EGR 224/Concept List/S24

The notes below are not meant to be comprehensive but rather to capture the general topics of covered during lectures in EGR 224 for Spring 2024. These notes are in no way a replacement for actively attending class.

Lecture 1 - 1/10 - Course Introduction, Nomenclature

• Circuit terms (Element, Circuit, Path, Branch and Essential Branch, Node and Essential Node, Loop and Mesh).
• Accounting:
  • # of Elements * 2 = total number of voltages and currents that need to be found using brute force method
  • # of Essential Branches = number of possibly-different currents that can be measured
  • # of Meshes = number of independent currents in the circuit (or generally Elements - Nodes + 1 for planar and non-planar circuits)
  • # of Nodes - 1 = number of independent voltage drops in the circuit
• Electrical quantities (charge, current, voltage, power)

Lecture 2 - 1/12 - Electrical Quantities

• Passive Sign Convention and Active Sign Convention and relation to calculating power absorbed and/or power delivered.
• Power conservation
• Kirchhoff's Laws
  • Number of independent KCL equations = nodes-1
  • Number of independent KVL equations = meshes
• Example of how to find $$i$$, $$v$$, and $$p_{\mathrm{abs}}$$ using conservation equations and how to check using extra conservation equations
• $$i$$-$$v$$ relationships of various elements (ideal independent voltage source, ideal independent current source, short circuit, open circuit, switch)
• Resistor symbol (and spring symbol)

Lecture 3 - 1/19 - Equivalents

• Resistance as $$R=\frac{\rho L}{A}$$
• $$i$$-$$v$$ relationship for resistors; resistance [$$\Omega$$] and conductance $$G=1/R$$ $$[S]$$
• $$i$$-$$v$$ for dependent (controlled) sources (VCVS, VCCS, CCVS, CCCS)
• Combining voltage sources in series; ability to move series items and put together
• Combining current sources in parallel; ability to move parallel items and put together
• Equivalent resistances
  • series and parallel
  • Examples/Req

Lecture 4 - 1/22 - Brute Force Method; Delta-Wye; Voltage Division Part 1

• Brute Force method
• Delta-Wye equivalencies (mainly refer to book)
• Voltage Division

Lecture 5 - 1/26 - Voltage Division Part 2, Current Division, and Node Voltage Division Part 1

• Voltage Re-Division
• Current Division and Re-Division
• Basics of NVM

Lecture 6 - 1/29 - Node Voltage Method

• NVM
  • Examples on Canvas
  • Labels:
    • Very Lazy: label ground, then make every other node a new unknown. Voltage sources, voltage measurements, and current measurements will provide additional equations.
    • Lazy: label ground, then label any node connected to ground if it has a voltage source or voltage measurement. Make every other node a new unknown. Voltage sources not connected to ground, voltage measurements not connected to ground, and current measurements will provide additional equations.
    • Smart: label ground; once a node gets labeled, if there is a voltage source or a voltage measurement anchored at that node, use the source or measurement to label the other node it is attached to. Current measurements will provide additional equations.
    • Really Smart: same as smart, only also use voltage drops across resistors with current measurements to relate node voltages.

Lecture 7 - 2/2 - Current Methods

• Examples on Canvas
• BCM
  • Labels:
    • Label each (essential) branch current, using as few unknowns as possible by incorporating current source and current measurement labels
• MCM
  • Labels:
    • Label each mesh current, understanding that current sources, current measurements, and voltage measurements will require additional equations.

Lecture 8 - 2/5 - Linearity and Superposition

• Definition of a linear system
• Examples of nonlinear systems and linear systems
  • Nonlinear system examples (additive constants, powers other than 1, trig):

$$\begin{align*} y(t)&=x(t)+1\\ y(t)&=(x(t))^n, n\neq 1\\ y(t)&=\cos(x(t)) \end{align*} $$

• Linear system examples (multiplicative constants, derivatives, integrals):

$$\begin{align*} y(t)&=ax(t)\\ y(t)&=\frac{d^nx(t)}{dt^n}\\ y(t)&=\int x(\tau)~d\tau \end{align*} $$

• Superposition
  • Redraw the circuit as many times as needed to focus on each independent source individually
  • Use combinations of Phm's Law, Voltaeg Division, and Current Division, rather than setting up and solving multiple equations
  • If there are dependent sources, you must keep them activated and solve for measurements each time - this likely means that superposition may not actually make solving things easier.

Lecture 9 - 2/9- Thévenin and Norton Equivalent Circuits

• Thévenin and Norton Equivalents
• Circuits with independent sources, dependent sources, and resistances can be reduced to a single source and resistance from the perspective of any two nodes
• Equivalents are electrically indistinguishable from one another
• Several ways to solve:
  • If there are only independent sources, turn independent sources off and find $$R_{eq}$$ between terminals of interest to get $$R_{T}$$. Then find $$v_{oc}=v_{T}$$ and recall that $$v_T=R_Ti_N$$
  • If there are both independent sources and dependent sources, solve for $$v_{oc}=v_T$$ first, then put a short circuit between the terminals and solve for $$i_{sc}=i_N$$. Recall that $$v_T=R_Ti_N$$
  • If there are only dependent sources, you have to activate the circuit with an external source.

