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 82pF 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 3 from 80m to 20m.

Core loss is highest at 20m, 0.134dB, which equates to 3W of core loss at 100W input.

Above, expected core temperature rise in free air is about 20°, quite modest.

]]>Before looking at the specifics of the Hirose U.FL connector, clean connectors work better and last longer. That should not be a revelation.

A can of IPA cleaner and a good air puffer are invaluable for cleaning connectors. The air puffer show has a valve in the right hand end, it doesn’t suck the dirt and solvent out of the connector and blow it back like most cheap Chinese puffers, this one was harder to find and expensive ($10!).

A clean tooth brush and / or small paint brush may prove useful. I would not put cotton buds, Q-tips or anything with free lint into the connector unless you are sure you can remove any lint (like with a strong jet of clean oil free compressed air).

The Hirose U.FL connector is widely used on miniature equipment for connecting RF coax cables.

I use them in lots of projects, mainly 2.4GHz WiFi and 915MHz LoRa. Above is an example where a 2.4GHz WiFi antenna is attached to an ESP8266 module in a project providing telemetry of dam level sensed with a 4-20mA pressure transducer.

The connector above has been on and off a development board more than 30 times, it is still in good working order. Visually, it cannot be faulted. The connector is on a Molex 105262 915MHz antenna which cost about $6 a few years ago, it is inexpensive hardware, but good quality.

If you take care of quality connectors they last more than 30 cycles, but if you do not, they might not last 10 cycles.

Above is a close up of the U.FL connector, it has two ears for lifting it off the male part without cocking it sideways (which will damage the connector).

Above, a DIY tool hooked under the ears. It snuggly fits so it cannot disengage, and it does not prevent expansion of the spring rim which must expand to disengage the male part. Note that it does not apply force to the cable or its crimp.

Above, the other end of my DIY tool is ground flat and square to the shaft, and allows pressure to be applied to the connector whilst without allowing it to cock sideways (which will damage the connector). Note that it does not apply force to the cable or its crimp.

This tool can reach down into confined spaces, so there is almost never an excuse to not use it.

With all that, IMHO opinion the use of U.FL connectors on boards like that above does not make a lot of sense for several reasons, but if you have one, and take care with the connectors as discussed above, you should get good value from the thing.

]]>90° methodas I will call it.

The reason why people make measurements at +/- 90 degrees on the smith chart is because the measurement accuracy using the shunt configuration when trying to measure the nominal value of an inductor or capacitor is highest at 0+j50 ohms (or 0-j50 ohms… OD).

To be clear, this is the phase of s11 or Γ being + or – 90° as applicable.

Is there something optimal when phase of s11 is + or – 90°?

Does the software / firmware / hardware give significantly more accurate response under such a termination?

Above is a diagram from a HP publication, slightly altered to suit the discussion.

V1 and V2 denote points of measurement proportional to the forward wave and reflected wave, think of uncorrected \(s_{11} \propto \frac{V_2}{V_1}\). Remember that s11 is a complex quantity, having amplitude and phase.

Different circuits and test configurations mean that the propagation time from the V1 sensor via the directional coupler to the terminals (shown as small circles) and connecting wires to the component Zx, reflection back along the path to the directional coupler and then to the V2 sensor is significant. That propagation time is manifest as a phase delay proportional to frequency.

Realise that |V1| will usually be relatively high, but |V2| may be very low, usability limited by the noise floor. Some detector circuits have weaknesses in phase measurement, more so when V2 is weak.

Above is a plot of the uncorrected phase of s11 from a NanoVNA with a SDR-kits test board OC termination and connected by about 300mm of RG400 coax to the NanoVNA.

In this case, |s11| is very close to unity, as it would be for measuring a high Q inductor or capacitor. Note there are no apparent glitches or even obvious noise on the measurement. There is no evidence of a range of phase of uncorrected s11 that should be avoided for accuracy reasons.

Now after calibration and application of correction, the phase of s11 above becomes approximately zero though the calibration frequency range.

If you were to connect a capacitor and find a frequency where the indicated phase of s11 is 90°, realise that the phase difference V2/V1 is larger, for example if that frequency was 50MHz, the phase of V2/V1 would actually be 90+63=153°.

So, what is the magic of displayed \(\angle s_{11}=90 °\)?

Why would you restrict the instrument to measure equivalent inductance at only one frequency that might not be relevant to the application (equivalent series inductance is not a frequency independent characteristic)?

