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LECTURE 19
- AC Generators
- 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
8/16/12 2
Alternating Current Generators (DEMO)
(N = 2 for this coil) 8/16/12 3
Alternating Current Generators
= t+$ position of n wrt B m $ 00 maximum positive 0
900 0 maximum
positive
1800 maximum
negative
2700 0 maximum
negative (^) 8/16/12 4
Alternating Current in a Resistor
8/16/12 5
Power Dissipated in a Resistor
Average value Peak value 8/16/12 6
Average value of cos^2 t
cos
sin
cos
cos d
cos 2 d
d
cos
d
1 cos 2
d
cos d
cos
Let t.
2 4 2 0 4 0 2 0 2 2 0 2 0 2 0 2 0 2 2
π π π π π π π
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Root-mean-square (rms) values
8/16/12 8
Standard Alternating Voltage in the US
+ peak
- peak
8/16/12 13
Inductors in AC Circuits
8/16/12 14
Inductors in AC Circuits
⎟ ⎠ ⎞ ⎜ ⎝ =^ ⎛^ ω −π ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ π θ= θ − = ω = ω 2 I I cos t , 2 Sincesin cos I I sin t V V cos t peak peak L L, peak 8/16/12 15
Relationship between Irms & Vrms
Irms = VL , rms XL
where XL = ω L is the inductive reactance.
- XL is similar to R in Irms = VR , rms R
- SI unit for XL : Ω (ohm)
- Average power delivered to an inductor in an ac circuit is zero. 8/16/12 16
Capacitors in AC Circuits
8/16/12 17
Capacitors in AC Circuits
⎟ ⎠ ⎞ ⎜ ⎝ ⎛ π = ω + ⎟ ⎠ ⎞ ⎜ ⎝ θ=−^ ⎛^ θ +π =− ω = ω 2 I I cos t , 2 Sincesin cos I I sin t V V cos t peak peak C C, peak 8/16/12 18
Relationship between Irms & VC,rms
Irms = VC , rms XC where XC =
ω C is the capacitive reactance
- XC is similar to R in Irms = VR , rms R
- SI unit for XC : Ω (ohm)
- Average power delivered to a capacitor in an ac circuit is zero. 8/16/12 19
Phasors
- 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
y y
y = r sin ω t
x = r cos ω t
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. 8/16/12 20
Phasors for R
- V in phase with I VR = RIR = ε m sin ω t
IR =
ε m
R
sin ω t
