Basic Circuit Analysis II

Spring 2014

Lecture Summary

**Lecture 1:** (1/15) Course Outline and introduction. Sinusoids:
properties and motivation for using in circuit analysis. Operations on
sinusoids.

**Lecture 2:** (1/17) Real sinusoids. Review complex
numbers. Complex sinusoids
and phasors. Steady state sinusoidal analysis (SSSA), impedence.
Procedure for finding output voltage/ current.

**Lecture 3:** (1/22)
SSSA for circuits, phasors, impedence, review KVL (mesh analysis), KCL
(node analysis). SSSA circuit examples (using matlab to solve equations.

**Lecture 4:** (1/24)
Sinusoids and
(matlab plots),. Review steps for Steady
State Sinusoidal Analysis (SSSA). Frequency response and computation using
circuit analysis. Transfer function and examples.

**Lecture 5:** (1/27)
Frequency response and computation using circuit analysis. Transfer function
and examples. Using matlab to plot transfer function.
Example, second order circuit,
bandpass filter, (frequency response).

**Discussion Section:** (1/28)
Homework problems, sinusoids, phasors, transfer function.
(will be posted).

**Lecture 6:** (1/29) Review calculation of transfer function
from circuit analysis. Thevenin and Norton equivalent circuits.
Example. Power of sinusoid and power from circuit elements. Maximum power
transfer.

**Lecture 7:** (1/31)
Review power of sinusoid and power from circuit elements and Maximum power
transfer.
Introduction to operational amplifiers.
Pin diagram and
internal circuitry. Study of input/output
behavior of op amps and applications, terminal characteristics.
PS 1 due.

**Lecture 8:** (2/3) Matrix and MATLAB review. Matrix addition,
multiplication, inversion. Solving systems of linear equations.
Introduction to operational amplifiers.
amplifier with and without feedback.

**Lecture 9:** (2/5) Opamp circuit application; inverting amplifier
without and with feedback. Ideal opamp characteristics; current into each
input is 0 and no voltage drop across inputs.
Passive and active circuits. Advantages
of active circuits; amplifier gains, don't need inductors, outputs
buffered (can build filters in cascade). SSSA with op amps. Example.
Introduction to Laplace Transform.

**Discussion Section:** (2/7)
Homework problems, sinusoidal steady state analysis, power,
transfer function.

**Lecture 10:** (2/10)
More realistic model for Opamp. Analysis of inverting amplifier with
feedback. Introduction to Laplace Transform. Motivation: alternate way
of analyzing circuits, generalizes sinusoidal steady state analysis, Laplace
Transforms can also handle initial conditions. Laplace transform definition.

**Lecture 11:** (2/12)
Laplace Transform pair; evaluating integral, properties, building tables.
Examples. Properties; linearity, time shift, time scaling. Using
properties to compute Laplace Transform. Impulse signal and properties.

**Discussion Section:** (2/14)
Homework problems, Laplace Transforms; computation by integrating,
properties, tables, and symbolic MATLAB.

**Lecture 12:** (2/19)
Impulse signal, sifting integral property, and Laplace Transform of
impulse signal.
Laplace transform and examples. Derivative property and derivation.
Application to circuits (capacitor and inductor relationships). Application
to solving differential equations. Initial Value Theorem and Final
Value Theorem. Derivation and examples.

**Lecture 13:** (2/21)
Laplace Transform, region of convergence, example. Inverse
Laplace Transform, using tables. Rational functions; poles,
zeros. Finding inverse Laplace Transform of rational functions;
making rational funciton proper, partial fraction expansion, taking
inverse Laplace Transform. Residue method for rational function with
simple poles. Example. PS 3 due.

**Lecture 14:** (2/24)
Laplace Transform and Inverse
Laplace Transform, using tables and properties. Rational functions; poles,
zeros. Finding inverse Laplace Transform of rational functions;
making rational funciton proper, partial fraction expansion, taking
inverse Laplace Transform. Residue method for rational function with
simple poles/ repeated poles. Complex poles. Examples.

**Lecture 15:** (2/26)
Transfer function. Poles and zeros. Region of convergence. Zero state response
(ZSR). Computing ZST; find transfer function, compute Laplace Transform
of input, find output, find inverse Laplace Transform. Examples. 2nd order
circuits; underdamped, critically damped, overdamped cases.

**Lecture 16:** (2/28)
Zero State Response (ZSR) using Laplace Transform. Examples. 2nd order
circuits; underdamped, critically damped, overdamped cases. Step response,
impulse response, arbitrary respons. Using Laplace Transform to find
solutions to Linear Constant Coefficient Differential Equations.

**Lecture 17:** (3/3)
Zero State Response (ZSR) using Laplace Transform. Alternate method of
computing ZSR using convolution integral. Convolution integral: definition,
commutative property. Computing convolution integral for RC circuit with
decaying exponential input. PS 4 due.
**Lecture 18:** (3/5)
Convolution procedure. Computing convolution integral for RC circuit with
pulse input. Step response and relation to impulse response. Natural
response and relation to step response. Using MATLAB to perform numeric
convolution.

