Chapter 30

Copyright © 2012 Pearson Education Inc. Modified Scott Hildreth – Chabot College 2016

PowerPoint® Lectures forUniversity Physics, Thirteenth Edition – Hugh D. Young and Roger A. Freedman

Lectures by Wayne Anderson

Chapter 30

Inductance

Introduction

• How does a coil induce a current in a neighboring coil?

• A sensor triggers the traffic light to change when a car arrives at an intersection. How does it do this?

Induction Loop Signals

Add soft iron core to electromagnet

increases B field

Add car above buried coil changes

B field & induces change in loop

Introduction

Energy through space for free??

• A puzzler!

Increasing current in

time

Creates increasing flux

INTO ring

Induce counter-

clockwise current and B field OUT

of ring


• But… If wire loop has resistance R, current around it generates energy! Power = i2/R!!


time

Induced current i around loop of

resistance R


• Yet…. NO “potential difference”!


timeInduced current i around loop of

resistance R


• Answer? Energy in B field!!


time

Increased flux induces EMF in

coil radiating power

Goals for Chapter 30 - Inductance

• Learn how current in one coil can induce Emf in another unconnected coil

• Relate induced Emf to rate of change of current

• To calculate energy in a magnetic field

Goals for Chapter 30

• Introduce circuit components

called INDUCTORS

• Analyze circuits containing resistors & inductors

• Describe electrical oscillations in circuits & why oscillations decay

Mutual inductance

• Mutual inductance: A changing current in one coil induces a current in a neighboring coil.

Increase current in coil 1

Increase B flux

Induce current in loop 2

Induce flux opposing change

Coil 1 Coil 2

i

Mutual inductance

• So… B2 in the second loop is proportional to i1 in the first loop.

• What does that proportionality constant depend upon?

Mutual inductance

• B2 is proportional to i1 and is affected by:

– # of windings in loop 1

– Area of loop 1

– Area of loop 2

• Define “M” as mutual inductance of coil 1 on coil 2

Mutual inductance

• Define “M” as mutual inductance on coil 2 from coil 1

• B2 = M21 I

• But what if loop 2 also has many turns?

Increase #turns in loop 2? =>

increase flux!

Mutual inductance

• IF you have N turns in coil 2,

each with flux B ,

total flux is Nx larger

• Total Flux = N2 B2

Mutual inductance

• B2 proportional to i1

So

N2 B = M21i1

M21 is the “Mutual Inductance”

[Units] = Henrys = Wb/Amp

Webers of Flux detected per Amp of generating current

in the other coil!

Henrys = ∙sec resistance in time??

Mutual inductance

• EMF induced in secondary loop(s) = - d (N2B2 )/dt

– Caused by change in flux of B field through secondary loops

– Created by the current in the primary loop(s)

• EMF2 = - N2 d [B2 ]/dt

Mutual inductance

• EMF2 = - N2 d [B2 ]/dt

But we defined mutual inductance M:

• N2 B = M21i1

so

• EMF2 = - M21d [i1]/dt

• But geometry is “shared”!

M21 = M12 = M

Mutual inductance

• Define Mutual inductance: A changing current in one coil induces a current in a neighboring coil.

• M = N2 B2/i1

M = N1 B1/i2

Mutual inductance examples

• Long solenoid with length l, area A, wound with N1 turns of wire

• N2 turns of secondary coil surround at its center. What is M?

• M = N2 B2/i1

• And B2 comes from B1 of first solenoid (B1 = oni1)

– n = N1/l

– B2 = B1A

• So…

– M = N2 (oi1 AN1/l) (1/ i1)



• M = N2 oi1 AN1/li1

– M = oAN1 N2 /l

• All geometry!

• Inductors are like capacitors

– YOU design them withfixed sizes, turns, lengths

– Application results in differentEMFs generated by differentchanges in current through them


• M = oAN1 N2 /l

• If N1 = 1000 turns, N2 = 10 turns, A = 10 cm2, l = 0.50 m

– M = 25 x 10-6 Wb/A

– M = 25 H


• Use this inductor (M = 25 H)N1 = 1000 turns, N2 = 10 turns, A = 10 cm2, l = 0.50 m

• Suppose i2 varies with time as = (2.0 x 106 A/s)t

• At t = 3.0 s, what is average flux through each turn of coil 1?

• What is induced EMF in solenoid?


• Suppose i2 varies with time as = (2.0 x 106 A/s)tN1 = 1000 turns, N2 = 10 turns, A = 10 cm2, l = 0.50 m

• At t = 3.0 s, i2 = 6.0 Amps

• M = N1 B1/i2 = 25 H

• B1= Mi2/N1 = 1.5x10-7 Wb

• Induced EMF in solenoid?

– EMF1 = -M(di2/dt)

– -50Volts

Self-inductance• Self-inductance: A varying current in a circuit induces an emf

in that same circuit.

• Always opposes the change!

• Define L = N B/i

• Li = N B

• If i changes in time:

• d(Li)/dt = NdB/dt = -EMF

or

• EMF = -Ldi/dt

Self-inductance• Self-inductance: A varying current in a circuit induces an emf

in that same circuit.

