Inductance and AC Circuits

• Mutual Inductance

• Self-Inductance

• Energy Stored in a Magnetic Field

• LR Circuits

• LC Circuits and Electromagnetic Oscillations

• LC Circuits with Resistance (LRC Circuits)

• AC Circuits with AC Source

• LRC Series AC Circuit

• Resonance in AC Circuits

• Impedance Matching

• Three-Phase AC

Inductance

• Induced emf in one circuit due to changes in the magnetic field produced by the second circuit is called mutual induction.

• Induced emf in one circuit associated with changes in its own magnetic field is called self-induction.

Inductance

2221

222122

2221222

dt

dN

NNdt

Nd

Unit of inductance: the henry, H:

1 H = 1 V·s/A = 1 Ω·s.

Mutual inductance: magnetic flux through coil2 due to current in coil 1

Induced emf due to mutual induction:

Mutual Inductance

121212 IMN

dt

dIM

dt

dN 1

2121

221

Mutual InductanceSolenoid and coil.

A long thin solenoid of length l and cross-sectional area A contains N1 closely packed turns of wire. Wrapped around it is an insulated coil of N2 turns. Assume all the flux from coil 1 (the solenoid) passes through coil 2, and calculate the mutual inductance.

Mutual Inductance

Reversing the coils.

How would the previous example change if the coil with turns was inside the solenoid rather than outside the solenoid?

A changing current in a coil will also induce an emf in itself:

Self-inductance: magnetic flux through the coil due to the current in the coil itself:

Self-Inductance

2222 LIN

dt

dIL

dt

dN 222

222

Self-Inductance

Solenoid inductance.

(a) Determine a formula for the self-inductance L of a tightly wrapped and long solenoid containing N turns of wire in its length l and whose cross-sectional area is A.

(b) Calculate the value of L if N = 100, l = 5.0 cm, A = 0.30 cm2, and the solenoid is air filled.

Self-Inductance

Direction of emf in inductor.

Current passes through a coil from left to right as shown. (a) If the current is increasing with time, in which direction is the induced emf? (b) If the current is decreasing in time, what then is the direction of the induced emf?

Self-Inductance

Coaxial cable inductance.

Determine the inductance per unit length of a coaxial cable whose inner conductor has a radius r1 and the outer conductor has a radius r2. Assume the conductors are thin hollow tubes so there is no magnetic field within the inner conductor, and the magnetic field inside both thin conductors can be ignored. The conductors carry equal currents I in opposite directions.

A circuit consisting of an inductor and a resistor will begin with most of the voltage drop across the inductor, as the current is changing rapidly. With time, the current will increase less and less, until all the voltage is across the resistor.

LR Circuits

LR Circuits

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LR Circuits

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LR Circuits

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If the circuit is then shorted across the battery, the current will gradually decay away:

LR Circuits

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LR Circuits

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LR CircuitsAn LR circuit.

At t = 0, a 12.0-V battery is connected in series with a 220-mH inductor and a total of 30-Ω resistance, as shown. (a) What is the current at t = 0? (b) What is the time constant? (c) What is the maximum current? (d) How long will it take the current to reach half its maximum possible value? (e) At this instant, at what rate is energy being delivered by the battery, and (f) at what rate is energy being stored in the inductor’s magnetic field?

Just as we saw that energy can be stored in an electric field, energy can be stored in a magnetic field as well, in an inductor, for example.

Analysis shows that the energy density of the field is given by

Energy Density of a Magnetic Field

Energy Stored in an Inductor

dt

diLiR

The equation governs the LR circuit is

Multiplying each term by the current i leads to

dt

diLiRii 2

inductor. thefrom dissipatedpower theis

battery. by the deliveredpower total theis 2Ri

i

Energy Stored in an Inductor

dt

diLi

dt

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Therefore, the third term represents the rate at which the energy is stored in the inductor

The total energy stored from i=0 to i=I is

2

0L 2

1LI

dt

diLiUI

Energy Density of a Magnetic Field

AB

LIU0

22

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1

The self-inductance of a solenoid is L=μ0nA2l. The magnetic field inside it is B=μ0nI. The energy stored thus is

Since Al is the volume of the solenoid, the energy per volume is

0

2

B 2B

u

This is the energy density of a magnetic field in free space.

LC Circuits and Electromagnetic Oscillations

An LC circuit is a charged capacitor shorted through an inductor.

Electromagnetic Oscillations

The current causes the charge in the capacitor to decreases so I=-dQ/dt. Thus the differential equation becomes

LC Circuits

0dt

dIL

C

Q

Across the capacitor, the voltage is raised by Q/C. As the current passes through the inductor, the induced emf is –L(dI/dt). The Kirchhof’s loop rule gives

02

2

LC

Q

dt

Qd

The charge therefore oscillates with a natural angular frequency

LC Circuits and Electromagnetic Oscillations

022

2

xdt

xd

The equation describing LC circuits has the same form as the SHO equation:

.LC

1

Electromagnetic Oscillations

tQQ cos0

The charge varies as

The current is sinusoidal as well:

2cossin

sin

00

0

tItI

tQdt

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Remark: When Q=Q0 at t=t0, we have φ=0.

LC Circuits and Electromagnetic Oscillations

The charge and current are both sinusoidal, but with different phases.

LC Circuits and Electromagnetic Oscillations

The total energy in the circuit is constant; it oscillates between the capacitor and the inductor:

LC Circuits and Electromagnetic Oscillations

LC circuit.

A 1200-pF capacitor is fully charged by a 500-V dc power supply. It is disconnected from the power supply and is connected, at t = 0, to a 75-mH inductor. Determine: (a) the initial charge on the capacitor; (b) the maximum current; (c) the frequency f and period T of oscillation; and (d) the total energy oscillating in the system.

LRC Circuits

Any real (nonsuperconducting) circuit will have resistance.

LRC Circuits

Adding a resistor in an LC circuit is equivalent to adding –IR in the equation of LC oscillation

0dt

dILIR

C

Q

Initially Q=Q0, and the switch is closed at t=0, the current is I=-dQ/dt. The differential equation becomes

02

2

LC

Q

dt

dQ

L

R

dt

Qd

LRC Circuits

The equation describing LRC circuits now has the same form as the equation for the damped oscillation:

022

2

xdt

dx

dt

xd

The solution to LRC circuits therefore is

t

L

RtQQ cos

2exp0

LRC Circuits

where ω02=1/LC.

The system will be underdamped for R2 < 4L/C, and overdamped for R2 > 4L/C. Critical damping will occur when R2 = 4L/C.

The damped angular frequency is

220 2

L

R

LRC Circuits

This figure shows the three cases of underdamping, overdamping, and critical damping.

LRC Circuits

Damped oscillations.

At t = 0, a 40-mH inductor is placed in series with a resistance R = 3.0 Ω and a charged capacitor C = 4.8 μF. (a) Show that this circuit will oscillate. (b) Determine the frequency. (c) What is the time required for the charge amplitude to drop to half its starting value? (d) What value of R will make the circuit nonoscillating?

• Mutual inductance:

• Energy density stored in magnetic field:

Summary of Chapter 30

• Self-inductance:

Summary of Chapter 30• LR circuit:

• Inductive reactance:

• Capacitive reactance:

.

.

Summary of Chapter 30

• LRC series circuit:

• Resonance in LRC series circuit:

.