Copyright © 2009 Pearson Education, Inc. Chapter 30 Inductance, Electromagnetic Oscillations, and AC Circuits

Copyright © 2009 Pearson Education, Inc.

Chapter 30Inductance, Electromagnetic Oscillations, and AC Circuits


Recap:

22

22

00

00

Ohm's Law: ;

1 1 1Capacitors: ;

2 2 2RC Circuit:

Charge: 1 ;

Discharge: ;

Inductors: ;

t t

t t

B B

VV IR P I R IV

R

QQ CV U CV QV

CRC

dQQ t Q e I t e

dt RdQ

Q t Q e I t edt R

N d dIL t N L

I dt dt


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.

30-4 LR Circuits

0

0

0

0

0

0 0

1 ;t

dI tV I t R L

dtdI t R

I t Vdt LI t

V LI t e

R R


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

30-4 LR Circuits

0

0

0

0

0

0

;t

dI tI t R L

dtdI t R

I tdt L

VI t

RV L

I t eR R


30-4 LR CircuitsExample 30-6: An 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?


30-5 LC Circuits and Electromagnetic Oscillations

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

2

0

0

0 0

0

1

0

10

0 ; 0 0

cos

sin s n

cos ;

i

Q t dI tL

C dtdQ t

I tdt

d Q tQ t

LC dt

Q t

Q t Q tLC

Q I t

Q t Q t

dQI t Q t I t

dt



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



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



Example 30-7: 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.


30-6 LC Oscillations with Resistance (LRC Circuit)

Any real (nonsuperconducting) circuit will have resistance.



Now the voltage drops around the circuit give

A current flowing through a resistor means energy is dissipated so the solution must die out over time.

The solutions to this equation are damped harmonic oscillations.



The system will be underdamped for R2 < 4L/C, and overdamped for R2 > 4L/C. Critical damping will occur when R2 = 4L/C. This figure shows the three cases of underdamping, overdamping, and critical damping.



The angular frequency for critical and under damped oscillations is given by

and the charge in the circuit as a function of time is

The over damped case, is more complicated but the solutions look like

( don’t worry

about this)

2 2

2 2

1 1' is imaginary in overdamped case;

4 4

R R

LC L L LC

2

20 2

1cosh '' ; ''

4

RtL

RQ Q e t

L LC

20 cos '

RtLQ Q e t



Example 30-8: 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?


Resistors, capacitors, and inductors have different phase relationships between current and voltage when placed in an ac circuit.

The current through a resistor is in phase with the voltage.

30-7 AC Circuits with AC Source


Therefore, the current through an inductor lags the voltage by 90°.


The voltage across the inductor is given by

or

.

Later

Earlier



The voltage across the inductor is related to the current through it:

The quantity XL is called the inductive reactance, and has units of ohms:

.



Example 30-9: Reactance of a coil.

A coil has a resistance R = 1.00 Ω and an inductance of 0.300 H. Determine the current in the coil if (a) 120-V dc is applied to it, and (b) 120-V ac (rms) at 60.0 Hz is applied.


Therefore, in a capacitor, the current leads the voltage by 90°.


The voltage across the capacitor is given by

.

Earlier

Later



The voltage across the capacitor is related to the current through it:

The quantity XC is called the capacitive reactance, and (just like the inductive reactance) has units of ohms:

.



Example 30-10: Capacitor reactance.

What is the rms current in the circuit shown if C = 1.0 μF and Vrms = 120 V? Calculate (a) for f = 60 Hz and then (b) for f = 6.0 x 105 Hz.



This figure shows a high-pass filter (allows an ac signal to pass but blocks a dc voltage) and a low-pass filter (allows a dc voltage to be maintained but blocks higher-frequency fluctuations).

large for small 1small for large

CX C


Analyzing the LRC series AC circuit is complicated, as the voltages are not in phase – this means we cannot simply add them. Furthermore, the reactances depend on the frequency.

30-8 LRC Series AC Circuit


We calculate the voltage (and current) using phasors – these are vectors representing the individual voltages.

Here, at t = 0, the current and voltage are both at a maximum. As time goes on, the phasors will rotate counterclockwise.



Some time t later, the phasors have rotated.



The voltage across each device is given by the x-component of each, and the current by its x-component. The current is the same throughout the circuit.



We find from the ratio of voltage to current that the “effective resistance,” called the impedance, of the circuit is given by

BUT – only an actual resistance dissipates energy. The inductor and capacitor store it then release it.




The phase angle between the voltage and the current is given by

The factor cos φ is called the power factor of the circuit.

or



Example 30-11: LRC circuit.

Suppose R = 25.0 Ω, L = 30.0 mH, and C = 12.0 μF, and they are connected in series to a 90.0-V ac (rms) 500-Hz source. Calculate (a) the current in the circuit, (b) the voltmeter readings (rms) across each element, (c) the phase angle , and (d) the power dissipated in the circuit.


The rms current in an ac circuit is

Clearly, Irms depends on the frequency.

30-9 Resonance in AC Circuits


We see that Irms will be a maximum when XC = XL; the frequency at which this occurs is

f0 = ω0/2π is called the resonant frequency.

30-9 Resonance in AC Circuits


30-10 Impedance MatchingWhen one electrical circuit is connected to another, maximum power is transmitted when the output impedance of the first equals the input impedance of the second.

The power delivered to the circuit will be a maximum when dP/dR2 = 0; this occurs when R1 = R2.


Summary of Chapter 30• LR circuit:

• Inductive reactance:

• Capacitive reactance:

.

.


Summary of Chapter 30

• LRC series circuit:

• Resonance in LRC series circuit:

.

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Copyright © 2009 Pearson Education, Inc. Chapter 30 Inductance, Electromagnetic Oscillations, and AC Circuits