I've been using the following simple circuit to determine the resonant frequency of LC tanks.

The forward-biased LED generates broadband RF noise, which is then amplified by the 2N3904 transistors. The amplified noise is magnetically coupled via link winding L_in into the unknown LC tank formed by L_test and C_test. Link winding L_in is simply one or a few turns on the unknown inductor L_test. Another link winding L_out couples the signal from the LC tank into a monitoring receiver.

At the resonant frequency of the LC tank (which is what we are trying to determine), a peak in the noise will be observed in the monitoring receiver. Therefore, to find the resonant frequency of the tank, we can either tune the monitoring receiver across its range searching for the noise peak, or we can use a variable capacitor for C_test and vary C_test while listening for a noise peak on the monitoring receiver.

As a particular example application, I am building a shortwave superheterodyne receiver with a ferrite rod antenna of unknown inductance, which will serve as the inductor in the front-end RF tank of the receiver. The RF tank is tuned with one gang of a dual-gang variable capacitor (the other gang will tune the LO tank). By coupling the noise generator into the front-end RF tank, I am easily able to determine the tuning range of the front-end RF tank.

Given the tuning range of the RF tank, I can then proceed to design the LO tank for the local oscillator, and also proceed to implement the ganged tuning such that the RF tank and the LO tank always maintain a fixed frequency offset as they are simultaneously tuned.

Also, if we know only one of either C_test or L_test, and want to find the other unknown value, then we can use known value in combination with the observed resonant frequency f to determine the unknown value, by solving the standard resonance formula f=1/(2*pi*sqrt(L*C)) for the unknown value. For example, if we know the value of C_test and the resonant frequency of the tank (as measured with the noise generator and a monitoring receiver), then we can determine the value of L_test. In practice, depending on the required accuracy, it may be necessary to take into account issues such as measurement uncertainty in the "known" value of C_test (which can be compensated for by taking several frequency measurements with several different and known C_test values and performing a least-squares fit to the data to determine the value of L that best explains the observed resonant frequency), or stray capacitance inherent to the inductor (which would need to be estimated and included in the resonant-frequency formula).

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