# L-meter

The usage of inductors in RF circuits is often intrinsic to its operation. Unfortunately, the measurement of small values of inductance is often difficult without specialized equipment. If, you have the ability to measure frequency, either directly with a frequency counter or indirect with an oscilloscope, this burden can easily be overcome.

The circuit diagram of the L-meter is shown above. It can be broken down into three sections. The circuit around Q1 from a Colpitts oscillator. The main frequency determining elements are C1, C2 and Lx together with stray capacitance in the circuit.
Q2 is the buffer stage. It isolates the oscillator from the load of the frequency counter or oscilloscope. The gate of Q2 is lifted to half the supply voltage. In this way the source has more headroom and the FET will be prevented from conduction on the positive half of the cycles even with larger input voltages.
Voltage regulator VR1 guarantees a stable voltage for the oscillator circuit so that fluctuations in battery voltage cannot change the oscillator frequency. VR1 has a drop down voltage of 3V, so the minimum required voltage at the input is 8V. Shunt regulator D1 (TL431) is used as voltage monitor in this circuit. The voltage at the reference terminal is compared with a 2.5V internal reference. If the voltage at the input comes above this value the regulator will start to conduct and will turn on the LED D2. This will happen at a voltage of 2.5V * (R9 + R8 + R6)/ R6 = 8.1V.

Calibration can be done in a number of ways. You can borrow an inductor with a known (measured) value from a fellow electronics enthusiast. Plug it into the circuit measure the frequency and calculate the in-circuit capacitance.
Another approach could be to buy some a couple of inductors from an electronics shop with a known value. Repeat the process above a couple of times and average the outcome.
A final approach might be to use a capacitor with a known value and inductor with a unknown value. This might sound counter-intuitive and illogical but actually works quit well !

Procedure
1. Connect the inductor Lx (with the unknown value) to the L-meter and measure the frequency. Let’s call this f1
2. Connect the inductor Lx(with the unknown value) to the L-meter, together with the capacitor (with the known value, let’s call it C1) and measure the frequency. Let’s call this f2

The value of inductor Lx can now be calculated with the following formula:

$L_x = { 1 \over{ C_1 4 \pi^2}} * ({ 1 \over {f_2^2}} – { 1 \over {f_1^2}} ) \tag {a}$ Cx, the capacitance in the circuit, can then be calculated with the following formula: $Cx = {1 \over { L_x 4 \pi^2 f_1^2} } \tag {b}$

For the calibration of my device I used a coil I had within hand’s reach. The coil form was 19mm PVC conduit with a length of 35mm. I put 20 turns of 0.8mm enameled copper wire over a length of 20mm. According to the calculator at hamwaves.com this should yield an inductance of 5.0uH.
The next item was MLCC capacitor marked with 100pF. Measured on digital multimeter, the actual capacitance was 102pF. Finally the procedure as described above was carried out which gave the following results:

\begin{align} C_1 & =102 pF\\ f_1 &= 3043 kHz\\ f_2 & = 2800 kHz \end{align} If we plug the numbers above in (a) we get a value for $$L_x =4.86 uH$$ Pretty close to the calculated value! If we the use formula (b) we get a value for $$C_x = 559pF$$.

The mathematical derivation for the calibration procedure outlined above is described below:

$f_1 ={ 1 \over {2 \pi \sqrt{LC_x} } } => C_x = {1 \over {L_x (2 \pi f_1)^2 }} \tag{1}$ $f_2 ={ 1 \over {2 \pi \sqrt{LC_2} } } => C_2 = {1 \over {L_x (2 \pi f_2)^2 }} \tag {2}$ $C_2 = C_x + C_1$ \begin{align}C_x + C_1 & = { 1 \over { L_x 4 \pi^2}} {1 \over {} f_2^2} \tag {3}\\ \\ C_x & = { 1 \over { L_x 4 \pi^2}} {1 \over {} f_1^2} \tag {4} \end{align} If we substract (4) from (3) we get : $C_1 = { 1 \over{ L_x 4 \pi^2}} * ({ 1 \over {f_2^2}} – { 1 \over {f_1^2}} ) \tag {5}$ Now we solve the equation for $$L_x$$ $L_x = { 1 \over{ C_1 4 \pi^2}} * ({ 1 \over {f_2^2}} – { 1 \over {f_1^2}} ) \tag {6}$ Now that we have $$L_x$$ we can use equation (4) to calculate $$C_x$$