Higher Engineering Mathematics, Sixth Edition

150 Higher Engineering Mathematics

339.4

Voltage

v (V)

67.9

2 67.9

2 339.4

v 5 339.4 sin 100 t 1 67.9 sin( 300 t 2 34 )

67.9 sin( 300 t 2 34 )

339.4 sin 100 t

Time t (ms)

(^1520)
5 10
Figure 14.36
Problem 22. A complex waveformvcomprises a
fundamental voltage of 240V rms and frequency
50Hz, together with a 20% third harmonic which
has a phase angle lagging by 3π/4rad at timet=0.
(a) Write down an expression to represent voltage
v. (b) Use harmonic synthesis to sketch the
complex waveform representing voltagevover one
cycle of the fundamental component.
(a) A fundamental voltage having an rms value of
240V has a maximum value, or amplitude of√
2 (240) i.e. 339.4V.
If the fundamental frequency is 50Hz then
angular velocity,ω= 2 πf= 2 π( 50 )= 100 πrad/s.
Hence the fundamental voltage is represented
by 339.4sin100πtvolts. Since the fundamen-
tal frequency is 50Hz, the time for one cycle
of the fundamental is given byT= 1 /f= 1 /50s
or 20ms.
The third harmonic has an amplitude equal to
20% of 339.4V, i.e. 67.9V. The frequency of
the third harmonic component is 3× 50 =150Hz,
thus the angular velocity is 2π(150), i.e.
300 πrad/s. Hence the third harmonic voltage
is represented by 67.9sin( 300 πt− 3 π/ 4 )volts.
Thus
voltage,v=339.4sin100πt
+67.9sin (300πt− 3 π/4)volts
(b) One cycle of the fundamental, 339.4sin100πt,
is shown sketched in Fig. 14.36, together with
three cycles of the third harmonic compon-
ent, 67.9sin( 300 πt− 3 π/ 4 )initially lagging by
3 π/4rad. By adding ordinates at intervals,
the complex waveform representing voltage is
produced as shown.
Now try the following exercise
Exercise 64 Further problems on harmonic
synthesis with complexwaveforms

1. A complex current waveform i comprises
   a fundamental current of 50Arms and fre-
   quency 100Hz, together with a 24% third
   harmonic, both being in phase with each other
   at zero time. (a) Write down an expression
   to represent currenti. (b) Sketch the complex
   waveform of current using harmonic synthesis
   over one cycle of the fundamental.
   [
   (a)i=( 70 .71sin628. 3 t
   
   
   • 16 .97sin1885t)A
   
   
   
   

]

2. A complex voltage waveformvis comprised
   of a 212.1Vrms fundamental voltage at a fre-
   quencyof50Hz,a30%secondharmoniccom-
   ponent lagging byπ/2rad, and a 10% fourth
   harmonic component leading by π/3rad.
   (a) Write down an expression to represent
   voltagev. (b) Sketch the complex voltage