• EMF = -Ldi/dt

• VOLTS = Henrys ∙ (Amps/Sec)

• Henrys = (Volts/Amps) ∙ sec

• Henrys = Ohm-seconds!

Inductors as circuit elements!

• Inductors ALWAYS oppose change:

• In DC circuits:

– Inductors maintain steady current flow even if supply varies

• In AC circuits:

– Inductors suppress (filter) frequencies that are too fast.

Potential across an inductor

• Potential across an inductor depends on rate of change of current through it.

• Self-induced EMF does not oppose current, but does opposes a change in current.

• If i doesn’t change in time, EMF of inductor = 0!

EMF = -Ldi/dt

Potential across an inductor• The potential across a

resistor drops in the direction of current flow

Vab = Va-Vb > 0

• The potential across an inductor depends on the rate of change of the current through it. • No change• Increasing current => Va > Vb• Decreasing current => Va < Vb


• No change in current: EMF of inductor = 0



• Increasing current: EMF of inductor against current

• Inductor acts like a temporary voltage source OPPOSITE to change.



• Inductor acts like a temporary voltage source OPPOSITE to change.

• This implies the inductor looks like a battery pointing the other way!

• Note Va > Vb!Direction of current flow from a battery oriented

this way


• Wait a minute?? Wasn’t it

• EMF = - Ldi/dt

• Yes! Sign depends upon di/dt, always opposing)

• Vab = Va – Vb = Ldi/dt

– If di/dt = 0, Vab = 0

– If di/dt >0, Vab > 0

• Va – Vb > 0 implies Va > Vb!

Vab = Ldi/dt

If current increases, induced EMF pushes back on circuit as if

inductor was a battery oriented

backwards!


• What if current was decreasing?

• Same result! The inductor acts like a temporary voltage source pointing OPPOSITE to the change.

• Now inductor pushes current in original direction

• Note Va < Vb!

The R-L circuit

• An R-L circuit contains a resistor and inductor and possibly an emf source.

• Start with both switches open

• Close Switch S1:

• Current flows

• Inductor resists flow

• Actual current less than maximum E/R

• E – i(t)R- L(di/dt) = 0

• di/dt = E /L – (R/L)i(t)

The R-L circuit

• Close Switch S1:

• E – i(t)R- L(di/dt) = 0

• di/dt = E /L – (R/L)i(t)

Boundary Conditions

• At t=0, di/dt = E /L

• i() = E /R

Solve this 1st order diff eq:

• i(t) = E /R (1-e -(R/L)t)

Current growth in an R-L circuit

• i(t) = E /R (1-e -(R/L)t)

• The time constant for an R-L circuit is = L/R.

• [ ]= L/R = Henrys/Ohm

• = (Tesla-m2/Amp)/Ohm

• = (Newtons/Amp-m) (m2/Amp)/Ohm

• = (Newton-meter) / (Amp2-Ohm)

• = Joule/Watt

• = Joule/(Joule/sec)

• = seconds!

Current growth in an R-L circuit

• i(t) = E /R (1-e -(R/L)t)

• The time constant for an R-L circuit is = L/R.

• [ ]= L/R = Henrys/Ohm

• EMF = -Ldi/dt

• [L] = Henrys = Volts /Amps/sec

• Volts/Amps = Ohms (From V = IR)

• Henrys = Ohm-seconds

• [ ]= L/R = Henrys/Ohm = seconds!

Vab = Ldi/dt

The R-L circuit

• E = i(t)R+ L(di/dt)

• Power in circuit = E I

• E i = i2R+ Li(di/dt)

• Some power radiated in resistor

• Some power stored in inductor

The R-L circuit example

• R = 175 ; i = 36 mA; current limited to 4.9 mA in first 58 s.

• What is required EMF

• What is required inductor

• What is the time constant?

The R-L circuit Example

• R = 175 ; i = 36 mA; current limited to 4.9 mA in first 58 s.

• What is required EMF

• What is required inductor

• What is the time constant?

• EMF = IR = (0.36 mA)x(175 ) = 6.3 V

• i(t) = E /R (1-e -(R/L)t)

• i(58s) = 4.9 mA

• 4.9mA = 6.3V(1-e -(175/L)0.000058)

• L = 69 mH

= L/R = 390 s

Current decay in an R-L circuit

• Now close the second switch!

• Current decrease is opposed by inductor

• EMF is generated to keep current flowing in the same direction

• Current doesn’t drop to zero immediately


• Now close the second switch!

• –i(t)R - L(di/dt) = 0

• Note di/dt is NEGATIVE!

• i(t) = -L/R(di/dt)

• i(t) = i(0)e -(R/L)t

• i(0) = max current before second switch is closed


• i(t) = i(0)e -(R/L)t


• Test yourself!

• Signs of Vab and Vbc when S1 is closed?

• Vab >0; Vbc >0

• Vab >0, Vbc <0

• Vab <0, Vbc >0

• Vab <0, Vbc <0


• Test yourself!


