RLC Circuits

Physics 102Professor Lee

CarknerLecture 25

Three AC Circuits Vmax = 10 V, f = 1Hz, R = 10

Vrms = 0.707 Vmax = (0.707)(10) = R = Irms = Vrms/R = Imax = Irms/0.707 = Phase Shift = When V = 0, I =

Vmax = 10 V, f = 1Hz, C = 10 F Vrms = 0.707 Vmax = (0.707)(10) = XC = 1/(2fC) = 1/[(2)()(1)(10)] = Irms = Vrms/XC = Imax = Irms/0.707 = Phase Shift = When V = 0, I = I max =

Three AC Circuits Vmax = 10 V, f = 1Hz, L = 10 H

Vrms = 0.707 Vmax = (0.707)(10) =

XL = 2fL = (2)()(1)(10) =

Irms = Vrms/XL =

Imax = Irms/0.707 = Phase Shift = When V = 0, I = I max =

For capacitor, V lags I

For inductor, V leads I

RLC Circuits

Z = (R2 + (XL - XC)2)½

The voltage through any one circuit

element depends only on its value of R, XC or XL however

RLC Circuit

RLC Phase

The phase angle can be related to the vector sum of the voltages

Called the power factor

RLC Phase Shift Also: tan = (XL - XC)/R The arctan of a positive number is positive so:

Inductance dominates

The arctan of a negative number is negative so:

Capacitance dominates

The arctan of zero is zero so:

Resistor dominates

Frequency Dependence

The properties of an RLC circuit depend not just on the circuit elements and voltage but also on the frequency of the generator

Frequency affects inductors and capacitors exactly backwards

High f means capacitors never build up much charge and so have little effect

High and Low f

For “normal” 60 Hz household current both XL and XC can be significant

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

High and Low Frequency

Today’s PAL a)       How would you change Vrms, R, C and

to increase the rms current through a RC circuit?

b)       How would you change Vrms, R , L and to increase the rms current through a RL circuit?

c)       How would you change Vrms, R , and to increase the current through an RLC circuit?

d)       What specific relationship between L and C would produce the maximum current through a RLC circuit?

LC Circuit

The capacitor discharges as a current through the inductor

This plate then discharges backwards through the inductor

Like a mass on a swing

LC Resonance

Oscillation Frequency

Since they are connected in parallel they must each have the same voltage

IXC = IXL

= 1/(LC)½

This is the natural frequency of the LC circuit

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

Will happen when Z is a minimum

Z = (R2 + (XL - XC)2)½

This will happen when = 1/(LC)½

Frequencies near the natural one will produce large current

Impedance and

Resonance

Resonance Frequency

Resistance and Resonance

Note that the current still depends on the resistance

So at resonance, the capacitor and inductor cancel out

Peak becomes shorter and also broader

Next Time

Read 22.1-22.4, 22.7 Homework, Ch 21, P 71, Ch 22, P

3, 7, 8