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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
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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
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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(6𝑡 − 37° )
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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
−𝑗37
Easy Way 1. Phasor 𝐼 =
= 1𝑒
1 =
4+𝑗6∗
2
4+𝑗3
2. 𝑖 𝑡 = cos(6𝑡 − 37° )
Remark
 Complex number help us simplify the computation. You do not need
to solve differential equations in AC circuits any more!
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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.
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Euler’s Identity
𝒆±𝒋𝒛 = 𝐬𝐢𝐧 𝒛 ± 𝒋 𝐜𝐨𝐬 𝒛
Resulting in
cos 𝒛 =
𝒆𝒋𝒛 +𝒆−𝒋𝒛
,
𝟐
sin 𝒛 =
𝒆𝒋𝒛 −𝒆−𝒋𝒛
𝟐𝒋
A sinusiod function can be equivalently transformed to a phasor:
𝑣 𝑡 = 𝑉𝑚 cos(𝜔𝑡 + 𝜙) ⟺ 𝐕 = 𝑉𝑚 ∠𝜙
Remark
 Frequency is separated from 𝑣 𝑡 by the transformation, which
remains unchanged in a LTI system once given from input.
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Algebraic Operations
Complex sinusiod related operations can be replaced by corresponding
simple phasor operations:
Complex operations in time domain
 A cos(𝜔𝑡 + 𝜑1 ) + B sin(𝜔𝑡 + 𝜑2 )
𝑑𝑣
 Differentiation:
𝑑𝑡
 Integration: 𝑣𝑑𝑡
Simple operations in phasor domain
⟺  A∠𝜑1 + B∠(𝜑2 − 90° )
⟺  Multiplication: 𝑗𝜔𝐕
𝐕
⟺  Division: 𝑗𝜔
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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.
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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.
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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
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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 → ∞.
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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.
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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
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Applications [2]
 Observe 𝑣𝑜 amplitude as the
frequency of the input 𝑣𝑖
changes. Notice it decreases
with frequency.
 Also observe 𝑣𝑜 shift as
frequency changes (phase).
 Need to study behavior of
networks for sinusoidal drive.
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Exercise 1
1. Determine the currents 𝑖1 (𝑡), 𝑖2 𝑡 and 𝑖3 (𝑡) using impedance
combination, nodal and mesh analysis.
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Exercise 1 - Impedance Combination
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Exercise 1 - Nodal Analysis
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Exercise 1 - Mesh Analysis
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Exercise 2
2. Determine the Thevenin and Notton equivalent circuit for the circuit
below, assuming that 𝜔 = 104 rad/s.
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Exercise 2 – Thevenin Equivalent
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Exercise 2 – Norton Equivalent
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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?
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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.
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Q&A
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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.
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