# RLC, Inductively Coupled Circuits

Cuthbert Nyack
Two inductively coupled RLC circuits are shown in the circuit below. Having 2 circuits gives 2 resonant frequencies whose separation depends on the value of the coupling inductor M. The Equation for the currents i1(primary loop) and i2( secondary loop) is given below:- Where a and b are:- The solution, from which the frequency response can be obtained is:- Resonance occurs at the 2 frequencies given by the following equation:- Applet below illustrates the dependence of the frequency response on the parameters M, C and R. L is assumed to be 1/C to keep the LC product equal to 1. The vertical scale for phase is from +pi at the top to -pi at the bottom. With HGain = 1 the horizontal frequency scale goes from 0 to 2rad/s.
Current i1 magnitude response is in green, phase is in red, real part is in orange and imaginary part in yellow.
Because of the small values of L and frequency used here, then R must be small to give a "sharp" peak. The sharpness of the peak changes with C because L also changes. When M is large the higher frequency resonance appears to be much broader, this is because the effective inductance at the higher is smaller and the Q is also smaller. As for the Capacitively coupled case, the oscillators are in phase at the lower frequency and out of phase at the higher frequency.

The applet below is similar to the one above but shows the solution for i2 in the secondary loop. For this applet the vertical scale for phase goes from +0.5pi at the top to -1.5pi at the bottom. For the secondary the coupling coefficient k = M/L and "critical" coupling occurs when k = M/L = 1/Q, which in this case reduces to M = R. This can be illustrated by the applet below. In both applets the value of M is restricted to be always less than L = 1/C.