Your Comments
This seemed very graphical and diagram based. What kind of problems would we see from this
topic?
This was one of the most confusing things I've ever seen. I did AP C in high school but I
don't think we did this topic? Or am I forgetting something?
What is reactance?
Can we discuss qualitatively why the current and voltage are out of phase for capacitors and
inductors? It's difficult to remember which is which without understanding why it works the
way it does.
I don't like having to do these prelectures the day of an exam...
This is the first time I didn't understand the prelecture at all. Please explain this!! What does
the phasor diagram do??
always wanted to learn about AC. Speaking of AC, can you play "Epic Rap Battles of History:
Thomas Edison vs Nikola Tesla" before class?
http://www.youtube.com/watch?v=gJ1Mz7kGVf0. Tesla is awesome!
THIS IS UNREAL @@.. who came up with the idea of a phasor diagram?
Set phasors to stun!
Electricity & Magnetism Lecture 20, Slide 1
Physics 212
Lecture 20
Today’s Concept:
AC Circuits
Maximum currents & voltages
Phasors: A Simple Tool
Electricity & Magnetism Lecture 20, Slide 2
Big Idea
Maximum Values (easy V=IR)
VRmax = Imax R
Imax = emax / Z
VLmax = Imax XL
VCmax = Imax XC
Value at specific time (phasors)
y component gives voltage
V-Inductor Leads current
V-Capacitor Lags current
Electricity & Magnetism Lecture 20, Slide 3
Resistors
e=V
maxsin(wt)
R
I = VR/R = Vmax/R sin(wt)
Amplitude = Vmax/R
Electricity & Magnetism Lecture 20, Slide 4
Capacitors
Q = CV = CVmaxsin(wt)
I = dQ/dt
e=V
maxsin(wt)
C
I = VmaxwC cos(wt)
Amplitude = Vmax/XC
90o
where XC = 1/wC
is like the “resistance”
of the capacitor
XC depends on w
Electricity & Magnetism Lecture 20, Slide 5
Inductors
L dI/dt = VL = Vmaxsin(wt)
e=V
maxsin(wt)
L
I = - Vmax/wL cos(wt)
Amplitude = Vmax/XL
90o
where XL = wL
is like the “resistance”
of the inductor
XL depends on w
Electricity & Magnetism Lecture 20, Slide 6
RL Clicker Question
An RL circuit is driven by an AC generator as shown in the figure.
XL = wL
L
R
As w -> 0, so does XL
As w -> 0,
resistance of circuit R
current gets bigger
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
Electricity & Magnetism Lecture 20, Slide 7
Summary
R
Imax = Vmax/R
VR in phase with I
Because resistors are simple
C
Imax = Vmax/XC
XC = 1/wC
VC 90o behind I
Current comes first since it
charges capacitor
Like a wire at high w
L
Imax = Vmax/XL
XL = wL
VL 90o ahead of I
Opposite of capacitor
Like a wire at low w
Electricity & Magnetism Lecture 20, Slide 8
CheckPoint 1c
f
A
B
C
D
The CURRENT is THE CURRENT
There is only 1 current in this circuit
Same everywhere in circuit
Electricity & Magnetism Lecture 20, Slide 9
Driven RLC Circuit
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
Imax R
C
emax
L
R
Vmax = Imax XL
V 90o ahead of I
Imax XC
Vmax = Imax R
V in phase with I
Always looks the same.
