Self-inductanceSelf-inductance

Self-inductanceSelf-inductance occurs when the changing occurs when the changing flux through a circuit arises from the circuit flux through a circuit arises from the circuit itselfitself• As the current increases, the magnetic flux As the current increases, the magnetic flux

through a loop due to this current also increasesthrough a loop due to this current also increases• The increasing flux induces an emf that opposes The increasing flux induces an emf that opposes

the currentthe current• As the magnitude of the current increases, the As the magnitude of the current increases, the

rate of increase lessens and the induced emf rate of increase lessens and the induced emf decreasesdecreases

• This opposing emf results in a gradual increase This opposing emf results in a gradual increase of the currentof the current

Self-inductance contSelf-inductance cont

The self-induced emf must be proportional The self-induced emf must be proportional to the time rate of change of the currentto the time rate of change of the current

• L is a proportionality constant called the L is a proportionality constant called the inductanceinductance of the device of the device

• The negative sign indicates that a changing The negative sign indicates that a changing current induces an emf in opposition to that current induces an emf in opposition to that changechange

t

iL

Self-inductance, finalSelf-inductance, final

The inductance of a coil depends on The inductance of a coil depends on geometric factorsgeometric factors

The SI unit of self-inductance is the The SI unit of self-inductance is the HenryHenry• 1 H = 1 (V 1 H = 1 (V · s) / A· s) / A

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For infinitely long solenoid

B = onIn: number of turns/m

Increase current through the coil from 0 to i in t.

Winding density: nLoop area: ALength of coil: ℓ

Causes flux change for each loop from = 0 to = AB where B = oni

Vind = - n ℓ /t = - n ℓ A(on)i/t = - n2oA ℓ (i/t)

L: self inductancePurely geometrical quantity

[L] = Henry

r = 1 cmℓ = 2.5 cmn = 10 turns/cm

= 0.01 m= 0.025 m= 1000 turns/m

L = n2oA ℓ = (1000)2(4 x 10-7)(x 0.0012)0.025 = 9.9 x 10-6 H = 9.9 H

Example 24.5A steady current of 5.0 A is flowing through a coil with 2 HSelf inductance. The current is suddenly stopped in 0.01 s. How large an emf induced in the coil?

Vind = -L (i/t)

i = if – ii = -5 A = - (2.0) (-5/0.01)= 1000 V

Inductors in an AC CircuitInductors in an AC Circuit

Consider an AC circuit Consider an AC circuit with a source and an with a source and an inductorinductor

The current in the The current in the circuit is impeded by circuit is impeded by the back emf of the the back emf of the inductorinductor

The voltage across the The voltage across the inductor always leads inductor always leads the current by 90° the current by 90° (T/4)(T/4)

AC

A

V

v= vosin(t)

v

i

v = -L i/ti = -io cos(t)

vosin(t) = Liosin(t) vo = Lio

i lags 90 deg behind v.

Inductive Reactance and Ohm’s Inductive Reactance and Ohm’s LawLaw

The effective resistance of a coil in The effective resistance of a coil in an AC circuit is called its an AC circuit is called its inductive inductive reactancereactance and is given by and is given by• XXLL = = L=2L=2ƒLƒL

When ƒ is in Hz and L is in H, XWhen ƒ is in Hz and L is in H, XLL will be in will be in ohmsohms

Ohm’s Law for the inductorOhm’s Law for the inductor• V = I XV = I XLL

Inductor in a CircuitInductor in a Circuit

Inductance can be interpreted as a Inductance can be interpreted as a measure of opposition to the rate of measure of opposition to the rate of change in the currentchange in the current• Remember resistance R is a measure of Remember resistance R is a measure of

opposition to the currentopposition to the current As a circuit is completed, the current As a circuit is completed, the current

begins to increase, but the inductor begins to increase, but the inductor produces an emf that opposes the produces an emf that opposes the increasing currentincreasing current• Therefore, the current doesn’t change from 0 Therefore, the current doesn’t change from 0

to its maximum instantaneouslyto its maximum instantaneously

RL CircuitRL Circuit

When the current When the current reaches its maximum, reaches its maximum, the rate of change and the rate of change and the back emf are zerothe back emf are zero

The time constant, The time constant, , , for an RL circuit is the for an RL circuit is the time required for the time required for the current in the circuit current in the circuit to reach 63.2% of its to reach 63.2% of its final valuefinal value

RL Circuit, contRL Circuit, cont

The time constant depends on R and The time constant depends on R and LL

The current at any time can be found The current at any time can be found byby

R

L

/te1R

I

Transformer

Iron Core

i1

V2

N1 N2

= - (i1/t)M21

Mutual inductance

AC

To ch. circuit

The RLC Series CircuitThe RLC Series Circuit

The resistor, The resistor, inductor, and inductor, and capacitor can be capacitor can be combined in a combined in a circuitcircuit

