Physics 2112
Unit 20
Outline:
 Driven AC Circuits
 Phase of V and I
 Conceputally
 Mathematically
 With phasors
Electricity & Magnetism Lecture 20, Slide 1
AC Generator
e = Vmaxsin(wdt)
Driving frequency
= natural frequency (wo)
Electricity & Magnetism Lecture 20, Slide 2
“Phase” between I and V
Simple Case - Resistors
IR = VR/R
R
Voltage goes up  current goes up
“In phase”  Phase angle = 0o
I= Vmax/R sin(wdt)
Amplitude = Vmax/R
Electricity & Magnetism Lecture 20, Slide 3
Capacitors
Q = CV = CVmaxsin(wt)
C
 I = VmaxwC cos(wt)
90o
Amplitude = Vmax/XC
where XC = 1/wC
is like the “resistance”
of the capacitor
XC depends on w
Unit 20, Slide 4
Inductors
dI
L = VL = Vmax sin(wt )
dt
L
Vmax
I =
cos(wt )
wL
90o
Amplitude = Vmax/XL
where XL = wL
is like the “resistance”
of the inductor
XL depends on w
Electricity & Magnetism Lecture 20, Slide 5
Phase Summary
R
Vmax
I=
sin(wd t )
R
C
I = Vmax Cw cos(wd t )
Vmax
=
sin(wd t  90o )
C
L
Vmax
I =
cos( wt )
wL
Vmax
=
sin(wt  90o )
L
V and I “in phase”
I “leads” V
I “lags” V
“ELI the ICE man”
Electricity & Magnetism Lecture 20, Slide 6
What does this look like together?
Notice phase relationships
Electricity & Magnetism Lecture 20, Slide 7
What does this look like together?
Capacitor and Inductor always 180o out of phase
Capacitor/Inductor and Resistor always 90o out of phase
Resistor is some unknown phase angle out of phase is
signal generator
Electricity & Magnetism Lecture 20, Slide 8
What about current?
Current is always the same through all elements (in
series)
Current and Voltage in phase across Resistor
Current and voltage out of phase by unknown phase angle
across signal generator
(We’ll find this “phase angle” later.)
Electricity & Magnetism Lecture 20, Slide 9
Reactance Summary
R
R
C
1
XC =
wC
L
X L = Lw
Doesn’t depend on w
w goes up, c goes down
w goes up, L goes up
Electricity & Magnetism Lecture 20, Slide 10
Example 20.1 (Inductor Reactance)
L
A 60Hz signal with a Vmax = 5V
is sent through a 50mH
inductor. What is the
maximum current, Imax,
through the inductor?
Electricity & Magnetism Lecture 20, Slide 11
Phasors
Think of same material graphically using
“phasors”
Phasor just thinks of sine wave as rotating
vector
Electricity & Magnetism Lecture 20, Slide 12
Circuit using Phasors
Represent voltage drops across
elements as rotating vectors
(phasors)
Imax XL
VL and VC 180o out of
phase
VL and VR 90o out of
phase
Remember VR and I in
phase
Imax R
Imax XC
Electricity & Magnetism Lecture 20, Slide 13
Make this Simpler
Imax XC
C
emax = Imax Z
f
emax
L
Imax(XL  XC)
Imax XL
R
Imax R
Imax R
Z = R 2  ( X L  X C )2
R
Impedance Triangle
X L  XC
tan (f ) =
R
Electricity & Magnetism Lecture 20, Slide 14
Summary
Imax XC
VCmax = Imax XC
VLmax = Imax XL
VRmax = Imax R
emax
C
emax
L
Imax XL
R
= Imax Z
Imax R
Imax = emax / Z
Z = R 2  ( X L  X C )2
Z= R
2
2


 X L  XC
X L  XC
tan (f ) =
R
f
R
Electricity & Magnetism Lecture 20, Slide 15
CheckPoint 1(A)
A RL circuit is driven
by an AC generator
as shown in the
figure.
The voltages across the resistor and
generator are.
A. always out of phase
B. always in phase
C. sometimes in phase and sometimes
out of phase
Electricity & Magnetism Lecture 20, Slide 16
CheckPoint 1(B)
A RL circuit is driven
by an AC generator
as shown in the
figure.
The voltages across the resistor and
inductor are.
A. always out of phase
B. always in phase
C. sometimes in phase and sometimes
out of phase
Electricity & Magnetism Lecture 20, Slide 17
CheckPoint 1(C)
A RL circuit is driven
by an AC generator
as shown in the
figure.
The phase difference between the
CURRENT through the resistor and
inductor
A. is always zero
B. is always 90o
C. depends on the value of L and R
D. depends on L, R and the
generator voltage
Electricity & Magnetism Lecture 20, Slide 18
Example 20.2 (LCR)
In the circuit to the right
• L=500mH
• Vmax = 6V
• C=47uF
• R=100W
V
C
L
R
What is the maximum current and phase angle if
w = 60rad/sec?
What is the maximum current and phase angle if
w = 400 rad/sec?
What is the maximum current and phase angle if
w = 206 rad/sec?
Electricity & Magnetism Lecture 20, Slide 19
What does this look like graphically?
Electricity & Magnetism Lecture 20, Slide 20
Point of confusion??
VL + VC + VR + e = 0
VL-max + VC-max + VR-max + e = 0
(Add like vectors)
I max
Vmax
=
Z
I
V
Z
(Imax and Vmax happen at different times.)
Electricity & Magnetism Lecture 20, Slide 21
CheckPoint 2(A)
A driven RLC circuit is
represented by the phasor
diagram to the right.
The vertical axis of the phasor
diagram represents voltage. When
the current through the circuit is
maximum, what is the potential
difference across the inductor?
A. VL = 0
B. VL = VL-max/2
C. VL = VL=max
CheckPoint 2(B)
A driven RLC circuit is represented by the above
phasor diagram.
When the capacitor is fully charged, what is the
magnitude of the voltage across the inductor?
A. VL = 0
B. VL = VL-max/2
C. VL = VL=max
Electricity & Magnetism Lecture 20, Slide 23
CheckPoint 2(C)
A driven RLC circuit is represented by the above
phasor diagram.
When the voltage across the capacitor is at its
positive maximum, VC = +VC-max, what is the
magnitude of the voltage across the inductor?
A. VL = 0
B. VL = VL-max/2
C. VL = VL=max
Electricity & Magnetism Lecture 20, Slide 24
Example 20.3
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
The maximum voltage for each component is related to its reactance and to the
maximum current.
The impedance triangle determines the relationship between the maximum
voltages for the components
 Strategic Analysis
Use Vmax and Imax to determine Z
Use impedance triangle to determine R
Use VCmax and impedance triangle to determine XL
Electricity & Magnetism Lecture 20, Slide 25