RLC Circuits PH 203 Professor Lee Carkner Lecture 24

RCL and AC   d = 2  f X C = 1/(  d C) X L =  d L  If you combine a resistor, capacitor and an inductor into one series circuit, they all will have the same current but all will have difference voltages at any one time  Voltages are all out of phase with each other

RLC Circuit

RLC Impedance   Called the impedance (Z) Z = (R 2 + (X L - X C ) 2 ) ½  The voltages for the inductor and capacitor are 180 degrees opposed and so subtract  V = IZ  Can think of Z as a generalized resistance for any AC circuit

Time Dependence   The instantaneous value (v, i)  The maximum value (V, I)  The root-mean-squared value (V rms, I rms )   However, the average of a sinusoidal variation is 0

Finding rms   Since power depends on I 2 (P =I 2 R) it does not care if the current is positive or negative  I rms = I/(2) ½ = I V rms = V/(2) ½ = V  The rms value is about 71% of the maximum

Phase Angle and Power Factor   They are separated by a phase angle  often written as: cos  = IR/IZ = R/Z   But I and V are out of phase and sometime they reinforce each other and sometimes they cancel out  Can write power as: P av = I rms V rms cos    We just need to know V and I through it at a given time

High and Low f   For high f the inductor acts like a very large resistor and the capacitor acts like a resistance-less wire   At low f, the inductor acts like a resistance-less wire and the capacitor acts like a very large resistor  No current through C, full current through L

Natural Frequency   Example: a swing   If you push the swing at all different random times it won’t   If you connect it to an AC generator with the same frequency it will have a large current

Resonance  This condition is known as resonance   Low Z, large I (I = V/Z) Z = (R 2 + (X L - X C ) 2 ) ½   This will happen when  d = 1/(LC) ½  Frequencies near the natural one will produce large current

Resistance and Resonance  Note that the current still depends on the resistance   at resonance, the capacitor and inductor cancel out  If we change R we do not change the natural frequency, but we do change the magnitude of the maximum current   Since the effect of L and C are smaller in any case

Next Time  Read  Problems: Ch 31, P: 45, 46, 61, Ch 32, P: 12, 14