Higher Engineering Mathematics, Sixth Edition

Irregular areas, volumes and mean values of waveforms 207

dd

b

y 1 y 2 y 3 y 4 y 5 y 6 y 7

y

dddd d

Figure 19.6

If the mid-ordinate rule is used to find the area under the
curve, then:

y=

sum of mid-ordinates

number of mid-ordinates

(

=

y 1 +y 2 +y 3 +y 4 +y 5 +y 6 +y 7

7

for Fig. 19.6

)

For asine wave, the mean or average value:

(i) over one complete cycle is zero (see Fig. 19.7(a)),

V

Vm

0

(a)

tt

V

Vm

0

(b)

V

Vm

0

(c)

t

Figure 19.7

(ii) over half a cycle is0.637×maximum value,or

( 2 /π)×maximum value,

(iii) of a full-wave rectified waveform (see Fig.

19.7(b)) is0.637×maximum value,

(iv) of a half-wave rectified waveform (see

Fig. 19.7(c)) is 0.318×maximum value,or

( 1 /π)maximum value.

Problem 5. Determine the average values over

half a cycle of the periodic waveforms shown in

Fig. 19.8.

01234

210

20

Voltage (V)

(a)

t(ms)

0

21

22

23

123456

3

2

1

Current (A)

(b)

t(s)

0

210

2468

10

Voltage (V)

(c)

t(ms)

Figure 19.8

(a) Areaundertriangularwaveform(a)forahalfcycle

is given by:

Area=^12 (base) (perpendicular height)

=^12 ( 2 × 10 −^3 )( 20 )

= 20 × 10 −^3 Vs

Average value of waveform

=

area under curve

length of base

=

20 × 10 −^3 Vs

2 × 10 −^3 s

=10V