Accurately measure high quality factor coils

Question

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.

Answer 1

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.

1. Zulauf and Rivas-Davila, Single-Turn Air-Core coils for High-Frequency Inductive Wireless Power Transfer

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.

2. Han et al., Evaluation of Magnetic Materials for Very High Frequency Power Applications

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.

Answer 2

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?