Electrical and Electronic Principles

COMPLEX WAVEFORMS & TRANSIENTS IN R-L-C CIRCUITS
[Course title]ELECTRICAL AND ELECTRONIC PRINCIPLES
8/27/19
ELECTRICAL AND ELECTRONIC PRINCIPLES

ELECTRICAL AND ELECTRONIC PRINCIPLES
Contents
Q1.a)................................................................................................................................................................... 4
i) Even function........................................................................................................................................... 4
Odd function................................................................................................................................................... 5
i) half-wave symmetry................................................................................................................................. 6
b)........................................................................................................................................................................ 8
Question 2.......................................................................................................................................................... 9
a).................................................................................................................................................................... 9
b).................................................................................................................................................................. 10
c)................................................................................................................................................................... 11
Q3. Preamble................................................................................................................................................... 13
i) Obtain the Fourier transform..................................................................................................................13
i).................................................................................................................................................................... 14
ii)................................................................................................................................................................... 14
iv).................................................................................................................................................................. 16
Q4.................................................................................................................................................................... 17
a) = 1×103 rad s-1...................................................................................................................17
b) = 2×103 rad s-1.................................................................................................................. 17
c) = 1×103 rad s-1................................................................................................................ 17
Q5.................................................................................................................................................................... 18
a)...................................................................................................................................................................... 18
a) 10 + 3t2 + sin4t......................................................................................................................................... 19
b) 4e-3t sin2t................................................................................................................................................. 19
b)...................................................................................................................................................................... 19
a).................................................................................................................................................................. 19
b).................................................................................................................................................................. 20
Q6.................................................................................................................................................................... 21
a).................................................................................................................................................................. 21
b).................................................................................................................................................................. 22
c)................................................................................................................................................................... 23
d).................................................................................................................................................................. 24

ELECTRICAL AND ELECTRONIC PRINCIPLES

ELECTRICAL AND ELECTRONIC PRINCIPLES
Q1.a)
i) Even function
The even function is the function which is symmetric about y axis that is when we take the
mirror image about the y axis, the positive and negative x axis function looks same.
The even function is written as f(x) = f (-x)
The following figure shows f(x) =x2, it is even function.

ELECTRICAL AND ELECTRONIC PRINCIPLES
Odd function
It Satisfies the property
F(x) =-f (-x), so it is symmetric about x axis.
Below is graph of odd function, f(x) =x3

ELECTRICAL AND ELECTRONIC PRINCIPLES
In consist of Fourier series, the integral property in the Fourier series for the even and odd
function gets simplifies or even cancelled out.
Some properties of odd and even functions are
1. The sum of two even functions is even, and of two odd ones odd.
2. The product of two even or two odd functions is even.
3. The product of an even and an odd function is odd.
For even function Fourier series will have only cos term, as the cos is even function.

ELECTRICAL AND ELECTRONIC PRINCIPLES
For odd function Fourier series will have only sin term.
Odd function Fourier series
Even function Fourier series
i) half-wave symmetry
f (t) =- f(t-M) for some M
In this function, the wave is shifted by one half of the period and then inverted as shown in
above graph.

ELECTRICAL AND ELECTRONIC PRINCIPLES
For half wave symmetry function
b)

ELECTRICAL AND ELECTRONIC PRINCIPLES
(a)and(c) is half wave symmetric function, so the terms given above will be used in Fourier
series.

ELECTRICAL AND ELECTRONIC PRINCIPLES
And zero amplitude for even dc term.
(b), (d), (e), (f) are even symmetric, so only cosine term is there in Fourier series.
Question 2
a)
α= π/3
V=230 V
F=50Hz
PBub=100 W
Power dissipated in bulb = Vor2/Rbulb

ELECTRICAL AND ELECTRONIC PRINCIPLES
Pbulb= Vs2/RBulb
RBulb = 2302/100 = 529 Ω
Vrms= (Vm/ √2 π) √[( π- α)+0.5sin(2 α)]
Vrms= (√(2)x230 / √(2 π)) √[( π- π/3 )+0.5sin(2 π/3)]
Vrms =206.115 V
Therefore, P0 = Vrms2/R = 206.115/529 = 80.3 Watts
b)
Angle on one half cycle so causing an asymmetrical waveform, as illustrated opposite. State the
effect, if any, this would have on the harmonic content of the waveform.
In this scenarioduring –ve half triac is not conducting properly conducting, the driving circuit
may have some problem, or it can have problem in inductive effect in load and source etc. the
conductivity of the switching device is not symmetric and possess some delay.
Some disadvantages are as follows
1. As the positive and negative half cycle is not symmetric there is some DC components in
supply current, Output current and output voltage.The average value of output voltage
must be zero, but it is not in this case. The DC component harmonics are there.
2. The control over the output voltage is not fulfilled.