In any event, the phase angle of the raw forward and reflected signals as sensed is likely to be significantly different to a corrected display and attempts to optimise the phase angle actually sensed is incredibly naive.

]]>Let’s explore the use of s21 shunt through to directly find the half power bandwidth of a series tuned circuit and calculate the Q from that and the resonant frequency (as demonstrated by online posters).

To eliminate most of the uncertainties of measurement, let’s simulate it in Simsmith.

The simulation has a series tuned circuit resonated at 3400kHz, and the source and plot are set to calculate |s21| in dB. Though the model specifies Q independent of frequency, the D block adjusts Q for a constant equivalent series resistance (ESR) which simplifies discussion of resonance and Q.

The markers are set to the frequencies where |s21| is 3dB higher that at the resonant frequency, and the bandwidth is 3601-3210=391kHz, which gives \(Q=\frac{f_0}{BW}=\frac{3400}{391}=8.7\).

But wait a minute, the simulation specifies Q=10!

Why is the measurement technique not giving the expected result?

It goes to understanding some basic concepts, and applying them to the measurement problem.

The response of a simple series resonant RLC circuit is well established, when driven by a constant voltage source the current is maximum where Xl=Xc (known as resonance) and falls away above and below that frequency. In fact the normalised shape of that response was known as the Universal Resonance Curve and used widely before more modern computational tools made it redundant.

Above is a chart of the Universal Resonance Curve from (Terman 1955). The chart refers to “cycles”, the unit for frequency before Hertz was adopted, and yes, these fundamental concepts are very old.

(Terman 1955) gives a general definition of Q that is widely accepted:

circuit Q is 2π(energy stored in the circuit)/(energy dissipated in circuit in one cycle) .

Q is easily assessed for a single reactor:

- inductor: energy stored is \(L=\frac{I_{pk}^2}{2}\) and energy lost per cycle is \(\frac{I^2R}{2 f}\) so \(Q=\frac{2 \pi f L}{R}=\frac{X_l}{R_s}\);
- capacitor: energy stored is \(\frac{C V_{pk}^2}{2}\) and energy lost per cycle is \(\frac{V^2}{2 f R_p}\) so \(Q=2 \pi f C R_p=\frac{R_p}{X_c}\).

A RLC series resonant circuit is nearly as easy as at the instance when the inductor current is maximum, capacitor voltage is zero and hence the energy stored is as per the simple inductor case and \(Q=\frac{X_l}{R_s}\).

Likewise for a parallel RLC resonant circuit, at the instant of maximum capacitor voltage, inductor current is zero and hence the energy stored is as per the simple capacitor case and \(Q=\frac{R_p}{X_c}\).

Note that this model is not valid where the inductor component (which is more completely a resonator itself) approaches its own self resonant frequency (SRF).

Q and bandwidth are related, high Q circuits have a narrow response, narrow bandwidth. Q is a factor used in normalising the Universal Resonance Curve (URC) above, and appears in several of the equations on the chart.

Lets focus in particular at the point on the URC corresponding to where |X|=R. When |X|=R, |Z| is 2^0.5 times that at resonance, and the current response is 2^-0.5 (or 0.707) times that at resonance. Since power=I^2*R, power at that point is half the maximum response. These are known at the half power points.

If you look at the URC, you will seen that the current response is 0.7 of maximum when a=0.5. Taking the case for a=0.5 (the half power points) and noting that Bandwidth (BW) is measured between the upper and lower points, BW=2*CyclesOffResonance, we can substitute into a=Q*(CyclesOffResonance/ResonantFrequency) to obtain BW=ResonantFrequency/Q. This is a well known relationship.

For a circuit where R is approximately constant with frequency, we can find the approximate BW by finding the frequencies at which |X|=R.

Lets repeat two important statements from the above:

- for a circuit where R is approximately constant with frequency, we can find the approximate BW by finding the frequencies at which |X|=R; and
- when |X|=R, |Z| is 2^0.5 times that at resonance, and the current response is 2^-0.5 (or 0.707) times that at resonance. Since power=I^2*R, power at that point is half the maximum response. These are known at the half power points.

This does not mention s21, though s21 is a means to a solution, but it is not by itself the solution.

We can use (12 term corrected) s21 shunt through measurement to find R, X, |Z| of the series circuit LC1. Remembering that s21 is a complex quantity, \(z_x=\frac{25 s_{21}}{1-s_{21}}\).

