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Chapter 1Complex Numbers and Phasors
Chapter Objectives:
Understand the concepts of sinusoids and phasors.
Apply phasors to circuit elements.
Introduce the concepts of impedance and admittance.
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Introduce the concepts of impedance and admittance.
Learn about impedance combinations.
Apply what is learnt to phase-shifters and AC bridges.
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)A second way of representing the complex number is by specifying the MAGNITUDE and r and the ANGLE θ in POLAR form.
z = x+ jy= z
POLAR FORM
=rθ θ∠ ∠
• x is the REAL part.
• y is the IMAGINARY part.
• r is the MAGNITUDE.
• φ is the ANGLE.
z = x+ jy= z =rθ θ∠ ∠ The third way of representing the complex number is the EXPONENTIAL form.
z = x+
EXPONENT
jy= z
IAL FORM
= jre θθ∠
Complex Numbers
A complex number may be written in RECTANGULAR FORM as: forms.
2 2 -1
z = x+ jy j= -1
cos y sin
z=
=tan
RECTANGULAR FORM
POLAR FORM
x r r
r
yr x y
x
θ θ
φ
θ
= =
∠
= +
j
2 2 -1
j
j
z= e EXPO
=tan
z = x + jy=
NE
= e
e =cos +j
NTIAL FORMr
yr x y
x
r r
φ
φ
φ
θ
φ
φ
= +
∠
j
j
sin
cos Re
Euler's Identity
Real part
Imaginary p
e
sin Im e art
φ
φ
φφ
φ
=
=
Complex Number Conversions
We need to convert COMPLEX numbers from one form to the other form.
z = (cos sin ) jx jy r re r jφφ φ φ= + = ∠ = +
2 2 1 Rectangula
z = (cos sin )
, ta r to Polar
Pola
n
cos , sin r to Rectangular
jx jy r re r j
yr x y
xx r y r
φφ φ φ
φ
φ φ
−
= + = ∠ = +
= + =
= =
Mathematical Operations of Complex Numbers
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
z + z =(x + x )+j(y + y )
z - z =(x -x )+j(y - y )
z z = r r +
ADDITION:
SUBTRACTION:
MULTIPLICATION:
DIVISION:
z r = -
φ φ
φ φ
∠
∠1 11 2
2 2
DIVISION:
RECIPROCAL:
SQUARE ROOT:
z r = -z r
1 1 = - z r
z=
COMPLEX CO
r 2
NJUGATE:z
φ φ
φ
φ
∗
∠
∠
∠
= jx jy r re φφ −− = ∠ − =
PhasorsA 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.
j
j
j
e =cos jsin
co
Euler's Identity
Real part
Imaginary pa
s Re e
s rtin Im e
φ
φ
φ
φ φφ
φ
± ±
=
=
(TimeDomain Repr.) (Phasor Domain Representation)
( ) Re (Converting Phasor back to time)
( ) cos( )m m
j t
v t V t V
v t e ω
ω φ φ= + ⇔ =
=
∠V
V
Given a sinusoidv(t)=Vmcos(ωt+φ).
Imaginary pas rtin Im eφ =
( )( ) cos( ) Re( ) Re( ) Re( )
PHA S .OR REP
j t j t j tm m
jm m
jmv t V t V e e
V
e e
V
V
e
ω φ ω ω
φ
φω φφ
+= + = = =
= = ∠ =
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
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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
°−∠= 406
°∠= 1404
Phasors
• Example 5:
• Transform the sinusoids corresponding to
phasors:
a)
b)
V 3010 °∠−=VA j12) j(5 −=I
Solution:
a) v(t) = 10cos(ωt + 210o) V
b) Since
i(t) = 13cos(ωt + 22.62o) A
°∠=∠+=+= − 22.62 13 )12
5( tan 512 j512 122I
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Phasor Diagrams
cos( )
sin(
Time
) 90
cos
Domain Representation Phasor Domain Re
( )
sin( ) 0
p.
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m m
m m
m m
m m
V t V
V t V
I t I
I t I
ω φ φω φ φω θ θω θ θ
+ ∠+ ∠ − °+ ∠+ ∠ − °
Time Domain Versus Phasor Domain
Differentiation and Integration in Phasor Domain Differentiating a sinusoid is equivalent to multiplying its corresponding phasor by jω.
( ) cos( ) Re
( )sin( ) cos( 90 )
= Re
j tm
m m
j t
v t V t e
dv t
d
V t V tdt
ev
j Jdt
ω
ω
ω θ
ω ω θ ω ω
ω ω
θ
= + =
= − + = − + + °
⇔ V V
V
Integrating a sinusoid is equivalent to dividing its corresponding phasor by jω.
(Time Domain) (Phasor Domain)
( ) cos( )
( ) sin( ) 90
V
m m
m m
v t V t V
v t V t V
dvJ
dt
vdtJ
ω φ φω φ φ
ω
ω
= + ⇔ = ∠= + ⇔ = ∠ − °
⇔
⇔∫
V
V
V
Adding Phasors Graphically
Adding sinusoids of the same frequency is equivalent to adding their corresponding phasors.
V=V1+V2
20cos(5 30 ) At − ° 1
5Ω 2 F
1H
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