ELECTRICAL AND ELECTRONIC PRINCIPLES
c)
f(wt)=0 for 0<wt< =π/2
=vsinwt for π/2< = wt< =π
= 0 for π< = wt< =3π/2
=vsinwt for 3π/2< = wt< =2π
Fourier coefficient

ELECTRICAL AND ELECTRONIC PRINCIPLES
Bn=0, as sin waveform is an even function
Therefore Fourier transform of the given waveform is

ELECTRICAL AND ELECTRONIC PRINCIPLES
Q3.
i)

ELECTRICAL AND ELECTRONIC PRINCIPLES
i)
Principle frequency can be obtained by seeing the frequency domain graph, the highest peak at
the lowest frequency.
So, the principal frequency is 53 Hz
ii)
For the data provided
Where , I1= rms value of the fundamental current
In= rms value of the highest harmonics given by
The total harmonics distortion may occur by many ways
This is given by
THD=
Irms= 52.7343 Hz
The harmonics can be calculated as follows
1st harmonic

ELECTRICAL AND ELECTRONIC PRINCIPLES
2nd Harmonic
Irms2=246.094
3rd Harmonic
Irms3=8π/2 = 246.094
Than Is2 can be written as ,
Is2=(6.253/2)A
= 351.563 Hz
The 5th, and 4th harmonics can be found in similar way
Then, Ipeak = 6.253 A
Substituting the values into THD equation
iv)
Attempt to synthesize the shape of the original waveform from its principal harmonics

ELECTRICAL AND ELECTRONIC PRINCIPLES
The amplitude of the waveform decreases as the frequency increases.
We can predict the original waveform using the above principle
For example
Q4.
a)
= 1×103 rad s-1
In this case of 2% tolerance
The settling time is given by
Ts=4xtime constant = 4
So, settling time Ts=8 msec.

ELECTRICAL AND ELECTRONIC PRINCIPLES
b)
= 2×103 rad s-1
Settling time, ts = 4/(x2xmsec
c)
= 1×103 rad s-1
Settling time, ts = 4/(2xmsec
IN this case as , so system is overdamped .there is no oscillation
The calculation above is used for the sketch as shown below

ELECTRICAL AND ELECTRONIC PRINCIPLES
Q5
a)
a)
10 + 3t2 + sin4t
Following the table given in this assignment , the laplace transform of each term in the equation
is shown below
10 = 10/s
3t2 = 6/s3
Sin4t = 4/(s2+16)
So, the laplace transform of the equation is
10/s +6/s3 +4/ (s2+16)

ELECTRICAL AND ELECTRONIC PRINCIPLES
b)
4e-3t sin2t
Sin (2t) = 2/ (s2+4)
And e-3t sin2t= 2/ ((s+3)2+4)
So, the laplace transform of the equation is given by
8/ ((s+3)2+4)
b)
a)
So, the inverse Laplace transform

ELECTRICAL AND ELECTRONIC PRINCIPLES
b)
Using table
Q6.

ELECTRICAL AND ELECTRONIC PRINCIPLES
a)
a) Laplace transform circuit is shown below

ELECTRICAL AND ELECTRONIC PRINCIPLES
b)
Current through RL
The transfer function is given by

ELECTRICAL AND ELECTRONIC PRINCIPLES
c)
When Ip = unit step
Ip(s)=1/s
Inverse Laplace transform of V(s) is

ELECTRICAL AND ELECTRONIC PRINCIPLES
d)
Estimate the time taken for the voltage vLto reach its steady state value if the current ipis subject
to a step change of 2 nA.
Voltage at time infinity.
V (t) at t=
lim(s-> 0)s.V(s) [Ip(s)=2n/s]
put s=0 in the following equation.
Where,

ELECTRICAL AND ELECTRONIC PRINCIPLES
= 2.35x1016
1/RLC = 4.24x10-17 sec
Now,
0.01/RL=[1-e-4.24x10-17t]
(0.01/5M)=2x10-9
Therefore,
t=4.7x104sec.