RLC Resonant Circuits

RLC Resonant CircuitsAndrew McHutchonApril 20, 20131 Capacitors and InductorsThere is a lot of inconsistency when it comes to dealing with reactances of complex components. The formatfollowed in this document is as follows. Theimpedance,Z, of a component or a circuit is defined as,Z=R+j X(1)whereRis the resistance,jis the imaginary unit, andXis the reactance. Note that the imaginary unit isoutsidethe reactance and there is a plus sign betweenRandj and Inductors are both components which can store energy: capacitors store it in an electric field andinductors in a magnetic field. Ideal capacitors and inductors are assumed to have zero resistance and so have apurely imaginary impedance,ZC=1j C= j CZL=j L(2)Following the convention in equation 1 we define the reactances to be,XC= 1 CXL= L(3)note that the minus sign is includedinsideXC, this is to ensure we get a plus sign in equation 1. You will oftensee capacitive reactance being quoted without this minus sign - be very careful!

L kZ C = R 1 1jR(X C + 1 X L) = R 1 + jR(!C 1!L) (9) (10) Figure 6 shows the magnitude and phase of the impedance of the circuit. Once again the resonant peak comes when X C = X L and hence the resonant frequency is given by the same relationship,! 0 = r 1 LC (11) The current gain can be calculated by looking at the current in each of the arms ...

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