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