Chapter 21 AC Circuits

DC vs AC

• DC = Direct current– Electrons flow constantly– Electrons only flow in one direction (negative to

positive)– Batteries provide DC current

DC vs AC

• AC = Alternating current– Electrons switch directions– “Pulsed current”– Home electricity– More efficient for power transmission over large

distances– USA uses 60 Hertz (60 cycles per second), many

other countries use 50 Hz

DC vs AC

DC•Electrons flow constantly•Electrons flow in only one direction•Batteries

AC•Electrons flow in short burst•Electrons switch directions (60 times a second)•House current

Faraday’s Experiment

• Trying to induce a current using magnetic fields

• No induced current in “Y” loop with a DC circuit

• Saw a current when opening and closing the switch (changing the magnetic field)

Electromagnetic Induction

An induced emf is produced by a changing magnetic field– Can move magnet or loop– Direction of motion controls direction of current– No movement, no current

Magnetic Flux (flow)

• B = Magnetic Flux

B = BAcos

B = Magnetic Field (T)

A = area passes through (m2)

= Angle ┴ to surface

If B ┴ to surface– Cos 0o = 1– Maximum flux

If B || to surface– Cos 90o = 0– No flux

Faraday’s Law of Induction

E = -NB

t

N = number of loops in a wire

Bt = change in magnetic flux over time

So why is it negative?

Lenz’s Law

An induced current’s magnetic field opposes the original change in flux

• Always tries to keep magnetic field inside loop constant.

• Use right-hand rule to predict direction of current.

Lenz’s Law: Ex 1

Why is the direction of the current as indicated?

• Area is decreasing

• Flux is decreasing

• Induced current points into paper through ring

Lenz’s Law: Ex 2

What will happen to the current if you allow the ring to relax to its original shape?

• Larger area

• Induced I will reverse direction

3 Ways to cause an emf

1. Change the magnetic field

2. Change area of loop

3. Rotate the loop (or magnet)

No flux Maximum flux

Lenz’s Law: Ex 3a

Predict the direction of the induced current in the following situations

• Counterclockwise current

• Magnet is going in (north in), need a current pointing north out through the loop

• No current

• Magnetic flux is || to the loop

• Magnetic field decreasing

• Counterclockwise current to increase it

• Decreasing flux

• Clockwise current induced

B

• Initially no flux

• Flux increases to left

• Counterclockwise current

Lenz’s Law: Ex 4

A square coil of 100 loops is quickly pulled from the magnetic field as shown in 0.10 s. Calculate the change in flux.

Bfinal =0

Binitial = BAcos0

Binitial = (0.60 T)(0.050m)2(1)

Binitial = 0.0015 Wb

= Bfinal – binitial

= 0 – 0.0015 Wb = -0.0015 Wb

What Voltage and current are produced in the loop (assume resistance = 100 )

E = -NB

t

E = -(100)(-0.0015 Wb) = 1.5 V

0.10 s

V = IR

I = V/R = 1.5 V/100 = 0.015 A (15 mA)

EMF in a Moving Conductor

• Slide a conducting bar on the wire loop

• Increasing area

• What direction is the induced current?

Motional EMF

E = B

t

E = BA

t

E = Blvt

t

E = Blv (assumes B ┴ to v)

Moving Conductor: Ex 1

An airplane travels at 1000 km/hr in a region where the earth’s magnetic field is 5 X 10-5T (vertical). What is the potential difference between the wing tips if they are 70 m apart?

1000 km/hr = 280 m/s

E = Blv

E = (5 X 10-5T )(70 m)(280 m/s) = 1.0 V

Moving Conductor: Ex 2

Blood contains charged ions. A blood vessel is 2.0 mm in diameter, the magnetic field is 0.080 T, and the blood meter registers a voltage of 0.10 mV. What is the flow velocity of the blood?