Above is the Simsmith model extended to calculate zx, the impedance of the series circuit LC1, and to plot some relevant quantities. Remember that the simulation Q is 10, and ESR is independent of frequency.

Note that your VNA may not be able to calculate zx in the s21 shunt thru configuration.

The chart is quite busy, it contains:

- the originally plotted |s21| as s21m in blue;
- R as zx.R in red;
- |X| as zx.I.M in yellow; and
- |Z| as zx.M in magenta.

We can calculate Q using the method used at the start of this article, and two methods based on the discussion of concepts.

Method | Bandwidth (kHz) | Calculated Q |

|s21| 3dB points | 391 | 8.7 |

\(|Z|=\sqrt{2}Z_0\) | 340 | 10.0 |

\(|X|=R\) | 340 | 10.0 |

The |s21| 3dB points clearly fails to reconcile, it is not based on sound concepts. It might give results that appear correct for some scenarios, but it is not soundly based.

The last two methods both reconcile with the simulated value of Q.

Measurements using a spectrum analyser and tracking generator to measure the equivalent of |s21| shunt through will also fail for the same reasons.

The simulation forced ESR to be independent of frequency as is done is most textbook discussion of the concepts discussed above.

Real world inductors at RF are not quite that simple, a first approximation is that \(R \propto \sqrt{f}\) by virtue of skin effect. Close wound solenoids and layered windings are complicated by Proximity Effect which increases R. Self resonance of the inductor (which is really of itself a resonator) causes a change in R at frequencies above about a tenth of self resonant frequency.

Let’s change the model to make \(R \propto \sqrt{f}\) to demonstrate a more practical scenario.

Note the curves are not so symmetric, a result of R varying with frequency.

We can calculate Q using the method used at the start of this article, and two methods based on the discussion of concepts.

Method | Bandwidth (kHz) | Calculated Q |

|s21| 3dB points | 391 | 8.7 |

\(|Z|=\sqrt{2}Z_0\) | 340 | 10.0 |

\(|X|=R\) | 342 | 9.9 |

Again, the |s21| 3dB points clearly fails to reconcile, it is not based on sound concepts. It might give results that appear correct for some scenarios, but it is not soundly based.

Even though the resonance response is a little skewed, methods two and three still produce a result that reconciles well with expected Q @ 3.4MHz. The skew warns the measurer to not measure only one half power point, but to measure both to offset some of the error due to the skewed response.

Measurement of real world series resonant circuits will be complicated by the above effects, parasitic inductances and capacitances of fixtures, and measurement error.

Application to inductors where the equivalent series inductance is not approximately independent of frequency is questionable, the response will not follow the URC (including the half power bandwidth inference), the broader inference of Q is not valid, it is possibly specious (ie superficially plausible, but wrong).

Measurement of Q is not quite a no-brainer, but application of sound concepts and an intelligent approach in experiment design should yield valid results.

Accurate measurement of s21 with a VNA system depends on the accuracy of the input impedance or Port 2, errors in which can be corrected by 12 term correction.

The measurements need to demonstrate the the inductor is sufficiently far below its SRF to be adequately described as a frequency independent inductance in series with some resistance.

Just as in the case above that you cannot determine half power bandwidth from |s21| 3dB points, the same applies to a parallel resonant circuit using s21 series through measurement.

Applying the information in this article to the parallel resonant circuit using s21 series through is left to the reader as an exercise.

- Terman, Frederick. 1955. Electronic and Radio Engineering – 4th ed. New York: McGraw-Hill.

There are occasions where it is not possible, or not convenient to locate the DUT at the reference plane. This article discusses the problem created, and some solutions that might give acceptable accuracy for the application at hand.

The discussion assumes the VNA is calibrated for nominal 50+j0Ω.

Above is a diagram of a configuration where the unknown Zl is not located exactly at the reference plane, but at some extension.

The problem created is that Vr50’/Vi50′ (s11) is not the same as if Zl was attached directly at the reference plane.

Let’s look at some possible solutions.

If the extension can be well characterised as a uniform transmission line, we can estimate the effect of the extension by calculation and approximately correct it.

The impedance transformation due to a section of uniform transmission line is \(Z_{in}=Z_0 \frac{Z_l+Z_0 tanh(\gamma l)}{Z_0+Z_l tanh(\gamma l)}\). If the transmission line parameters are known, the the impedance ‘measured’ at the reference plane can be corrected to the actual DUT location.

