Download Loop and Nodal Analysis and Op Amps

Circuits II
EE221
Instructor: Kevin D. Donohue
Course Introduction, Website
Resources, and Phasor Review
Course Policies Procedures
Introduce Instructor and Teaching Assistant
Review Syllabus
Expectations and Workload
Team Project
Relevance of Course
Course goal: Develop problem solving
skills useful for designing (electrical)
systems involving information/power.
 Circuits: A connection of components
with electrical properties typically
arranged to process information or
transfer power.
 Entropy and Enthalpy ?

Circuits in your head
Circuit elements used to describe neural membrane
15 
20 mH
Z
10 
http://www.mindcreators.com/NeuronModel.htm
Relevance of Course
Electromagnetics:
Antennas, Circuit Boards,
Remote Sensors, Optics
and Lasers
Electronics:
Power:
Motors, Generators,
Transmission lines,
Conversion
Signals and Systems:
Amplifiers, Filters,
Signal Processors,
Sensors, Digital,
Computer
Communications, Control, Signal
Processing, Computer
EE221, EE211
Circuits
Course Outcomes:
1.
2.
3.
4.
5.
6.
Perform AC steady-state power analysis on single-phase
circuits.
Perform AC steady-state power analysis on three-phase
circuits.
Analyze circuits containing mutual inductance and ideal
transformers.
Derive transfer functions (variable-frequency response)
from circuits containing independent sources, dependent
sources, resistors, capacitors, inductors, operational
amplifiers, transformers, and mutual inductance elements.
Derive two-port parameters from circuits containing
impedance elements.
Use SPICE to compute circuit voltages, currents, and
transfer functions.
Course Outcomes:
7.
8.
9.
Describe a solution with functional block diagrams (topdown design approach).
Work as a team to formulate and solve an engineering
problem.
Use computer programs (such as MATLAB and SPICE) for
optimizing design parameters and verify design
performance.
Web Sites of Interest
Matlab Resources
Manuel on Matlab Basics
http://www.mathworks.com/access/helpdesk/help/pdf_doc/matlab/getstart.pdf

Download PDF on “Getting Started” and read sections on Introduction
through Matrices and Arrays (Pages 1-1 to 2-19).
MATLAB Tutorials:
http://www.mathworks.com/academia/student_center/tutorials/index.html
 A graphic description to step through basic exercises in Matlab. Should
have Matlab open while going through this so you can try the examples.
Consider it homework this week to go through the interactive tutorial (about 2
hours). Nothing to hand in for it.
Octave (a Free Matlab Clone)
http://www.gnu.org/software/octave/
Web Sites of Interest
B2SPICE
Demos and Free Lite Version
http://www.beigebag.com/demos.htm
Students can download a free Lite Version on their own PCs. The
Lite version has some functional limits but saved files that can be
opened with university’s full version.
Within the B2SPICE program itself are simple tutorial (under the
help menu) to get student started with using the basic function of
the program.
Phasor Review
What is a complex number and why is
it used to solve electrical engineering
problems?
 What is a phasor? Who introduced it
to the profession? Why is it popular?

The Sinusoidal Function
Terms for describing sinusoids:
x(t )  X m sin(t   )  X m sin(2ft   )
Maximum Value,
Amplitude, or
Magnitude
Phase
Frequency
in cycles/second or
Hertz (Hz)
Radian Frequency
in Radian/second
1
1
0.8
0.4
0.4
0.2
0.2
0
2 

sin  2t 

5 

-0.2
-0.4
-0.6
0
-0.2
-0.4
-0.6
-0.8
-0.8
-1
-1
-1.2566
-0.2566
0.7434
1.7434
2.7434
Radians
3.7434
4.7434
.2
0.6
Amplitude
Amplitude
sin( 2t )
.4
0.6
0.8
-0.4
-0.2
0
0.2
0.4
0.6
0.8
Seconds
1
1.2
1.4
1.6
Trigonometric Identities


cos(t )  sin t  
2



 cos(t )  cos t   (or 180 )


sin(t )  cos t  
2



 sin(t )  sin t   (or 180  )

sin(   )  sin( ) cos( )  cos( ) sin( )
Radian to degree conversion
cos(   )  cos( ) cos( )  sin( ) sin( )
Degree to radian conversion
multiply by /180
multiply by 180/
X m sin  t     X m cos( ) sin(  t )  X m sin(  ) cos( t )

1   B  
A cos( t )  B sin(  t )  A  B cos t  tan 
 
 A 

2
2
Complex Numbers
Each point in the complex number plane can be
represented in a Cartesian or polar format.
a  jb  r exp( j )  r
IM

r  a2  b2
a
r
b
RE
b

1
  tan  
a
a  r cos( )
b  r sin( )
Complex Arithmetic
Addition:
(a  jb )  (c  jd )  (a  c)  j (b  d )
Multiplication and Division:
(r )(v )  rv(   )
r r
 (   )
v v
Simple conversions:
j  190  ,
1
  j,
j
- 1  180 
Euler’s Formula
Show: exp( j )  cos( )  j sin(  )
A series expansion ….
j  2 j  3  4 j  5
exp( j )  1 





1! 2!
3!
4!
5!
cos( )  1 
sin(  ) 

1!
2
2!


4 6
4!

6!

3 5 7
3!

5!

7!

Complex Forcing Function
Consider a sinusoidal forcing function given as a complex
function:
X m exp( j (t   ))  X m cos(t   )  jX m sin(t   )



Based on the concept of orthogonality, it can be shown that for
a linear system, the real part of the forcing function only
affects the real part of the response and the imaginary part
of the forcing function only affect the imaginary part of the
response.
For a linear circuit excited by a sinusoidal function, the steadystate response everywhere has the same frequency. Only the
magnitude and phase of the response can change.
A useful factorization:
X m exp( j (t   ))  X m exp( j ) exp( jt )
Mechanical Analogy
Electrical: Energy transfers between electric field
(capacitor) and magnetic field (inductor)
Mechanical: Energy transfers between gravitational
field and elasticity of spring.
http://www.youtube.com/watch?v=T7fRGXc9SBI
Note: Every part of the spring moves at the same
frequency, only the phase and magnitude of the
oscillation changes. The same is true for a linear
RLC circuit.
Phasors
Sinusoidal function notation for linear circuits can be
more efficient if the exp(-jt) is dropped, leaving
the magnitude and phase quantities maintained via
phasor notation:
Time Domain
x(t )  A cos(t   )
x(t )  A sin(t   )
Examples …
Frequency Domain
X̂  A

ˆ
X  A   90


Phasors Examples
Find the equivalent impedances Z for the circuits
below at a frequency of 60 Hz:
15 
20 mH
Z
10 
Show Z = 8.1+j*5.5 = 9.8334.33
.1 mF