Higher Engineering Mathematics

TRIGONOMETRIC WAVEFORMS 165

B

Now try the following exercise.

Exercise 72 Further problems on harmonic

synthesis with complex waveforms

1. A complex current waveform icomprises
   a fundamental current of 50 A rms and fre-
   quency 100 Hz, 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 com-
   plex waveform of current using harmonic
   synthesis over one cycle of the fundamental.
   [
   (a)i=(70.71 sin 628. 3 t
   
   
   • 16 .97 sin 1885t)A
   
   
   
   

]

2. A complex voltage waveform v is com-
   prised of a 212.1 V rms fundamental voltage
   at a frequency of 50 Hz, a 30% second har-
   monic component lagging byπ/2 rad, and
   a 10% fourth harmonic component leading
   byπ/3 rad. (a) Write down an expression to
   represent voltagev. (b) Sketch the complex
   voltage waveform using harmonic synthesis
   over one cycle of the fundamental waveform.
   [
   (a)v=300 sin 314. 2 t
   +90 sin (628. 3 t−π/2)
   +30sin(1256. 6 t+π/3) V

]

3. A voltage waveform is represented by:
   v= 20 +50 sinωt
   +20sin(2ωt−π/2) volts.

Draw the complex waveform over one

cycle of the fundamental by using harmonic

synthesis.

4. Write down an expression representing a
   currentihaving a fundamental component
   of amplitude 16 A and frequency 1 kHz,
   together with its third and fifth harmonics
   being respectively one-fifth and one-tenth the
   amplitude of the fundamental, all compo-
   nents being in phase at zero time. Sketch the
   complex current waveform for one cycle of
   the fundamental using harmonic synthesis.
   [
   i=16 sin 2π 103 t+ 3 .2 sin 6π 103 t
   
   
   • 1 .6 sinπ 104 tA
   
   
   
   

]

5. A voltage waveform is described by

v=200 sin 377t+80 sin

(

1131 t+

π

4

)

+20 sin

(

1885 t−

π

3

)

volts

Determine (a) the fundamental and harmonic

frequencies of the waveform (b) the percent-

age third harmonic and (c) the percentage

fifth harmonic. Sketch the voltage waveform

using harmonic synthesis over one cycle of

the fundamental.

⎡

⎣

(a) 60 Hz, 180 Hz, 300 Hz

(b) 40%

(c)10%

⎤

⎦