This is not a common feature, but it appears in Rigexpert’s Antscope (1) as the add/subtract cable feature.

The accuracy depends on the accuracy of the transmission line model and the line characterisation, it needs to be verified, it is not an insignificant issue.

If we can assume that port extension conductors form a uniform transmission line with negligible loss, we can use the relationship that \(Z_{in}=Z_0 \frac{Z_l+\jmath Z_0 tan(\beta l)}{Z_0+\jmath Z_l tan(\beta l)}\).

We can correct the measurement by adjusting for the extension round trip phase delay due to βl, so at each measurement frequency we add that phase delay back into the phase of ‘measured’ s11.

Accuracy depends on Zo=50+j0Ω and negligible loss, though acceptable results might be obtained for small departures. Note that negligible loss infers a short line section.

This facility is often provided in VNAs and PC client programs.

This can also be used with good effect to approximately compensate for a test fixture that does not exactly meet the conditions specified previously. The error may be small if:

- the line section is electrically very short; and
- Zl >>Zo; or
- Zl<<Zo.

If we assume that port extension conductors form a uniform transmission line with negligible loss, we can use the relationship that \(Z_{in}=Z_0 \frac{Z_l+\jmath Z_0 tan(\beta l)}{Z_0+\jmath Z_l tan(\beta l)}\).

Let’s look in more detail at the two cases.

\(Z_{in}=Z_0 \frac{Z_l+\jmath Z_0 tan(\beta l)}{Z_0+\jmath Z_l tan(\beta l)}\). \(For Z_l>>Z_0, Z_{in} \to Z_0 \frac{1}{\jmath tan(\beta l)}\) so we can derive an equivalent length at Zo: \(Z_{0_1} \frac{1}{tan(\beta_1 l_1)} \approx Z_{0_2} \frac{1}{tan(\beta_2 l_2)}\).

For small values of \(\beta l, tan(\beta l)=\beta l\), and noting that \(\frac{\beta_1}{\beta_2}=\frac{v_{f2}}{v_{f1}}\) (where vf is the velocity factor), \(l_2 \approx \frac{Z_{0_2}v_{f2}}{Z_{0_1}v_{f1}}l_1\).

To calculate e-delay, Z_{02} and v_{2} will be 50 and 1 respectively, and \(edelay=\frac{l}{c_0 v_f}=\frac{l \cdot \text{1e12}}{299792458} \text{ps}\).

So, it turns out that when Zl dominates the expression for Zin, that an electrically short uniform transmission line port extension of some Zo other than 50Ω can be offset as in the expression above.

The appropriate e-delay (as it is often known) can be discovered by measurement of an OC at the end of the extension, and adjusting it until the phase of s11 is 0° independent of frequency.

Whilst the assumptions might seem quite restrictive, this technique can be used with good utility for suitable DUT and fixture, eg finding the high impedance self resonant frequency of an inductor / resonator.

\(Z_{in}=Z_0 \frac{Z_l+\jmath Z_0 tan(\beta l)}{Z_0+\jmath Z_l tan(\beta l)}\). \(For Z_l<<Z_0, Z_{in} \to Z_0 \jmath tan(\beta l)\) so we can derive an equivalent length at Zo: \(Z_{0_1} tan(\beta_1 l_1) \approx Z_{0_2} tan(\beta_2 l_2)\).

For small values of \(\beta l, tan(\beta l)=\beta l\), and noting that \(\frac{\beta_1}{\beta_2}=\frac{v_{f2}}{v_{f1}}\) (where vf is the velocity factor), \(l_2 \approx \frac{Z_{0_1}v_{f2}}{Z_{0_2}v_{f1}}l_1\).

To calculate e-delay, Z_{02} and v_{2} will be 50 and 1 respectively, and \(edelay=\frac{l}{c_0 v_f}=\frac{l \cdot \text{1e12}}{299792458} \text{ps}\).

So, it turns out that when Zo dominates the expression for Zin, that an electrically short uniform transmission line port extension of some Zo other than 50Ω can be offset as in the expression above.

The appropriate e-delay (as it is often known) can be discovered by measurement of an SC at the end of the extension, and adjusting it until the phase of s11 is 180° independent of frequency (easiest with a wrapped phase plot if available).

Whilst the assumptions might seem quite restrictive, this technique can be used with good utility for suitable DUT and fixture, eg finding the low impedance resonant frequency and input impedance of a transmission line section.

