ch 9 sinusoids and phasors l.
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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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1 introduction
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
2. Sinusoids
  • Consider the sinusoidal voltage
    • T: period of the sinusoid

Electric Circuit, 2007

sinusoids 2
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
Sinusoids (3)
  • Let us examine the two sinusoids
  • Trigonometric identities

Electric Circuit, 2007

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

where

Electric Circuit, 2007

example 9 1
Example 9.1
  • Find the amplitude, phase, period, and frequency of the sinusoid

Electric Circuit, 2007

example 9 2
Example 9.2
  • Sol)

Electric Circuit, 2007

3 phasors
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

Electric Circuit, 2007

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

Electric Circuit, 2007

phasors 3
Phasors (3)
  • Euler’s identity

with

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

Electric Circuit, 2007

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

Electric Circuit, 2007

phasors 5
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
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
Example 9.3
  • Evaluate these complex numbers
  • Sol)
  • a)
  • then
  • Taking the square root

Electric Circuit, 2007

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

Electric Circuit, 2007

example17
Example
  • Example 9.6
  • Example 9.7
  • Using the phasor approach, determine the current i(t)

Electric Circuit, 2007

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

Electric Circuit, 2007

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

Electric Circuit, 2007

example 5 6
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

5 impedance and admittance
5. Impedance and Admittance
  • 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 ,

Electric Circuit, 2007

impedance and admittance 2
Impedance and Admittance (2)
  • Impedance = Resistance + j Reactance
  • where
  • Adimttance Y is the reciprocal of impedance, measured in siemens (S)
  • Admittance = Conductance + j Susceptance

Electric Circuit, 2007

example 9 9
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

Electric Circuit, 2007

6 kirchhoff s law in the frequency domain
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

Electric Circuit, 2007

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

Electric Circuit, 2007

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

Electric Circuit, 2007

example 9 10
Example 9.10
  • Find the input impedance of the circuit with w=50 rad/s
  • Sol)
  • The input impedance is

Electric Circuit, 2007

example 9 11
Example 9.11
  • Determine vo(t) in the circuit
  • Sol)
  • Time domain  frequency domain
  • Voltage-division principle

Electric Circuit, 2007