E = Blv

v = E /Bl

v = (1.0 X 10-4 V)

(0.080 T)(0.0020m)

v = 0.63 m/s

Electric Generators (Dynamo)

• Generator is the inverse of a motor

• AC Generator shown

• Rotation through magnetic field induces I

• Current flows first one way, then the other

• Segments ab and cd are moving conductor

• (Side segments have force in wrong direction)

E = Blv┴

v┴ = vsin

E = 2NBlvsin

• Can consider angular rotation

= t

v = r = (h/2) h = length of ad or bc

E = 2NBlvsinE = 2NBl(h/2) sint lh = Area

E = NBAsint

Remember

= 2f

f = frequency (Hertz)

(radians/s)

• Over 99% of electricity in US produced by generators– Coal/oil/gas plants– Wind power– Nuclear– Water

• 60 Hz in US and Canada

• 50 Hz in some others

Generator: Ex 1

A 60-Hz generator rotates in a 0.15 T magnetic field. If the area of the coil is 0.020 m2, how many loops must it contain for a peak output of 170 V?

E = NBAsintassume t = 90

E = NBAE /Ba = 2f = 260Hz) = 377 s-1

DC Generator

• Split ring commutator

• Many windings smooth out the current

Alternator

• Engine turns the rotor

• Magnetic field produced

• Current induce in stationary stator coils

Counter EMF

• Counter (back) emf – as a motor turns, an emf is induced that opposes the motion (Lenz’s law)

• Counter emf is less than the external voltage when under a load

• The slower a motor rotates, the less counter emf

Counter EMF: Ex 1

The windings of a DC motor have a resistance of 5.0 . When the motor reaches full speed, the counter emf is 108 V. What is the current when the motor is just starting up, and when it reaches full speed (voltage = 120 V)

Just starting up (almost no counter emf)

V = IR

I = V/R = 120 V/5.0 = 24 A

At full speed (V = 120 V – 108 V = 12 V)

I = V/R = 12 V/5.0 = 2.4 A

Current is VERY high at start– Lights may dim when refrigerator starts– Lights dim if on when starting a car

Counter EMF: Ex 2

If a blender or drill jams (motor can’t turn), the device may burn out. Why?

– No counter emf– Current can be very high– Wires may heat up

Torque: Eddy Currents

Generators also have a counter torque

• Counter-torque

• Only produced when drawing current

• Larger the current, larger the counter torque– Law of conservation of energy – more mechanical

input needed to produce larger currents

Eddy Currents

• Any conductor moving through magnetic field will have eddy currents

• Electrons in atoms are moving as the metal moves

• Magnetic field induced to oppose the change

• Consider moving wheel below

• Eddy currents Force opposes rotation

• Braking a train car

• Resistance for an exercise bike

• Can produce a lot of heat

Transformers

• Increase or decrease AC voltage

• TV – increase voltage for picture tube

• Power packs – decrease voltage

• Utility poles – decrease voltage for house

YEAH!!! MY FAVORITE TOPIC!!!!

• Two coils linked by soft iron core

• Can be intertwined

• Flux from primary induces a current in the secondary (99% efficient)

• Vary number of loops to control voltage

VS = NS

VP NP

• Step-up Transformer – Increases voltage

• Step-down Transformer – Decreases voltage

• POWER can’t increase (can’t get something for nothing)

P = VI

PP = PS

VPIP = VSIs

IS = NP

IP NS

Transformers: Ex 1

A transformer for a radio reduces the voltage from 120 V to 9.0 V. The secondary has 30 turns and the radio draws 400 mA. Calculate the turns in the primary.

VS = NS

VP NP

NP = NSVP = (30)(120V) = 400 turns

VS 9V

Calculate the current in the primary

IS = NP

IP NS

IP = ISNS = (0.400A)(30) = 0.030 A (30 mA)

NP (400)

Calculate the power transformed

P = IV

P = (0.030 A)(120 V) = 3.6 W

(can use either primary or secondary)

Transformers: Ex 2

An average of 120 kW of power is sent to a small town 10 km from the power plant. The transmission lines for a total resistance of 0.40 . Calculate the power lost to resistance if the power is transmitted at 240 V vs. 24,000 V.

