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Physics 212
Lecture 20
AC Circuits
Maximum currents & voltages
Phasors: A Simple Tool
Physics 212 Lecture 20, Slide 1
Music
Who is the Artist?
A)
B)
C)
D)
E)
Derek Trucks Band
Robert Plant & Alison Kraus
CCR
Emmylou Harris & Gregg Allman
Led Zeppelin
Theme of the week
Bluegrass fiddlers (Mark O’Connor, Alison Kraus) doing something different !!
Physics 212 Lecture 20, Slide 2
Exam Tonight
• Regular exam 7:00
• Conflict exam 5:15 in 100 MSEB
– Covers material in Lectures 9 – 18
– Bring your ID: Rooms determined by discussion section (see link)
• Final Exam
– “Regular”: Fri. May 4, 1:30; “Conflict”: Fri. May 11, 8:00
– Sign up for the conflict in the gradebook if you wish
Physics 212 Lecture 18, Slide 3
Your Comments
“For the first time, I can actually say that I don’t hate circuits!!!!
What an accomplishment :)"
Change
today – driven
circuit: Ea.c.
“I thought the checkpoints were difficult! I
understood the prelecture concept. Everything is
pretty much tied to KVR. However I don’t
completely get the correlation between maximum
voltage or currents across inductors, resistors and
capacitors.”
“This stuff is very confusing. I like the phasor
diagrams though.”
THE KEY TECHNIQUE:
Draw the phasor diagram
(and impedance triangle)
for each problem!
“I don’t get how to read the phasor diagram”
“You putting a lecture on the same day as an exam just killed a puppy.”
“I could really use some AC to cool off my brain before this exam...”
“Set phasers to maximum stun.”
05
Physics 212 Lecture 20, Slide 4
Resistors
e = Vmaxsin(wt)
R
I = VR/R = Vmax/R sin(wt)
Amplitude = Vmax/R
Physics 212 Lecture 20, Slide 5
Capacitors
Q = CV = CVmaxsin(wt)
I = dQ/dt
e = Vmaxsin(wt)
C
I = VmaxwC cos(wt)
Amplitude = Vmax/XC
90o
where XC = 1/wC: “Reactance”
is like the “resistance”
of the capacitor
XC depends on w
Physics 212 Lecture 20, Slide 6
Inductors
L dI/dt = VL = Vmaxsin(wt)
e = Vmaxsin(wt)
L
I = - Vmax/wL cos(wt)
Amplitude = Vmax/XL
90o
where XL = wL: “Reactance”
is like the “resistance”
of the inductor
XL depends on w
Physics 212 Lecture 20, Slide 7
CheckPoint 1a
An RL circuit is driven by an AC generator as shown in the figure
The
A voltages across the resistor and generator are
B Always out of phase
A)
B) Always in phase
C Sometimes in and sometimes out of phase
C)
“Resistor and inductor are out of phase, and generator is determined by vector sum, thus out of
phase”
“The phasors for resistor and generator are in phase.”
“it depends on the voltage and resistance”
Physics 212 Lecture 20, Slide 9
CheckPoint 1a
An RL circuit is driven by an AC generator as shown in the figure
Draw Voltage Phasors
Imax XL
emax
Imax R
The
A voltages across the resistor and generator are
B Always out of phase
A)
B) Always in phase
C Sometimes in and sometimes out of phase
C)
Physics 212 Lecture 20, Slide 10
CheckPoint 1b
An 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 and sometimes out of phase
“Voltage across resistor always lags the voltage across the inductor by 90 degrees.”
“Their voltage is scaled by the current produced by the generator”
“Dependent on time”
Physics 212 Lecture 20, Slide 11
CheckPoint 1b
An RL circuit is driven by an AC generator as shown in the figure
Draw Voltage Phasors
Imax XL
emax
Imax R
The voltages across the resistor and inductor are
A) Always out of phase B) Always in phase
C) Sometimes in and sometimes out of phase
Physics 212 Lecture 20, Slide 12
CheckPoint 1c
An 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
“Current = current, it’s the same current because it’s the same circuit.”
“The phase difference will remain constant at 90 degrees.”
