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40
Signals and Spectra Chap. 2
V dc = 8v(t)9 = 8V cos v 0 t9
= 1 T 0 /2
V cos v 0 t dt = 0
(2–8)
T 0 L T 0 /2
where v 0 = 2p/T 0 and f 0 = 1/T 0 = 60 HZ. Similarly, I dc = 0. The instantaneous power is
p(t) = 1V cos v 0 t2(I cos v 0 t) = 1 2 VI11 + cos 2 v 0 t2
(2–9)
The average power is
P = h 1 2 VI11 + cos 2 v 0 t2i
= VI T 0 /2
11 + cos 2 v
2T 0 t2 dt
0 L
T 0 /2
= VI
(2–10)
2
As can be seen from Eq. (2–9) and Fig. 2–3c, the power (i.e., light emitted) occurs in pulses at a
rate of 2f 0 = 120 pulses per second. (In fact, this lamp can be used as a stroboscope to “freeze”
1
mechanically rotating objects.) The peak power is VI, and the average power is
2 VI , where V is
the peak voltage and I is the peak current. Furthermore, for this case of sinusoidal voltage and
sinusoidal current, we see that the average power could be obtained by multiplying V> 12 with
I> 12. See Example2_02.m.
RMS Value and Normalized Power
DEFINITION.
The root-mean-square (RMS) value of w(t) is
W rms = 38w 2 (t)9
(2–11)
THEOREM.
If a load is resistive (i.e., with unity power factor), the average power is
P = 8v2 (t)9
R
= I 2 rms R = V rms I rms
where R is the value of the resistive load.
= 8i 2 (t)9R = V 2
rms
R
(2–12)
Equation (2–12) follows from Eq. (2–7) by the use of Ohm’s law, which is v(t) = i(t)R, and
Eq. (2–11).
Continuing Example 2–2, V rms = 120 V. It is seen from Eq. (2–11), when sinusoidal
waveshapes are used, that V rms = V/12 and I rms = I/12. Thus, using Eq. (2–12), we see that
1
the average power is
2 VI , which is the same value that was obtained in the previous discussion.
The concept of normalized power is often used by communications engineers. In this concept,
R is assumed to be 1 Æ, although it may be another value in the actual circuit. Another way of