Lecture no. 1 on Chapter 9 of Alexander Sadiku_Spring 2024).ppt

Chapter 9 – Sinusoidal Alternating
Waveforms and the concept of Phasors
Fundamentals of Electric Circuits
Alexander and Sadiku

Chapter Objectives:
 Understand the concepts of sinusoids and phasors.
 Apply phasors to circuit elements.
 Introduce the concepts of impedance and admittance.
 Learn about impedance combinations.
 Apply what is learnt to phase-shifters and AC bridges.

Introduction
 Alternating waveforms
The term alternating indicates only that the waveform alternates between two
prescribed levels (positive and negative ) in a set time sequence (called the
time period).
Alternating Circuits- Circuits driven by alternating voltage or, current
sources are called ac circuits.

Sinusoidal ac Voltage
Characteristics and Definitions
 Generation
An ac generator (or alternator) powered by water power, gas, or
nuclear fusion is the primary component in the energy-conversion
process.
 The energy source turns a rotor (constructed of alternating
magnetic poles) inside a set of windings housed in the stator (the
stationary part of the dynamo) and will induce voltage across the
windings of the stator.
dt
d
N
e



Sinusoidal ac Voltage Characteristics and
Definitions
 Generation
 Wind power and solar power energy are receiving increased
interest from various districts of the world.
The turning propellers of the wind-power station are connected directly to
the shaft of an ac generator.
Light energy in the form of photons can be absorbed by solar cells.
Solar cells produce dc, which can be electronically converted to ac
with an inverter.
 A function generator, as used in the lab, can generate and control
alternating waveforms.

 Definitions
 Waveform: The path traced by a quantity, such as voltage, plotted
as a function of some variable such as time, position, degree,
radius, temperature and so on.
 Instantaneous value: The magnitude of a waveform at any
instant of time; denoted by the lowercase letters (e1, e2 ).
Peak amplitude: The maximum value of the waveform as
measured from its average (or mean) value, denoted by the
uppercase letters Em (source of voltage) and Vm (voltage drop
across a load).

 Definitions
Peak value: The maximum instantaneous value of a function
as measured from zero-volt level.
 Peak-to-peak value: Denoted by Ep-p or Vp-p, the full voltage
between positive and negative peaks of the waveform, that is,
the sum of the magnitude of the positive and negative peaks.
Periodic waveform: A waveform that continually repeats itself
after the same time interval.

 Definitions
Period (T): The time interval between successive repetitions of a
periodic waveform (the period T1 = T2 = T3), as long as successive
similar points of the periodic waveform are used in determining T
 Cycle: The portion of a waveform contained in one period of time
Frequency: (Hertz) the number of cycles that occur in 1 s
Hz)
(hertz,
1
T
f 

 Defined polarities and direction
 The polarity and current direction will be for an instant in time in the
positive portion of the sinusoidal waveform.
 In the figure, a lowercase letter is employed for polarity and current
direction to indicate that the quantity is time dependent; that is, its
magnitude will change with time.

The Sine Wave
Why Sinusoids are used in AC Circuit Analysis?
 Almost all natural phenomena have sinusoidal characteristics e.g. the motion of a
pendulum, the vibration of a string, the ripples on the ocean texture, and the natural
response to underdamped second-order systems.
 Signals in the sinusoidal form are simple to generate and transmit. Electrical
generators rotate, so they produce voltage and current that is sinusoidal. It is the form
of voltage generated throughout the world and supplied to homes, factories,
laboratories, and so on. It is the dominant form of signal in the communications and
electric power industries.
 Using Fourier analysis, any practical periodic signal can be represented as a sum of
sinusoids.
 Mathematically, a sinusoid is easy to handle- The derivative and integral of a sinusoid
are also sinusoids.

The Sine Wave
 A sinusoidal forcing function produces both transient response and steady-state
response, much like the step function.
 The transient response dies out with time so that only the steady-state response
remains. When the transient response has become negligibly small compared with
the steady-state response, we say that the circuit is operating at sinusoidal steady-
state.
 The sinusoidal waveform is the only alternating waveform whose shape is
unaffected by the response characteristics of R, L, and C elements.
 The voltage across (or current through) a resistor, coil, or capacitor is sinusoidal in
nature.
 The unit of measurement for the horizontal axis is the degree. A second unit of
measurement frequently used is the radian (rad)- defined as the quadrant of a
circle where the distances subtended on the circumference equals the radius
of the circle.



