# RLC Capacitively Coupled Resonance

Cuthbert Nyack
Two capacitively 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 capacitor C1. The Equation for the currents i1(first loop) and i2( second 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:- Applet below illustrates the dependence of the frequency response on the parameters C1, 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. Note that 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 second loop. For this applet the vertical scale for phase goes from +0.5pi at the top to -1.5pi at the bottom.