3/5/07 U.W. Physics 104, Spring 2007 1

Lecture 12: Inductors, AC

Power, Circuits

• AC Power

– Transformers

– Transmission

• Inductors

– RL Circuits

– LCR Circuits• Impedance

• Resonance

3/5/07 U.W. Physics 104, Spring 2007 2

Generators

vv

•

x

!"

r

Flux is decreasing at moment shown

3/5/07 U.W. Physics 104, Spring 2007 3

Generators and EMF

#1 = B L v sin(!)

v = "rvv

•

x

!"

r=w/2

#1 = "r B L sin(!) = #

2

# = #1

+ #2

= 2"r B L sin(!)

# = A B " sin(!)

# = A B " sin("t) #t

1

2

"AB

$"AB

Two of the sides (width w) do not contribute

but two with lengths (L) do

N loops: # = NAB " sin("t)

3/5/07 U.W. Physics 104, Spring 2007 4

Direct versus Alternating Current

• Chemical power

– Battery

– Constant voltagedrop

– Constant current

• Circular motion

– Generator

– Sinusoidal timevariation

3/5/07 U.W. Physics 104, Spring 2007 5

Sinusoidal Pattern

• Circular motion

– Sinusoidalpattern

– Frequency, f

– Period, T = 1 / f

– Goes through 0twice in oneperiod

3/5/07 U.W. Physics 104, Spring 2007 6

AC Power

• Voltage, current and power depend on time

• We are interested in averages (f = 60Hz)

3/5/07 U.W. Physics 104, Spring 2007 7

AC Source

V(t) = V0 sin(2(f t)

– V0 = maximum voltage

– f = frequency (cycles/second)

• V(t) = 24 sin(8( t)

v

t

+24

-240.25 0.52(f t = 8(t

f = 4 Hz

(1/4) seconds/cycle

RMS: Root Mean Square Vrms=Vmax/)23/5/07 U.W. Physics 104, Spring 2007 8

Power Transmission

A generator produces 1.2 Giga watts of power, which it transmits to a

town 7 miles away through power lines with a total resistance 10-5

ohms. How much power is lost in the lines if the energy is

transmitted at 120 Volts?

P = IV Power delivered by generator through lines

I = P/V = 1.2x109 W/120 V = 107 Amps in lines!

P = I2R Power lost in lines

= 1014 10-5 = 1 Giga Watt Lost in Lines!

Large current is the problem. Since P=IV, use high

voltage and low current to deliver power.

If V = 12,000 Volts, only 1 Kilo Watt will be lost!

3/5/07 U.W. Physics 104, Spring 2007 9

Transformers

• Key to Modern electrical system

• Produce from 120 volts AC

– Arbitrarily small voltages.

– Arbitrarily large voltages.

• Nearly 100% efficient

3/5/07 U.W. Physics 104, Spring 2007 10

Transformers

Increasing current in primary createsan increase in flux throughprimary and secondary.

%#

(primary) (secondary)

iron

VsVp

Same &'/&t

Energy conservation: IpVp = IsVs

R

3/5/07 U.W. Physics 104, Spring 2007 11

Transformers Question?

(primary) (secondary)

iron

VsVp R

A 12 Volt battery is connected to a transformertransformer that has a 100 turn primary coil, and200 turn secondary coil. What is the voltageacross the secondary after the battery has beenconnected for a long time?

1) Vs = 0 2) Vs = 6 3) Vs = 12 4) Vs = 24

Transformers depend on a

change in flux so they only

work for alternating

currents!

3/5/07 U.W. Physics 104, Spring 2007 12

RL Circuit• Upon turn-on current rise is inhibited by

L, but eventually rises to V/R

• Upon turn-off current fall to zero is

slowed down by presence of L

– Time constant, * = L/R

3/5/07 U.W. Physics 104, Spring 2007 13

Resistors in AC circuit• Ohm’s Law Vr = I R always true.

– Vmax = ImaxR

I

t

vr

t

R

Frequency

Re

sis

tan

ce

– Voltage across resistor is “IN PHASE” with

current through the resistor.

3/5/07 U.W. Physics 104, Spring 2007 14

Capacitors in AC circuit• Vc = Q/C always true.

– However, voltage & current are not constant

– Vmax = ImaxXc

– Capacitive Impedance: XC = 1/(2(fC)

vC

t

I

t

C

Frequency

Imp

ed

an

ce

– When capacitor is uncharged there is no voltage across it

– When capacitor is charged, no current flows

– Voltage across capacitor

“Lags” current.

3/5/07 U.W. Physics 104, Spring 2007 15

Inductors in AC circuit

I

t

vL

t

L

• VL = -L(&I)/(&t) always true.

