Complex Waveforms and Transients in R-L-C Circuits

Running head: COMPLEX WAVEFORMS 1
Complex Waveforms and Transients in R-L-C Circuits
Name
Institution

COMPLEX WAVEFORMS 2
Complex Waveforms and Transients in R-L-C Circuits
Question 1
Part a (i)
An even function, f, is a function that is symmetric about the y-axis. That is if the graph of the
function is reflected along the y-axis then it remains the same. We say that f is even, if, for all x
and –x in the domain f ( x ) =f (−x ). For instance, f ( x )=x2 is an even function as shown in figure
1.
Figure 1: A graph of f ( x )=x2 Figure 2: A graph of f ( x )=x3
On the other hand, an odd function is the function that satisfies the property f (−x)=f ( x). For
instance,f ( x )=x3 as shown in figure 2. In relation to Fourier series, we often come across
integration of sine and cosine functions. The odd or even property help in simplification of
integration.
Part a (ii)

COMPLEX WAVEFORMS 3
Half wave symmetry is a property of a signal that implies that, when it is shifted by half of its
period then the resulting signal resembles the negative of the initial signal. That is,
f ( t−π ) =−f (t) as illustrated in figure 3.
Figure 3: Half wave symmetry
Part b
By inspection , a∧c are neither even nor odd. However, they are wave symmetric. Therefore, the
even coefficients of cosine and sine terms would have zero amplitude. Similarly, figure
b , d , e ,∧f are even symmetric resulting to only cosine terms in their Fourier expressions.
Question 2
Part a
V rms=230 v , α = π
3 , Dissipated power=100 w ,∧frequency =50 Hz
Peak voltage of the AC source, V s =V rms × √ 2=230× √ 2=325.2691V
V rms across theload= √ V s
2 π
2
{(π −α )−1/2(sin 2 π −sin 2α )}
V rms=V s √ 1
2 π {(π− π
3 )−1/2 ¿ ¿

COMPLEX WAVEFORMS 4
V rms=325.2691 √ 1
2 π { 2 π
3 −1/2 ¿ ¿
V rms=325.2691 √ 1
2 π { 2 π
3 −1/2 ¿ ¿
V rms across theload =325.2691× 0.63423=206.296 V
Resistance of bulb¿ V rms
power rating
2
= 230
100
2
=529Ω
Power dissipated¿ V rmsload
2
R =206.296
529
2
=80.45 w
Part b
Any distortions leading to asymmetrical waveforms will increase the harmonic content of the
waveform.
Part c
A sketch of the waveform is shown in figure 4
Figure 4: A sketch of the waveform