# Analysing harmonic content of ac voltage waveforms

An a.c. voltage, V, comprises of a fundamental voltage of 100 V rms at a frequency of 120 Hz, a 3rd harmonic which is 20% of the fundamental, a 5th harmonic which is 10% of the fundamental and at a phase angle of 1.2 radians lagging.

(i) Write down an expression for the voltage waveform.

(ii) Sketch the waveforms of the harmonic components.

(iii) Determine the voltage at 20 ms.

(iv) Given an ideal V = 100 V rms, what is the percentage error at 20 ms?

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#### Solution Preview

(i) The fundamental is 100 V rms thus the amplitude A0 is

A0 = 100*SQRT(2) = 141.4

Frequency of the fundamental is f0 = 120 Hz; assuming a sinusoid we can express the fundamental as (assuming zero initial phase)

F0 = A0*cos(2*PI*f0t) = 141.4*cos(240*PI*t)

Third harmonic has amplitude 20% of the fundamental so its amplitude is A3 = 0.2*A0

A3 = 0.2*141.1 = 28.3

As it is the third harmonic its frequency is f3 = 3*f0 = 3 x 120 = 360 Hz

Thus the third harmonic may be expressed as

F3 = A3*cos(2*PI*f3*t) = 28.3cos(720*PI*t)

Similarly the 5th harmonic has amplitude A5 which is 10% of the fundamental so we can say

A5 = 0.1*A0 = 0.1 x 141.4

A5 = ...

#### Solution Summary

Analysing an ac waveform interms of its components, a fundamnetal of frequency 120 Hz, amplitude 100V rms, a thrid harmonic 20% of the fundamental, a 5th harmonic 10% of the fundamental with lagging phase of 1.2 radians