LECTURE 19
Alternating Current Generators (DEMO)
•  AC Generators
(N = 2 for this coil)
•  AC Circuits
•  Start by considering simple circuits
with one element (R, C, or L) in
addition to the driving emf.
•  It will lead to Oscillations and Driven RLC
circuits
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Alternating Current in a Resistor
Alternating Current Generators
 = t+$
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position of n
wrt B
2
m
$
00
maximum
positive
0
900
0
maximum
positive
1800
maximum
negative
0
2700
0
maximum
negative
3
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4
1
Power Dissipated in a Resistor
Average value of cos2t
Let θ = ωt .
cos 2 θ =
∫
2π
0
cos 2 θdθ
∫
2π
0
2π
0
1 + cos 2θ
dθ
2
2π
1 2π
1 2π
1 4π
dθ + ∫ cos 2θdθ π + ∫ cos γdγ
∫
0
0
2
4 0
cos 2 θ = 2
=
2π
2π
4π
1
π + sin γ
π+0
4
0
cos 2 θ =
=
2π
2π
1
∴cos 2 θ =
2
Peak value
Average value
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dθ
=
∫
5
Root-mean-square (rms) values
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6
Standard Alternating Voltage in the US
+peak
-peak
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7
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8
2
How “Standard” is 120 VAC?
AC Power Distribution
Nikola Tesla
•  AC power can travel at high voltages
and low amps, therefore smaller
power loss
•  Tesla liked 60 Hz and 240 V
•  Standard in Europe was defined by a
German company AEG ( monopoly)
who chose 50Hz (20% less efficient in
generation, 10-15% less efficient in
transmission)
•  Originally Europe was also 110V, but
they changed to reduce power loss
and voltage drop for the same copper
diameter
http://www.teslasociety.com/
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9
The Power Grid
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10
Using rms values: summary
Using rms values of current and voltage
allows you to use the familiar dc
formulas, such as V = IR and P = I 2 R.
One ac ampere is said to flow in a circuit
if it produces the same joule heating as
one ampere of dc current under the
same conditions.
At your house the peak voltage will be
170 V
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11
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12
3
Inductors in AC Circuits
Inductors in AC Circuits
VL = VL ,peak cos ωt
I = I peak sin ωt
π⎞
⎛
Since sin θ = cos⎜ θ − ⎟,
2⎠
⎝
π⎞
⎛
I = I peak cos⎜ ωt − ⎟
2⎠
⎝
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13
Relationship between Irms & Vrms
I rms =
VL, rms
XL
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14
Capacitors in AC Circuits
where X L = ω L is the inductive reactance.
1. X L is similar to R in I rms =
VR, rms
R
.
2. SI unit for X L : Ω (ohm)
3. Average power delivered to an inductor in an ac circuit is zero.
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15
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16
4
Capacitors in AC Circuits
Relationship between Irms & VC,rms
I rms =
VC = VC,peak cos ωt
VC, rms
XC
where XC =
1
is the capacitive reactance
ωC
I = − I peak sin ωt
1. XC is similar to R in I rms =
π⎞
⎛
Since sin θ = − cos⎜ θ + ⎟,
2⎠
⎝
π⎞
⎛
I = I peak cos⎜ ωt + ⎟
2⎠
⎝
VR, rms
R
.
2. SI unit for XC : Ω (ohm)
3. Average power delivered to a capacitor in an ac circuit is zero.
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17
Phasors
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18
Phasors for R
•  A phasor is a “vector” whose magnitude is the maximum value
of a quantity (eg V or I) and which rotates counterclockwise in a
2-d plane with angular velocity . Recall uniform circular
motion:
The projections of r
(on the vertical y axis)
execute sinusoidal
oscillation.
x = r cos ω t
y
y
$
x
y = r sin ω t
•  V in phase with I
Angular speed: phasors rotate counter clockwise about the origin with an
angular speed of .
Length: represents the amplitude of the AC quantity
Projection: on the vertical axis represents the value of the AC quantity at
time t.
Rotation angle: phase of the AC quantity at time t.
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IR =
19
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εm
sin ω t
R
VR = RI R = ε m sin ω t
20
5
Phasors for C
(
cos (ω t )
I C = ω Cε m sin ω t + 90°
= ω Cε m
VC =
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Phasors for L
•  V lags I by 90
)
V leads I by 90
VL = L
Q
= ε m sin ω t
C
dI L
= ε m sin ω t
dt
εm
sin ω t − 90°
ωL
ε
= − m cos ω t
ωL
IL =
21
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(
)
22
6