RLC, Inductively Coupled Circuits
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
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
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.
Return to main page
Return to page index
COPYRIGHT © 1996 Cuthbert A. Nyack.