 57.3
57.296
rad
1

The Sine Wave
 If we define x as the number of intervals of r (the radius) around
the circumference of a circle, then
C = 2r = x • r and we find x = 2
Therefore, there are 2 rad around a 360° circle, as shown in
the figure.

The Sine Wave
 The quantity  is the ratio of the circumference of a circle to
its diameter.
 For 180° and 360°, the two units of measurement are
related as follows:

The Sine Wave
 The sinusoidal
wave form can be
derived from the
length of the vertical
projection of a radius
vector rotating in a
uniform circular
motion about a fixed
point.

The Sine Wave
The velocity with which the radius vector rotates about the
center, called the angular velocity, can be determined from
the following equation:

The Sine Wave
The angular velocity () is:
Since () is typically provided in radians per second, the angle 
obtained using  = t is usually in radians.
The time required to complete one revolution is equal to the
period (T) of the sinusoidal waveform. The radians
subtended in this time interval are 2.
or
t

 
T


2
 f

 2


General Format for the Sinusoidal
Voltage or Current
 The basic mathematical
format for the sinusoidal
waveform is:
where:
Am is the peak value of the
waveform
 is the unit of measure for the
horizontal axis

General Format for the Sinusoidal Voltage
or Current
 The equation  = t states that the angle  through which the rotating
vector will pass is determined by the angular velocity of the rotating
vector and the length of time the vector rotates.
 For a particular angular velocity (fixed ), the longer the radius vector
is permitted to rotate (that is, the greater the value of t ), the greater will
be the number of degrees or radians through which the vector will pass.
 The general format of a sine wave can also be as:

General Format for the Sinusoidal
Voltage or Current
For electrical quantities such as current and voltage, the general
format is:
i = Imsint = Imsin
e = Emsint = Emsin
where: the capital letters with the subscript m represent the amplitude, and the
lower case letters i and e represent the instantaneous value of current and
voltage, respectively, at any time t.

Phase / Phase Relations
 Phase of an ac quantity at any instant is the angle φ (in
radian or, in degree) travelled by the phasor
representing that ac quantity upto the instant of
consideration measured from reference.
 Other way, phase of a particular value of an alternating
quantity is the fractional part of time period or cycle
through which the quantity has advanced from the
selected zero position of reference.
 For a Sine wave- The maximum positive value (+ Vm)
occurs at T/4 second or π/2 radians. We say that phase
of maximum positive value is T/4 second or π/2 radians.
 For a Cosine wave- The maximum positive value (+
Vm) occurs at 0 second or zero radians. We say that
phase of maximum positive value is 0 second or zero
radians.

Phase of Sinusoids
 A periodic function is one that satisfies v(t) = v(t + nT), for all
t and for all integers n.
f
1
T
2 f
Hz ;
 Only two sinusoidal values with the same frequency can be
compared by their amplitude and phase difference.
 If phase difference is zero, they are in phase; if phase difference is
not zero, they are out of phase.

Phase of Sinusoids
 The terms lead and lag are used to indicate the relationship between two
sinusoidal waveforms of the same frequency plotted on the same set of axes.
 The cosine curve is said to lead the sine curve by 900
.
 The sine curve is said to lag the cosine curve by 900
.
 900
is referred to as the phase difference between the two waveforms.
When determining the phase measurement we first note that each sinusoidal
function has the same frequency, permitting the use of either waveform to
determine the period.
 Since the full period represents a cycle of 360°, the following ratio can be
formed:

Phase of Sinusoids
 Consider the sinusoidal voltage having phase φ,
v(t) Vm sin( t )
• v2 LEADS v1 by phase φ.
• v1 LAGS v2 by phase φ.
• v1 and v2 are out of phase.