The current in the The current in the circuit is the same circuit is the same at any time and at any time and varies sinusoidally varies sinusoidally with timewith time

Current and Voltage Current and Voltage Relationships in an RLC CircuitRelationships in an RLC Circuit

The instantaneous The instantaneous voltage across the voltage across the resistor is in phase resistor is in phase with the currentwith the current

The instantaneous The instantaneous voltage across the voltage across the inductor leads the inductor leads the current by 90°current by 90°

The instantaneous The instantaneous voltage across the voltage across the capacitor lags the capacitor lags the current by 90°current by 90°

AC

A i= iosin(t)

VLVC

v = vc + vL + vR

VR

i

vR

vL

vC

i= iosin(t)

vo = Xio sin(t)Rio

i lags 90 deg behind v.

cos(t)Lio

- cos(t)(1/C)io

V lags 90 deg behind i.

v = vc + vL + vR

= -(1/C)io cos(t) + Lio cos(t) + Rio sin(t)= {(L-1/C)io} cos(t) + Rio sin(t)

= {(L-1/C)2 + R2}1/2(io) sin(t+)

XRLC

vo = Xio

XRLC = {(L-1/C)2 + R2}0.5

= 1/(LC)0.5

R

io = vo/XRLC

Resonance: L = 1/C

RLC Series Resonance Circuit

AC

A i= iosin(t)

V

vo

RLC Parallel Resonance Circuit

= 1/(LC)0.5

Power in an AC CircuitPower in an AC Circuit

No power losses are associated with No power losses are associated with capacitors and pure inductors in an AC capacitors and pure inductors in an AC circuitcircuit• In a capacitor, during one-half of a cycle In a capacitor, during one-half of a cycle

energy is stored and during the other half the energy is stored and during the other half the energy is returned to the circuitenergy is returned to the circuit

• In an inductor, the source does work against In an inductor, the source does work against the back emf of the inductor and energy is the back emf of the inductor and energy is stored in the inductor, but when the current stored in the inductor, but when the current begins to decrease in the circuit, the energy is begins to decrease in the circuit, the energy is returned to the circuitreturned to the circuit

Power in an AC Circuit, contPower in an AC Circuit, cont

The average power delivered by the The average power delivered by the generator is converted to internal generator is converted to internal energy in the resistorenergy in the resistor• PPavav = IV = IVRR = I = IV cos V cos

• cos cos is called the is called the power factorpower factor of the of the circuitcircuit

Phase shifts can be used to maximize Phase shifts can be used to maximize power outputspower outputs

Resonance in an AC CircuitResonance in an AC Circuit

ResonanceResonance occurs at occurs at the frequency, ƒthe frequency, ƒoo, , where the current has where the current has its maximum valueits maximum value• To achieve maximum To achieve maximum

current, the impedance current, the impedance must have a minimum must have a minimum valuevalue

• This occurs when XThis occurs when XLL = = XXCC

LC2

1ƒo

Resonance, contResonance, cont Theoretically, if R = 0 the current would be Theoretically, if R = 0 the current would be

infinite at resonanceinfinite at resonance• Real circuits always have some resistanceReal circuits always have some resistance

Tuning a radioTuning a radio• A varying capacitor changes the resonance A varying capacitor changes the resonance

frequency of the tuning circuit in your radio to frequency of the tuning circuit in your radio to match the station to be receivedmatch the station to be received

Metal DetectorMetal Detector• The portal is an inductor, and the frequency is set The portal is an inductor, and the frequency is set

to a condition with no metal presentto a condition with no metal present• When metal is present, it changes the effective When metal is present, it changes the effective

inductance, which changes the current which is inductance, which changes the current which is detected and an alarm soundsdetected and an alarm sounds

Simple Radio

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Summary of Circuit Elements, Summary of Circuit Elements, Impedance and Phase AnglesImpedance and Phase Angles

James Clerk MaxwellJames Clerk Maxwell

Electricity and Electricity and magnetism were magnetism were originally thought to originally thought to be unrelatedbe unrelated

in 1865, James Clerk in 1865, James Clerk Maxwell provided a Maxwell provided a mathematical theory mathematical theory that showed a close that showed a close relationship between relationship between all electric and all electric and magnetic phenomenamagnetic phenomena

Maxwell’s Starting PointsMaxwell’s Starting Points

Electric field lines originate on positive Electric field lines originate on positive charges and terminate on negative charges and terminate on negative chargescharges

Magnetic field lines always form closed Magnetic field lines always form closed loops – they do not begin or end anywhereloops – they do not begin or end anywhere