]]>This applies to the specific combination of versions of firmware and software client, do not assume it applies to other combinations.

DiSlord NanoVNA-D v1.1.00 firmware supports a scan_bin command where bit 3 of the outmask field is used to request raw measurement data, ie uncorrected measurements.

NanoVNA-App-v1.1.209-OD10 supports exploitation of that capability when it recognises that firmware version and command support.

Above, NanoVNA-App-v1.1.209-OD10 has a dropdown list to choose calibration mode.

- If you choose None or APP, it makes requests raw measurements.
- If you choose APP, it requests raw measurements and applies the current software calibration data set to correct the measured data.
- If you choose VNA, it requests corrected measurements from the NanoVNA using the current calibration set.

If you use NanoVNA-App-v1.1.209-OD10 to make a calibration set, when you measure each of the calibration configurations, it requests raw measurements from the NanoVNA (you do not need to disable correction in the NanoVNA). Further the calibration dataset is independent of the calibration setting on the NanoVNA, or future calibrations performed on the NanoVNA.

For avoidance of doubt, the calibration files made in NanoVNA-App-v1.1.209-OD10 are specific the the hardware and firmware in a specific NanoVNA and cannot be safely applied to another hardware instance, or a repaired or modified instance, or a firware upgrade (which may alter the inherent raw measurement calibration).

It appears that NanoVNA-App-v1.1.209-OD10 is adaptive and will work in this way with any NanoVNA (V1) firmware that returns “scan_bin” or “scan” in the list of commands given by the help command as in the following log excerpt, but the firmware needs to honor the outmask NO_CALIBRATION bit (b3).

16.133 tx: help 16.201 rx: help 16.201 rx: Commands: scan scan_bin data frequencies freq sweep power offset bandwidth time sd_list sd_read sd_delete saveconfig clearconfig dump touchcal touchtest pause resume cal save recall trace marker edelay capture vbat tcxo reset smooth usart_cfg usart vbat_offset transform threshold help info version color 16.201 rx: ch> 16.201 tx: version 16.201 rx: version 16.201 rx: 1.1.00

The following is a trace of scan commands for corrected and raw data from NanoVNA-App’s comms monitor, only the first five lines of each response are shown. The response is freq,s11.r,s11.i,n and outmask.

The load is an open circuit, theoretically s11 should be 1+j0.

53.810 tx: scan 29000000 300000000 51 3 53.819 rx: scan 29000000 300000000 51 3 53.899 rx: 29000000 1.000473261 0.004341440 53.910 rx: 34420000 0.999537408 -0.008618354 53.919 rx: 39840000 0.998404224 -0.023998050 53.929 rx: 45260000 0.996850176 -0.038681508 53.939 rx: 50680000 0.995249920 -0.052829040 48.150 tx: scan 29000000 300000000 51 11 48.150 rx: scan 29000000 300000000 51 11 48.270 rx: 29000000 0.661236416 -0.055518208 48.280 rx: 34420000 0.659682816 -0.064852924 48.280 rx: 39840000 0.658275584 -0.073653920 48.280 rx: 45260000 0.656932800 -0.082290024 48.280 rx: 50680000 0.655571648 -0.090402656

The response is freq, s11.r, s11.i, n and outmask.

- When the outmask has bit 3 false (the first group), measurement is corrected and the s11 result is close to 1+j0.
- When the outmask has bit 3 true (the first group), measurement is raw (ie not corrected) and the s11 result is not at all close to 1+j0.

You could do the same thing with a terminal emulator, here is Putty.

In summary, for the specific combination of DiSlord NanoVNA-D v1.1.00 and NanoVNA-App-v1.1.209-OD10:

- NanoVNA-App is able to request raw (ie uncorrected) measurements;
- it does so when it uses its internal calibration mode APP or when calibration mode is None, or when building a new NanoVNA-App internal calibration data set;
- NanoVNA-APP reads corrected measurements from the NanoVNA only in the VNA calibration mode, and it uses whatever profile is active at the time on the NanoVNA.

Some other firmware versions and PC client programs would seem to use layer on layer of calibration, interpolation, correction in some situations, and recalibrating the NanoVNA makes all the software based calibrations that depended on it invalid (they probably still work, but produce errored results).

]]>Let me say I am leery of built in features that invite users to perform something they do not understand, and may misinterpret the outcome.

