ac circuit physics 202, lecture 20physics 202, lecture 20 today’s topics !! ac circuits with ac...
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Physics 202, Lecture 20
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
AC Circuit
"!Find out current i and voltage difference !VR, !VL, !VC.
Notes:
•! Kirchhoff’s rules still apply !
•! A technique called phasor analysis is convenient.
i
!Vmax Sin("t)
Resistors in an AC Circuit
"!!V-IR=0 at any time
i
iR=!V/R=Imax sin"t, Imax=!Vmax/R
#!The current through an resistor is in phase with the voltage across it Phasor view
Function view
Inductors in an AC Circuit
"! !V - Ldi/dt =0
! iL=Imax sin("t-#/2)
Imax=!Vmax/XL,
XL= "L " inductive reactance #!The current through an indictor is
90o behind the voltage across it. Phasor view
Function view
i
Capacitors in an AC Circuit
"! !V - q/C=0, dq/dt =i
i
! iL=Imax sin("t+#/2)
Imax=!Vmax/XC,
XC= 1/("C) " capacitive reactance #!The current through a capacitor is
90o ahead of the voltage across it. Phasor view
Function view
Summary of Phasor Relationship
IR and !VR in phase IL 90o behind !VL
!VL 90o ahead of IL
IC 90o ahead of !VC
!VC 90o behind IC
Note that we set !V w.r.t. I
RLC Series In AC Circuit
"! The current at all point in a series circuit
has the same amplitude and phase
(set it be i=Imaxsin"t)
#!!vR= ImaxR sin("t + 0)
!vL= ImaxXLsin("t + #/2)
! !vC= ImaxXCsin("t - #/2)
Voltage across RLC:
!vRLC = !vR + !vL + !vR
= ImaxRsin("t)
+ ImaxXLsin("t + #/2)
+ ImaxXCsin("t - #/2) =!Vmaxsin("t+$)
i
how to get them?
Phasor Technique
"! The phasor of !vRLC = vector sum of phasors for !vR ,
!vL , !vC.
!
"Vmax
= "VR
2+ ("V
L#"V
C)
2
= Imax
R2
+ (XL# X
C)
2
)(tan 1
R
XXCL
!=
!"
Note: XL= "L, XC=1/("C)
!
"V = "Vmax sin(#t + $)
Current And Voltages in a Series RLC Circuit
!Vmax Sin("t +$)
i= Imax Sin("t)
!vR=(!VR)max Sin("t)
!vL=(!VL)max Sin("t + #/2)
!vC=(!VC)max Sin("t - #/2)
22
maxmax )(CLXXRIV !+="
)(tan 1
R
XXCL
!=
!"
Quiz: Can the voltage amplitudes across
each components , (!VR)max , (!VL)max , (!VC)max larger than
the overall voltage amplitude !Vmax ?
!
"V = "Vmax sin(#t + $)
Impedance
"!For general circuit configuration:
! !V=!Vmaxsin(wt+$) , !Vmax=ImaxZ
! Z: is called Impedance.
! e.g. RLC circuit :
"! In general impedance is a complex number. Z=Zei$.
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)
!V
i
22 )(CLXXRZ !+=
Z
Resonances In Series RLC Circuit
"!The impedance of an AC circuit is a function of ".
$! e.g Series RLC:
•! when "="0 = (i.e. XL=XC )
" 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
2222 )1
()(C
LRXXRZCL
!! "+="+=
LC
1
Summary of Impedances and Phases
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