Physics 102: Lecture 12
AC Circuits
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R
Physics 102: Lecture 12, Slide 1
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Review: Generators and EMF
Voltage across generator:
 = w A B sin(q)
 = w A B sin(wt)
 = Vmax sin(wt)
1
•
w
v
2
x
q
v
r
Vmax 
Frequency = How fast its spinning
Amplitude = Maximum voltage
Physics 102: Lecture 12, Slide 2
t
-Vmax
20
AC Source
V(t) = Vmax sin(wt)=Vmax sin(2pf t)
Vmax = maximum voltage
f = frequency (cycles/second)
+24
V(t) = 24 sin(8p t)
2pf t = 8pt
-24
f = 4 Hz
T=(1/4)seconds/cycle
0.25
0.5
RMS: Root Mean Square Vrms=Vmax/√2
Physics 102: Lecture 12, Slide 3
RMS?
V(t) = Vmax sin(2pf t)
+Vmax
-Vmax
Square:
Mean:
Vmax2 / 2
Vmax2
square Root:
Vmax / √2
RMS: Root Mean Square Vrms=Vmax/√2
Physics 102: Lecture 12, Slide 4
Preflight 12.1, 12.2
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R
I(t) = 10 sin(377 t)
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Find Imax
Well… We know that the maximum value
sine is 1. So the maximum current is 10!
Imax = 10 A
Find Irms
Just like Vrms=Vmax/√2 …
Irms=Imax/√2
Physics 102: Lecture 12, Slide 5
=10/√2 A = 7.07 A
Resistors in AC circuit
VR = I R
always true – Ohm’s
Law
R
• VR,max = ImaxR
• Voltage across resistor is “IN PHASE” with current.
Resistance (R)
– VR goes up and down at the
same times as I does.
I
t
VR
Frequency does not
affect Resistance!
Frequency
Physics 102: Lecture 12, Slide 6
t
Capacitors in AC circuit
VC = Q/C
always true
• VC,max = ImaxXC
• Capacitive Reactance:
C
XC = 1/(2pfC)
• Voltage across capacitor “LAGS” current.
Reactance (XC)
– VC goes up and down
just after I does.
Frequency does
affect Reactance!
Frequency
Physics 102: Lecture 12, Slide 7
I
t
VC
t
Inductors in AC circuit
VL = -L(DI)/(Dt) always true
• VL,max = ImaxXL
• Inductive Reactance:
L
XL = 2pfL
• Voltage across inductor “LEADS” current.
Reactance (XL)
– VL goes up and down
just before I does.
I
t
VL
Frequency does
affect Reactance!
Frequency
Physics 102: Lecture 12, Slide 8
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ACT/Preflight 12.4, 12.5
The capacitor can be ignored when…
(a) frequency is very large
(b) frequency is very small
The inductor can be ignored when…
(a) frequency is very large
(b) frequency is very small
Physics 102: Lecture 12, Slide 9
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AC Circuit Voltages
An AC circuit with R= 2 W, C = 15 mF, and L = 30 mH
has a current I(t) = 0.5 sin(8p t) amps. Calculate the
maximum voltage across R, C, and L.
VR,max = Imax R = 0.5  2 = 1 Volt
VC,max = Imax XC= 0.5  1/(8p0.015) = 1.33 Volts
VL,max = Imax XL = 0.5  8p0.03 = 0.38 Volts
Physics 102: Lecture 12, Slide 10
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R
C
ACT: AC Circuit Voltages
An AC circuit with R= 2 W, C = 15 mF, and L = 30 mH
has a current I(t) = 0.5 sin(8p t) amps. Calculate the
maximum voltage across R, C, and L.
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R
Now the frequency is increased so I(t) = 0.5 sin(16p t).
Which element’s maximum voltage decreases?