Complex Numbers
 A complex number may be written in RECTANGULAR FORM as:
 
RECTANGULAR FORM
z = x+ jy j= -1, x=Re z , y=Im(z)
• x is the REAL part.
• y is the IMAGINARY part.
• r is the MAGNITUDE.
• φ is the ANGLE.
 A second way of representing the complex number is by specifying the
MAGNITUDE, r and the ANGLE θ in POLAR form.
z = x+ jy= z
POLAR FORM
=r
 
 
 The third way of representing the complex number is the EXPONENTIAL
form.
z = x+
EXPONENT
jy= z
IAL FORM
= j
re 



 A complex number may be written in RECTANGULAR FORM as:
Complex Numbers

 We need to convert COMPLEX numbers from one form to the other form.
2 2 1
Rectangular to Polar
Polar to Rectangu
z = (cos sin )
, tan
cos , sin lar
j
x jy r re r j
y
r x y
x
x r y r

  

 


     
  
 
Complex Numbers’ Conversion

 Mathematical operations on complex numbers may require conversions from
one form to other form.
1 2 1 2 1 2
1 2 1 2 1 2
1 2 1 2 1 2
1 1
1 2
2 2
z + z =(x + x )+j(y + y )
z - z =(x -x )+j(y - y )
z z = r r +
ADDITION:
SUBTRACTION:
MULTIPLICATION:
DIVISION:
RECIPROCAL:
SQUARE ROOT:
z r
= -
z r
1 1
= -
z r
z=
COMPLEX CO
r
2
NJUGATE: z
 
 







 r j
x jy re 
 
   
Mathematical Operations of Complex
Numbers

Lecture no. 1 on Chapter 9 of Alexander Sadiku_Spring 2024).ppt

Phasors
(TimeDomain Repr.) (Phasor Domain Representation)
( ) Re{ } (Converting Phasor back to time)
( ) cos( )
m m
j t
v t V t V
v t e 
  
   


V
V
 A phasor is a complex number that represents the amplitude and phase of a
sinusoid.
 Phasor is the mathematical equivalent of a sinusoid with time variable dropped.
 Phasor representation is based on Euler’s identity.
 Given a sinusoid v(t)=Vmcos(ωt+φ).
 
 
j
j
j
e =cos jsin
co
Euler's Identity
Real part
Imaginary pa
s Re e
s rt
in Im e



 






( )
( ) cos( ) Re( ) Re( ) Re( )
PHAS .
OR REP
j t j t j t
m m
j
m m
j
m
v t V t V e e
V
e e
V
V
e
   


 


    
   
V
V

Phasors
 Given the sinusoids i(t)=Imcos(ωt+φI) and v(t)=Vmcos(ωt+ φV) we can obtain the
phasor forms as:

Phasors
 Amplitude and phase difference are two principal
concerns in the study of voltage and current sinusoids.
 Phasor will be defined from the cosine function in all
our proceeding study. If a voltage or current
expression is in the form of a sine, it will be changed to
a cosine by subtracting from the phase.
 Example
 Transform the following sinusoids to phasors:
 i = 6cos(50t – 40o
) A
 v = –4sin(30t + 50o) V
Solution:
a. I A
b. Since –sin(A) = cos(A+90o
);
v(t) = 4cos (30t+50o
+90o
) = 4cos(30t+140o
) V
Transform to phasor => V V



 40
6


 140
4

Phasors
Solution:
a) v(t) = 10cos(t + 210o
) V
b) Since
i(t) = 13cos(t + 22.62o
) A







 
22.62
13
)
12
5
(
tan
5
12
j5
12 1
2
2
I
 Example 5:
 Transform the sinusoids corresponding to phasors:
a)
b)
V
30
10 



V
A
j12)
j(5 

I

Phasors(Contd.)
 The addition of sinusoidal voltages and currents will frequently be
required in the analysis of ac circuits.
One lengthy but valid method of performing this operation is to place
both sinusoidal waveforms on the same set of axes and add
algebraically the magnitudes of each at every point along the abscissa.
 Long and tedious process with limited accuracy.
A shorter method uses the rotating radius vector.
The radius vector, having a constant magnitude (length) with one end
fixed at the origin, is called a phasor when applied to electric circuits.

Phasors as Rotating Vector
 Consider an alternating current represented by the equation i = Im sinωt. Take a line OP to
represent to scale the maximum value Im. Imagine the line OP (or **phasor, as it is called) to be
rotating in anticlockwise direction at an angular velocity ω rad/sec about the point O.
 Measuring the time from the instant when OP is horizontal, let OP rotate through an angle θ (=
ωt) in the anticlockwise direction. The projection of OP on the Y-axis is OM.
 OM = OP sin θ = Im sin ωt = i, the value of current at that instant
 Hence the projection of the phasor OP on the Y-axis at any instant gives the value of current at
that instant.
 Thus when θ = 90º, the projection on Y-axis is OP (= Im) itself. That the value of current at this
instant (i.e. at θ or ωt = 90º) is Im can be readily established if we put θ = 90º in the current
equation.

