Lectures 16 & 17
Sinusoidal Signals, Complex
Numbers, Phasors, Impedance
& AC Circuits
Nov. 7 & 9, 2011
Material from Textbook by Alexander & Sadiku and Electrical Engineering:
Principles & Applications, A. R. Hambley is used in lecture slides.
Sinusoids and Phasors
Chapter 9
9.1
9.2
9.3
9.4
9.5
9.6
9.7
Motivation
Sinusoids’ features
Phasors
Phasor relationships for circuit elements
Impedance and admittance
Kirchhoff’s laws in the frequency domain
Impedance combinations
3
How to determine
v(t) and i(t)?
vs(t) = 10V ->
AC
How can we apply what we have learned before to
determine i(t) and v(t)?
4
War of the Currents
•  Edison’s DC power
was first
–  Good for lights &
motors
–  Ease of generation
–  Not easy transmit long
distances
vs
•  Westinghouse &
Tesla (AC power)
–  Needed math skils
–  Needed generator
and other technology
–  Easy for long distance
transmission
See http://en.wikipedia.org/wiki/
War_of_Currents
Sinusoids
•  A sinusoid is a signal that has the form of the sine or
cosine function.
•  A general expression for the sinusoid,
where
Vm = the amplitude of the sinusoid
ω = the angular frequency in radians/s (ω = 2 π f)
Ф = the phase
6
Sinusoids
A periodic function is one that satisfies v(t)
= v(t + nT), for all t and for all integers n.
•  Only two sinusoidal values with the same
frequency can be compared by their amplitude
and phase difference.
•  If phase difference = zero, they are in phase; if7
phase difference ≠zero, they are out of phase.
Sinusoid Examples
Example
Given a sinusoid,
, calculate its
amplitude, phase, angular frequency, period, and
frequency.
Solution:
Amplitude = 5, phase = –60o, angular frequency
= 4π rad/s, Period = 0.5 s, frequency = 2 Hz.
8
Sinusoid Example
Example 2
Find the phase angle between
and
, does i1 lead or lag i2?
Solution:
Since sin(ωt+90o) = cos ωt
therefore, i1 leads i2 155o
9
EE 101 Schedule
Version 11-8-11
Class
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
Lecture
1
2
3
4
5
6
7
8
22
23
24
24
25
18
19
20
21
22
26
27
23
24
9
10
11
12
13
14
15
16
17
Date
9-23-11
9-26-11
9-28-11
9-30-11
10-3-11
10-5-11
10-7-11
10-10-11
10-12-11
10-14-11
10-17-11
10-19-11
10-21-11
10-24-11
10-26-11
10-28-11
10-31-11
11-2-11
11-4-11
11-7-11
11-9-11
11-11-11
11-14-11
11-16-11
11-18-11
11-21-11
11-23-11
11-25-11
11-28-11
11-30-11
Topic
Introduction
Fundamentals of Electrical Engineering
Circuit Laws, Voltage & Current Dividers
Node/Loop Analysis
Node/Loop Analysis
Thévenin Equivalent Circuits
Norton Equivalent Circuits
Amplifiers
Review for Midterm 1
Midterm 1
Op Amps
Op-Amp Circuits
Op-Amp Circuits
Inductance and Capacitance
First Order Transient Response
RC/RL Circuits, Time Dependent Op Amp Circuits
Second Order Transient Response
Review for Midterm 2
Midterm 2
Sinusoidal Signals, Complex Numbers, Phasors
Phasor Circuits
Veteran’s Day
AC Power, Thevenin
Fourier Analysis, Transfer Function, Decibels, Low Pass Filters
Bode Plot, Series & Parallel Resonance. High Pass Filters
2nd Order Filters, Active Filters, Resonances
Magnetic Circuits, Materials
Thanksgiving Day
Mutual Inductance & Transformers
AC Power Engineering
12-2-11 Review for Final at normal class time and place
Final Exam Thursday, December 8th (noon to 3 pm)
Reading Ahead
A & S Ch 1
A & S Ch 2
A& S Ch 3
Homework
Review Math
Quiz
Pre-Req
Hmwk 1 Due
1
A & S Ch 4
Hmwk 2 Due
A & S Ch 5
Hmwk 3 Due
2
A & S Ch 6
A & S Ch 7
Hmwk 4 Due
3
A & S Ch 8
Hmwk 5 Due
4
A & S Ch 9
Hmwk 6 Due
A & S Ch 10
A & S Ch 18 + Ch 17
A & S Ch 17
Hmwk 7 Due
A & S Ch 13
Hmwk 9 Due
Parts of A&S Chs 11&12
Hmwk 8 Due
Hmwk 10 Due
5
6
Resources to Learn Complex Numbers
& other Topics for Engineering
•
•
•
•
•
•
•
•
•
•
•
•
http://www.khanacademy.org/ The Kahn Academy has lessons and
worked examples on a multitude of math topics, including complex
numbers
Here are some complex number videos at the Kahn Academy
http://www.khanacademy.org/video/introduction-to-i-and-imaginarynumbers
http://www.khanacademy.org/video/calculating-i-raised-to-arbitraryexponents
http://www.khanacademy.org/video/i-as-the-principal-root-of--1--alittle-technical
http://www.khanacademy.org/video/complex-numbers--part-1
http://www.khanacademy.org/video/complex-numbers--part-2
There are also exercises for practice at the Kahn Academy site
Others sites for useful links for students
http://www.engineerguy.com/
http://www.youtube.com/education
http://www.apple.com/education/itunes-u
Phasors
•  A phasor is a complex number
that represents the amplitude
and phase of a sinusoid.
