Physics 202, Lecture 19
Today’s Topics
  AC Circuits with AC Source
  AC Power Source
  Phasor
  Resistors, Capacitors and Inductors in AC Circuit
  RLC Series In AC Circuit
  Impedance
AC Power Source
  ΔV = ΔVmax Sin(ωt+φ0) = ΔVmax Sin(ωt)
ω: angular frequency
ω=2πf
T=2π/ω
t=0
AC Circuit: Series RLC
Phasor
  A sinusoidal function x= Asinφ can be represented
graphically as a phasor vector with length A and
angle φ (w.r.t. to horizontal)
Asinφ
 Find out current i and voltage difference ΔVR, ΔVL, ΔVC.
i
ΔVmax Sin(ωt)
Initial phase at t=0
(usually set φ0=0)
A
φ
Notes:
•  Kirchhoff’s rules still apply !
•  A technique called phasor analysis is convenient.
1
A Phasor is Like a Graph
  One graphical representation is
using rectangular coordinates.
  The voltage is on the vertical axis.
  Time is on the horizontal axis.
 The phasor is drawn similar to polar coordinate.
  The radial coordinate represents the amplitude of the
voltage.
  The angular coordinate is the phase angle.
  The vertical axis coordinate of the tip of the phasor
represents the instantaneous value of the voltage.
  The horizontal coordinate does not represent anything.
 Alternating currents can also be represented by phasors.
(A). Resistors in an AC Circuit
 ΔV(t) – i(t) R = 0 at any time
i
Function view
iR=ΔV/R=Imax sinωt, Imax=ΔVmax/R
 The current through an resistor is
in phase with the voltage across it
(B). Inductors in an AC Circuit
  ΔV – L di/dt = 0
(C). Capacitors in an AC Circuit
  ΔV - q/C=0, dq/dt =i
i
i
Function view
 iL=Imax sin(ωt-π/2)
Imax=ΔVmax/ ωL =ΔVmax/XL,
XL= ωL  inductive reactance
 The current through an indictor is
90o behind the voltage across it.
Phasor view
Phasor view
Function view
 iL=Imax sin(ωt+π/2)
Imax=ΔVmax/ [1/(ωC) ]=ΔVmax/XC,
XC= 1/(ωC)  capacitive reactance
 The current through a capacitor is
90o ahead of the voltage across it.
Phasor view
2
Summary of Phasor Relationship
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)
i
Voltage across RLC:
IR and ΔVR in phase
|IR|=|ΔVR|/R
IL 90o behind ΔVL
ΔVL 90o ahead of IL
|IL|=|ΔVL|/XL
IC 90o ahead of ΔVC
ΔVC 90o behind IC
|IC|=|ΔVC|/XC
Phasor Technique (again)

ΔvRLC = ΔvR + ΔvL + ΔvR
= ImaxRsin(ωt) + ImaxXLsin(ωt + π/2) + ImaxXCsin(ωt - π/2)
=ΔVmaxsin(ωt+φ)
how to get them?
Current And Voltages in a Series RLC Circuit
The phasor of ΔvRLC = vector sum of phasors for ΔvR ,
ΔvL , ΔvC.
ΔvR=(ΔVR)max Sin(ωt) = R i(t)
ΔvL=(ΔVL)max Sin(ωt + π/2)
ΔvC=(ΔVC)max Sin(ωt - π/2)
i= Imax Sin(ωt)
ΔVmax Sin(ωt +φ)
Note: XL= ωL, XC=1/(ωC)
Can the voltage amplitudes across each components ,
(ΔVR)max , (ΔVL)max , (ΔVC)max larger than the overall
voltage amplitude Δvmax ? (each one of them)
3
Impedance
ΔV
Summary of Impedances and Phases
Z
 For general circuit configuration:
ΔV=ΔVmaxsin(wt+φ) , ΔVmax=Imax|Z|
Z: is called Impedance.
i
e.g. RLC circuit :
 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)
Comparison
Between Impedance and Resistance
Resistance
Impedance
Symbol
R
Z
Application
Circuits with only R
Circuits with R, L, C
Value Type
Real
Complex: Z=|Z|eiφ
I - ΔV Relationship
ΔV=IR
ΔV=IZ, ΔVmax=Imax|Z|
In Series:
R=R1+R2+R3+…
Z=Z1+Z2+Z3+…
In Papallel:
1/R=1/R1+1/R2+1/R3+…
1/Z=1/Z1+1/Z2+1/Z3+…
4