– Vmax = ImaxXL

– Inductive Impedance: XL = 2(f L

Frequency

Imp

ed

an

ce

– Voltage across inductor

“Leads” current.

3/5/07 U.W. Physics 104, Spring 2007 16

Impedance

R is independent of f

R

3/5/07 U.W. Physics 104, Spring 2007 17

Impedance

R is independent of f

XL increases with f

R

XL

3/5/07 U.W. Physics 104, Spring 2007 18

Impedance

XC increases

with 1/f

RXL

XC

R is independent of f

XL increases with f

3/5/07 U.W. Physics 104, Spring 2007 19

ImpedanceR is independent of f

XL increases with f

XC increases

with 1/f

Z: XL and XC subtractR

XL XC

Z

f0

3/5/07 U.W. Physics 104, Spring 2007 20

I0(XL-XC)

Impedance Triangle

• Voltage across source must

equal sum of voltage across all of

the elements at all times.

• VL leads by 90o and VC lags by

90o, and are 180o out of phase

and they tend to cancel

I0 XL=VL

I0 XC=VC

I0R

V generator=I 0

Z

• V0 (t) = VR (t) +VC (t) +VL (t)

• Impedance Z = sqrt(R2 +(XL – XC)2)

• Peak V0 = I0 Z

• tan(+) = (XL – XC)/R

+

3/5/07 U.W. Physics 104, Spring 2007 21

Generators in AC circuit

• VG+VL+VR+VC = 0 always true.

– Vmax = Imax Z

– Total Impedance: Z = (R2+ (XL2- XC

2))1/2

– Voltage across generators sometimes leads and

sometimes lags current.

– When impedance is

smallest, i.e., XL = XC,

power is largest

L

R

C

Frequency

Po

we

r

f0

3/5/07 U.W. Physics 104, Spring 2007 22

Power

• The voltage generator supplies power,and the resistor dissipates it.

• Capacitor and Inductor store andrelease energy.

• P = IV varies with values of R, C, L.

– Purely resistive R=Z, only dissipation in R

– For R!Z, energy stored/extracted from LC

• Average power, P = Irms Vrms cos(+)where cos(+)=R/Z

3/5/07 U.W. Physics 104, Spring 2007 23

AC Summary

Resistors: VR=IR In phase

Capacitors: VC=I XC: Xc = 1/(2(f C) Lags

Inductors: VL=I XL: XL = 2(f L Leads

Generator V0 = IZ: Z=sqrt(R2 +(XL-XC)2)

Can lead or lag tan(+) = (XL-XC)/R

Power is only dissipated in resistor:

P=IV0 cos(+)

3/5/07 U.W. Physics 104, Spring 2007 24

Preflight 1• A power outlet puts out 60 Hz AC. Which of the

following statements is true

– Voltage goes to zero only 60 times per second

– Current goes to zero only 60 times per second

– Power output goes to zero only 60 times per s

– Voltage , current and power output stay constant

– Voltage, current and power go to zero 120 times per second

One period is a complete 2(revolution - sine or cosine go

to zero twice, so does sin2.

3/5/07 U.W. Physics 104, Spring 2007 25

Preflight 2• Household electric outlet puts out 120 V 60 Hz AC

that is used to power a light bulb. The peak voltage

difference across the bulb is:

– 84 V

– 120 V

– 170 V

– Depends on the resistance of the bulb.

3/5/07 U.W. Physics 104, Spring 2007 26

Preflight 3• Power is transferred from the power plant to your

house through high voltage power lines because:

– Generators at power plants operate at high voltages.

– It will decrease power loss

– The power company wants to discourage people from

climbing on the lines

3/5/07 U.W. Physics 104, Spring 2007 27

Preflight: LCR 1An alternating current circuit consists of a resistor and

capacitor and inductor connected in series to a generator.

Which of the following statements is true.

– As the frequency of the generator becomes infinitely

large, the capacitor can be ignored

– As the frequency of the generator is decreased to zero,

the capacitor can be ignored.

L

R

C

The current flows through the

capacitor as it is being charged

and discharged. If the charging

and discharging are happening at

rapid rate, the current is freely

flowing through the capacitor -

equivalent to a wire!

3/5/07 U.W. Physics 104, Spring 2007 28

Preflight: LCR 2An alternating current circuit consists of a resistor and

capacitor and inductor connected in series to a generator.

Which of the following statements is true.

– As the frequency of the generator becomes infinitely

large, the inductor can be ignored

– As the frequency of the generator is decreased to zero,

the inductor can be ignored.

L

R

C

The current flow through the

inductor is inhibited due to self

inductance which is proportional

to the rate of change of current. If

the current flow is almost

constant the inductor simply acts

like a wire!