Chapter 32

Inductance

Introduction

• In this chapter we will look at applications of induced currents, including:– Self Inductance of a circuit– Inductors as circuit elements.– Mutual Inductance between two circuits– RL, LC, and RLC Combinations

32.1 Self-Inductance

• It is important in this section that we make sure to distinguish between the physical source of emf and current in a circuit, and the emf and current that are induced from magnetic fields. – emf and current for those caused by a battery or

power supply– Induced emf and induced current for those caused

by changing magnetic fields.

32.1

• Consider a simple switched resistor circuit.– When the switch is closed, the current does not

immediately flow at its maximum value, but increases with time.

– This creates a magnetic flux increasing with time, and therefore an induced emf.

32.1

– From Lenz’s law we know the induced emf will oppose the changing flux, and so is opposite to the direction of the battery emf.

– This is also called a “back emf” similar to what is found in a motor coil.

– This self induced emf is given the symbol εL

32.1

• Consider a second example of wire coil wrapped around a cylindrical core.

• With the direction of the current, the B field points to the left.

32.1

• As the current increases, so does the B Field, giving an induced emf in the opposite direction.

32.1

• As the current and B field decrease, inducing an emf with the direction of the current.

32.1

• A self-induced emf is always proportional to the time rate of change of current.

• L is the proportionality constant called “inductance” and depends on the geometry and physical properties of the coil.

dt

dILL

32.1

• By combining this expression with Faraday’s Law

• We get an expression for the inductance L, of a coil (solenoid/toroid) assuming the same magnetic flux passes through each turn

dt

dN B

L

I

NL B

32.1

• We can also describe it simply as

• Remember that Resistance is a measure of opposition to current (R = ΔV/I)

• Inductance is a measure of opposition to change in current.

dtdIL L

32.1

• The unit for inductance is the henry (H) equal to a volt-second per amp.

• Quick Quiz p. 1005• Examples 32.1-32.2

A

sV 1 H 1

32.2 RL Circuits

• If a circuit contains a coil (ex: solenoid) the inductance of the coil prevents the current from changing instantaneously.

• If this element has a large inductance, it is called and inductor and shows up in circuit schematics as a coil.

• An inductor acts to resist changes to the current.

32.2

• Consider a circuit with a battery, resistor, inductor, and switch. An RL Circuit.

• What happens when the switch is closed at t = 0?

• The current increases, anda back emf is induced in the inductor.

32.2

• Going around the circuit (Kirchoff’s Loop rule), the potential difference is given as

• The solution to this differential equation gives I as a function of time.

or

0dt

dILIR

LRteR

I /1 /1 te

RI

R

L

32.2

• Where I = 0 and t = 0 and the time constant τ = L/R

• This exponential function increases asymptotically to the maximum current ε/R.

32.2

• Taking the first derivative of the equation gives the rate of change of current.

• We see that this is at its maximum value at t = 0. (Also from the slope of the current plot)

/teLdt

dI

32.2

• ADD SWITCH EXAMPLE

32.2

• Quick Quizzes p. 1009• Example 32.3

32.3 Energy in a Magnetic Field

• Since the induced emf resists an instantaneous current, the battery must provide more energy in circuits with an inductor. – Part of that energy is dissipated in the restistor.– The remainder is stored in the magnetic field

within the inductor.

32.3

• The rate at which energy stored in a inductor is given as

• We can integrate to get the total energy stored

• Note this is similar to the energy in a capacitor

• We can also determine the energy density for a given inductor geometry, like a solenoid.

22

1 LIU

22

1 VCU

dt

dILI

dt

dU

32.3

• In example 32.1 we found the inductance of a solenoid to be

• And the B field in a solenoid is given as

• By substituting into the energy equation for L and I

AnL o2

nIB o

2

22

122

1

n

BAnLIU

oo

32.3

• This can be simplified to

• And since Al is the volume of the solenoid, the energy per unit volume is

AB

Uo2

2

oB

B

A

Uu

2

2

32.3

• Again we see similarities with E-fields

• Quick Quiz p. 1012• Example 32.4

22

1 Eu oE

32.4 Mutual Inductance

• Often times, an emf can be induced in a circuit because of changing currents in other nearby circuits, a process called mutual induction.

• Let’s look at two wound coils of wire.• The current in one wire creates a magnetic field that pass through the other coil.

32.4

• The magnetic flux through coil two from coil one is given as Φ12.

