Sinusoids and Phasors

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Chapter 9

• Sinusoids and Phasors

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• Thus far our analysis only concentrates on dc
  circuits.
• Now we begin the analysis in which the source is
  time varying.

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Sinusoids

• Sinusoid is a signal that has the form of the
  sine and cosine function.
• Vm the amplitude of the sinusoid
• ? the angular frequency of the sinusoid
• ?t the argument of the sinusoid

Sinusoidal voltage
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Sinusoids

• Period of the sinusoid (T)
• T 2p/? ? Measured in seconds.
• Cyclic frequency (f)
• f 1/T ? Measured in Hertz.
• ?/2p
• Thus ? 2pf

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Sinusoids

• General expression for the sinusoid

Argument of the sinusoid
phase
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Sinusoids

• i.e. two sinusoids
• v2 starts first in time.
• v2 leads v1 by Ø
• Ø ? 0
• v2 and v1 are out of phase

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Sinusoids

• Sinusoid can be expressed as sine or cosine form.
• When comparing sinusoids, better to express both
  in sine or both in cosine form.
• This can be achieved by Trigonometric identities.

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Sinusoids

• From these identities, it can be shown that

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Sinusoids

• Graphical approach _at_ alternative approach to
  trigonometric identities.

Angle -ve clockwise ve counterclockwise
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Sinusoids

• Graphical technique can be used to add two
  sinusoids of the same frequency when one in sine
  and the other in cosine forms
• Where

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Sinusoids

• i.e. add the two sinusoids.
• Thus

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Sinusoids

• Example 1
• Given the sinusoid, calculate its amplitude,
  phase,?, period and frequency.

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Sinusoids

• Example 2
• Calculate the phase angle between v1 and v2.
  State which sinusoid is leading.
• Note When comparing sinusoid, express them in
  the same form.

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Phasors

• Sinusoids are easily expressed in terms of
  phasors.
• A phasor is a complex number that represents the
  amplitude and phase of the sinusoid.
• Phasors are written in bold face.
• Before completely define and apply phasors to
  circuit analysis, we have to be familiar with
  complex numbers.

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Phasors

• Complex number can be written in rectangular
  form
• z x jy
• Where
• x real part of z
• y imaginary part of z
• Also can be written as

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Phasors

• Thus z can be represented in three ways

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Phasors

• Given x and y
• Or given r and Ø
• xrcos Ø and yrsin Ø
• Operations on complex s refer page 376 and 377
  in text book

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Phasors

• Given
• Thus, phasor representation of the sinusoid is

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Phasors

• For example, given
• and
• Thus the phasor diagram is

• Make sure to convert sine to cosine form so that
  the sinusoid can be written as the real part of
  the complex number.
• Sinusoid phasor transformation? refer table 9.1
  page 379.

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Phasors

• Differentiation and integration

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Phasors

• Example 1
• Evaluate the following complex numbers

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Phasors

• Example 2
• Express these sinusoids as phasors

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Phasors

• Example 3
• Find the sinusoids corresponding to these phasors

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Phasors

• Example 4
• If v1 -10sin(?t30o) and v2 20cos(? t-45o),
  find vv1v2

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Phasors

• Example 5
• Find the voltage v(t) in a circuit described by
  the integrodifferential equation using phasor
  approach.

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Phasor Relationship for Circuit Elements -
Resistor

• Resistor
• Given current through resistor is
• and VIR (Ohms Law) in phasor form
• But phasor representation of the current is
• Hence VRI

Phasor domain
I
Ohms Law in phasor form
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Phasor Relationship for Circuit Elements -
Resistor
Time domain
Phasor domain
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Phasor Relationship for Circuit Elements -
Inductor

• Inductor
• Given current through an inductor
• In time domain v L(di/dt)
• but
• Thus
• Where I

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Phasor Relationship for Circuit Elements -
Inductor
The current and voltage are 90o out of phase
(voltage leads current by 90o)
Time domain
Phasor domain
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Phasor Relationship for Circuit Elements -
Capacitor

• Capacitor
• Given voltage through an inductor
• In time domain i C(dv/dt)
• but
• Thus Where V
• And

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Phasor Relationship for Circuit Elements -
Capacitor
The current and voltage are 90o out of phase
(current leads voltage by 90o)
Time domain
Phasor domain
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Phasor Relationship for Circuit Elements
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Phasor Relationship for Circuit Elements

• Example 1
• If voltage v6cos(100t-30) is applied to a 50µF
  capacitor, calculate the current through the
  capacitor.

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Impedance and Admittance

• Previously
• VRI Vj?LI VI/(j?C)
• Or V/I R, V/Ij?L, V/I1/(j?C)
• Ohms Law in phasor form
• ZV/I R, j?L, 1/(j?C)
• Z frequency dependent quantity.
• Known as impedance.
• Measured in Ohms (O)

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Impedance and Admittance

• Two cases to consider
• ? 0
• ZL0 ? short circuit
• ZC? ? open circuit
• ? ? ?
• ZL ? ? open circuit
• ZC0 ? short circuit

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Impedance

• Impedance can also be expressed in rectangular
  form as
• ZRjX
• Where
• R Real Z (resistance)
• X Imaginary Z (reactance)
• all measured in ohms
• In polar form
• Where and
• Or and

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Admittance

• Reciprocal of impedance and are measured in
  Siemens (S).
• Y 1/Z I/V
• In rectangular form
• Y GjB
• Where
• G Re Y ? conductance (S)
• B Im Y ? susceptance (s)
• and
• GjB 1/(RjX)

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Impedance and Admittance

• Example 1
• Determine v(t) and i(t) for the following circuit.

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Kirchoffs Laws in the Frequency Domain

• KCL and KVL both holds true in frequency (phasor)
  domain.
• Series Configurations
• Impedance
• ZeqZ1Z2
• Admittance
• 1/Yeq1/Y11/Y2
• Voltage Division Rule
• V1Z1V/(Z1Z2)
• V2Z2V/(Z1Z2)

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Kirchoffs Laws in the Frequency Domain

• Parallel Configurations
• Impedance
• 1/Zeq1/Z11/Z2
• Admittance
• YeqY1Y2
• Current Division Rule
• I1Z2I/(Z1Z2)
• I2Z1I/(Z1Z2)

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Kirchoffs Laws in the Frequency Domain

• DeltaY Transformation
• ? to y conversion
• Z1 ZbZc/(ZaZbZc)
• Z2 ZaZc/(ZaZbZc)
• Z3 ZaZb/(ZaZbZc)
• Y to ? conversions
• Za (Z1Z2 Z2 Z3 Z1Z3)/Z1
• Zb (Z1Z2 Z2 Z3 Z1Z3)/Z2
• Zc (Z1Z2 Z2 Z3 Z1Z3)/Z3

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Kirchoffs Laws in the Frequency Domain

• Example 1
• Determine the input impedance of the circuit at
  ?10rad/s.

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Kirchoffs Laws in the Frequency Domain

• Example 2
• Determine v0

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Kirchoffs Laws in the Frequency Domain

• Example 3
• Find I.