•  It can be represented in one of
the following three forms:
a.  Rectangular
b.  Polar
c.  Exponential
where
12
Phasor
Mathematic operation of complex number:
1.  Addition
2.  Subtraction
3.  Multiplication
4.  Division
5.  Reciprocal
6.  Square root
7.  Complex conjugate
8.  Euler’s identity
13
Phasor
Example
•  Evaluate the following complex numbers:
a.
b.
Solution:
a. –15.5 + j13.67
b. 8.3 + j2.2 (See Appendix A p.10)
14
Phasor
•  Transform a sinusoid to and from the time domain to
the phasor domain:
(time domain)
(phasor domain)
•  Amplitude and phase difference are two principal
concerns in the study of voltage and current sinusoids.
•  Phasor will be defined from the cosine function in all our
further study. If a voltage or current expression is in the
form of a sine, it will be changed to a cosine by
subtracting from the phase.
•  Frequency ω = 2 π f is implicit, i.e. understood from other
information.
15
Phasor
Example
Transform the following sinusoids to phasors:
i = 6cos(50t – 40o) A
v = – 4sin(30t + 50o) V
Solution:
a. I
A (frequency is understood to be ω = 50 r/s
b. Since –sin(A) = cos(A+90o);
v(t) = 4cos (30t+50o+90o) = 4cos(30t+140o) V
Transform to phasor => V
V
16
Phasor Example
Example:
Transform the phasors corresponding to sinusoids:
a.
b.  I = j(12 + j5) A
€
Solution:
a)  v(t) = 10cos(ωt + 210o) V (phases are typically > 0)
5
) = 13∠ 22.62°
12
b)  Since I = 12 + j5 = 12 2 + 5 2 ∠ tan −1 (
i(t) = 13cos(ωt + 22.62o) A
17
€
Phasor Notation
The differences between v(t) and V:
•
•
•
v(t) is instantaneous or time-domain representation
V is the frequency or phasor-domain representation.
v(t) is time dependent, V is not.
v(t) is always real with no complex term, V is
generally complex.
Note: Phasor analysis applies
; when
it is applied to two or more sinusoid signals only if they have
the same frequency.
18
Phasor
Relationship between differential, integral operation
in phasor listed as follow:
19
Phasor
Example
Use phasor approach, determine the current i(t) in a
circuit described by the integro-differential equation.
Answer:
I got i(t) = 2.24 cos(2t + 208o) A
See what you get for this.
20
Phasor
•  In-class exercise, we can derive the differential equations for the
following circuit in order to solve for vo(t) in phase domain Vo.
•
However, the derivation may sometimes be very tedious.
Is there any quicker and more systematic methods to do it?
21
Phasors & Circuits
The answer is YES!
Instead of first deriving the differential equation
and then transforming it into phasor to solve
for Vo, we can transform all the RLC
components into phasor first, then apply the
KCL laws and other theorems to set up a
phasor equation involving Vo directly.
22
Phasor Relationships
for Circuit Elements
Resistor:
Inductor:
Capacitor:
23
Phasor Relationships
for Circuit Elements
Summary of voltage-current relationship
Element
Time domain
Frequency domain
R
L
C
24
Phasor Relationships
for Circuit Elements
Example
If voltage v(t) = 6 cos(100t – 30o) is applied to a 50 µF
capacitor, calculate the current, i(t), through the
capacitor.
Answer:
i(t) = 0.03 cos(100t + 60o) mA
25
Impedance and Admittance
•  The impedance Z of a circuit is the ratio of the phasor
voltage V to the phasor current I, measured in ohms Ω.
where R = Re, Z is the resistance and X = Im, Z is the
reactance. Positive X is for L and negative X is for C.
•  The admittance Y is the reciprocal of impedance,
measured in siemens (S).
26
Impedance and Admittance
Impedances and admittances of passive elements
Element
Impedance
Admittance
R
L
C
27
Impedance and Admittance
28
Impedance and Admittance
After we know how to convert RLC components
from time to phasor domain, we can transform
a time domain circuit into a phasor/frequency
domain circuit.
Hence, we can apply the KCL laws and other
theorems to directly set up phasor equations
involving our target variable(s) for solving.
29
Impedance and Admittance
Example
Refer to Figure below, determine v(t) and i(t).
Answers: i(t) = 1.118cos(10t – 26.56o) A; v(t) = 2.236cos(10t + 63.43o) V
30
Amplitude & Phase Relationships
•  vs = 5 cos (10 t) = 5 cos (10 t + 360)
•  i(t) = 1.11 cos(10 t – 26.6°) = 1.11 cos (10 t + 333.4)
•  v(t) = 2.236 cost (10 t + 63.4°)
Kirchhoff’s Laws
in the Frequency Domain
•  Both KVL and KCL are hold in the phasor
domain or more commonly called frequency
domain.
•  Moreover, the variables to be handled are
phasors, which are complex numbers.
•  All the mathematical operations involved are
now in complex domain.
32
Impedance Combinations
•  The following principles used for DC circuit
analysis all apply to AC circuit.
•  For example:
a.  voltage division
b.  current division
c.  circuit reduction
d.  impedance equivalence
e.  Y-Δ transformation
33
Impedance Combinations
Example
Determine the input impedance of the circuit in figure below
at ω =10 rad/s.
Answer: Zin = 32.5 – j73.7
34