Chapter 9 Complex Numbers and Phasors

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Chapter 9Complex Numbers and Phasors

• 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.

Huseyin BilgekulEENG224 Circuit Theory
IIDepartment of Electrical and Electronic
EngineeringEastern Mediterranean University
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Complex Numbers

• A complex number may be written in RECTANGULAR
  FORM as

• A second way of representing the complex number
  is by specifying the MAGNITUDE and r and the
  ANGLE ? in POLAR form.

• The 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.
• f is the ANGLE.

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Complex Numbers

• A complex number may be written in RECTANGULAR
  FORM as forms.

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Complex Number Conversions

• We need to convert COMPLEX numbers from one form
  to the other form.

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Mathematical Operations of Complex Numbers

• Mathematical operations on complex numbers may
  require conversions from one form to other form.

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Phasors

• 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 Eulers
  identity.
• Given a sinusoid v(t)Vmcos(?tf).

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Phasors

• Given the sinusoids i(t)Imcos(?tfI) and
  v(t)Vmcos(?t fV) we can obtain the phasor forms
  as

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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(A90o) v(t) 4cos
(30t50o90o) 4cos(30t140o) V Transform
to phasor gt V V
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Phasors

• Example 5
• Transform the sinusoids corresponding to
  phasors
• a)
• b)

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Phasor as Rotating Vectors
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Phasor Diagrams

• The SINOR
• Rotates on a circle of radius Vm at an
  angular velocity of ? in the counterclockwise
  direction

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Phasor Diagrams
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Time Domain Versus Phasor Domain
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Differentiation and Integration in Phasor Domain

• Differentiating a sinusoid is equivalent to
  multiplying its corresponding phasor by j?.

• Integrating a sinusoid is equivalent to
  dividing its corresponding phasor by j?.

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Adding Phasors Graphically

• Adding sinusoids of the same frequency is
  equivalent to adding their corresponding phasors.
• VV1V2

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Solving AC Circuits

• We can derive the differential equations for the
  following circuit in order to solve for vo(t) in
  phase domain Vo.

• 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.