Phasors as Rotating Vector
 If we plot the projections of the phasor on the Y-axis versus its angular position point-by-point,
a sinusoidal alternating current wave is generated as shown in Fig. 11.47. Thus the phasor
represents the sine wave for every instant of time.
 The following points are worth noting :
 (i) The length of the phasor represents the maximum value and the angle with axis of
reference (i.e., X-axis) indicates the phase of the alternating quantity i.e. current in this case.
 (ii) The phasor representation enables us to quickly obtain the numerical values and, at the
same time, have a picture before the eye of the events taking place in the circuit. Thus in the
position of the phasor OP shown in Fig. 11.47, the instantaneous value is OM, the phase is θ
and frequency is ω/2π.
 (iii) A phasor diagram permits addition and subtraction of alternating voltages or currents with
a fair degree of ease.

Phasors
 Phasors algebra for sinusoidal quantities is applicable only for
waveforms having the same frequency.

Phasor
Diagrams
 The SINOR
Rotates on a circle of radius Vm at an angular velocity of ω in the counterclockwise
direction
j t
e 
V

Phasor Diagrams
cos( )
sin(
Time
) 90
cos
Domain Representation Phasor Domain Re
( )
sin( ) 0
p.
9
m m
m m
m m
m m
V t V
V t V
I t I
I t I
  
  
  
  
 
   
 
   

Time Domain Versus Phasor Domain

Differentiation and Integration in Phasor
Domain
(TimeDomain) (Phasor Domain)
( ) cos( )
( ) sin( ) 90
V
m m
m m
v t V t V
v t V t V
dv
J
dt
vdt
J
  
  


    
      



V
V
V
 Differentiating a sinusoid is equivalent to multiplying its corresponding phasor
by jω.
 Integrating a sinusoid is equivalent to dividing its corresponding phasor by jω.

Adding Phasors
Graphically
 Adding sinusoids of the same frequency is equivalent to
adding their corresponding phasors.
V=V1+V2

1
5

2 F
1
H
10
Practice Problem 9.7

We can derive the differential equations for the following circuit in
order to solve for vo(t) in phase domain Vo.
2
0
0
2
5 400
20 sin(4 15 )
3 3
o
o
d v dv
v t
dt dt
   
However, the derivation may sometimes be very
tedious.
Is there any quicker and more systematic methods to do it?
 Instead of first deriving the differential equation and then
transforming it into phasor to solve for Vo, we can transform all the
RLC components into phasor first, then apply the KCL laws and
other theorems to set up a phasor equation involving Vo directly.
Solving AC Circuits

Applications
(120 V at 60 Hz) versus (220 V at 50 Hz)
 In North and South America the most common available ac supply is
120 V at 60 Hz, while in Europe and the Eastern countries it is 220 V
at 50 Hz.
 Technically there is no noticeable difference between 50 and 60
cycles per second (Hz).
The effect of frequency on the size of transformers and the role it plays
in the generation and distribution of power was also a factor.
The fundamental equation for transformer design is that the size of the
transformer is inversely proportional to frequency.
 A 50 HZ transformer must be larger than a 60 Hz (17% larger)

Applications
120 V versus 220 V
Higher frequencies result in concerns about arcing, increased losses
in the transformer core due to eddy current and hysteresis losses, and
skin effect phenomena.
Larger voltages (such as 220 V) raise safety issues beyond those of
120 V.
 Higher voltages result in lower current for the same demand, permitting the
use of smaller conductors.
 Motors and power supplies, found in common home appliances and
throughout the industrial community, can be smaller in size if supplied with a
higher voltage.

Applications
Dangers of high-frequency supplies
 Remember a microwave cooks meat with a frequency of 2.45 GHz at 120
V.
 Standing within 10 feet of a commercial broadcast band AM transmitter
working at 540 kHz would bring disaster.
 Bulb savers
 When the light bulb is first turned on, the filament must absorb the
heavy current caused by the light switch being turned on.
 To save the filament from this surge an inductor is placed in series with
the bulb to choke out the spike down the line.

Lecture no. 1 on Chapter 9 of Alexander Sadiku_Spring 2024).ppt

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