Lots of the discussion ran to explaining why measurement of a sample of coax would be out by 5% or more, lots of pseudo tech discussion about age related, contamination related, quality related explanations for the measurement, things which might cause the measurer to condemn the sample, to discard it.

Well, you would want to be pretty confident in yourself to make that call, given that the explanation might well be measurement error.

I don’t use this facility, so I am quite unfamiliar with it, and there is no documentation, so one make make an informed guess as to how to use it.

Let’s measure…

The DUT is a length of 50 year old RG213 with NCV jacket, with good quality N(M) connectors and measures 9.860m to the outside of the connector shells. Note that the reference plane for the male connectors is approximately 10mm @ VF=1 inside from the outside plane of the shell.

The cable velocity factor has previously been carefully measured: NanoVNA – measuring cable velocity factor – demonstration – coax. Note that VF (0.66) is not totally independent of frequency, but the error is small in this case, and it is rounded to 1 part in 66, +/- 0.8% rounding error.

A word about N type connectors.

The ‘standard’ reference plane on N connectors is shown in the diagram above. For the purpose of this article, length measurements were made between the reference planes at both ends of the cable.

The N connector maintains Zo=50Ω through the connector, though a 9.3mm section of the path is usually air dielectric which needs to be taken into account for mm accuracy as needed for some applications (eg tuned line lengths used in repeater filters).

For the measurement of resonances here, a short circuit (SC) line termination is used as it provides the best accuracy of Zo in the region of that termination, albeit the last 9mm is at VF=1.

A first approximation model for capturing the effect of the location of the SC and OC in calibration parts is simply the propagation delay of the reflected wave, often known as port extension in VNAs and client software. This should be a quite adequate correction for quality termination parts below 100MHz.

Above are three N type SC terminations:

- commercial N(F)-SC;
- commercial N(M)-SC
- DIY N(M)-SC.

The red arrows on the diagram above show the common location of the short circuit on SC termination parts. The N(F)-SC is at the reference plane, the N(M)-SC is about 5mm or 16.7ps (one way, or port extension=33.4ps) from the reference plane.

The DIY N(M)-SC has its shorting plate further out than common commercial parts, it measures 54ps greater one way delay than one with the short at the red arrow above (port extension=87.4ps).

In the case of the cable measurements made for this article, the N(F)-SMA(M) adapter has to be included.

Above is a screenshot of e-delay adjusted to compensate an adapter and SC termination for flat wrapped phase of s11 response . Wrapped phase is a convenient display as it does not flick between + and – 180 due to measurement noise, but the normal (or wrapped) s21 phase can be used as shown in the later NanoVNA screenshot.

The NanoVNA was OSL calibrated out the outside of a SMA connector saver permanently attached to the Port.

To measure the cable with N connectors, an N(F)-SMA(M) adapter was attached, and a precision N(M) short attached to that adapter.

Above, the green trace displays phase of s11, and e-delay was adjusted until it was flat at the higher frequencies. This provides a fairly good calibration for the fixture offset and the reference plane is now the SC plane in the precision N(M) short, which is 9mm outside the N(F) reference plane.

Having calibrated the fixture, a precision N(F) short was attached to one end of the cable, and the other end attached to the N jack on the adapter on the VNA.

The reported 9.912m needs adjustment for the location of the reference planes, we expect it should measure 9.920m.

The NanoVNA measurement at 9.920 is 60mm longer than the physically measured 9.860m, 0.61% error.

That would seem good enough for such a simple test.

The display shows Loss=0.128dB. What does this mean? Is it a calculation of matched line loss (MLL), at what frequency? It is about the number one might expect at the first quarter wave resonant frequency ~5MHz… but it is a number without context.

Results with cable sections using UHF series connectors, especially with loose coupling sleeves may not be as reliable. The loose collar will cause unpredictable / unreliable results, install a UHF(F)-UHF(F) adapter to fix the problem. Likewise for plugs of similar construction (eg SMA).

]]>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.

]]>The other property of an inductor that if often sought is the Q factor (or simply Q). Q factor derives from “quality factor”, higher values of Q are due to lower resistance for the same inductance… so you might regard them as a higher quality inductor, lower loss relatively, and in resonant circuits, higher Q inductors yielded a narrower response.

Let’s visit the Q factor and measurements / plots of Q.

Q factor derives from the work of Kenneth S Johnson, the term having been formally used in 1920 to refer to the ratio of reactance to effective resistance of an inductor.