At 240 V

P = IV

I = P/V = 120,000 W/240 V = 500 A

Ploss = I2R

Ploss = (500 A)2 (0.40 ) = 100 kW

At 24,000 V

P = IV

I = P/V = 120,000 W/24,000 V = 5 A

Ploss = I2R

Ploss = (5 A)2 (0.40 ) = 10 W

• Transformers only work on ac

• dc only produces a secondary voltage when switch is opened or closed

Microphones

• Coil moves in and out of magnetic field with sound

• emf induced in the coil

• Current is then sent to speakers, recorders, etc..

Tape Heads

Recording

• Changing current in coil creates magnetic field

• Magnetizes the metal on the tape

Playback

• Changing magnetic field from tape induces current in coil

• Digital tape only has 1’s and 0’s

Seismograph

• Magnet moves and creates current in coil

• Current translated into signal for eart’s movement

Self-Inductance: Solenoids

• Solenoid (inductor) – coil of wire (choke coil)

• L = inductance of the coil (Henry’s)

• As current increases in an inductor, an induced emf is created

• Induced emf retards the increase of current (like a back emf)

• Usually want to avoid inductance– Resistors are wound in two directions to cancel the

inductance

• Acts as a resistor for alternating current (impedance)

• Ex– dc current can burn out a transformer– ac has self-inductance (impedance) that limits the

current

Inductance of a solenoid

L = 0N2A

l

0 = 4 X 10-7 T m/A

N = number of turns

A = cross-sectional area

l = length of solenoid

Inductance: Ex 1

Calculate the inductance of a solenoid with 100 turns, a length of 5.0 cm, and a cross sectional area of 0.30 cm2.

L = 0N2A

l

L = (4 X 10-7 T m/A)(100)2(3 X 10-5m2)

(0.05 m)

L = 7.5 X 10-6 H or 7.5 H

Inductance: Ex 2

The same solenoid is now filled with an iron cores ( = 4000 0). Calculate the inductance

L = (4000)(7.5 X 10-6H)

L = 0.030 H or 30 mH

LR Circuits

• Electromagnets

• Radio tuners

• L is inductance

• R is resistance of inductor and any other resistance

• Initially very low impedance

• Impedance rises with current

I = V (1-e-t/)

R

= time constant (time to read 63% of max)

= L

R

Turning on current– Current rises quickly, then levels off

Turning off current– Opposite shape

– I = Imax e-t/

LR Circuits: Ex 1

A solenoid has an inductance of 87.5 mH and a resistance of 0.250 . Find the time constant.

= L/R

= 87.5 X 10-3 H/0.250 W = 0.350 s

How long will it take for the current to reach half of its maximum value?

Imax = V/R t = ln2

I = ½ V/R t = (0.350 s)(0.693)

1 V = V (1-e-t/) t = 0,243 s

2 R R

½ = 1-e-t/

e-t/ ½

et/ln(et/ ln2

AC Current

• Current increases and decreases sinusoidally

I = Iocos2ft

Vrms = Vo/√2

Irms = Io/√2

1. Resistors and AC

• Current changes with voltage

• I and V are in phase

• Energy is lost as heat

P = IV = Irms2R = Vrms

2/R

2. Inductors and AC

• Current lags the voltage by 90o

• 90o is ¼ of a cycle (360o)

• No power is lost to heat

• Energy is held in magnetic field, than returned to source

V = IXL

XL = inductive reactance

XL = 2fL

Inductors and AC: Ex 1

A coil has a resistance of 1.00 and an inductance of 0.300 H. What current is in the coil at 120 V dc?

In dc frequency = 0 so XL = 0

V = IR

I = V/R = 120 V/ 1.00 = 120 A

What current is in the coil in the voltage is 120 V (rms) at 60.0 Hz?

XL = 2fL

XL = 2(60.0 Hz)(0.300 H) = 113

(ignore the 1 of resistance, not in phase)

V = IXL

I= V/XL

I = 120 V/113 = 1.06 A (much lower)

3. Capacitors and AC

• DC - no current flows once plates are charged

• AC – Current flows constantly (plates charge and discharge)

• Current leads voltage by 90o

• No power loss to heat

V = IXC

XC = 1

2fC

(the larger the capacitance, the more charge it can hold: less retarding of current flow)

Capacitors act as filters– Prevent spikes in current flow– Keep current steady

Capacitors and AC: Ex 1

What are the peak and rms currents of a circuit if C = 1.0 F and Vrms = 120 V. The frequency is 60.0 Hz.

