Physics 202, Lecture 18
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
„ Resonances In Series RLC Circuit (Demo only)
‰ ΔV = ΔVmax Sin(ωt+φ0) = ΔVmax Sin(ωt)
ω: angular frequency
ω=2πf
Initial phase at t=0
(usually set φ0=0)
T=2π/ω
t=0
Thursday:
„ More on RLC resonance
„ Review of “Wave Motion” (Ch.16)
AC Circuit: Series RLC
‰ Find out current i and voltage difference ΔVR, ΔVL, ΔVC.
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
„
AC Power Source
i
ΔVmax Sin(ωt)
φ
Notes:
• Kirchhoff’s rules still apply !
• A technique called phasor analysis is convenient.
1
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
ÎThe current through an resistor is
in phase with the voltage across it
Phasor view
Capacitors in an AC Circuit
Function view
Î iL=Imax sin(ωt-π/2)
Imax=ΔVmax/ ωL =ΔVmax/XL,
XL= ωL Æ inductive reactance
ÎThe current through an indictor is
90o behind the voltage across it.
Phasor view
Summary of Phasor Relationship
‰ ΔV - q/C=0, dq/dt =i
i
Function view
Î iL=Imax sin(ωt+π/2)
Imax=ΔVmax/ [1/(ωC) ]=ΔVmax/XC,
XC= 1/(ωC) Æ capacitive reactance
ÎThe current through a capacitor is
90o ahead of the voltage across it.
IR and ΔVR in phase
|IR|=|ΔVR|/R
Phasor view
IL 90o behind ΔVL IC 90o ahead of ΔVC
ΔVL 90o ahead of IL ΔVC 90o behind IC
|IL|=|ΔVL|/XL
|IC|=|ΔVC|/XC
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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)
ÎΔvR= ImaxR sin(ωt + 0)
ΔvL= ImaxXLsin(ωt + π/2)
Δ C= ImaxXCsin(ωt
Δv
i ( t - π/2)
/2)
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 (
X L − XC
)
R
Voltage across RLC:
ΔvRLC = ΔvR + ΔvL + ΔvR
= ImaxRsin(ωt) + ImaxXLsin(ωt + π/2) + ImaxXCsin(ωt - π/2)
=ΔVmaxsin(ωt+φ)
Note: XL= ωL, XC=1/(ωC)
ΔV = ΔVmax sin(ωt + φ )
how to get them?
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 ? (Discuss with your TA)
Impedance
‰ For general circuit configuration:
ΔV=ΔVmaxsin(wt+φ) , ΔVmax=Imax|Z|
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)
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Summary of Impedances and Phases
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+…
Resonances In Series RLC Circuit
(Demo Only Today)
‰ The impedance of an AC circuit is a function of ω.
1
ƒ e.g Series RLC:
2
2
2
Z = R + ( X L − X C ) = R + (ωL −
ωC
)2
1
• when ω=ω0 =
(i.e. XL=XC )
LC
Æ lowest
l
t impedance
i
d
Æ largest
l
t currentÎ
tÎ resonance
‰ For a general AC circuit, at resonance:
ƒ Impedance is at lowest
ƒ Phase angle is zero. (In phase)
ƒ Imax is at highest
ƒ Power consumption is at highest
See demo.
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