Lecture 21 Sinusoids and Phasors

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Lecture 21Sinusoids and Phasors

• ECE 205
• Prof. Ali Keyhani

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Phasors

• Phasor is a complex number representing the
  amplitude and phase angle of a sinusoid
• Eulers relationship
• Eulers relationship applied to general sinusoid

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Phasors

• Phasor representation of sinusoid v(t)

Phasor Diagram
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Phasors

• Complex exponential (rotating phasor)

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Phasor Properties

• Additive Property
• Note This result applies only if sinusoids
  have the same frequency

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Phasor Properties

• Derivative Property
• Note Differentiating a sinusoid changes the
  amplitude by a factor ? and shifts the phase
  angle by 90.

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Example 1

• Construct the phasors representing the following
  signals
• By using the additive property find the sum of
  the waveforms

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Example 1

• Solution

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Phasor Diagram
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Complex Numbers
A complex number is a quantity in the form of
Where a and b are real numbers, and
a real part b imaginary part
is called the conjugate of z
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Complex Numbers
A complex number can also be written in phasor
form
where
Magnitude (or norm)
- Angle (or phase)
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Complex Numbers
i
Conversion between two forms
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Complex Numbers Operation
Addition / Subtraction
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Complex Numbers Operation
Multiplication
A complex number times its conjugate ? the square
of its magnitude
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Complex Numbers Operation
Division
Addition and subtraction can be easily done in
regular form. While multiplication and division
are a little bit complicated.
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Complex Numbers Operation
Multiplication
(13)
Division
(14)
Multiplication and division are much easier to be
done in phasor form.