Ch.9 Sinusoids and Phasors

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Ch.9 Sinusoids and Phasors. 1. Introduction. AC is more efficient and economical to transmit over long distance Sinusoid is a signal that has the form of the sine or cosine function Sinusoidal current = alternating current (ac) Nature is sinusoidal Easy to generate and transmit

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Ch.9 Sinusoids and Phasors

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Ch.9 Sinusoids and Phasors

1. Introduction
• AC is more efficient and economical to transmit over long distance
• Sinusoid is a signal that has the form of the sine or cosine function
• Sinusoidal current = alternating current (ac)
• Nature is sinusoidal
• Easy to generate and transmit
• Any practical periodic signal can be represented by a sum of sinusoids
• Easy to handle mathematically

Electric Circuit, 2007

2. Sinusoids
• Consider the sinusoidal voltage
• T: period of the sinusoid

Electric Circuit, 2007

Sinusoids (2)
• Periodic function
• Satisfies f(t) = f(t+nT), for all t and for all integers n
• Hence
• Cyclic frequency f of the sinusoid

Electric Circuit, 2007

Sinusoids (3)
• Let us examine the two sinusoids
• Trigonometric identities

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Sinusoids (4)
• Graphical approach
• Used to add two sinusoids of the same frequency

where

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Example 9.1
• Find the amplitude, phase, period, and frequency of the sinusoid

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Example 9.2
• Sol)

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3. Phasors
• Phasor is a complex number that represents the amplitude and phase of a sinusoid
• Provides a simple means of analyzing linear circuits excited by sinusoidal sources
• Complex number

with

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Phasors (2)
• Operations of complex number
• Subtraction:
• Multiplication:
• Division:
• Reciprocal:
• Square Root:
• Complex Conjugate:

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Phasors (3)
• Euler’s identity

with

• Given a sinusoid
• Thus, where
• Plot of the

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Phasors (4)
• Phasor representation of the sinusoid v(t)

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Phasors (5)
• Derivative & integral of v(t)
• Derivative of v(t)
• Phasor domain representation of derivative v(t)
• Phasor domain rep. of Integral of v(t)

Electric Circuit, 2007

Phasors (6)
• Summing sinusoids of the same frequency
• Differences between v(t) and V
• v(t) is time domain representation, while V is phasor domain rep.
• v(t) is time dependent, while V is not
• v(t) is always real with no complex term, while V is generally complex
• Phasor analysis
• Applies only when frequency is constant
• Applies in manipulating two or more sinusoidal signals only if they are of the same frequency

Electric Circuit, 2007

Example 9.3
• Evaluate these complex numbers
• Sol)
• a)
• then
• Taking the square root

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Example
• Example 9.4
• Transform these sinusoids to phasors
• Example 9.5
• Find the sinusoids represented by these phasors

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Example
• Example 9.6
• Example 9.7
• Using the phasor approach, determine the current i(t)

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4. Phasor Relationships for Circuit Elements
• Voltage-current relationship
• Resistor: ohm’s law
• Phasor form
• Inductor
• Phasor form

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Phasor Relationships for Circuit Elements(2)
• Inductor
• The current lags the voltage by 90o.
• Capacitor:
• Phasor form
• The current leads the voltage by 90o.

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Example 5.6
• The voltage v=12cos(60t+45o) is applied to a 0.1H inductor. Find the steady-state current through the inductor
• Sol)
• Converting this to the time domain,

Electric Circuit, 2007

• Voltage-current relations for three passive elements
• Ohm’s law in phasor form
• Imdedance Z of a circuit is the ratio of the phasor voltage to the phasor current I, measured in ohms
• When ,
• When ,

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• Impedance = Resistance + j Reactance
• where
• Adimttance Y is the reciprocal of impedance, measured in siemens (S)
• Admittance = Conductance + j Susceptance

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Example 9.9
• Find v(t) and i(t) in the circuit
• Sol)
• From the voltage source
• The impedance
• Hence the current
• The voltage across the capacitor

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6. Kirchhoff’s law in the frequency domain
• For KVL,
• Then,
• KVL holds for phasors
• KCL holds for phasors
• Time domain
• Phasor domain
• KVL & KCL holds in frequency domain

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7. Impedance Combinations
• Consider the N series-connected impedances
• Voltage-division relationship

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Impedance Combinations (2)
• Consider the N parallel-connected impedances
• Current-division relationship

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Example 9.10
• Find the input impedance of the circuit with w=50 rad/s
• Sol)
• The input impedance is

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Example 9.11
• Determine vo(t) in the circuit
• Sol)
• Time domain  frequency domain
• Voltage-division principle

Electric Circuit, 2007