Lecture 10 - 2/12 - Capacitors and Inductors

• Intro to capacitors and inductors
• Basic physical models
• Basic electrical models
• Energy storage
• Continuity requirements
• Finding circuit equation models
• DCSS equivalents

Lecture 11 - 2/16 - Initial Conditions and Finding Equations

• DCSS equivalents
• Finding values just before and just after circuit changes
  • For $$t=0^+$$, can model inductor as independent current source and capacitor as independent voltage source
• Using Node Voltage Method to get model equations

Lecture 12 - 2/19 - First-Order Circuits (constant forcing functions)

• First-order switched circuits with constant forcing functions
• Sketching basic exponential decays
• Using the Node Voltage Method to get model equation
• (intro to complex numbers)

Lecture 13 - 2/23 - Test 1

Test

Lecture 14 - 2/26 - ACSS and Phasors

• Overview of Calculator Tips
• Solving ACSS using just trig gets complex very quickly - we will use complex analysis to simplify the process
• At the heart of complex analysis is an understanding of Complex Numbers
• Motivation for phasors
• Reminder: a phasor is a complex number whose magnitude represents the amplitude of a single frequency sinusoid and whose angle represents the phase of a single frequency sinusoid
• EGR_224/Spring_2022/Sandbox
• Impedance: a ratio of phasors (though not a phasor itself)
  • $$\mathbb{Z}=R+jX$$ where $$R$$ is resistance and $$X$$ is reactance
  • Admittance $$\mathbb{Y}=\frac{1}{\mathbb{Z}}=G+jB$$ where $$G$$ is conductance and $$B$$ is susceptance
  • For common elements:
    • $$\mathbb{Z}_R=R$$
    • $$\mathbb{Z}_L=j\omega L$$
    • $$\mathbb{Z}_R=\frac{1}{j\omega C}$$
  • Impedances add in series and admittances add in parallel

Lecture 15 - 3/1 - More ACSS and Phasors

• Analytical and numerical Phasor Analysis
• RLC circuits, transfer functions, and differential models

Lecture 16 - 3/4 - Bode Plots

• Decibel - based on the base-10 logarithm of a power ratio, multiplied by 10 (to get the deci- part)
  • We are assuming power is related to voltage or current squared, and the log of a square is twice the log, so $$H_{dB}=20\,\log_{10}|\mathbb{H}(j\omega)|$$
• Bode plots
  • Magnitude plot is $$20\,\log_{10}|\mathbb{H}(j\omega)|$$ versus $$\omega$$ with $$\omega$$ on a log scale
  • Angle plot is $$\angle \mathbb{H}(j\omega)$$ versus $$\omega$$ with $$\omega$$ on a log scale
• Translation between decibels and magnitudes
  • 1 $$\leftrightarrow$$ 0 dB
  • 10 $$\leftrightarrow$$ 20 dB
  • 100 $$\leftrightarrow$$ 40 dB
  • 0.1 $$\leftrightarrow$$ -20 dB
  • 2 $$\leftrightarrow$$ $$\approx$$ 6 dB
  • 5 $$\leftrightarrow$$ $$\approx$$ 14 dB
  • 0.25=1/2/2 $$\leftrightarrow$$ 0-6-6 dB=-12 dB
  • 8=2*2*2=$$2^3$$ $$\leftrightarrow$$ 6+6+6 dB = 3 * 6 dB = 18 dB
  • $$\sqrt{2}$$=$$2^{1/2}$$ $$\leftrightarrow$$ $$\frac{1}{2}$$ 6 dB = 3 dB
• 3 dB below maximum for a transfer function amplitude is critical because that indicates where the half-power frequency is.
• Four main terms in transfer functions of real signals:
  • $$\mathbb{H}(j\omega)=K\,\color{red}{(j\omega)^p}\,\color{green}{\Pi_k(1+j\frac{\omega}{\omega_{cut}})^{n_k}}\,\color{blue}{\Pi_l((j\omega)^2+2\zeta_l\omega_{n_l}(j\omega)+(\omega_{n_l})^2)^{m_l}}$$
  • We looked at constants, $$j\omega$$, and $$1+j\frac{\omega}{\omega_{cut}}$$

Lecture 17 - 3/8 - More Bode Plots

• Amplified RC circuit high-pass filter
• Band-pass filter with corner location
• Notch-filter with double-corner correction

Lecture 18 - 3/18 - 1st and 2nd Order Filters

• Low-pass filter
• General homogeneous solution for 2nd order (undamped, underdamped, critically damped, overdamped)

Lecture 19 - 3/22 - LTI Systems

• Step functions and integrals
• Impulse functions
• Singularity Functions
• Integrals with steps and impulses
• Linearity review
• Time Invariance
• Impulse response and step response
• Convolution

Lecture 20 - 3/25 - More Convolution

• [Convolution Shortcuts]

Lecture 21 - 3/29 - Laplace Transforms

• [1]
• Derivation from LTI
• Basic LT pairs (impulse, step, exponential, sinusoids, exponential sinusoids)
• Basic LT properties (frequency shift, time shift)
• Meaning and importance of Region of Convergence and pole-zero plot

Lecture 22 - 4/1 - Laplace Transform Applications

• Basic LT properties (integral, derivative, frequency derivative)
• Partial fraction expansion and short-cut
• LT of semi-periodic signals

Lecture 24 - 4/8 - Unilateral Laplace Transform Applications

• Unilateral Laplace and derivative property
• Solving differential equations with non-zero initial conditions
• Solving circuits with initial energy