1) VR,max
2) VC,max
3) VL,max
Physics 102: Lecture 12, Slide 11
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Summary so far…
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• I = Imaxsin(2pft)
• VR = ImaxR sin(2pft)
• VR in phase with I
R
VR
I
• VC = ImaxXC sin(2pft–p/2)
1
1
• VC lags I 𝑋𝐶 =
=
2𝜋𝑓𝐶 𝜔𝐶
• VL = ImaxXL sin(2pft+p/2)
• VL leads I
Physics 102: Lecture 12, Slide 12
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VC
𝑋𝐿 = 2𝜋𝑓𝐿 = 𝜔𝐿
t
VL
Kirchhoff: generator
voltage
Vgen
L
R
C
Write down Kirchhoff’s Loop Equation:
Vgen(t) = VL(t) + VR(t) + VC(t) at every instant of time
I
However …
Vgen,max  VL,max+VR,max+VC,max
Maximum reached at different
times for R, L, C
We solve this using phasors
Physics 102: Lecture 12, Slide 13
VR
t
VC
VL
A reminder about sines and cosines
y
Recall: y coordinates
of endpoints are
• asin(q + p/2)
• asin(q)
• asin(q – p/2)
a
q+p/2
a
q
x
q-p/2
a
Physics 102: Lecture 12, Slide 14
Graphical representation of voltages
I = Imaxsin(2pft) (q = 2pft)
VL = ImaxXL sin(2pft + p/2)
VR = ImaxR sin(2pft)
VC = ImaxXC sin(2pft – p/2)
ImaxXL
q+p/2
L
R
ImaxR
q
C
q-p/2
ImaxXC
Physics 102: Lecture 12, Slide 15
Phasor Diagrams: A Detailed Example
• I = Imaxsin(2pft)
• VR = VR,maxsin(2pft)
t = 1 f=1/12
2pft = p/6
p/6
VR,maxsin(p/6)
Length of vector = Vmax across that component
Vertical component = instantaneous value of V
Physics 102: Lecture 12, Slide 16
Phasor Diagrams
• I = Imaxsin(p/3)
• VR = VR,maxsin(p/3)
t=2
2pft = p/3
VR,maxsin(p/3)
p/3
Length of vector = Vmax across that component
Vertical component = instantaneous value of V
Physics 102: Lecture 12, Slide 17
Phasor Diagrams
• I = Imaxsin(p/2)
• VR = VR,maxsin(p/2)
VR,max
t=3
2pft = p/2
VR,maxsin(p/2)=V0
p/2
Length of vector = Vmax across that component
Vertical component = instantaneous value of V
Physics 102: Lecture 12, Slide 18
Phasor Diagrams
• I = Imaxsin(4p/6)
• VR = VR,maxsin(4p/6)
t=4
2pft = 4p/6
VR,maxsin(4p/6)
4p/6
Length of vector = Vmax across that component
Vertical component = instantaneous value of V
Physics 102: Lecture 12, Slide 19
Phasor Diagrams
• I = Imaxsin(p)
• VR = VR,maxsin(p)
VR,maxsin(p)=0
t=6
2pft = p
VR,max
p
Length of vector = Vmax across that component
Vertical component = instantaneous value of V
Physics 102: Lecture 12, Slide 20
Phasor Diagrams
• I = Imaxsin(8p/6)
• VR = VR,maxsin(8p/6)
t=8
2pft = 8p/6
8p/6
VR,maxsin(8p/6)
Length of vector = Vmax across that component
Vertical component = instantaneous value of V
Physics 102: Lecture 12, Slide 21
Phasor Diagrams
• I = Imaxsin(10p/6)
• VR = VR,maxsin(10p/6)
t = 10
2pft = 10p/6
10p/6
VR,maxsin(10p/6)
Length of vector = Vmax across that component
Vertical component = instantaneous value of V
Physics 102: Lecture 12, Slide 22
AC circuit summary
Kirchoff’s Loop Equation always holds true:
Vgen = VL + VR + VC
However, Vgen,max  VL,max+VR,max+VC,max
Maximum reached at different times for R, L, C
I
VR
t
VR in phase with I
L
R
VL VC lags I
VC
VL leads I
Phasors represent instantaneous voltages
Physics 102: Lecture 12, Slide 23
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