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Physics 212
Lecture 20
AC Circuits
Maximum currents & voltages
Phasors: A Simple Tool
Physics 212 Lecture 20, Slide 1
Main Point 1
First, we introduced the driven LCR circuit by adding a generator which serves to
eliminate the resistive damping by sustaining voltage and current oscillations at the
driving frequency of the generator.
Physics 212 Lecture 20, Slide 2
Main Point 2
Second, we introduced phasors, a graphical representation of the voltages and
phases in the circuit that allows us to efficiently solve for the maximum current
and the phase between the driving voltage and the current in the one loop circuit.
In particular, we represent the maximum voltage across each element as a vector
that rotates counterclockwise at the driving frequency with fixed relative phases
as shown. Both the phase between the driving voltage and the current, f, and the
maximum value of the current, Im, can then be obtained from the impedance
triangle as shown.
Physics 212 Lecture 20, Slide 3
Resistors
e = emaxsin(wt)
R
I = VR/R= Vmax/R sin(wt)
Physics 212 Lecture 20, Slide 4
Capacitors
Q = VC = Cemaxsin(wt)
I = dQ/dt
e = emaxsin(wt)
C
I = VmaxwC cos(wt)
Physics 212 Lecture 20, Slide 5
Physics 212 Lecture 20, Slide 6
Inductors
LdI/dt = VL = emaxsin(wt)
e = emaxsin(wt)
L
I = - Vmax/wL cos(wt)
Physics 212 Lecture 20, Slide 7
Physics 212 Lecture 20, Slide 8
RL ACT
An RL circuit is driven by an AC generator as shown in the
figure.
L
Imax = Vmax/XL
XL = wL
R
For what driving frequency w of the generator will the
current through the resistor be largest
A) w large
B) Current through R doesn’t depend on w
C) w small
Physics 212 Lecture 20, Slide 9
Physics 212 Lecture 20, Slide 10
Summary:
How do I remember all this ?
R
Imax = Vmax/R
V in phase with I
Because resistors are simple
C
Imax = Vmax/XC
XC = 1/wC
L Imax = Vmax/XL
XL = wL
V 90o behind I
Current comes first since it
charges capacitor
Like a wire at high w
V 90o ahead of I
Opposite of capacitor
Like a wire at low w
Physics 212 Lecture 20, Slide 11
Makes sense to write everything in
terms of I since this is the same
everywhere in a one-loop circuit:
Phasors make this
simple to see
Imax XL
Vmax = Imax XC
V 90o behind I
C
emax
L
R
Imax R
Vmax = Imax XL
V 90o ahead of I
Vmax = Imax R
V in phase with I
Imax XC
Always looks the same.
Only the lengths will
change
Physics 212 Lecture 20, Slide 12
Physics 212 Lecture 20, Slide 13
Summary:
Imax XC
C
VCmax= Imax XC
emax
VLmax= Imax XL
Imax XL
R
VRmax= Imax R
emax = Imax Z
L
Imax R
Imax = emax / Z
Z  R   X L  XC 
2
X L  XC
tan f  
R
2
(XLX C)
f
R
Physics 212 Lecture 20, Slide 14
Checkpoint 1a
A
B
C
Physics 212 Lecture 20, Slide 15
Physics 212 Lecture 20, Slide 16
Checkpoint 1b
A
B
C
Physics 212 Lecture 20, Slide 17
Checkpoint 1c
A
B
C
D
Physics 212 Lecture 20, Slide 18
Physics 212 Lecture 20, Slide 19
Checkpoint 2a
A
B
C
Physics 212 Lecture 20, Slide 20
Physics 212 Lecture 20, Slide 21
Checkpoint 2b
A
B
C
Physics 212 Lecture 20, Slide 22
Checkpoint 2c
A
B
C
Physics 212 Lecture 20, Slide 23
Physics 212 Lecture 20, Slide 24
Calculation
Consider the harmonically driven series LCR circuit shown.
Vmax = 100 V
Imax = 2 mA
VCmax = 113 V
The current leads generator voltage by 45o
L and R are unknown.
C
V ~
L
R
What is XL, the reactance of the inductor, at this frequency?
Conceptual Analysis
Strategic Analysis
Physics 212 Lecture 20, Slide 25
Physics 212 Lecture 20, Slide 26
Physics 212 Lecture 20, Slide 27