Higher Engineering Mathematics

IRREGULAR AREAS, VOLUMES AND MEAN VALUES OF WAVEFORMS 221

C

Problem 6. Determine the mean value of cur-

rent over one complete cycle of the periodic

waveforms shown in Fig. 20.9.

Figure 20.9

(a) One cycle of the trapezoidal waveform (a) is

completed in 10 ms (i.e. the periodic time is

10 ms).

Area under curve=area of trapezium

=^12 (sum of parallel sides) (perpendicular

distance between parallel sides)

=^12 {(4+8)× 10 −^3 }(5× 10 −^3 )

= 30 × 10 −^6 As

Mean value over one cycle

=

area under curve

length of base

=

30 × 10 −^6 As

10 × 10 −^3 s

=3mA

(b) One cycle of the sawtooth waveform (b) is
completed in 5 ms.

Area under curve =^12 (3× 10 −^3 )(2)

= 3 × 10 −^3 As

Mean value over one cycle

=

area under curve

length of base

=

3 × 10 −^3 As

5 × 10 −^3 s

=0.6 A

Problem 7. The power used in a manufactur-

ing process during a 6 hour period is recorded at

intervals of 1 hour as shown below.

Time(h) 0123456

Power (kW) 0 14 29 51 45 23 0

Plot a graph of power against time and, by using

the mid-ordinate rule, determine (a) the area

under the curve and (b) the average value of the

power.

The graph of power/time is shown in Fig. 20.10.

Figure 20.10

(a) The time base is divided into 6 equal intervals,

each of width 1 hour. Mid-ordinates are erected

(shown by broken lines in Fig. 20.10) and

measured. The values are shown in Fig. 20.10.

Area under curve=(width of interval)

×(sum of mid-ordinates)

=(1)[7. 0 + 21. 5 + 42. 0

+ 49. 5 + 37. 0 + 10 .0]

=167 kWh(i.e. a measure

of electrical energy)

(b) Average value of waveform

=

area under curve

length of base

=

167 kWh

6h

=27.83 kW