AC Circuit
Physics 202, Lecture 19
 Find out current i and voltage difference ΔVR, ΔVL, ΔVC.
Today’s Topics
  AC Circuits with AC Source
  Resistors, Capacitors and Inductors in AC Circuit
  RLC Series In AC Circuit
  Impedance
  Resonances In Series RLC Circuit
i
ΔVmax Sin(ωt)
Notes:
•  Kirchhoff’s rules still apply !
•  A technique called phasor analysis is convenient.
Resistors in an AC Circuit
 ΔV-IR=0 at any time
Inductors in an AC Circuit
  ΔV - Ldi/dt =0
i
i
Function view
iR=ΔV/R=Imax sinωt, Imax=ΔVmax/R
 The current through an resistor is
in phase with the voltage across it
Phasor view
Function view
 iL=Imax sin(ωt-π/2)
Imax=ΔVmax/XL,
XL= ωL  inductive reactance
 The current through an indictor is
90o behind the voltage across it.
Phasor view
1
Capacitors in an AC Circuit
Summary of Phasor Relationship
  ΔV - q/C=0, dq/dt =i
Note that we set ΔV w.r.t. I
i
Function view
 iL=Imax sin(ωt+π/2)
Imax=ΔVmax/XC,
XC= 1/(ωC)  capacitive reactance
 The current through a capacitor is
90o ahead of the voltage across it.
IR and ΔVR in phase
Phasor view
RLC Series In AC Circuit
  The current at all point in a series circuit
has the same amplitude and phase
(set it be i=Imaxsinωt)
 ΔvR= ImaxR sin(ωt + 0)
ΔvL= ImaxXLsin(ωt + π/2)
ΔvC= ImaxXCsin(ωt - π/2)
IL 90o behind ΔVL
ΔVL 90o ahead of IL
IC 90o ahead of ΔVC
ΔVC 90o behind IC
Phasor Technique

i
The phasor of ΔvRLC = vector sum of phasors for ΔvR ,
ΔvL , ΔvC.
ΔVmax = ΔVR2 + (ΔVL − ΔVC ) 2
= Imax R 2 + (X L − X C ) 2
φ = tan −1 (
Voltage across RLC:
ΔvRLC = ΔvR + ΔvL + ΔvR
= ImaxRsin(ωt)
+ ImaxXLsin(ωt + π/2)
+ ImaxXCsin(ωt - π/2) =ΔVmaxsin(ωt+φ)
X L − XC
)
R
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Note: XL= ωL, XC=1/(ωC)
ΔV = ΔVmax sin(ωt + φ )
how to get them?
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2
Current And Voltages in a Series RLC Circuit
ΔvR=(ΔVR)max Sin(ωt)
ΔvL=(ΔVL)max Sin(ωt + π/2)
ΔvC=(ΔVC)max Sin(ωt - π/2)
ΔVmax = I max R 2 + ( X L − X C ) 2
i= Imax Sin(ωt)
ΔVmax Sin(ωt +φ)
φ = tan −1 (
X L − XC
)
R
ΔV = ΔVmax sin(ωt + φ )
Quiz: Can the voltage amplitudes across
each components , (ΔVR)max , (ΔVL)max , (ΔVC)max larger than
the overall voltage amplitude ΔVmax ?
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Impedance
 For general circuit configuration:
ΔV=ΔVmaxsin(wt+φ) , ΔVmax=ImaxZ
Z: is called Impedance.
e.g. RLC circuit : Z =
ΔV
Z
i
R 2 + ( X L − X C )2
 In general impedance is a complex number. Z=Zeiφ.
It can be shown that impedance in series and parallel
circuits follows the same rule as resistors.
Z=Z1+Z2+Z3+… (in series)
1/Z = 1/Z1+ 1/Z2+1/Z3+… (in parallel)
(All impedances here are complex numbers)
Resonances In Series RLC Circuit
 The impedance of an AC circuit is a function of ω.
  e.g Series RLC: Z = R 2 + ( X − X )2 = R 2 + (ωL − 1 )2
L
C
ωC
1
•  when ω=ω0 =
(i.e. XL=XC )
LC
 lowest impedance  largest current resonance
 For a general AC circuit, at resonance:
  Impedance is at lowest
  Phase angle is zero. (In phase)
  Imax is at highest
  Power consumption is at highest
3
Summary of Impedances and Phases
4