**Discussion Section:** (2/14)
Review for exam.

**Lecture 19:** (3/7)
Convolution procedure. Example (natural response of RC circuit). Ways
of getting zero state response; Laplace transform, convolution, properties.
Convolution properties; commutative, associative, distributive, time shift.

**Lecture 20:** (3/10)
Convolution example. Total response; Zero state response, zero input response.
Using Laplace Transform to get total response. Handling initial conditions;
voltage across capacitor, current through inductor. Example.

**Exam 1:** (3/12)

**Lecture 21:** (3/14)
Total response, zero input response. Handling initial conditions. Examples.
PS 5 due.

**Lecture 22:** (3/17)
Total response, zero input response. Handling initial conditions. Example.
Introduction to state space methods.

**Discussion Section:** (3/19)
Total response and convolution integral.

**Lecture 23:** (3/21)
Introduction and motivation to state space methods. Replace capacitors with
current sources and inductors with voltage sources. Conduct circuit
analysis to get first order vector differential equations with output equation.
Identify A,b,c,and d. Example.

**Lecture 24:** (3/31)
State space methods. Replace capacitors with
current sources and inductors with voltage sources. Conduct circuit
analysis to get first order vector differential equations with output equation.
Identify A,b,c,and d. Examples.

**Lecture 25:** (4/2)
State space method. Example. Getting Laplace Transform of output from A, b,
c, d. Get transfer function from state space solution. PS 6 due.

**Lecture 26:** (4/4)
Review state space equations. Getting transfer function, impulse response,
step response, and frequency response from state space realization. Getting
total response (direct using Laplace Transform, state space method, linear
constant coefficient differential equations). Getting state space realization
from linear constant coefficient differential equations. Example. Review
first order systems (transfer function): lowpass filter, circuit
realization, pole, impulse response, step response.

**Lecture 27:** (4/7)
First order system: high pass filter, circuit realization, pole/zero, impulse
response, step response. Second order system: low pass filter, circuit
realization, poles, step response (over damped, critically damped, under
damped), frequency response. Repeat for high pass filter, bandpass filter.

**Lecture 28:** (4/9)
Bounded input/ bounded output stability. Determining stability, region of
convergences (ROC), ROC must include jw axis. Example of unstable circuit.
Introduction to filter design, frequency selective filters. Ideal lowpass
filter, approximation to ideal frequency response, tolerance (passband,
stopband). How to design filters; design stage to approximate transfer
function (look at magnitude), realization using cascade of filters.
PS 7 due.

**Lecture 29:** (4/11)
Introduction to filter design, frequency selective filters. Approximation
to ideal frequency response; use approximation filter given specifications,
circuit implementation using a case of biquad filters and first order
filter. Bode plots; finding an approximation based on poles and zeros.
Magnitude and phase plots. Magnitude plots using dB scale. Simple
pole plots.

**Lecture 30:** (4/14)
Bode plots; finding an approximation based on poles and zeros. Simple
poles, pair of complex poles. Find plots of magnitude and phase
for rational transfer functions. Filter design; approximation
to ideal frequency response. Introduction to Butterworth filters.

**Lecture 31:** (4/16)
Butterworth filter design. Lowpass filters. Properties; monotonic decreasing,poles on unit circle, high frequency rolloff, Maximally flat in passband
(first 2n-1 derivatives at 0 are 0). Pole structure for even and odd order
filter. Circuit implementation using biquads and first order circuits in
series.
PS 8 due.

**Discussion Section:** (4/17)
Convolution, total response, state space mathods.

**Lecture 32:** (4/21)
Frequency selective filter design. Design approximating magnitude of
transfer function. Butterworth filter design, MATLAB commands. Butterworth
filter properties, filter implementation using biquads and possibly
first order filter in cascade. Chebyshev 1 filter design.

**Discussion Section:** (4/22)
Convolution, total response, state space mathods.

**Exam 2:** (4/23)

**Lecture 33:** (4/25)
Review step response; initial value theorem, Final value theorem. Magnitude
and frequency scaling. Frequency selective filters; lowpass, highpass, bandpass
filters. MATLAB commands. Implementing Sallen
Key biquads with equal capacitance values. Implementing first order circuits.

**Lecture 34:** (4/28)
Exam 2 discussion; state space methods, zero input response.
Introduction to Fourier Transforms. Motivation and definition. Fourier
Transform Pair. Conditions
for existence of Fourier Transform. Example; left sided and right sided
decaying exponential signals.

**Lecture 35:** (4/30)
Fourier Transform pair. Properties; linearity, time shift, scaling, duality,
convolution. Examples; impulse signal, sinusoid signal, pulse signal,
sinc signal.
PS 9 due.

**Lecture 36:** (5/2)
Fourier Transform pair, properties. Circuit examples; left sided signal,
sinusoid signal. Introduction to periodic signals. Frequency content
of periodic signals. Passing periodic signals through circuits.

**Discussion Section:** (5/2)
Filter design, state space, zero input response.

**Discussion Section:** (5/5)
Filter design, Fourier Transforms, solving circuit problems using Fourier
Transforms.

**Discussion Section:** (5/15)
Review for final.