Lectures 18
Sinusoidal (AC) Steady State
Circuit Analysis
Nov. 14, 2011
Material from Textbook by Alexander & Sadiku and Electrical Engineering:
Principles & Applications, A. R. Hambley is used in lecture slides.
EE 101 Schedule
Version 11-14-11
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. 14 + bit of 18
Hmwk 7 Due
Hmwk 8 Due
A & S Ch 13
Parts of A&S Chs 11&12
5
Hmwk 9 Due
Hmwk 10 Due
6
Sinusoidal Steady-State Analysis
Chapter 10 in Alexander & Sadiku
10.1
10.2
10.3
10.4
10.5
10.6
Basic Approach
Nodal Analysis
Mesh Analysis
Superposition Theorem
Source Transformation
Thevenin and Norton Equivalent Circuits
3
Steps to Analyze AC Circuits:
1.
2.
3.
Transform the circuit to the phasor or frequency
domain.
Solve the problem using circuit techniques (nodal
analysis, mesh analysis, superposition, etc.).
Transform the resulting phasor to the time domain.
Time to Freq.
Solve for
variables in
Freq.
Freq. to Time
4
Kirchhoff’s Laws in Phasor Form
We can apply KVL directly to phasors.
The sum of the phasor voltages equals
zero for any closed path.
KVL
The sum of the phasor currents entering a
node must equal the sum of the phasor
currents leaving.
KCL
Circuit Analysis Using
Phasors and Impedances
1. Replace the time descriptions of the
voltage and current sources with the
corresponding phasors. (All of the
sources must have the same frequency.)
2. Replace inductances by their complex
impedances ZL = jωL. Replace
capacitances by their complex
impedances ZC = 1/(jωC). Resistances
have impedances equal to their
resistances.
3. Analyze the circuit using any of the
techniques studied earlier in Chapters 1 - 4,
performing the calculations with complex
arithmetic.
Phasor Relationships
for Circuit Elements
Summary of voltage-current relationship
Element
Time domain
Frequency domain
R
L
C
8
Example of Series RLC
Time t
Domain
Find i(t):
Frequency, ω = 500 r/s is understood
Frequency ω
or Phasor
Domain
Phasors in RLC Series Example
Find the phasor
voltage across each
element:
Nodal Analysis
Example
Using nodal analysis, find V1 and V2 in the circuit below.
After Transformation to
Phasor Domain
Answer:
V1(t) = 38.7 sin(2t + 69.7°) V
V2(t) = 6.75 sin(2t + 165.7°) V
11
Mesh Analysis
Example
Find Io in the following figure using
mesh analysis.
6
After Transformation to
Phasor Domain
30
Answer: Io = 3.58 at an angle of 65.4° A
12
Superposition Theorem
When a circuit has sources operating at different
frequencies,
•  The separate phasor circuit for each
frequency must be solved independently,
and
•  The total response is the sum of timedomain responses of all the individual
phasor circuits.
13
Superposition Theorem
Example
Calculate vo in the circuit of figure shown below using
the superposition theorem.
Vo = 4.631 sin(5t – 81.12°) + 1.051 cos(10t – 86.24°) V
14
Source Transformation
15
Source Transformation
Example
Find Io in the circuit of figure below using the
concept of source transformation.
Io = 3.288 at an angle of 99.46° A
16
Thevenin and Norton Equivalent Circuits
Thevenin transform
Norton transform
17
Step-by-step Thévenin/
Norton-Equivalent-Circuit
Analysis
1. Perform two of these:
a. Determine the open-circuit voltage Vt = voc.
b. Determine the short-circuit current In = isc.
c. Zero the sources and find the Thévenin
impedance Zt looking back into the
terminals.
2. Use the equation Vt = Zt In to compute
the remaining value.
3. The Thévenin equivalent consists of a
voltage source Vt in series with Zt .
4. The Norton equivalent consists of a
current source In in parallel with Zt .
Thevenin Equivalent Example
Example
Find the Thevenin equivalent at terminals a–b of
the circuit below.
Zth =12.4 – j3.2 Ω
VTH = 18.97 at an angle of -51.57° V
20