Download Inductance - Chabot College

Chapter 30
Inductance
PowerPoint® Lectures for
University Physics, Thirteenth Edition
– Hugh D. Young and Roger A. Freedman
Lectures by Wayne Anderson
Copyright © 2012 Pearson Education Inc.
Energy through space for free??
• A puzzler!
Creates
increasing flux
INTO ring
Increasing
current in
time
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Energy through space for free??
• A puzzler!
Creates
increasing flux
INTO ring
Increasing
current in
time
Copyright © 2012 Pearson Education Inc.
Induce
counterclockwise
current and
B field OUT
of ring
Energy through space for free??
• But… If wire loop has resistance R, current
around it generates energy! Power = i2/R!!
Increasing
current in
time
Induced current
i around loop of
resistance R
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Energy through space for free??
• Yet…. NO “potential difference”!
Increasing
current in
time
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Induced current
i around loop of
resistance R
Energy through space for free??
• Answer? Energy in B field!!
Increasing
current in
time
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Increased flux
induces EMF in
coil radiating
power
Goals for Chapter 30 - Inductance
• To learn how current in one coil can induce an
emf in another unconnected coil
• To relate the induced emf to the rate of change
of the current
• To calculate the energy in a magnetic field
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Goals for Chapter 30
• Introduce circuit components called
INDUCTORS
• Analyze circuits containing resistors and
inductors
• Describe electrical oscillations in circuits and
why the oscillations decay
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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?
• Why does a coil of metal
behave very differently from a
straight wire of the same metal?
• We’ll learn how circuits can be
coupled without being
connected together.
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Mutual inductance
• Mutual inductance: A changing current in one coil
induces a current in a neighboring coil.
Increase current
in coil 1
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Mutual inductance
• Mutual inductance: A changing current in one coil
induces a current in a neighboring coil.
Increase B flux
Increase current
in coil 1
Copyright © 2012 Pearson Education Inc.
Mutual inductance
• Mutual inductance: A changing current in one coil
induces a current in a neighboring coil.
Increase B flux
Induce flux
opposing change
Increase current
in coil 1
Copyright © 2012 Pearson Education Inc.
Induce current in
loop 2
Mutual inductance
• EMF induced in single loop 2 = - d (B2 )/dt
–
Caused by change in flux through second loop of B field
–
Created by the current in the single loop 1
• So… B2 is proportional to i1
• What does that proportionality constant depend upon?
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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
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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!
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Mutual inductance
• IF you have N turns in
coil 2,
each with flux B ,
total flux is Nx larger
• Total Flux = N2 B2
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Mutual inductance
• B2 proportional to i1
So
N2 B = M21i1
M21 is the “Mutual Inductance”
[Units] = Henrys = Wb/Amp
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Mutual inductance
• EMF2 = - N2 d [B2 ]/dt
and
•
N2 B = M21i1
so
• EMF2 = - M21d [i1]/dt
• Because geometry is
“shared”
M21 = M12 = M
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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
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Mutual inductance examples
• Long solenoid with length l, area A, wound with N1 turns of wire
• N2 turns surround at its center. What is M?
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Mutual inductance examples
• M = N2 B2/i1
• We need B2 from the first solenoid (B1 = moni1)
• n = N1/l
• B2 = B1A
• M = N2 moi1 AN1/li1
• M = moAN1 N2 /l
• All geometry!
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Mutual inductance examples
• M = moAN1 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 mH
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Mutual inductance examples
• Using same system (M = 25 mH)
• Suppose i2 varies with time as = (2.0 x 106 A/s)t
• At t = 3.0 ms, what is average flux through each turn of coil 1?
• What is induced EMF in solenoid?
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Mutual inductance examples
• Suppose i2 varies with time as = (2.0 x 106 A/s)t
• At t = 3.0 ms, i2 = 6.0 Amps
• M = N1 B1/i2 = 25 mH
• B1= Mi2/N1 = 1.5x10-7 Wb
• Induced EMF in solenoid?
–
EMF1 = -M(di2/dt)
–
-50Volts
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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 = NdB/dt = -EMF
or
• EMF = -Ldi/dt
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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.
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Potential across an inductor
• The potential across an
inductor depends on the
rate of change of the
current through it.
• The self-induced emf
does not oppose current,
but opposes a change in
the current.
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Vab = -Ldi/dt
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
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Magnetic field energy
• 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!
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Magnetic field energy
• The energy stored in an inductor is U = 1/2 LI2.
• The energy density in a magnetic field (Joule/m3) is
•
u = B2/2m0
(in vacuum)
•
u = B2/2m
(in a magnetic material)
• Recall definition of m0 (magnetic permeability)
•
B = m0 i/2pr (for the field of a long wire)
•
m0 = Tesla-m/Amp
• [u] = [B2/2m0] = T2/(Tm/Amp) = T-Amp/meter
Copyright © 2012 Pearson Education Inc.
Magnetic field energy
• The energy stored in an inductor is U = 1/2 LI2.
