Page 118 - Schaum's Outline of Theory and Problems of Electric Circuits

WAVEFORMS AND SIGNALS
CHAP. 6]
6.6 THE AVERAGE AND EFFECTIVE (RMS) VALUES 107

A periodic function f ðtÞ, with a period T, has an average value F avg given by
ð ð
1 T 1 t 0 þT
F avg ¼h f ðtÞi ¼ f ðtÞ dt ¼ f ðtÞ dt ð11Þ
T 0 T t 0
The root-mean-square (rms) or effective value of f ðtÞ during the same period is defined by
ð 1=2
1 t 0 þT 2
F eff ¼ F rms ¼ f ðtÞ dt ð12Þ
T
t 0
2 2
It is seen that F eff ¼h f ðtÞi.
Average and effective values of periodic functions are normally computed over one period.

EXAMPLE 6.9 Find the average and effective values of the cosine wave vðtÞ¼ V m cos ð!t þ Þ.
Using (11),
ð
1 T V m
T
V avg ¼ V m cos ð!t þ Þ dt ¼ ½sinð!t þ ފ ¼ 0 ð13Þ
0
T 0 !T
and using (12),
ð ð
1 T 1 T
2
2
2
2
2
V eff ¼ V m cos ð!t þ Þ dt ¼ V m ½1 þ cos 2ð!t þ ފ dt ¼ V m =2
T 0 2T 0
p ffiffiffi
from which V eff ¼ V m = 2 ¼ 0:707V m (14)
Equations (13) and (14) show that the results are independent of the frequency and phase angle . In other words,
the average of a cosine wave and its rms value are always 0 and 0.707 V m , respectively.
EXAMPLE 6.10 Find V avg and V eff of the half-rectified sine wave

V m sin !t when sin !t > 0
vðtÞ¼ ð15Þ
0 when sin !t < 0
From (11),
ð
1 T=2 V m T=2
V avg ¼ V m sin !tdt ¼ ½ cos !tŠ 0 ¼ V m = ð16Þ
T 0 !T
and from (12),
ð ð
1 T=2 1 T=2
2 2 2 2 2
V eff ¼ V m sin !tdt ¼ V m ð1 cos 2!tÞ dt ¼ V m =4
T 0 2T 0
from which V eff ¼ V m =2 (17)
EXAMPLE 6.11 Find V avg and V eff of the periodic function vðtÞ where, for one period T,

V 0 for 0 < t < T 1
vðtÞ¼ Period T ¼ 3T 1 ð18Þ
V 0 for T 1 < t < 3T 1
V 0 V 0
We have V avg ¼ ðT 1 2T 1 Þ¼ (19)
3T 3
2
2
and V eff ¼ V 0 ðT 1 þ 2T 1 Þ¼ V 0 2
3T
from which V eff ¼ V 0 (20)
The preceding result can be generalized as follows. If jvðtÞj ¼ V 0 then V eff ¼ V 0 .