Download Chapter 33 - University of Utah Physics

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1
Chapter 33
Alternating-Current Circuits
Summary
log v
Definitions
b
In AC circuits that contain inductors
and capacitors, it is useful to define the
inductive reactance X L and the capacitive reactance X C as
X L ; vL
(33.10)
1
vC
(33.18)
XC ;
where v is the angular frequency of the
AC source. The SI unit of reactance is
the ohm.
The impedance Z of an RLC series AC circuit is
(33.25)
Z ; "R 2 1 1 X L 2 X C 2 2
This expression illustrates that we cannot simply add the resistance
and reactances in a circuit. We must account for the applied voltage
and current being out of phase, with the phase angle f between the
current and voltage being
f 5 tan 21 a
XL 2 XC
b
R
(33.27)
The sign of f can be positive or negative, depending on whether X L is
greater or less than X C . The phase angle is zero when X L 5 X C .
Concepts and Principles
The rms current and rms voltage in
an AC circuit in which the voltages and
current vary sinusoidally are given by
I rms 5
I max
"2
(33.4)
5 0.707I max
DVmax
5 0.707 DVmax
(33.5)
"2
where I max and DVmax are the maximum
values.
DVrms 5
If an AC circuit consists of a source and a resistor, the current is in
phase with the voltage. That is, the current and voltage reach their
maximum values at the same time.
If an AC circuit consists of a source and an inductor, the current
lags the voltage by 90°. That is, the voltage reaches its maximum
value one-quarter of a period before the current reaches its maximum value.
If an AC circuit consists of a source and a capacitor, the current
leads the voltage by 90°. That is, the current reaches its maximum
value one-quarter of a period before the voltage reaches its maxiObjective Questions
1021
mum value.
The average power delivered by the source in an RLC circuit is
Pavg 5 I rms DVrms cos f
(33.31)
An equivalent expression for the average power is
Pavg 5 I
2
rms
R
The rms current in a series RLC circuit is
I rms 5
(33.32)
The average power delivered by the source results in increasing internal energy in the resistor. No power loss occurs in an ideal inductor
or capacitor.
A series RLC circuit is in resonance when the inductive reactance
equals the capacitive reactance. When this condition is met, the rms
current given by Equation 33.34 has its maximum value. The resonance frequency v 0 of the circuit is
v0 5
1
"LC
(33.35)
The rms current in a series RLC circuit has its maximum value when
the frequency of the source equals v 0, that is, when the “driving” frequency matches the resonance frequency.
Objective Questions
DVrms
"R 1 1 X L 2 X C 2 2
2
(33.34)
AC transformers allow for easy changes
in alternating voltage according to
Dv 2 5
N2
N1
Dv 1
(33.41)
where N1 and N 2 are the numbers of
windings on the primary and secondary
coils, respectively, and Dv1 and Dv 2 are
the voltages on these coils.
1. denotes answer available in Student Solutions Manual/Study Guide
1. An inductor and a resistor are connected in series
across an AC source as in Figure OQ33.1. Immediately
but its peak voltage remains the same. (e) None of those
statements is correct.