• Vab >0; Vbc >0

• WHY?

• Current increases suddenly, so inductor resists change


• Test yourself!


• Vab >0; Vbc >0

• WHY?

• Current still flows around the circuit counterclockwise through resistor

• EMF generated in L is from c to b

• So Vb> Vc!


• Test yourself!

• Signs of Vab and Vbc when S2 is closed, S1 open?

• Vab >0; Vbc >0

• Vab >0, Vbc <0

• Vab <0, Vbc >0

• Vab <0, Vbc <0


• Test yourself!

• Signs of Vab and Vbc when S2 is closed, S1 open?

• Vab >0; Vbc >0

• Vab >0, Vbc <0

• WHY?

• Current still flows counterclockwise

• di/dt <0; EMF generated in L is from b to c!So Vb < Vc!

Magnetic field energy

• Inductors store energy in the magnetic field:

U = 1/2 LI2

• Units: L = Henrys (from L = N B/i )

• N B/i = B-field Flux/current through inductor that creates that flux

Wb/Amp = Tesla-m2/Amp

• [U] = [Henrys] x [Amps]2

• [U] = [Tesla-m2/Amp] x [Amps]2 = Tm2Amp

• But F = qv x B gives us definition of Tesla

• [B] = Teslas= Force/Coulomb-m/s = Force/Amp-m


• Inductors store energy in the magnetic field:

U = 1/2 LI2

• [U] = [Tesla-m2/Amp] x [Amps]2 = Tm2Amp

• [U] = [Newtons/Amp-m] m2Amp = Newton-meters = Joules = Energy!


• The energy stored in an inductor is U = 1/2 LI2.

• The energy density in a magnetic field (Joule/m3) is

• u = B2/20 (in vacuum)

• u = B2/2 (in a magnetic material)

• Recall definition of 0 (magnetic permeability)

• B = 0 i/2r (for the field of a long wire)

0 = Tesla-m/Amp

• [u] = [B2/20] = T2/(Tm/Amp) = T-Amp/meter


• The energy stored in an inductor is U = 1/2 LI2.

• The energy density in a magnetic field (Joule/m3) is

• u = B2/20 (in vacuum)

• [u] = [B2/20] = T2/(Tm/Amp) = T-Amp/meter

• [U] = Tm2Amp = Joules

• So… energy density [u] = Joules/m3

Calculating self-inductance and self-induced emf

• Toroidal solenoid with area A, average radius r, N turns.

• Assume B is uniform across cross section. What is L?

Calculating self-inductance and self-induced emf

• Toroidal solenoid with area A, average radius r, N turns.

• L = N B/i

• B = BA = (oNi/2r)A

• L = oN2A/2r (self inductance of toroidal solenoid)

• Why N2 ??

• If N =200 Turns, A = 5.0 cm2,

r = 0.10 m

L = 40 H

The L-C circuit

• An L-C circuit contains an inductor and a capacitor and is an oscillating circuit.

• Initially capacitor fully charged; close switch

• Charge flows FROM capacitor, but inductorresists that increased flow.

• Current builds in time.

• At maximum current, charge flow now decreases through inductor

• Inductor now resists decreased flow, and keeps pushing charge in the original direction

i

The L-C circuit


• Initially capacitor fully charged; close switch

• Charge flows FROM capacitor, but inductorresists that increased flow.

• Current builds in time.

• Capacitor slowly discharges

• At maximum current, no charge is left on capacitor; current now decreases through inductor

• Inductor now resists decreased flow, and keeps pushing charge in the original direction

i

The L-C circuit


The L-C circuit


• Now capacitor fully drained;

• Inductor keeps pushing charge in the original direction

• Capacitor charge builds up on other sideto a maximum value

• While that side charges, “back EMF” fromcapacitor tries to slow charge build-up

• Inductor keeps pushing to resist that change.

i

The L-C circuit


The L-C circuit


• Now capacitor charged on opposite side;

• Current reverses direction! System repeatsin the opposite direction

i

The L-C circuit


Electrical oscillations in an L-C circuit

• Analyze the current and charge as a function of time.

• Do a Kirchoff Loop around the circuit in the direction shown.

• Remember i can be +/-

• Recall C = q/V

• For this loop:

-Ldi/dt – qC = 0


• -Ldi/dt – qC = 0

• i(t) = dq/dt

• Ld2q/dt2 + qC = 0

• Simple Harmonic Motion!

• Pendulums

• Springs

• Standard solution!

• q(t)= Qmax cos(t+)

where 1/(LC)½


• q(t)= Qmax cos(t+)

• i(t) = - Qmax sin(t+)

(based on this ASSUMED direction!!)

• 1/(LC)½ = angular frequency

The L-C circuit


Electrical and mechanical oscillations

• Table 30.1 summarizes the analogies between SHM and L-C circuit oscillations.

The L-R-C series circuit

• An L-R-C circuit exhibits damped harmonic motion if the resistance is not too large.

Date post:	21-Mar-2016
Category:	Documents
Upload:	alvis
View:	32 times
Download:	0 times

Chapter 30

Documents