Electric Circuits
By
Prof. Dr. Mousa Abd-Allah
Dr. Ayman Yousef
Dr. Amr Ameen
Contents
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CH1: Sinusoids, Phasors and resonance
CH2: Sinusoidal steady-State Analysis
CH3: AC Power Analysis
CH4: Three Phase Circuits
CH5: Transient Response of Circuits
CH6: Fourier Analysis and Circuit Applications
CH7: Two-Port Circuits
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References
q
q
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Mahmood Nahvi, Joseph A. Edminister, “ Electric Circuits” ,
Schaum’s Outlines, Mc Graw Hill. ISBN: 9780071633727,
2011.
James W. Nilsson, Susan A. Riedel, “ Electric Circuits” ,
Prentice Hall, New Jersey, ISBN:9780137050512-0137050518,
2011.
Jhon Bird, “ Electrical Theory and Technology” , Elsevier,
Oxford, ISBN:9780080549798-0080549799, 2007.
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Chapter
I
Sinusoids , Phasors
and Resonance
By
Dr. Ayman Yousef
Sinusoids
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Alternating (AC) Sinusoidal
Waveforms
Period of a Waveform
l
Period (T): The time interval
between successive repetitions of a
periodic waveform (time required for
completing one full cycle).
l
One period occupies exactly 360º of a
sine waveform. Period
frequency
l
Cycle: The portion of a waveform
contained
in one period of time.
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(Sec)
6
Alternating (AC) Sinusoidal
Waveforms
Frequency of a Waveform
l
Frequency (f): The number of cycles that occur in 1 sec
(number of cycles that is completed each second.)
(Hertz)
l
This figure shows four cycles per second, or a waveform
that has a frequency of 4 Hz.
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Sinusoids
ØA SINUSOID is a signal that has the form of the sine or cosine function.
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Instantaneous Value of a Wave
Instantaneous value: The magnitude of a waveform at any instant of
time; denoted by the lowercase letters (v1, v2).
Peak amplitude: The maximum value of the waveform as measured
from reference horizontal line (the greater value of Instantaneous value),
denoted by the uppercase letters Vm.
where φ v is the phase angle
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Phase of Sinusoids
l
OA represents a vector
that is free to rotate
anticlockwise about 0
at an angular velocity
of ω rad/s. A rotating
vector is known as a
phasor.
l
Any quantity which
varies sinusoidally can
thus be represented as
a phasor.
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Rotating of single phasor
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Rotating of Two phasors
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Phase of Sinusoids
Ø Only two sinusoidal values with the same frequency can be
compared by their amplitude and phase difference.
Ø If phase difference is zero, they are in phase; if phase difference is
not zero, they are out of phase.
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Phase of Sinusoids
Ø 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.
ØThe cosine curve is said to lead the sine curve by 90° .
ØThe sine curve is said to lag the cosine curve by 90° .
Ø90 is referred to as the phase angle between the two waveforms.
ØWhen 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.
ØSince the full period represents a cycle of 360° .
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Phase of Sinusoids
ØConsider the sinusoidal voltage having phase φ ,
• v2 LEADS v1 by phase φ .
• v1 LAGS v2 by phase φ .
• v1 and v2 are out of phase.
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Average Value of a Wave
Suppose a time-varying function f(t) is defined on the interval a ≤ t ≤ b.
The area A, under the graph of f(t) is given by the integral
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Average Value of Sine Wave
T
1
Vav = ∫ v(t )dt
T0
The average value of only half a cycle of the Sine wave is given by:
The average value of the full cycle of
the wave will be zero because the sine
wave symmetrical about zero.
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Root-mean-square value
of sin wave
l
The R.M.S. (or effective) value of an alternating voltage or current (AC) is
the value which would produce the same amount of heat energy in a resistor
as a direct voltage or current (DC) of the same magnitude.
R.M.S.
(3)
Root
(2)
Mean
(1)
square
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Ex.1: Find the amplitude, phase, period, and frequency
of the sinusoid: v(t)=12cos(50t+10o)
Solution
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Ex.2 : Calculate the phase angle between
v1 = -10 cos(ωt+50o) & v2 = 12 sin(ωt-10o)
state which sinusoid is leading
Solution
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Complex Numbers
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Complex Numbers
ØA complex number may be written in RECTANGULAR FORM as:
ØA second way of representing the complex number is by specifying the
MAGNITUDE (r) and the ANGLE (θ ) in POLAR form.
ØThe third way of representing the complex number is the EXPONENTIAL form.
• x is the REAL part.
• y is the IMAGINARY part.
• r is the MAGNITUDE.
• φ is the ANGLE.
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Complex Numbers
ØA complex number may be written in RECTANGULAR FORM as:
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Complex Number Conversions
ØMathematical operations on complex numbers may require conversions
from one form to other form.
ØConverting Rectangular to Polar:
ØConverting Polar to Rectangular :
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Mathematical Operations of Complex
Numbers
ØMathematical operations for conversions complex numbers from one
form to other form.
ØAddition and Subtraction must be on the Rectangular form.
ØMultiplication and Division must be on the Polar form.
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Adding Phasors Graphically
ØAdding sinusoids of the same frequency is equivalent to adding their
corresponding phasors.
V=V1+V2
V1 = V1∠ − 45o v
V2 = V2 ∠30o v
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Phasors
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Phasors
Ø The phasor is a complex number that carries the amplitude
and phase angle information of a sinusoidal function.
Ø Phasor is the mathematical equivalent of a sinusoid with time
variable dropped.
Ø Phasor will be defined from the cosine function in all our
proceeding study.
Ø 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.
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Phasors
ØPhasor representation is based on Euler’s formula.
ØEuler's formula indicates that sinusoids can be represented
mathematically by the sum of two complex-valued functions
Given a sinusoid v(t)=Vmcos(ω t+φ )
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Phasors
Ø Given the sinusoids i(t)=Imcos(ω t+φ I) and v(t)=Vmcos(ω t+ φ
V)
we can obtain the phasor forms as:
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Phase Difference of a Sinusoidal Waveform
l
The current, i is lagging the voltage, v by angle Φ and in our example
above this is 30o.
l
So the difference between the two phasors representing the two sinusoidal
quantities is angle Φ and the resulting phasor diagram will be.
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Ex. 3 : Transform these sinusoids to phasors:
(a) i = 6 cos(50t-40o)
(b) v = - 4 sin(30t+50o)
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Ex.4: Given i1(t)=4cos(ωt+30o) A, and
i2(t)=5sin(ωt-20o) A , find their sum.
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Phasor Relationships
for
Circuit Elements
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Phasor Relationships for Resistor
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Phasor Relationships for Resistor
Time Domain
Frequency Domain
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Phasor Relationships for Inductor
Where the Inductive reactance is:
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Phasor Relationships for Inductor
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Phasor Relationships for Capacitor
Where the capacitive reactance is:
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Phasor Relationships for Capacitor
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Phasor Relationships for Circuit
Elements
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Phasor Relationships for Circuit
Elements
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