• The energy density in a magnetic field (Joule/m3) is
•
u = B2/2m0
(in vacuum)
• [u] = [B2/2m0] = T2/(Tm/Amp) = T-Amp/meter
• [U] = Tm2Amp = Joules
• So… energy density [u] = Joules/m3
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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?
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Calculating self-inductance and self-induced emf
• Toroidal solenoid with area A, average radius r, N turns.
• L = N B/i
• B = BA = (moNi/2pr)A
• L = moN2A/2pr (self inductance of toroidal solenoid)
• Why N2 ??
• If N =200 Turns, A = 5.0 cm2,
r = 0.10 m
L = 40 mH
Copyright © 2012 Pearson Education Inc.
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.
Copyright © 2012 Pearson Education Inc.
Potential across an inductor
• The potential across an
inductor depends on the
rate of change of the
current through it.
• The self-induced emf
does not oppose current,
but opposes a change in
the current.
Copyright © 2012 Pearson Education Inc.
Potential across an inductor
• The potential across an
inductor depends on the
rate of change of the
current through it.
• The self-induced emf does
not oppose current, but
opposes a change in the
current.
• The inductor acts like a
temporary voltage source
pointing OPPOSITE to the
change.
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Potential across an inductor
• The inductor acts like a
temporary voltage source
pointing OPPOSITE to the
change.
• This implies the inductor
looks like a battery
pointing the other way!
• Note Va > Vb!
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Direction of current flow
from a battery oriented
this way
Potential across an inductor
• 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!
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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)
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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)
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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! 
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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
Vab = -Ldi/dt
• EMF = -Ldi/dt
• [L] = Henrys = Volts /Amps/sec
• Volts/Amps = Ohms (From V = IR)
• Henrys = Ohm-seconds
• [ ]= L/R = Henrys/Ohm = seconds! 
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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
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The R-L circuit example
•
R = 175 W; i = 36 mA; current
limited to 4.9 mA in first 58 ms.
•
What is required EMF
•
What is required inductor
•
What is the time constant?
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The R-L circuit Example
•
R = 175 W; i = 36 mA; current
limited to 4.9 mA in first 58 ms.
•
What is required EMF
•
What is required inductor
•
What is the time constant?
•
EMF = IR = (0.36 mA)x(175W )
= 6.3 V
•
i(t) = E /R (1-e -(R/L)t)
•
i(58ms) = 4.9 mA
•
4.9mA = 6.3V(1-e -(175/L)0.000058)
•
L = 69 mH
•
 = L/R = 390 ms
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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
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Current decay in an R-L circuit
• 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
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Current decay in an R-L circuit
• i(t) = i(0)e -(R/L)t
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Current decay in an R-L circuit
• 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
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Current decay in an R-L circuit
• 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
• WHY?
•
Current increases
suddenly, so inductor
resists change
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Current decay in an R-L circuit
• Test yourself!
• Signs of Vab and Vbc when
S1 is closed?
•
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!
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Current decay in an R-L circuit
• 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
Copyright © 2012 Pearson Education Inc.
Current decay in an R-L circuit
• 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!
Copyright © 2012 Pearson Education Inc.
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 inductor
resists 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
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i
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 inductor
resists 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
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i
The L-C circuit
• An L-C circuit contains an inductor and a capacitor and is an
oscillating circuit.
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The L-C circuit
• An L-C circuit contains an inductor and a capacitor and is an
oscillating circuit.
• Now capacitor fully drained;
• Inductor keeps pushing charge in the
original direction
• Capacitor charge builds up on other side
to a maximum value
i
• While that side charges, “back EMF” from
capacitor tries to slow charge build-up
• Inductor keeps pushing to resist that change.
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The L-C circuit
• An L-C circuit contains an inductor and a capacitor and is an
oscillating circuit.
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The L-C circuit
• An L-C circuit contains an inductor and a capacitor and is an
oscillating circuit.
• Now capacitor charged on opposite side;
• Current reverses direction! System repeats
in the opposite direction
i
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The L-C circuit
• An L-C circuit contains an inductor and a capacitor and is an
oscillating circuit.
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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
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Electrical oscillations in an L-C circuit
• -Ldi/dt – qC = 0
• i(t) = dq/dt
• Ld2q/dt2 + qC = 0
• Simple Harmonic Motion!
• Pendulums
• Springs
• Standard solution!
• q(t)= Qmax cos(wt+f)
where w = 1/(LC)½
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Electrical oscillations in an L-C circuit
• q(t)= Qmax cos(wt+f)
• i(t) = - w Qmax sin(wt+f)
(based on this ASSUMED
direction!!)
• w = 1/(LC)½ =
angular frequency
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The L-C circuit
• An L-C circuit contains an inductor and a capacitor and is an
oscillating circuit.
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Electrical and mechanical oscillations
• Table 30.1 summarizes the analogies between SHM and L-C
circuit oscillations.
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The L-R-C series circuit
• An L-R-C circuit exhibits
damped harmonic motion
if the resistance is not too
large.
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