Depending on the fixture, it might be possible to use the same part used in the SOLT calibration.

We can measure Zc using a s11 reflection measurement and calculate Zc as \(Z_c=50 \frac{1+S11}{1-S11}\).

Now we can measure s21 with Zc in series from Port 1 to Port 2. Zs is given by \(Z_s=\frac{Z_l-s21\left(Z_c+Z_l \right)}{s21-1}\).

Having determined Zs and Zl, we can now measure s21 of an unknown Zu in series from Port 1 to Port 2 and calculate \(Z_u=\frac{Z_s+Z_l}{S_{21}}-\left(Z_s+Z_l\right)\).

A small ferrite cored test inductor was measured with a ‘bare’ nanoVNA SOLT calibrated, firstly using s11 reflection.

Above is the test fixture.

Above is the R,X,|Z| plot from the s11 reflection measurement.

Above is a calculation of Z at 7.90MHz from the saved .s1p file, it reconciles with the cursor data on the plot.

Above is a calculation of Z using the values from a series-through sweep saved .s2p file using Agilent’s formula.

Above is a calculation of Z correcting for Zs and Zl as discussed in this article.

In this case there is not a lot of difference in the values obtained. Measurement noise is an issue, all measurements were single captures with no averaging (the averaging function in the firmware does not have the expected outcome so it is not used).

Above is calculation of a simple s11 reflection measurement.

Above is calculation of a simple s21 series-through measurement. The R value is significantly lower.

Above is calculation of enhanced s21 series-through measurement. There is a small difference in both R and X compared to the simple s21 method.

Note there is greater departure of calculated Zs and Zl from the ideal 50+j0Ω

Above is a screenshot from Simsmith basically to plot a very expanded view of the Smith chart of the s11 reflection measurement Port 2. There is measurement noise, but the underlying response is a curve is reasonably clear and while it doesn’t exactly follow a curve of constant G, it is a reasonable first approximation and suggests the cause appears to be a small shunt capacitance… around 2pF. A short series section of low Zo transmission line has quite similar effects.

So even though Port 2 has very good InsertionVSWR at 7MHz, it is somewhat poorer at 30MHz.

The study charted departure from ideal of Port 2 impedance. Though not charted, departure of Port 1 impedance was calculated at 29.7MHz, and departure in both Port 1 and Port 2 impedances impacts the simple s21 series-through measurement.

Online experts opine that s11 reflection measurement is not capable of good accuracy above perhaps 500Ω, and that Agilent’s s21 series-through configuration yields much better results… but that is not borne out in the 7MHz example which is typical of a high impedance ferrite cored common mode choke.

The Zs Zl corrected measurement is a lot more complicated to perform, and gave very similar results to Agilent’s s21 series-through method at lower frequencies.

The fixture is all important in obtaining good results. There may be cases where it is easier to build a good s21 series-through fixture than a good s11 reflection fixture, particularly for larger parts, and that might drive a preference for s21 series-through measurement.

Further, the enhanced algorithm may provide improved accuracy, especially for high Zu in fixtures where departure from ideal is unavoidable.

The technique described here might give improved accuracy for VNAs where 12 term correction is not available.

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

Mornhinweg ferrite core measurements – #31 discussed his measurements of a #31 suppression sleeve.

Above are his measurements of a FB-61-6873 sleeve. Essentially there are two measurements at each frequency, and the expected flux density B is in the ratio of approximately 2:1. He has fitted a straight line on a log/log graph to the measurements at each frequency. The similarity of the slopes is not unexpected, and is a tribute to his experiment design, execution and calculations.

The first thing to consider is the slope of the the fitted line.

If the material was linear in its characteristic, we might expect the core heating to increase as the square of B. The blue line has such a slope and can be used to compare the slope of the red lines. By eye, the measurements show slightly higher slope, and no conclusions can be drawn about linearity as there are only two points for each frequency.

Lets estimate the core loss of a 2 turn inductor (ignoring conductor loss) using complex permeability from the datasheet.

