RLC Circuits

PH 203

Professor Lee Carkner

Lecture 24

RCL and AC

d = 2fXC = 1/(dC)

XL = dL 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 = (R2 + (XL - XC)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 (Vrms, Irms)

However, the average of a sinusoidal variation is 0

Finding rms

Since power depends on I2 (P =I2R) it does not care if the current is positive or negative

Irms = I/(2)½ = 0.707 I

Vrms = V/(2)½ = 0.707 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:

Pav = IrmsVrms 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 = (R2 + (XL - XC)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 32.1-32.5 Problems: Ch 31, P: 45, 46, 61, Ch 32, P:

12, 14