Rotating Generators and Faraday’s Law

B B dA B A B A cos

o t 0

B B A cos t

Bd d B A cos tdt 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 tR 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 tR R

AC Power

peak

2 2 2P I R I cos t R

2peak2

av peak

V1 1P P I R2 2 R

T / 2

2

T / 2

1 1cos t dtT 2

?

2 1 1cos cos 22 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 R2

2 2peak rms

V V1P2 R R

Inductive Circuits

L L peakdIV V cos t Ldt

L peak L peakpeak

L

V VI sin t sin t I cos t

L X 2

Lpeak

L

VIX

LX LInductive Reactance

peak cos t LdIV Ldt

Capacitive Circuits

peak cos t CQ CV

C peakC peak peak

C

VI CV sin t sin t I cos t

X 2

C

peakC

VIX

C1XC

Capacitive Reactance

Voltage transformers

Bs s

dV Ndt

Psolenoid o P

N I A

BP P

dV Ndt

SB P

P S

Vd Vdt N N

S PN N step up transformer

P SN N step down transformer

Current in transformers

SS P

P

NV VN

Pr imary 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 0C dt

2

2

d Q Q 0dt LC

Kirchhoff Loop Equation: Solution:

maxQ Q cos t

1LC

I t 0 0 maxQ(t 0) Q

Energy in an LC circuit

22

2maxE

Q1 QU cos t2 C 2C

maxQ Q cos t 1LC

2 2 2

2 2 2max maxB

L Q Q1U LI sin t sin t2 2 2C

maxdQI Q sin tdt

2 2 2

2 2max max maxE B

Q Q QU U cos t sin t2C 2C 2C

max maxI Q

Active Figure 32.17

(SLIDESHOW MODE ONLY)

LRC Circuits

Q dIRI L 0C dt

Kirchhoff Loop Equation:

Solution:

2

2

d Q dQ QL R 0dt dt C

btmaxQ Q e cos ' t

2

2

1 R'LC 4L

R

b 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 RLC 4L

2

2

1 RLC 4L

2

2

1 RLC 4L

24L RC

24L RC

24L RC

Driven RLC Circuit

app peak

2

app peak2

dI QV cos t L IR 0dt C

d Q dQ 1L R Q V cos tdt dt C

Phasor Diagrams

1 L CX Xtan

R

22L CZ R X X

Resonance

L CX X

22L C

Vapp peakI cos tR X X

1LC

Power:

2app peak app peak app peak2

av peak peak app peak2

V V V1 1 1 R 1P I R R I V cos2 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 oLQR

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?