RLC Circuits Physics 102 Professor Lee Carkner Lecture 25

Three AC Circuits   V max = 10 V, f = 1Hz, R = 10   V rms =  V max = (0.707)(10) =  R =  I rms =  V rms /R =  I max = I rms /0.707 =  Phase Shift =  When V = 0, I =   V max = 10 V, f = 1Hz, C = 10 F   V rms =  V max = (0.707)(10) =  X C = 1/(2  fC) = 1/[(2)(  )(1)(10)] =  I rms =  V rms /X C =  I max = I rms /0.707 =  Phase Shift =  When V = 0, I = I max =

Three AC Circuits   V max = 10 V, f = 1Hz, L = 10 H   V rms =  V max = (0.707)(10) =  X L = 2  fL = (2)(  )(1)(10) =  I rms =  V rms /X L =  I max = I rms /0.707 =  Phase Shift =  When V = 0, I = I max =

For capacitor, V lags IFor inductor, V leads I

RLC Circuits  Z = (R 2 + (X L - X C ) 2 ) ½   The voltage through any one circuit element depends only on its value of R, X C or X L 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  = (X L - X C )/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 X L and X C 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 V rms, R, C and  to increase the rms current through a RC circuit?  b) How would you change V rms, R, L and  to increase the rms current through a RL circuit?  c) How would you change V rms, 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 IX C = IX L  = 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 = (R 2 + (X L - X C ) 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 , 22.7  Homework, Ch 21, P 71, Ch 22, P 3, 7, 8