We should keep in mind that suppression sleeves are not controlled principally for permeability in manufacture (they are controlled for Z at some frequency, see Using complex permeability to design with Fair-rite suppression products), so predictions based on published permeability curves have additional uncertainty.

Above is a plot of calculated core loss vs frequency, and Mornhinweg’s measurements at 21V impressed. The nature of the responses are somewhat similar, importantly the dip around 7MHz exists in both measured and calculated responses.

Though #61 is a low loss material at 1MHz, the loss turns upwards at the low end in the above plot simply because there is not much magnetising impedance Zm, and so magnetising current Im is high, and Im^2R grows quickly with decreasing frequency.

Above is a chart comparing calculated core loss for 2t and 4t on the core, it can be seen that increased Zm leads to lower core loss. The 4t case has 50% the volts/turn, 50% the flux density, 25% of the loss.

The matter of control of suppression product parameters has been mentioned and could be one of the factors giving rise to a difference to the datasheet.

However, there are other issues with #61 operated at high power.

Above is the B-H curve for the material, and at 100° saturation is above 1000gauss (100mT). The measurements at 21V have B ranging from 11mT @ 1.8MHz to 0.4mT @ 50MHz, it is always well under saturation.

Whilst we might then think that small signal analysis applies, lets look at the temperature sensitivity of the material.

We can see from the chart above that the loss increases with core temperature, more so at the lower frequency / higher B range. The DUT temperature varied from 21.5 to 29.4° @ 1.8MHz, not a very great range but enough to cause a 10% increase in loss at the end of the test. This issue is more relevant to the final application where the core might well operate at higher temperatures.

Another factor is that of conductor loss which is captured in the measurements. Though we might dismiss conductor loss out of hand, at the lowest frequency the real component of Zm (ie the equivalent core loss resistance) is very low and wire RF resistance may be significant.

- Mornhinweg, M. 2019. Ferrite core loss in HF power applications.

- Ferrite permeability interpolations
- 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

- inductive; and
- suppression.

Sometimes the same dimensioned cores are available in both categories with different part numbers and possibly different prices, implying some real difference in behavior, eg 5943003801 and 2643803802 are both FT240-43 sized cores.

Material datasheets often contain a note like this from the #43 datasheet:

Characteristic curves are measured on standard Toroids (18/10/6 mm) at 25°C and 10 kHz unless otherwise indicated. Impedance characteristics are measured on standard shield beads (3.5/1.3/6.0 mm) unless otherwise indicated.

I sought to clarify my interpretation of this clause by asking Fair-rite …whether the published material permeability curves / tables apply to suppression product. Can I use the published permeability curves / tables to predict inductor impedance reliably for suppression products?

Fair-rite’s Michael Arasim advised…

Yes the published permeability curve can be used to predict impedance. There will be some variance in the shape of the curve due to individual part size as well as process and material variation. The level of this variance will change depending on the individual material but, the curves themselves are all produced using the same sized toroidal core for each material. One thing of note; Inductive rated parts in theory will be controlled more tightly to adhere to the complex permeability curve since their acceptance criteria is generally going to be inductance and loss factor based. Impedance rated parts are accepted based on a minimum impedance at select test frequencies. Since the impedance is a complex value influenced by both the inductive and resistive components of the complex permeability; In theory you could see more variance in each component and still hit the impedance rating. Prior to production, all lots of our materials are screened to ensure that they will adhere (within a tolerance) to the published material data.

So, the following notes and tools are applicable to Fair-rite inductive and suppression products, but one might expect more variation in the real and imaginary components of impedance with suppression products, and loss in cores used for transformers.

- 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

My own experience is that the difference is not huge, but it explains the somewhat wider variation observed with suppression products and the need to verify designs by measurement.

]]>I have been asked several times recently about the assumed emissivity and the accuracy questioned, I assume this has been discussed online somewhere.