Only the lengths will change
Electricity & Magnetism Lecture 20, Slide 10
The Voltages still Add Up
Imax XC
But now we are adding vectors:
C
emax
L
Imax XL
R
Imax XL
Imax XL
Imax R
emax
Imax R
Imax R
Imax XC
Imax XC
Imax R
Imax XC
Imax XL
emax
Electricity & Magnetism Lecture 20, Slide 11
Make this Simpler
Imax XC
C
emax
L
Imax XL
R
Imax XL
Imax XL
Imax R
emax
Imax R
Imax XC
Imax R
Imax XC
Electricity & Magnetism Lecture 20, Slide 12
Make this Simpler
Imax XC
C
emax
L
Imax XL
R
Imax XL
Imax R
emax = Imax Z
Imax R
Imax(XL - XC)
Imax R
Imax XC
Electricity & Magnetism Lecture 20, Slide 13
Make this Simpler
Imax XC
C
emax
L
Imax XL
R
Imax R
emax = Imax Z
Imax(XL - XC)
Imax R
Electricity & Magnetism Lecture 20, Slide 14
Make this Simpler
Imax XC
C
emax = Imax Z
f
emax
L
Imax(XL - XC)
Imax XL
R
Imax R
Imax R
(XL - XC)
R
Impedance Triangle
X L - XC
tan (f ) =
R
Electricity & Magnetism Lecture 20, Slide 15
Summary
Imax XC
VCmax = Imax XC
C
VLmax = Imax XL
emax
VRmax = Imax R
emax
L
Imax XL
R
= Imax Z
Imax R
Imax = emax / Z
Z = R 2  ( X L - X C )2
Z = R  X L - X C 
2
X L - XC
tan (f ) =
R
2
f
(XL - XC)
R
Electricity & Magnetism Lecture 20, Slide 16
Example: RL Circuit Xc = 0
emax
L
Imax XL
R
Imax R
Imax XL
emax
Imax R
Electricity & Magnetism Lecture 20, Slide 17
CheckPoint 1a
Draw Voltage Phasors
Imax XL
emax
Imax R
A
B
C
Electricity & Magnetism Lecture 20, Slide 18
CheckPoint 1b
Draw Voltage Phasors
Imax XL
emax
Imax R
A
B
C
Electricity & Magnetism Lecture 20, Slide 19
CheckPoint 2a
IXL
e
IR
e
IR
What does the voltage phasor diagram look like
when the current is a maximum?
IXc
IXL
IX
c Slide 22
Electricity & Magnetism Lecture 20,
CheckPoint 2b
IXL
IXc
A driven RLC circuit is represented by the phasor diagram below.
e
IR
IR
IXL
e
IXc
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
What does the voltage phasor diagram look like
when the capacitor is fully charged?
Electricity & Magnetism Lecture 20, Slide 23
CheckPoint 2C
IXL
IXc
A driven RLC circuit is represented by the phasor diagram below.
e
IR
IR
IXL
e
IXc
When the voltage across the capacitor is at its positive maximum, V C = +VC,max, what
is the voltage across the inductor ?
A) VL = 0
B) VL = VL,max
C) VL = -VL,max
What does the voltage phasor diagram look like when
the voltage across capacitor is at its positive maximum?
Electricity & Magnetism 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
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
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?
Compare XL and XC at this frequency:
A) XL < XC
B) XL = XC
C) XL > XC
VL
D) Not enough information
This information is determined from the phase
Current leads voltage
VL = ImaxXL
VC = ImaxXC
IR
45o
VR (phase of current)
VC
V
V
leads
Electricity & Magnetism Lecture 20, Slide 26
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?
What is Z, the total impedance of the circuit?
A) 70.7 kW
B)
50 kW
C) 35.4 kW
D) 21.1 kW
Vmax 100V
Z=
=
= 50k W
I max 2mA
Electricity & Magnetism Lecture 20, Slide 27
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 ~
R
What is XL, the reactance of the inductor, at this frequency?
Z = 50kW
sin(45) =.707
What is R?
A)
70.7 kW
L
B) 50 kW
C)
35.4 kW
D)
21.1 kW
cos(45) =.707
Determined from impedance triangle
R
45o
(XC - XL)
R
cos(45) =
Z
R = Z cos(45o)
= 50 kW x 0.707
= 35.4 kW
Electricity & Magnetism Lecture 20, Slide 28
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 ~
R
Z = 50kW
What is XL, the reactance of the inductor, at this frequency?
A) 70.7 kW
We start with the
impedance triangle:
R
45o
Z
B) 50 kW
C) 35.4 kW
XC - X L
= tan 45 = 1
R
L
D)
21.1 kW
R = 35.4kW
XL = XC - R
What is XC?
VCmax = ImaxXC
(XC - XL)
XL = 56.5 kW - 35.4 kW
113
XC =
= 56.5k W
2
Electricity & Magnetism Lecture 20, Slide 29