• We can define the mutual inductance of the two coils as

• This property of the pair of coils depends on both geometries and their orientations with respect to each other.

1

12212 I

NM

32.4

• If the I1 varies with time, then the emf induced in coil 2 is given as

• Similarly, if the current I2 varies with time, then the emf induced in coil 1 is

dt

dIM 1

122

dt

dIM 2

211

32.4

• It can be shown that

• So the emf induced in each coil can be written

and

• In mutual induction, the emf induced in one coil is always proportional to the rate of change of current in the other coil.

MMM 2112

dt

dIM 1

2 dt

dIM 2

1

32.4

• Quick Quiz p. 1014• Example 32.6

32.5 Oscillations in an LC Circuit

• An simple LC circuit consists of a initially charged capacitor, an inductor, and a switch. • When the switch is closed, we find that both

the current in the circuit and the charge on the capacitor oscillate between maximum positive and negative values.

32.5

• The oscillations should continue indefinitely if we assume– Zero resistance in the circuit (zero loss to internal

energy)– Zero loss to the radiation of energy.

• If we look at the energy involved, we can see a lot of similarities to a simply harmonic oscillator.

32.5

• The harmonic oscillator transitions from Uelastic with +A, to K, back to Uelastic with -A.

• The LC oscillator transitions from fully charged capacitor

• To maximum current and therefore magnetic field in the inductor

• Back to a fully charged capacitor with opposite polarity.

C

QU

2

2max

22

1 LIU

C

QU

2

2max

32.5

• The circuit swings past a “simple discharge” because of the inductor’s resistance to changing current.– At Qmax, I = zero– At Imax, Q = zero– At –Qmax. I = zero

32.5

• Capacitor is fully charged, switch is closed.

32.5

• Current reaches maximum value as capacitor is fully discharged.

32.5

• Current continues to flow recharging the Cap with opposite polarity.

32.5

• Capacitor discharges to maximum current flowing in the opposite direction.

32.5

• Current slows to zero as the circuit reaches its original state.

32.5

• An “oscillation” in the charge on the capacitor of this type is represented by a differential equation, the solution to which is a trig function (sine/cosine).

• Where ω is the angular frequency and is the phase constant.

tQQ cosmax

LC

1

02

2

C

Q

dt

QdL

32.5

• ω represents the natural frequency of the circuit, and as such has applications in resonance.

• Determining the phase angle based on initial conditions. – At t = 0• I = 0• Q = Qmax

– So the phase angle must be, = 0.

32.5

• Since current is the rate of charge flow, we can take dQ/dt to find it as a function of time.

tQdt

dQI sinmax

tII sinmax

32.5

32.5

• Again recognize that the total energy in the system is the sum of electrical energy and magnetic energy stored in the cap and inductor.

LC UUU

22

1

2

2LI

C

QU

tLItC

QU 22

max212

2max sincos

2

32.5

32.5

• See Table 32.1 p 1021 for analogies between physical oscillation and electrical oscillation.

• Quick Quizzes p. 1019• Example 32.7

32.6 The RLC Circuit

• When studying the LC circuit, we recognize the analogy to an harmonic oscillator for ideal conditions (no resistance).

• With a resistor added to the circuit, we get a dissipation of energy, and so the oscillations will not continue indefinitely.

32.6

• The analogue to the RLC circuit is a damped oscillator.

32.6

• The presence of resistance in the circuit adds an additional term to the differential equation describing the circuit.

• When R = 0, we have the ideal LC circuit. • As the value of R increases we go through the

various types of damping.

02

2

C

Q

dt

dQR

dt

QdL

32.6

• When R is small, the damping is light. The solution to the differential is

• Where ωd is

• Note the value of ω, when R = 0.

teQQ dL

Rt cos2max

21

2

2

1

L

R

LCd

32.6

• Note the combination of the exponential decay and the sinusoidal oscillation.

• The amplitude of the maximum charge decreases within the decay envelope.

32.6

• As the value of R increases we approach critical damping where the circuit reaches equilibrium as fast as possible without oscillating.

• For values of R > Rc, we have overdamped conditions, returning slowly to equilibrium.

CLRc /4

32.6

• Overdamped

32.6

• Understanding how the presence of resistance affects the circuit is essential because the LC circuit is just an ideal case, all “LC” circuits have some damping.

• Applications– Variable Tuning (ie. Radio station frequencies)– Signal Filtering

The End…

• No seriously, its over…

• Don’t make this weird…