copyright © 2009 pearson education, inc. lecture 10 – ac circuits

Copyright © 2009 Pearson Education, Inc.

Lecture 10 – AC Circuits


Current from a battery flows steadily in one direction (direct current, DC). Current from a power plant varies sinusoidally (alternating current, AC).

25-7 Alternating Current (Chapter 25)


The voltage varies sinusoidally with time:

as does the current:

25-7 Alternating Current

,,


Multiplying the current and the voltage gives the power:



Usually we are interested in the average power:


.


The current and voltage both have average values of zero, so we square them, take the average, then take the square root, yielding the root-mean-square (rms) value:




Example 25-13: Hair dryer.

(a) Calculate the resistance and the peak current in a 1000-W hair dryer connected to a 120-V line. (b) What happens if it is connected to a 240-V line in Britain?


Chapter 30Inductance, Electromagnetic Oscillations, and AC Circuits


•Energy Stored in a Magnetic Field

•LR Circuits

• LC Circuits with Resistance (LRC Circuits)

• AC Circuits with AC Source

•LRC Series AC Circuit

• Resonance in AC Circuits

Units of Chapter 30


Energy Stored in a Magnetic Field

• Energy U stored in an inductor of inductance L when it is carrying a current I

• U = ½ LI2

• The energy is stored in the magnetic field of the inductor.

• Similarly, energy stored in a capacitor C when potential difference across it is V

• U = ½ CV2

• The energy is stored in the electric field of the capacitor.


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


30-4 LR Circuits

The sum of potential differences around the loop gives

Integrating gives the current as a function of time:

The time constant of an LR circuit is

.

. .


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

30-4 LR Circuits

.


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?


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

.



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

.



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.


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 what are called 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 voltages across each device are 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




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


Power in Inductor, Capacitor, Resistor and AC Circuits

• Power is dissipated only by a resistance; none is by inductance & capacitance.



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


Summary of Chapter 30• LR circuit:

• Inductive reactance:

• Capacitive reactance:

.

.


Summary of Chapter 30

• LRC series circuit:

• Resonance in LRC series circuit:

.

copyright © 2009 pearson education, inc. lecture 10 – ac circuits

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