Though initially used in the context of an L-R circuit, it quickly gained wider application.

(Terman 1955) gives a more general definition of Q that is widely accepted:

circuit Q is 2π(energy stored in the circuit)/(energy dissipated in circuit in one cycle) .

Q is easily assessed for a single reactor:

- inductor: energy stored is \(L=\frac{I_{pk}^2}{2}\) and energy lost per cycle is \(\frac{I^2R}{2 f}\) so \(Q=\frac{2 \pi f L}{R}=\frac{X_l}{R_s}\);
- capacitor: energy stored is \(\frac{C V_{pk}^2}{2}\) and energy lost per cycle is \(\frac{V^2}{2 f R_p}\) so \(Q=2 \pi f C R_p=\frac{R_p}{X_c}\).

A RLC series resonant circuit is nearly as easy as at the instance when the inductor current is maximum, capacitor voltage is zero and hence the energy stored is as per the simple inductor case and \(Q=\frac{X_l}{R_s}\).

Likewise for a parallel RLC resonant circuit, at the instant of maximum capacitor voltage, inductor current is zero and hence the energy stored is as per the simple capacitor case and \(Q=\frac{R_p}{X_c}\).

BW=ResonantFrequency/Q. This is a well known relationship, and can be used to find the Q of a component by resonating it with a very low loss capacitor finding the half power bandwidth when driven by a constant current or constant voltage source as appropriate.

For a practical air cored inductor well below SRF, it was stated earlier that \(X \propto f\) and \(R \propto \sqrt f\), the latter is by virtue of skin effect. It is a first approximation, and does not account for proximity effect that becomes more significant where the space between turns is less than the wire diameter. This simple model is used in lots of analysis tools.

Measuring R and X for high Q inductors is a challenge, especially for low end VNAs, and there are other issues of the measurement fixture which all cause noisy measurement results, so for clarity, this article uses a mathematical model.

Let’s look at Simsmith calculation of Q for our 20µH inductor assuming Q @ 5MHz of 200 (realistic), \(X \propto f\) and \(R \propto \sqrt f\). The latter means that \(Q \propto \sqrt f\).

The accuracy of the Simsmith models is a little affected by the step size of the frequency increments used. We do not need to obsess over this, the answers are very close to ideal and demonstrate the behavior being discussed.

Let’s calculate Q inferred by Z.

That is all good. This looks like we can infer Q from Z over this frequency range (assuming that the inductor does not exhibit a self resonance, ie Cse=0).

Now let’s resonate that 20µ inductor with Q 283 at 10MHz with a lossless capacitance… so circuit Q is unchanged.

Above is the voltage response when it is driven with a constant current source. We will move the cursor off resonance to find the lower half power point.

Ok, the lower half power point is 10-0.018MHz, and if we assume symmetry, the half power bandwidth is 36kHz, and we can calculate circuit Q as \(Q=\frac{f_r}{BW}=\frac{10.0}{0.036}=278\) which reconciles with the expected value.

Let’s calculate Q inferred by Z for the resonated circuit, \(Q=\frac{|X|}{R}\)… this is how VNA Q plots are often calculated.

Wow, Q @ 10MHz is zero!

Of course it is zero, at resonance, \(X=0\) so \(Q=\frac{|X|}{R}=\frac{0}{R}=0\).

Q inferred by Z for the resonant circuit by \(Q=\frac{|X|}{R}\) is wrong above about f_{0}/5.

We established earlier that the test inductor could be characterised as having an equivalent shunt capacitance of 1.336pF to ‘explain’ the measured self resonance at 30.8MHz.

Above is a Simsmith model of that equivalent circuit modelled up to 40MHz. Just as in the case above of a resonant circuit, the Q inferred by Z for the self resonant circuit by \(Q=\frac{|X|}{R}\) is wrong above about SRF/5.

The fundamental definition of Q is in terms of energy, higher Q implies a lower loss inductor.

Whilst at lower frequencies, inductance of an air cored solenoid might be frequency independent, the effective resistance is usually not, and so Q is not usually frequency independent for that scenario.

Characterising an inductor as having certain inductance and Q is rather meaningless unless the frequency is also specified.

Just as the case where deriving L from inductor Z is unsafe, so it deriving Q from inductor Z by \(Q=\frac{X}{|R|}\) as is often done in VNAs (though both may be acceptable well below SRF).

Continued at NanoVNA-H4 – inductor challenge – part 6.

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