# Accurately measure high quality factor coils

What would be the most suitable way to measure the quality factor of high-Q coils under the following conditions?

Specifications

1. Frequency: 5 MHz to 15 MHz, the target application is wireless charging
2. Coil type: air core, constructed using hollow copper tubes, size is around 15 cm
3. Range of the measurements: Inductance $$\L\approx 1 \mu H\$$, Quality factor $$\Q> 1000\$$ (estimated from the simulations) and resistance $$\R=\frac{\omega L}{Q}\approx 30~m \Omega\$$ (at $$\5 MHz\$$)

What I have: VNA, low-frequency LCR meter (only up to 1MHz), other laboratory equipment such as Oscilloscopes, DC supply, etc. I don't have access to an Impedance analyser working at the target frequency range.

My approach: I first measure coil inductance at low frequency using the LCR meter and then add series capacitor (NP0 type - resistance of the capacitors is expected to be negligible) to tune the coil to the desired frequency. Next, coil impedance is measured using VNA (impedance parameters). Finally, use curve-fitting to estimate the coil resistance. Currently, estimated resistance is more than three times the expected values.

Identified problems in my approach are 1. Coil resistance is very small and the input impedance is not matched with the VNA at the resonance, therefore, I am not sure how accurate is the reading. 2. In this method, I cannot directly get the frequency response as the resistance is a function of the frequency - This also affects the curve fitting.

The Question: Is there any better way to measure high-Q coils accurately. I came across a book from Matlab, which I don't have access I am not sure if there is any useful information there.

• I would do much like you have done, only I would rely on the fact that the Q-factor of the coil is directly proportional to the loss (the purely real resistance) of the coil at its resonant frequency, and that this could be thought of as being the "direct" way of measuring it. I would, as you have done, create a series LC circuit using the best ceramic capacitor I could get my hands on (and making the leads as short as possible), measure the value of the capacitor using the LCR meter, measure the characteristic impedance of the LC circuit using the VNA and tune it using more caps to get it to – Vinzent Feb 14 '20 at 17:03
• ..to get it to 50 ohm (at the resonant frequency) and then I would measure the loss of the LC circuit at its resonant frequency. If I really wanted to make a big deal out of getting a super accurate result, then I would take a few more inductors and capacitors (you would have to figure out how many you would need depending on the number of unknowns in your model) and then I would measure them all up against each other (all combinations of Lx and Cy), and create a system of equations to solve for the complex impedances of them all. – Vinzent Feb 14 '20 at 17:10
• How did you estimate the Expected resistance to be 0.03 ohms? – Andy aka Feb 14 '20 at 18:23
• @Andyaka, Through finite element simulations. Also, there are some research articles that reports similar results with the measurements. – Pojj Feb 14 '20 at 20:41
• Were skin effects modelled? – Andy aka Feb 14 '20 at 20:45

## 2 Answers

I know two papers in the academic literature (IEEE) regarding measurement of Q factors in your frequency range, maybe they can help you. Both use mica or porcelain capacitors in the resonance circuit to obtain the lowest possible capacitor ESR.

The first one uses a VNA (E5061B) to measure the impedance magnitude of a resonance tank consisting of the inductor and a parallel capacitor. The impedance is measured around the resonance frequency and then they use the 3-dB method to find Q of the resonance tank, $$\ Q_T \$$

\begin{align} Q_T = \frac{\omega}{\Delta \omega_{3dB}} \end{align}

Then the quality factor of the coil is back-calculated

\begin{align} Q_L = \frac{Q_C Q_T}{Q_C - Q_T} \end{align}

For reference, they report errors of 1-4 mohm between simulation and measurement resulting in loss of up to 100 points in quality for the experimental setup.

In the second paper, a RF power amplifier feeds a sinusoidal current into a series resonance of the inductor and a capacitor: ground -> amplfier -> L -> C -> ground. The input (ground to amplifier output) and the output (voltage over capacitor) are measured using ground referenced oscilloscope probes. The peak ratio is found manually by tuning the frequency and then calculating the quality factor:

\begin{align} Q_L \approx \frac{V_{out}}{V_{in}} \end{align}

However, $$\Q_C\$$ must be much larger than $$\Q_L\$$ to have good accuracy and this will be difficult in your case.

• @Thanks. These would be worth trying. – Pojj Feb 20 '20 at 14:36

A coil 1 uH and 30 mOhm should be measurable with a vector analyzer, at least between 1 to 10 MHz, check this image of the specs of a E5061B where I marked in orange the area that should be interesting for you: You can find the rest of the specs here here.

Are you sure you did a correct de-embedding?

• @KenGimes, Thanks, The link is useful. Then it is clear that the accuracy depends on the type of equipment. My one is ROHDE & SCHWARZ. – Pojj Feb 20 '20 at 14:36
• I'm glad that I could be of assistance. I've worked with Rohde & Schwarz equipment before, they're quite good. Anyway, there are many Vector Network Analysers out there, each with their own specs. One other thing you might to check is the meurement method, the plot above is for 1-2 shunt-thru S parameter measurement. – Ken Grimes Feb 20 '20 at 16:38