Page 375 - Schaum's Outline of Theory and Problems of Applied Physics

360 ALTERNATING-CURRENT CIRCUITS [CHAP. 29

POWERFACTOR
The power absorbed in an ac circuit is given by

P = IV cos φ

where I and V are effective values and φ is the phase angle between voltage and current. The quantity cos φ is
the power factor of the circuit. At resonance, φ = 0, cos φ = 1, and the power absorbed is a maximum. The
power factor in an ac circuit is equal to the ratio between its resistance and its impedance:
R R
Power factor = cos φ = =
Z R + (X L + X C ) 2
2
◦
Power factors are often expressed as percentages, so a phase angle of, say, 25 would give rise to a power factor
◦
of cos 25 = 0.906 = 90.6 percent.
Ac power sources are usually specified in terms of apparent power, the product of V eff and I eff , and are
measured in voltamperes (VA). The true power P = IV cos φ is not always specified because for a circuit
with power factor less than 1, a power greater than the true power P must be supplied. Thus a power factor of
90.6 percent means that an apparent power of 1 VA must be supplied for each 0.906 W of true power consumed
by the circuit. To obtain a true power of 1 W, an apparent power of 1/0.906 = 1.104 VA must be supplied.

SOLVED PROBLEM 29.14
A coil of unknown resistance and inductance draws 4 A when it is connected to a 12-V dc power source
and 3 A when it is connected to a 12-V, 100-Hz power source. (a) Find the values of R and L.(b)How
much power is dissipated when the coil is connected to the dc source? (c) When it is connected to the ac
source?
(a) There is no inductive reactance when direct current passes through the coil, so its resistance is

V 1 12 V
R = = = 3
I 1 4A
At f = 100 Hz the impedance of the circuit is

V 2 12 V
Z = = = 4
I 2 3A
2 2
2
and so since Z = R + (X − X ) and X C = 0 here,
L C

2
2
2
2
X L = Z − R = (4 ) − (3 ) = 2.65
Hence the inductance of the coil is
X L 2.65
L = = = 4.22 mH
2π f (2π)(100 Hz)
2
2
(b) P 1 = I R = (4A) (3 ) = 48 W
1
2
2
(c) P 2 = I R = (3A) (3 ) = 27 W
2
SOLVED PROBLEM 29.15
A 50-µF capacitor, a 0.3-H inductor, and an 80- resistor are connected in series with a 120-V, 60-Hz
power source (Fig. 29-11). (a) What is the impedance of the circuit? (b) How much current flows in it?
(c) What is the power factor? (d) How much power is dissipated by the circuit? (e) What must be the
minimum rating in volt amperes of the power source?