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592 ⏐⏐⏐ THE BASIC ELEMENTS AND PHASORS
P
i
+
v
–
Load
FIG. 14.28
Determining the power delivered in a
sinusoidal ac network.
there comes a point with increasing frequency when the element will
take on inductive characteristics. It also points out that the frequency of
application defines the type of capacitor (or inductor) that would be
applied: Electrolytics are limited to frequencies up to perhaps 10 kHz,
while ceramic or mica can handle frequencies beyond 10 MHz.
The expected temperature range of operation can have an important
impact on the type of capacitor chosen for a particular application.
Electrolytics, tantalum, and some high-k ceramic capacitors are very
sensitive to colder temperatures. In fact, most electrolytics lose 20% of
their room-temperature capacitance at 0°C (freezing). Higher temperatures
(up to 100°C or 212°F) seem to have less of an impact in general
than colder temperatures, but high-k ceramics can lose up to 30% of
their capacitance level at 100°C compared to room temperature. With
exposure and experience, you will learn the type of capacitor employed
for each application, and concern will arise only when very high frequencies,
extreme temperatures, or very high currents or voltages are
encountered.
14.5 AVERAGE POWER AND POWER FACTOR
For any load in a sinusoidal ac network, the voltage across the load and
the current through the load will vary in a sinusoidal nature. The questions
then arise, How does the power to the load determined by the
product v·i vary, and what fixed value can be assigned to the power
since it will vary with time?
If we take the general case depicted in Fig. 14.28 and use the following
for v and i:
v � Vm sin(qt � vv) i � Im sin(qt � vi) then the power is defined by
p � vi � Vm sin(qt � vv)Im sin(qt � vi) � VmIm sin(qt � vv) sin(qt � vi) Using the trigonometric identity
cos(A � B) � cos(A � B)
sin A sin B ����
2
the function sin(qt � vv) sin(qt � vi) becomes
sin(qt � vv) sin(qt � vi) �
cos[(qt � v v) � (qt � v i)] � cos[(qt � v v) � (qt � v i)]
�������
2
cos(vv � vi) � cos(2qt � vv � vi) �����
2
so that Fixed value Time-varying (function of t)
⎫
⎪
⎪
⎬
⎪
⎪
⎭
V mI m
⎫
⎪
⎬
⎪
⎭
V mI m
p � � � cos(vv � vi) � � � � cos(2qt � vv � vi) 2
2 �
A plot of v, i, and p on the same set of axes is shown in Fig. 14.29.
Note that the second factor in the preceding equation is a cosine
wave with an amplitude of V mI m/2 and with a frequency twice that of
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