When first measuring ferrites with non-contact thermometers, I performed some experiments to discover whether the default emissivity ε=0.95 applied. It would be convenient if it did, and permit use of some instruments that do not allow adjustment of ε.

In the past, I have compared the reading of non-contact thermometers with several K thermocouple meters and a Thermomelt indicator, and observed insignificant difference (ie less than the variance of repeated measurements).

The following experiment is a thermal pic of a FT240-43 core on the black plastic case of the instrument. The setup has had hours to stabilise thermally.

Above is a combined thermal image and faint visual image. This instrument has only one readout point, and by moving it around, only 0.1° variation was observed between the background and the core.

The shape of the core cannot be reliably discerned in the thermal image, the variation is mostly measurement noise… but lets assume there was 0.1° difference between the black plastic case and the core when they are both at 23° and calculate the implied ε that would account for that indication.

Above, the calculated actual ε that would account for 0.1° error is 0.949. Such an error in ε does not translate to the same error in T at different T.

Above, calculating the actual temperature of a measurement when the actual ε is 0.949 and the instrument is calibrated for 0.95 gives 100°, the error is not significant in a three digit precision answer.

If we find what value of ε is required to cause a -1° error at 100°, it is 0.940… well below the estimated 0.949.

I sleep easy that the thermographs I have presented of ferrite cored inductors and transformers do not have significant error due to the assumed black body emissivity of the ferrite.

]]>The calculator has been revised to include 45° chamfers of a specified length on all four corners. If the chamfer angle differs, the error is very small in the range 30-60°. If the corners are radiused, use the radius as the chamfer length, the error is very small.

We do not need to obsess over these errors as they will usually be dwarfed by manufacturing tolerances.

The calculation of ΣA/l for the sharp corner model is fairly simple.

\(\int _{ir}^{or}\frac{w}{2 \pi r}dr\)

\(=\frac{w}{2 \pi }\left(\ln \left(or\right)-\ln \left(ir\right)\right)\)

To implement the chamfer adjustment, the ΣA/l component of the missing material is calculated.

Firstly the inner chamfers (which are simpler).

\(\int _{ir}^{ir+cr}\:\frac{r-ir}{2 \pi r}dr\)

\(=\frac{1}{2\pi}\left(cr-ir\left(\ln \left(ir+cr\right)-\ln \left(ir\right)\right)\right)\)

Then the outer chamfers are calculated.

\(\int _{or-cr}^{or} \frac{\left(r-\left(or-cr\right)\right)}{2 \pi r}dr\)

\(=\frac{1}{2\pi }\left(cr\left(\ln \left(or\right)-\ln \left(-cr+or\right)\right)-or\left(\ln \left(or\right)-\ln \left(-cr+or\right)\right)+cr\right)\)

The final ΣA/l is the first quantity less the two missing components.

Implemented in javascript in the calculator…

aol=width/2/pi*Math.log(od/id)*1e-3; aol=aol-(cr-ir*(Math.log(ir+cr)-Math.log(ir)))/(2*pi)*1e-3*2; aol=aol-(cr*(Math.log(or)-Math.log(-cr+or))-or*(Math.log(or)-Math.log(-cr+or))+cr)/(2*pi)*1e-3*2;

Let’s look at some examples.

Above is a calculation for the popular FT240-43. Without the chamfer adjustment, ΣA/l would be 0.001091 (which is the value given by Fair-rite in the datasheet), adjusting for chamfer the reduction is 2.5%.

Above is an example calculation of ΣA/l and Al. the calculated ΣA/l is less than 1% less than if the chamfer were ignored. The difference may be greater on some cores, especially very small cores.

Above is an example calculation for a very small core with radiused corners. The chamfer approximation reduces ΣA/l and Al by about 2%… again the manufacturing tolerances dwarf the adjustment.

Sizing the adjustment is an accurate way to determine if it is significant or not, and 2% accuracy does not have a lot of application to ferrites with 20% tolerance… but it does become more important when trying to characterise the ferrite material based on core dimensions.

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