Inductance Meter for Small Inductances

M.A. Haidekker, May 2023

MOTIVATION

In the context of my permanent-magnet MRI project, I realized that the RF transmit and receive subsystem requires inductors with very small inductances. Air coils of less than 1μH are commonly needed Most of them need to be hand-wound, because the special geometries of, for example, the RF transmit coil, do not exist. A general idea of the coil's inductance can be obtained from Wheeler's formula for single-layer coils (radius \(r\), length \(l\), number of turns \(n\) ),

\[L = \frac{r^2 \cdot n^2}{9 r + 10 l } \]

However, the geometry plays a major role, and minor deviations from the ideal coil geometry can impact the inductance.

Inductance meters can be purchased. Commonly, these are combined R-L-C meters that are based on a balanced bridge. The bridge is excited with a low-frequency AC signal. One specific R-L-C meter used in this lab uses 120Hz and 1kHz. While it is a useful instrument for most applications, is becomes very inaccurate for inductors below approximately 5μH. Below 2μH, no reasonable value is displayed. The reason for the inaccuracy is that the impedance at low frequencies is very low (for example, 1μH at 120Hz has only 0.00075j Ω). Even a sensitive Wheatstone bridge would be challenged.

The obvious remedy for small inductances is to increase the frequency. These inductors are, after all, intended as radiofrequency coils. This thought leads to the next step: Why not use a resonant tank circuit? The principle is simple: Build an L-C oscillator with the unknown coil (the D.U.T. = device under test) as one part of the tank circuit. The frequency of its oscillation is

\[f = \frac{1}{2 \pi \sqrt{L C} } \]

which can be solved for L if the frequency is measured and C is known.

RF OSCILLATORS

There are many well-known and commonly-used oscillator circuits, prominent among them the Hartley, Meissner, Colpitts and Pierce oscillators. Out of these, the Colpitts oscillator is favorable, because it can be realized with one single frequency-determining inductor, which is an obvious design requirement. A basic schematic of a Colpitts oscillator can be seen in Figure 1A. The tank circuit is formed with L1 and the capacitive voltage divider C1a and C1b, whose effective series capacitance determines the tank frequency. Less common is the emitter-coupled oscillator (Figure 1B) that will be explained further below.

Figure 1: Principle of two oscillators. (A) Colpitts oscillator. Q1 is the gain element, and feedback with 180° phase shift occurs between collector and base. A characteristic feature of the Colpitts oscillator is the capacitive voltage divider C1a, C1b, whose series capacitance determines the resonant frequency. L2 is large (mH range) and provides the large-signal connection of the collector to the supply voltage. R1 and R2 determine the base operating point, and R3 serves as large-signal stabilizing element. C3 bypasses R3 for high frequencies and provides the small-signal connection of the emitter to ground. (B) Emitter-coupled oscillator. This oscillator is composed of a differential amplifier (Q1, Q2) whose outputs are coupled to their opposite inputs. Feedback with 180° phase shift exists between the collector of Q1 and the base of Q2, and the resonant tank impedance is placed at that node.

The idea was first tested with the Colpitts oscillator. For the schematic in Figure 1A, values of 10nF were used for C1a and C1b; 100nF for C2 and C3; furthermore, R1=22k, R2=4.7k, R3=560Ω, and L2=10mH. The transistor was a 2N2369A small-signal transistor with a high transition frequency of 500MHz. Tests were run with inductors from 1μH to 47μH, and the measured frequency showed an excellent match with the calculated resonance frequency. However, the Colpitts oscillator showed undesirable behavior across the inductance range. The oscillation amplitude grows with decreasing frequency, and below approximately 500kHz the waveform at the collector is severely distorted, because the transistor is driven into saturation. At the other end, the oscillation stops and restarts periodically due to the lower loop gain. For any limited frequency range, this behavior can be fixed by optimizing the operating point and gain, but a wide frequency range is unattainable with a single operating point. The closely-related Clapp oscillator was tested in a similar fashion, but the same undesirable behavior was observed.

Next, the emitter-coupled oscillator was examined. The first design was built with the 2N2369A transistor as well. For C1, 1nF was chosen, and the common emitter current source, which is responsible for the difference behavior, was approximated by a 10kΩ resistor. The circuit was operated at 12V. In this circuit, too, the measured frequency showed an excellent match with the calculated resonance frequency. Its major drawback was a very strong dependency of the amplitude on the frequency and an overall very small amplitude of less than 0.1V p-p at frequencies above 1MHz. Correspondingly, waveform distortion was observed below 250kHz as the amplitude exceeds the linear range of the difference stage.

IMPROVING THE EMITTER-COUPLED OSCILLATOR

For the emitter-coupled oscillator, the DC component at the collector of Q1 (close to VCC) needs to be removed, and the signal must be drawn from that node with high impedance. Figure 2 shows the added output stage with Q4. At the same time, the amplitude was made variable by providing a voltage controlled current source (Q3). The base voltage of Q3, adjustable with P1, translates into the emitter voltage and the current through R1. Because \(I_B \ll I_C \), the same current is the combined emitter current of Q1 and Q2. The amplitude of the AC oscillation is proportional to the joint emitter current. In this configuration, an amplitude-stabilizing scheme can be designed in which Q3 reduces its collector current with increasing amplitude. Included in Figure 2 is a single-transistor amplifier stage (Q5), which increases the amplitude of the AC signal approximately 20-fold.

Figure 2: Improved emitter-coupled oscillator. The AC signal is drawn from the oscillating node with the emitter follower Q4, and a voltage amplifier stage around Q5 follows. The current source in the differential stage was made variable with Q3. The joint emitter current and thus the AC signal amplitude is adjustable with P1.

Acknowledgements: Significant experimentation with these circuits was performed by the MRI team in CSEE 4790, Spring 2023, and I wish to express my thanks to Hunter Bradford, Annie Jones, J.R. Nelson, Anthony Petti and Brad Stanley. You did a really great job with the class-E amplifier and with the inductance meter!

With these prerequisites, we can take a look at the circuit details in Part 2 of this article.