756
SUMMARY
Phasors and Alternating Currents
(Section 22.1) An ac source produces an emf that varies sinusoidally
with time. A sinusoidal voltage or current can be represented by a
phasor—a vector that rotates counterclockwise with constant angular velocity v equal to the angular frequency of the sinusoidal quantity. Its projection on the horizontal axis at any instant represents the
instantaneous value of the quantity. For a sinusoidal current i with
maximum value I, the phasor is given by i 5 I cos vt (Equation 22.2).
In power calculations, it is useful to use the root-mean-square
(rms) value: Irms 5 I "2 (Equation 22.3).
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Resistance and Reactance
(Section 22.2) If the current is given by i 5 I cos vt (Equation 22.2)
and the voltage v between two points is v 5 V cos 1 vt 1 f 2 , then f
is called the phase angle of the voltage relative to the current.
The voltage across a resistor R is in phase with the current, and
the voltage and current amplitudes are related by VR 5 IR
(Equation 22.6). The voltage across an inductor L leads the current
with a phase angle of f 5 90°; the voltage and current amplitudes
are related by VL 5 IXL (Equation 22.12), where XL 5 vL (Equation 22.11) is the inductive reactance of the inductor. The voltage
across a capacitor C lags the current with a phase angle f 5290°;
the voltage and current amplitudes are related by VC 5 IXC
(Equation 22.17), where XC 5 1 vC (Equation 22.16) is the
capacitive reactance of the capacitor.
v
I
Phasor
vt
O
i 5 I cos vt
R, X
XL
XC
R
v
O
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The Series R–L–C Circuit
(Section 22.3) In a series R–L–C circuit, the voltage and current
amplitudes are related by V 5 IZ (Equation 22.21), where Z is the
impedance of the circuit: Z 5 "R 2 1 3 vL 2 1 1 vC 2 4 2 (Equation 22.22). The phase angle f of the voltage relative to the current
is given by
i
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f 5 arctan
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vL 2 1 vC
.
R
(22.23)
V 5 IZ
VL 5 IXL
d
a
I
2q
C
R
q
b
f
VL 2 VC
vt
VR 5 IR
O
c
VC 5 IXC
L
v, i, p
1
P 5 2 VI cos f
Power in Alternating-Current Circuits
(Section 22.4) The average power input P to an ac circuit is I
times one-half the component of the voltage that is in phase with
the current, or P 5 12 VI cos f 5 VrmsIrms cos f (Equation 22.29),
where f is the phase angle of voltage with respect to current.
Power is dissipated only through the resistor. For circuits containing only capacitors and inductors, f 5 690° and the average
power is zero. The quantity cos f is called the power factor.
p
f
v
Graphs of p, v, and i versus
time for an arbitrary
combination of resistors,
t inductors, and capacitors.
The average power is
positive.
v
i
Series and Parallel Resonance
(Sections 22.5 and 22.6) The current in a series R–L–C circuit
reaches a maximum, and the impedance reaches a minimum, at an
angular frequency v0 5 1 1 LC 2 1/2 known as the resonance angular frequency. This phenomenon is called resonance. At resonance, the voltage and current are in phase and the impedance Z is
equal to the resistance R. The smaller the resistance, the sharper is
the resonance peak. In an R–L–C parallel circuit, the total current
attains a minimum, and the impedance attains a maximum, at the
resonance angular frequency v0 .
I (A)
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200 V
The lower a circuit’s resistance, the
higher and sharper is the resonance
peak in the current near the
resonance angular frequency v0.
500 V
2000 V
O
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v (rad s)