Lecture 11 Electromagnetic Oscillations and Alternating Current Ch. 31

• Topics

– Generators– Transformers– LC Circuit Qualitatively – Electrical and Magnetic energy oscillations– Alternating current– Pure C and L circuits– Series RLC circuit– Power and Transformers

• Demos

• Elmo• Polling

dAnBm ˆ⋅=•r

φ

AdBrr

⋅=dAB θcos=

θ

n̂

B

€

ε =−dΦ

dt= −d(BAcosθ)

dt= −BA

dcosθ

dt= BAsinθ

dθ

dt

= BAω sinθ but θ =ωt so dθ

dt=ω

€

ε =BAω sinωt

€

ε =εm sinωt

Where is the rotational angular frequency of the generator

f and f= 60 Hz

Coil of wireWhat is a Generator?

Driven RLC Series Circuit

Series RLC• Show Generator voltage vs time on scope.

•Vary frequency• Show voltage across resistor compared to signal voltage

•How is signal voltage related to voltage across resistor•Vary frequency - ELI the ICE man

• Show voltage across inductor compared to resistor. •Voltage across resistor is in phase with current through resistor• Does voltage across inductor lead or lag the current

• Show voltage across capacitor compared to resistor. •Voltage across resistor is in phase with current through resistor• Does voltage across the capacitor lead or lag the current

• Show phase angle is related to impedance in circuit and is frequency dependent

• Show resonance by varying frequency.• Need table of numbers for some frequency and at resonance.

Series LCR circuit

f

Hz

ωL

Ohms

1/ωC

Ohms

R

Ohms

Z

Ohms

I=V/Z

Amps Deg

VR

Volts

VL VC

100 2.7 1600 10 1600 0.006 -89.6 .06 .016

9.6

1000 27 160 10 133 0.075 -85.7 .75 2.0 12

2445 65.3 65.3 10 10 1 0 10 65.3

65.3

10000 270 16 10 270 0.037 87.7 0.37 10 0.6

φ =tan−1(ωL −

1

ωCR

)

φ =tan−1(2.7 − 1600

10) = −89.6

€

=2πf

L = 4.2 mH

C =1.0 μF€

Z = R2 + (XL − XC )2

€

XL = 2πfL

XC =1

2πfCV=10 Volts

P=Irms2 R

Effective Power

Irms =I

2rms=root-mean-square

Peak value

Low Pass FilterHigh Pass Filter

Chapter 13 Problem 17

In an oscillating LC circuit, L = 28.0 mH and C = 7.50 µF. At time t = 0 the current is 9.50 mA, the charge on the capacitor is 3.20 µC, and the capacitor is charging.(a) What is the total energy in the circuit?(b) What is the maximum charge on the capacitor?(c) What is the maximum current?(d) If the charge on the capacitor is given by q = Q cos(ωt + ϕ), what is the phase angle ϕ?(e) Suppose the data are the same, except that the capacitor is discharging at t = 0. What then is ϕ?

Chapter 31 Problem 21

In an oscillating LC circuit with C = 62.0 µF, the current as a function of time is given by I = (1.10) sin(2500t + 0.670), where t is in seconds, I in amperes, and the phase angle in radians.

(a) How soon after t = 0 will the current reach its maximum value?(b) What is the inductance L?(c) What is the total energy?