“the phase is dependent on L and R”
“current = v/R”
Physics 212 Lecture 20, Slide 13
CheckPoint 1c
An RL circuit is driven by an AC generator as shown in the figure
The CURRENT is THE CURRENT
Imax XL
f
emax
Imax R
The phase difference between the CURRENT through the
f is the phase between
resistor and inductor
generator and current
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
Physics 212 Lecture 20, Slide 14
Review
R
Imax = Vmax/R
VR in phase with I
Because resistors are simple
C
Imax = Vmax/XC
XC = 1/wC
“Reactance”
L
Imax = Vmax/XL
XL = wL
“Reactance”
VC 90o behind I
Current comes first since it
charges capacitor
Like a wire at high w
VL 90o ahead of I
Opposite of capacitor
Like a wire at low w
Physics 212 Lecture 20, Slide 15
The 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
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 16
Imax XC
The voltages still add up
C
emax
Now we are adding three
vectors:
Imax XL
L
R
Imax R
Imax XL
Imax R
emax
Imax R
Imax R
Imax XC
Imax XC
Imax XL
Imax XC
Imax XL
emax
Physics 212 Lecture 20, Slide 17
Imax XC
Making this simpler…
C
emax
Imax XL
L
Imax XL
R
Imax R
Imax XL
emax
Imax R
Imax XC
Imax R
Imax XC
Physics 212 Lecture 20, Slide 18
Imax XC
Making this simpler…
C
emax
L
Imax XL
R
Imax R
Imax XL
emax = Imax Z
Imax R
Imax(XL-XC)
Imax R
Imax XC
Physics 212 Lecture 20, Slide 19
Imax XC
Making this simpler…
C
emax
L
Imax XL
R
Imax R
emax = Imax Z
Imax(XL-XC)
Imax R
Physics 212 Lecture 20, Slide 20
Imax XC
Making this simpler…
C
emax
Imax XL
R
emax = Imax Z
Imax R
Imax(XL-XC)
f
L
Imax R
(XL-XC)
f
Impedance Triangle
R
X L  XC
tan f  
R
Physics 212 Lecture 20, Slide 21
Imax XC
Summary:
C
VCmax= Imax XC
emax
VLmax= Imax XL
L
Imax XL
R
VRmax= Imax R
Imax R
emax = Imax Z
Imax = emax / Z
Z  R   X L  XC 
2
X L  XC
tan f  
R
2
XL-XC)
f
R
Physics 212 Lecture 20, Slide 22
Example: RL Circuit Xc=0
emax
L
Imax XL
R
Imax R
Imax XL
emax
Imax R
Physics 212 Lecture 20, Slide 23
CheckPoint 2a
A driven RLC circuit is represented by the phasor diagram below.
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
“when current is max the Xl vector is zero”
“There is a voltage drop across the resistor too.”
“The current should be max when the voltage is also at max.”
Physics 212 Lecture 20, Slide 24
CheckPoint 2a
A driven RLC circuit is represented by the phasor diagram below.
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
What does the voltage
phasor diagram look
like when the current
IXL
is a maximum?
IXeL
IR
e
IR
IXc
IXc
Physics 212 Lecture 20, Slide 25
CheckPoint 2b
CheckPoint 2b
A driven RLC circuit is represented by the phasor diagram below.
A
When
the capacitor is fully charged, what is the magnitude of the voltage across the
B
inductor?
C VL = 0
A)
B) VL = VL,max/2
C) VL = VL,max
“The voltage across the capacitor would be at a maximum, so the voltage across the inductor
would be 0”
“half voltage because there is a resistor and capacitor”
“its negative but the magnitude is the same as its max”
Physics 212 Lecture 20, Slide 26
CheckPoint 2b
A driven RLC circuit is represented by the phasor diagram below.
IXc
IXL
e
What does the voltage
phasor diagram look
like when the capacitor
is fully charged?
IR
IXc
When the capacitor is fully charged, what is the magnitude of the voltage across the
e
IXL
inductor?
A) VL = 0
B) VL = VL,max/2
C) VL = VL,max
IR
Physics 212 Lecture 20, Slide 27
CheckPoint 2c
CheckPoint 2c
A driven RLC circuit is represented by the phasor diagram below.
When the voltage across the capacitor is at its positive maximum, VC = +VC,max, what
is the voltage across the inductor ?
A) VL = 0
B) VL = VL,max
C) VL = -VL,max
“all energy in capacitor”
“The voltages have to be equal.”
“The capacitor and inductor voltage phasors point in opposite directions by definition.”
Physics 212 Lecture 20, Slide 28
CheckPoint 2c
CheckPoint 2c
A driven RLC circuit is represented by the phasor diagram below.
IXc
IXL
e
What does the voltage
phasor diagram look
like when the voltage
across capacitor is at
its positive maximum?
IR
IXc
When the voltage across the capacitor is at its positive maximum, VC = +VC,max, what
e
is the voltage across the inductorIX?L
A) VL = 0
B) VL = VL,max
C) VL = -VL,max
IR
Physics 212 Lecture 20, Slide 29
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
Physics 212 Lecture 20, Slide 30
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
(D) Not enough information
• This information is determined from the phase
–
Current leads voltage
45o
VL
VL = ImaxXL
VC = ImaxXC
VR (phase of current)
V
VC
IR
V
leads
Physics 212 Lecture 20, Slide 31
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 k
(B) 50 k
(C) 35.4 k
(D) 21.1 k
Vmax 100V
Z

 50 k 
I max 2 mA
Physics 212 Lecture 20, Slide 32
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
Z = 50k
What is XL, the reactance of the inductor, at this frequency?
sin(45)=.707
What is R?
(A) 70.7 k
cos(45)=.707
(B) 50 k
(C) 35.4 k
(D) 21.1 k
• Determined from impedance triangle
R
45o
Z=50k
(XC-XL)
R
cos(45) 
Z
R = Z cos(45o)
= 50 k x 0.707
= 35.4 k
Physics 212 Lecture 20, Slide 33
Calculation
C
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.
V ~
R
Z = 50k
What is XL, the reactance of the inductor, at this frequency?
(A) 70.7 k
(B) 50 k
We start with the
impedance triangle:
R
45o
Z
(C) 35.4 k
XC  X L
 tan 45  1
R
L
R = 35.4k
(D) 21.1 k
XL = X C - R
What is XC ?
(XC-XL)
VCmax = ImaxXC
XL = 56.5 k – 35.4 k
113
XC 
 56.5k
2
Physics 212 Lecture 20, Slide
34