A varying magnetic field induces an emf A varying magnetic field induces an emf and hence an electric field (Faraday’s Law)and hence an electric field (Faraday’s Law)

Magnetic fields are generated by moving Magnetic fields are generated by moving charges or currents (Ampcharges or currents (Ampère’s Law)ère’s Law)

Maxwell’s PredictionsMaxwell’s Predictions Maxwell used these starting points and a Maxwell used these starting points and a

corresponding mathematical framework to corresponding mathematical framework to prove that prove that electric and magnetic fields play electric and magnetic fields play symmetric roles in naturesymmetric roles in nature

He hypothesized that a changing electric field He hypothesized that a changing electric field would produce a magnetic fieldwould produce a magnetic field

Maxwell calculated the speed of light to be Maxwell calculated the speed of light to be 3x103x1088 m/s m/s

He concluded that visible light and all other He concluded that visible light and all other electromagnetic waves consist of fluctuating electromagnetic waves consist of fluctuating electric and magnetic fields, with each electric and magnetic fields, with each varying field inducing the othervarying field inducing the other

Hertz’s Confirmation of Hertz’s Confirmation of Maxwell’s PredictionsMaxwell’s Predictions

Heinrich Hertz was Heinrich Hertz was the first to the first to generate and generate and detect detect electromagnetic electromagnetic waves in a waves in a laboratory settinglaboratory setting

James Clerk Maxwell’s Equations(1867)

Pre-Maxwell

JB

t

BE

B

E

o

o

0

t

EJB

t

BE

B

E

ooo

o

0

E-field comes out from p-charge andterminates at negative charge.

B-field cannot do like E-field.No magnetic monopole!

Faraday’s law

Ampere’s law

The velocity of transverse undulations in our hypothetical medium, calculated from the electromagnetic experiments, agrees so exactly with the velocity of light calculated fromthe optical experiments, that we can scarcely avoid the inference that light consists in the transverse undulation of same medium which is the cause of electric and magnetic Phenomena.

)/104)(2/1085.8(

1127212 ANmNC

voo

2.9986 x 108 m/s

Hertz’s Experiment(1887)

Hertz’s Basic LC CircuitHertz’s Basic LC Circuit When the switch is When the switch is

closed, oscillations closed, oscillations occur in the current occur in the current and in the charge on and in the charge on the capacitorthe capacitor

When the capacitor is When the capacitor is fully charged, the total fully charged, the total energy of the circuit is energy of the circuit is stored in the electric stored in the electric field of the capacitorfield of the capacitor• At this time, the current At this time, the current

is zero and no energy is is zero and no energy is stored in the inductorstored in the inductor

LC Circuit, contLC Circuit, cont As the capacitor discharges, the energy stored As the capacitor discharges, the energy stored

in the electric field decreasesin the electric field decreases At the same time, the current increases and At the same time, the current increases and

the energy stored in the magnetic field the energy stored in the magnetic field increasesincreases

When the capacitor is fully discharged, there When the capacitor is fully discharged, there is no energy stored in its electric fieldis no energy stored in its electric field• The current is at a maximum and all the energy The current is at a maximum and all the energy

is stored in the magnetic field in the inductoris stored in the magnetic field in the inductor The process repeats in the opposite directionThe process repeats in the opposite direction There is a continuous transfer of energy There is a continuous transfer of energy

between the inductor and the capacitorbetween the inductor and the capacitor

Hertz’s Experimental ApparatusHertz’s Experimental Apparatus

An induction coil is An induction coil is connected to two connected to two large spheres large spheres forming a capacitorforming a capacitor

Oscillations are Oscillations are initiated by short initiated by short voltage pulsesvoltage pulses

The inductor and The inductor and capacitor form the capacitor form the transmittertransmitter

Hertz’s ExperimentHertz’s Experiment

Several meters away from the Several meters away from the transmitter is the receivertransmitter is the receiver• This consisted of a single loop of wire This consisted of a single loop of wire

connected to two spheresconnected to two spheres• It had its own inductance and It had its own inductance and

capacitancecapacitance When the resonance frequencies of When the resonance frequencies of

the transmitter and receiver the transmitter and receiver matched, energy transfer occurred matched, energy transfer occurred between thembetween them

Hertz’s ConclusionsHertz’s Conclusions

Hertz hypothesized the energy Hertz hypothesized the energy transfer was in the form of wavestransfer was in the form of waves• These are now known to be These are now known to be

electromagnetic waveselectromagnetic waves Hertz confirmed Maxwell’s theory by Hertz confirmed Maxwell’s theory by

showing the waves existed and had showing the waves existed and had all the properties of light wavesall the properties of light waves• They had different frequencies and They had different frequencies and

wavelengthswavelengths