Irregular areas, volumes and mean values of waveforms 207
d
d
b
y 1 y 2 y 3 y 4 y 5 y 6 y 7
y
d
d
d
d
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 a sine wave, the mean or average value:
(i) over one complete cycle is zero (see Fig. 19.7(a)),
V
V m
0
(a)
t
t
V
V m
0
(b)
V
V m
0
(c)
t
Figure 19.7
(ii) over half a cycle is 0.637 × maximum value, or
(2/π ) × maximum value,
(iii) of a full-wave rectified waveform (see Fig.
19.7(b)) is 0.637 × 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.
0
1 2
3 4
210
20
Voltage (V)
(a)
t (ms)
0
21
22
23
1 2
5 6
3
4
3
2
1
Current (A)
(b)
t (s)
0
210
2 4
6 8
10
Voltage (V)
(c)
t (ms)
Figure 19.8
(a) Area under triangular waveform (a) for a half cycle
is given by:
Area =
1
2 (base) (perpendicular height)
=
1
2 (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
= 10 V