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Discussion 7
CLASS X
NOV. XX
Overview
 Phasors in Electrical Engineering (EE)
 Phasors ⇌ Sinusoid
 Circuit Models in Frequency Domain
 Generalized Circuit Analysis
1
Phasors in Electrical Engineering (EE)
Why phasors are useful in EE? It is a long story…
 Any practical periodic function can be represented as sum of
sinusoids (FT)
 For LTI system, sinusoids input results in sinusoids outputs
 Phasors can perfectly represent sinusoids mathematically and
what’s more, they can simplify the computation
 With the help of phasors, DC circuit analysis methods can be
generalized to AC circuits
2
Motivation for Using Phasors
A voltage source 5cos(6t) is connected in series with a 4Ω resistor and a
0.5H inductor. Compute the current i(t) which flows through all three
devices. [1]
di
0.5
dt
Hard Way 1. KVL: 5 cos 6t = 4i +
(1)
2. the trial solution is i(t) = A cos(6t) + B sin(6t)
(2)
3. Substituting (2) to (1) gives
[5−4A−3B]cos(6t)+[−4B+3A]sin(6t)=0
4. Set t = 0 and t = π, solving 5 = 4A + 3B and 0 = 3A − 4B.
5. A = 0.8 and B = 0.6, i(t) = 0.8 cos(6t) + 0.6 sin(6t)
6. i(t) = cos(6 − 37° )
3
Motivation for Using Phasors
A voltage source 5cos(6t) is connected in series with a 4Ω resistor and a
0.5H inductor. Compute the current i(t) which flows through all three
devices.
°
5
5
−37
Easy Way 1. Phasor  =
= 1
1 =
4+6∗
2
4+3
2.   = cos(6 − 37° )
Remark
 Complex number help us simplify the computation. You do not need
to solve differential equations in AC circuits any more!
4
Phasors ⇌ Sinusoid*
Phasor can perfectly represent sinusiod in the following aspects :
 A sinusiod function can be transformed to a phasor, and they
are totally equivalent mathematically
 Sinusiod related operations can be replaced by corresponding
phasor operations
*: “⇌” means “equivalent to” here; the left/right side can represent each other.
5
Euler’s Identity
± =   ±   
Resulting in
cos  =
 +−
,

sin  =
 −−

A sinusiod function can be equivalently transformed to a phasor:
  =  cos( + ) ⟺  =  ∠
Remark
 Frequency is separated from   by the transformation, which
remains unchanged in a LTI system once given from input.
6
Algebraic Operations
Complex sinusiod related operations can be replaced by corresponding
simple phasor operations:
Complex operations in time domain
 A cos( + 1 ) + B sin( + 2 )

 Differentiation:

 Integration: 
Simple operations in phasor domain
⟺  A∠1 + B∠(2 − 90° )
⟺  Multiplication: 

⟺  Division: 
7
Impedance Model in Phasor Domain
 The current through the capacitor
will increase with higher frequency,
even if the voltage amplitude stays
constant.
 It takes a higher voltage to push the
same current amplitude through an
inductor at higher frequency.
 Just the same as in time domain.
8
Impedance Model in Phasor Domain
Remarks
 The impedance of any circuit element is two numbers, a
magnitude and a phase, at every frequency value.
 Both the magnitude and the phase may be frequency dependent.
 When we describe the impedance, we need to specify at what
frequency we are describing the impedance.
9
AC Circuits Analysis
Steps to analyze AC circuits:
1. Transform the circuit to the phasor
2. Solve the problem using:
 KCL/KVL in frequency domain
 Series/Parallel Impedance Combinations/∆-Y Trans.
 Nodal/Mesh Analysis
 Superposition Theorem
 Source Transformation
 Thevenin/Norton Equivalent Circuits
3. Transform the results in frequency domain to the time domain
10
Transient vs. Steady-State Analysis
Sinusoid steady-state analysis: solve steady-state (t → ∞) response
which contains purely sinusoidal, without the transient exponential
terms which decays to 0 as t → ∞.
11
Applications
Did you ever buy audio equipment, like microphone? If you looked at
the specifications for audio equipment you would probably find the
following:
 If the unit is a speaker set, you'll find separate frequency
responses for the different speakers like the mid-range, or the
woofer and the tweeter.
 If the unit is a microphone, you'll find a frequency response that
tells you how the unit responds to different frequencies.
12
Applications
Given a linear system or circuit described mathematically,
 Be able to compute a frequency response for the system
 Be able to predict an output signal from a given input sinusoidal
signal
13
Applications [2]
 Observe  amplitude as the
frequency of the input 
changes. Notice it decreases
with frequency.
 Also observe  shift as
frequency changes (phase).
 Need to study behavior of
networks for sinusoidal drive.
14
Exercise 1
1. Determine the currents 1 (), 2  and 3 () using impedance
combination, nodal and mesh analysis.
15
Exercise 1 - Impedance Combination
16
Exercise 1 - Nodal Analysis
17
Exercise 1 - Mesh Analysis
18
Exercise 2
2. Determine the Thevenin and Notton equivalent circuit for the circuit
below, assuming that  = 104 rad/s.
19
Exercise 2 – Thevenin Equivalent
20
Exercise 2 – Norton Equivalent
21
Some After-Class Questions for you
 When capacitors or inductors used in AC circuits, is the current and voltage
peak at the same time? Why or why not?
 In a linear system, does the frequency of a sinusoid convey information?
 Can phasor analysis be performed on multiple frequencies circuits?
22
Some After-Class Questions for you
 Phasor analysis gives the transient response or steady-state response?
 What's the physical meaning of the imaginary component of impedance?
 Find an intuitive instance, and explain to a high school student why do we
need frequency domain analysis.
23
Q&A
24
REFERENCES
[1] COMPLEX NUMBERS AND PHASORS, Professor Andrew E. Yagle, EECS 206
Instructor, Fall 2005, Dept. of EECS, The University of Michigan, Ann Arbor, MI
48109-2122
[2] Anant Agarwal and Jeffrey Lang, course materials for 6.002 Circuits and
Electronics, Spring 2007. MIT OpenCourseWare, Massachusetts Institute of
Technology.
25