XC = 1 = 1

2fC 2(60 Hz)(1.0 X 10-6F)

XC = 2700

Vrms = Vo/√2

Vo = Vrms√2

Vo = (120 V)(√2) = 170 V

Io = Vo/XC

Io = 170 V/2700

Io = 63 mA

Irms = Vrms/XC

Irms = 120 V/2700 = 44 mA

Resistor Current and Voltage in phase

V = IR

Inductor Current lags the voltage by 90o

XL = 2fL

Capacitor Current leads the voltage by 90o

XC = 1 2fC

LRC Circuits

• Most circuits have L, R, and C’s in them

• Voltages across R, L and C are not in phase

• Vo VRo + VLo + Vco

• Currents are in phase

Phasor Diagrams

• Can analyze voltages on a graph

• Can resolve the vectors and create a triangle

• As time goes on, phasor diagram rotates

• Resolved voltage changes

= Phase Angle (angle at which voltage is out of phase from current)

• Resolving the vectors gives the instantaneous voltage

Impedance (Z)

• Impedance = total resistance to the flow of current (from L, R, and C)

• Electrical devices are often impedance matched (tuner connected to an amplifier)

Z = √R2 + (XL – XC)2

Z = √R2 + (2fL – 1/ 2fC)2

Vrms = IrmsZ

Vo = IoZ

tan = XL – XC

R

cos = R

Z

Power dissipated (lost) to impedance

Pave = I2rmsZcos

Pave = IrmsVrmscos

cos = Power factor

cos = 1 for pure resistor

cos = 0 for pure inductor or capacitor

Impedance: Ex 1

A circuit has R=25.0 , L = 30.0 mH, and C = 12.0 F. Calculate the impedance of the circuit if they are connected to a 90.0 V ac(rms), 500 Hz source. Also calculate the phase angle.

XL = 2fL

XL = 2(500 Hz)(0.030 H) = 94.2

XC = 1/2fC

XC = 1/2(500 Hz)(12 X 10-6F) = 26.5

Z = √R2 + (XL – XC)2

Z = √(25.0)2 + (94.2 – 26.5 )2

Z = 72.2

tan = XL – XC

R

tan = 94.2 – 26.5 25.0

= 69.7o

Calculate the rms current

V = IZ

Irms = Vrms/Z = 90.0 V/ 72.2

Irms = 1.25 A

Calculate the voltage drop across each element

(VR)rms = IrmsR = (1.25 A)(25.0 ) = 31.2 V

(VL)rms = IrmsXL = (1.25 A)(94.2 ) = 118 V

(VC)rms = IrmsXC = (1.25 A)(26.5 ) = 33.1 V

Voltages do not add to 90.0 V (out of phase)

Calculate the power loss in the circuit

Pave = IrmsVrmscos

Pave = (1.25 A)(90.0 V)cos(69.7o)

Pave = 39.0 W

Resonance in AC: Oscillators

• LC circuits have a frequency for maximum current

• Used for tuning radio stations

• Resonant frequency

fo = 1

2 √LC

• When R is small, pure LC circuit

• When switch closed– Capacitor discharges– Current creates magnetic field (energy stored in

inductor’s magnetic field)– Current then flows back to plates (stored in

capacitor’s electric field)– Electromagnetic oscillation

• Tuners vary either L or C to tune in a station

• Can also be used to broadcast (radio, cellphone)

Resonant Frequency: Ex 1

A radio tunes in a station at 980 kHz at a capacitance of 3 F. What is the inductance of the circuit?

fo = 1

2 √LC

ANS: 8.8 X 10 -9 H or 8.8 pH

Energy Stored in a Magnetic Field

For a solenoid:

U = ½ B2 Al

o

Note that Al is the volume of the cylinder (could be the volume of other shapes)