rotating generators and faraday’s law 0 for n loops of wire
TRANSCRIPT
Rotating Generators and Faraday’s Law
B B dA B A B A cos
o t 0
B B A cos t
Bd dB A cos t
dt dt
B A sin t
N B A sin t
For N loops of wire
Alternating Current
oV t V
oV t V sin t
oo
V t VI(t) sin t I sin t
R R
oI t I
AC Generator and a Resistor
peak cos t
R R peakV V cos t
R peakRR peak
VVI cos t I cos t
R R
AC Power
peak
2 2 2P I R I cos t R
2peak2
av peak
V1 1P P I R
2 2 R
T / 2
2
T / 2
1 1cos t dt
T 2
?
2 1 1cos cos 2
2 2
Root Mean Square (rms)
peakV t V cos t peakI(t) I cos t
2peak2 V
V2
2peak peak2
rms
V VV V
2 2
2peak2 I
I2
2peak peak2
rms
I II I
2 2
2 2peak rms
1P I R I R
2
2 2peak rms
V V1P
2 R R
Inductive Circuits
L L peak
dIV V cos t L
dt
L peak L peakpeak
L
V VI sin t sin t I cos t
L X 2
Lpeak
L
VI
X LX L
Inductive Reactance
peak cos t L
dIV L
dt
Capacitive Circuits
peak cos t CQ CV
C peakC peak peak
C
VI CV sin t sin t I cos t
X 2
Cpeak
C
VI
X C
1X
C
Capacitive Reactance
Voltage transformers
Bs s
dV N
dt
Psolenoid o P
NI A
BP P
dV N
dt
SB P
P S
Vd V
dt N N
S PN N step up transformer
P SN N step down transformer
Current in transformers
SS P
P
NV V
N
Primary SecondaryP P
P P S SV I V I
P P S SN I N I
PI
SI
Actually currents are 180 degrees out of phase
Example: transformers
PI
SI
p
p
s
V 110V
N 916
N 100
What is Vs ?
LC Circuits
Q dIL 0
C dt
2
2
d Q Q0
dt LC
Kirchhoff Loop Equation: Solution:
maxQ Q cos t
1
LC
I t 0 0 maxQ(t 0) Q
Energy in an LC circuit
22
2maxE
Q1 QU cos t
2 C 2C
maxQ Q cos t 1
LC
2 2 2
2 2 2max maxB
L Q Q1U LI sin t sin t
2 2 2C
max
dQI Q sin t
dt
2 2 2
2 2max max maxE B
Q Q QU U cos t sin t
2C 2C 2C
max maxI Q
Active Figure 32.17
(SLIDESHOW MODE ONLY)
LRC Circuits
Q dIRI L 0
C dt
Kirchhoff Loop Equation:
Solution:
2
2
d Q dQ QL R 0
dt dt C
btmaxQ Q e cos ' t
2
2
1 R'
LC 4L
Rb
2L
Damped RLC Circuit
• The maximum value of Q decreases after each oscillation– R < RC
• This is analogous to the amplitude of a damped spring-mass system
Active Figure 32.21
(SLIDESHOW MODE ONLY)
LRC Circuits
• Underdamped
• Critically Damped
• Overdamped
R
t2L
oQ Q e cos ' t
2
2
1 R'
LC 4L
2
2
1 R
LC 4L
2
2
1 R
LC 4L
2
2
1 R
LC 4L
24LR
C
24LR
C
24LR
C
Driven RLC Circuit
app peak
2
app peak2
dI QV cos t L IR 0
dt C
d Q dQ 1L R Q V cos t
dt dt C
Phasor Diagrams
1 L CX Xtan
R
22L CZ R X X
Resonance
L CX X
22L C
Vapp peakI cos t
R X X
1
LC
Power:
2app peak app peak app peak2
av peak peak app peak2
V V V1 1 1 R 1P I R R I V cos
2 2 Z 2 Z Z 2
av rms app rmsP I V cos
Power Factor
What is Power factor at Resonance?
More Resonance
2 2app rms
av 22 2 2 2 2o
V RP
L R
o oLQ
R
41. An emf of 96.0 mV is induced in the windings of a coil when the current in a nearby coil is increasing at the rate of 1.20 A/s. What is the mutual inductance of the two coils?
49. A fixed inductance L = 1.05 μH is used in series with a variable capacitor in the tuning section of a radiotelephone on a ship. What capacitance tunes the circuit to the signal from a transmitter broadcasting at 6.30 MHz?
55. Consider an LC circuit in which L = 500 mH and C = 0.100 μF. (a) What is the resonance frequency ω0? (b) If a resistance of 1.00 kΩ is introduced into this circuit, what is the frequency of the (damped) oscillations? (c) What is the percent difference between the two frequencies?
LC Demo
R = 10 C = 2.5 FL = 850 mH
1. Calculate period2. What if we change
C = 10 F3. Underdamped?4. How can we
change damping?