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