CHAP. 6]

6.6

107

WAVEFORMS AND SIGNALS

THE AVERAGE AND EFFECTIVE (RMS) VALUES A periodic function f ðtÞ, with a period T, has an average value Favg given by ð ð 1 T 1 t0 þT f ðtÞ dt ¼ f ðtÞ dt Favg ¼ h f ðtÞi ¼ T 0 T t0

ð11Þ

The root-mean-square (rms) or eﬀective value of f ðtÞ during the same period is deﬁned by  ð t0 þT 1=2 1 Feff ¼ Frms ¼ f 2 ðtÞ dt T t0

ð12Þ

2 It is seen that Feff ¼ h f 2 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 vðtÞ ¼ Vm cos ð!t þ Þ. Using (11), ð 1 T V V cos ð!t þ Þ dt ¼ m ½sinð!t þ ÞT0 ¼ 0 Vavg ¼ T 0 m !T

ð13Þ

and using (12), 2 ¼ Veff

1 T

ðT

Vm2 cos2 ð!t þ Þ dt ¼

0

1 2T

ðT

Vm2 ½1 þ cos 2ð!t þ Þ dt ¼ Vm2 =2

0

pﬃﬃﬃ Veff ¼ Vm = 2 ¼ 0:707Vm

from which

Equations (13) and (14) show that the results are independent of the frequency and phase angle . the average of a cosine wave and its rms value are always 0 and 0.707 Vm , respectively. EXAMPLE 6.10

Find Vavg and Veff of the half-rectiﬁed sine wave  Vm sin !t when sin !t > 0 vðtÞ ¼ 0 when sin !t < 0

(14) In other words,

ð15Þ

From (11), Vavg ¼

1 T

ð T=2 Vm sin !t dt ¼ 0

Vm ½ cos !tT=2 ¼ Vm = 0 !T

ð16Þ

and from (12), 2 ¼ Veff

1 T

ð T =2

Vm2 sin2 !t dt ¼

0

from which EXAMPLE 6.11

We have

ð T=2

Vm2 ð1  cos 2!tÞ dt ¼ Vm2 =4

0

Veff ¼ Vm =2 Find Vavg and Veff of the periodic function vðtÞ where, for one period T,  V0 for 0 < t < T1 vðtÞ ¼ Period T ¼ 3T1 V0 for T1 < t < 3T1 Vavg ¼ 2 ¼ Veff

and

1 2T

V0 V0 ðT  2T1 Þ ¼ 3T 1 3

(17)

ð18Þ

(19)

V02 ðT þ 2T1 Þ ¼ V02 3T 1

from which The preceding result can be generalized as follows.

Veff ¼ V0 If jvðtÞj ¼ V0 then Veff ¼ V0 .

(20)

Mahmood_Nahvi_eBook_Schaum_s_Outlines_Theory_An
Mahmood_Nahvi_eBook_Schaum_s_Outlines_Theory_An