ECE 110/Concept List/F25

Lecture 3 - 9/2 - Course Introduction, Nomenclature; Electrical Quantities

• 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)
• Passive Sign Convention and Active Sign Convention and relation to calculating power absorbed and/or power delivered.
• Kirchhoff's Laws
  • KVL: $$\Sigma_{Loop}v_{DROP}=0$$
    • Number of independent KVL equations = meshes
  • KCL: $$\Sigma_{Mesh*}i_{LEAVING}=0$$
    • *or any closed geometric shape; may be larger or smaller than a node
    • Number of independent KCL equations = nodes-1
• Example of how to find $$i$$, $$v$$, and $$p_{\mathrm{abs}}$$ using conservation equations and how to check using extra conservation equations
• Power conservation

Lecture 4 - 9/4 - Sources and Resistors; Conservation Equations Applied

• $$i$$-$$v$$ relationships of various elements
  • ideal independent voltage source, including short circuit
  • ideal independent current source, including open circuit
  • switch
• Resistor symbol (and spring symbol)
• 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)
• Conservation Equations Applications:
  • Label every unlabeled element current and voltage (source values and required measurements for controlled sources should already be labeled)
    • Use passive sign convention for passive elements (resistors so far); can use active or passive for others (in class: we used passive)
  • # of unknowns = (2*# Elements) - # sources
  • Use accounting from the previous lecture:
    • Number of independent KVL equations = meshes
    • Number of independent KCL equations = nodes-1
    • New: Number of Ohm's Law equations (Element equations) = number of passive elements

Lecture 5 - 9/9 - Equivalent Circuits and Division

• Combining voltage sources in series; ability to move series items and put together
  • Parallel voltage sources must have same voltage or bad things happen
• Combining current sources in parallel; ability to move parallel items and put together
  • Series current sources must have same current or bad things happen
• Order of items in series does not matter to outside world; order of items in parallel does not matter to outside world
• Equivalent resistances
  • series and parallel
  • Examples/Req
• Voltage Division and Re-division
• Current Division and Re-Division

Lecture 6 - 9/11 - More Equivalents and Division; Node Voltage Method

• Delta-Wye Networks
• Basics of NVM
• NVM
  • Labels:
    • Really 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.
  • #KCL = #nodes - 1 - #v. sources
  • Write KCL at nodes not touching a voltage source; if there are voltage sources, look at nodes on either side or source to make a supernode
  • Power:
    • $$p_{abs,R}$$ can be found with $$v^2/R$$
    • $$p_{del,i}$$ can be found with active $$vi$$
    • $$p_{del,v}$$ requires finding actively measured $i$ through voltage source

Lecture 7 - 9/16 - Current Methods

• Examples on Canvas
• BCM
  • Labels:
    • Label each (essential) branch current, using as few unknowns as possible by incorporating current sources as well as current measurement and voltage labels (if the latter is on a resistor)
• MCM
  • Labels:
    • Label each mesh current, understanding that current sources, current measurements, and voltage measurements will require additional equations.
• #KVL = #meshes-#i. sources
  • Write KVL for mesh not including a current source; if there are currentsources, look at supermesh (loop) on either side of source
• Power:
  • • $$p_{abs,R}$$ can be found with $$i^2R$$
    • $$p_{del,v}$$ can be found with active $$vi$$
    • $$p_{del,i}$$ requires finding actively measured $v$ across current source

Lecture 8 - 9/18 - Computational Methods

• NVM / MCM / BCM example
• Using Maple to set up and solve simultaneous equations

Lecture 9 - 9/23 - 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 Ohm's Law, Voltage 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 for circuits with one or more dependent sources.

Lecture 10 - 9/25 - 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 11 - 9/30 - Operational Amplifiers

• Model using two resistors and a VCVS
• Without feedback, only really good as a comparator
• Feedback from output to inverting input makes circuit more useful
• Ideal op-amp assumptions are about the op-amp, not the circuit: $$A\rightarrow\infty$$, $$r_i\rightarrow\infty$$, $$r_o\rightarrow 0$$
• Using ideal op-amp with a circuit with feedback from output to inverting input leads to a very useful circuit with implications of:
  • No voltage drop between the input terminals
  • No current entering the input terminals
  • Still possible to have current at the output terminal!

Lecture 12 - 10/2 - More Op-Amp Circuits

• Various configurations that are directly or nearly-directly from the circuit developed in Lecture 11:
  • Non-inverting
    • Buffer / Voltage follower as a specific instance
  • Inverting
    • Inverting summation with multiple source / resistor pairs attached to inverting input
    • Mixing board conversations
  • Difference
    • Voltage divider on non-inverting side
• Creating a circuit to produce a proscribed linear combination of input voltages

Lecture 15 - 10/16 - Capacitors and Inductors

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

Lecture 16 - 10/21 - 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
• Finding values as $$t\rightarrow\infty$$

Lecture 17 - 10/23 - First-Order Circuits

• First-order switched circuits with constant forcing functions
• Sketching basic exponential decays
• Using methods (NVM, MCM, BCM) to get model equations

Lecture 18 - 10/28 - Sinusoids and Complex Numbers

• Solving even a simple differential equation with a sinusoidal input is somewhat complicated
• At the heart of complex analysis is an understanding of Complex Numbers
• Overview of Calculator Tips

Lecture 19 - 10/30 - ACSS and Phasors

• Solving ACSS using just trig gets complex very quickly - we will use complex analysis to simplify the process - this is a motivation for phasors
• 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
• 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 20 - 11/4 - More ACSS and Phasors

• To use phasors to solve ACSS,
  • Replace functions of t with their phasor representation
  • Replace $$\frac{d}{dt}$$ with $$j\omega$$
  • Solve for the output phasor as a function of the input phasor
• Find transfer function $$\mathbb{H}(j\omega)$$ as a ratio of an output phasor and an input phasor
  • Use transfer function to note that $$\mathbb{V}_{out}=\mathbb{H}(j\omega)\,\mathbb{V}_{in}$$ (input, output, or both could also be currents)
  • If given numerical values, can use those to get actual magnitudes and phases for output and convert to time

Lecture 21 - 11/6 - More Phasors

• Once in the phasor domain, use KVL, KCL, NVM, MCM, BCM, and whatever else from resistive circuits to get relationships between input phasors and output phasors
• Note that $$\frac{d}{dt}$$ in the time domain is the same as multiplying a phasor by $$j\omega$$ in the frequency domain - this will allow us to use frequency techniques to back out differential equations
• In class looked at LR circuit in DCSS, after switching a DC source, and in ACSS

Lecture 23 - 11/13 - Filters

• An $$RC$$ circuit can be both a high-pass filter (resistor voltage) or a low-pass filter (capacitor voltage)
• An $$LR$$ circuit can be both a high-pass filter (inductor voltage) or a low-pass filter (resistor voltage)
• The "cut-off frequency" of a filter is the frequency at which the magnitude of the transfer function is $$\frac{1}{\sqrt{2}}\mathbb{H}_{max}$$ - this is also known as the "half-power frequency"
  • Basic $$RC$$ circuit: $$\omega_{cutoff}=\frac{1}{RC}$$
  • Basic $$LR$$ circuit: $$\omega_{cutoff}=\frac{R}{L}$$
• Basic circuits have maximum magnitudes of 1 for the transfer function; can use op-amps to get others.

Lecture 24 - 11/18 - Digital Logic 1

• Introduction to binary
• Billion $$\neq$$ Billion
• Introduction to boolean operators
• Basic operations: not, and, or, xor
• Truth tables
• DeMorgan's Laws

Lecture 25 - 11/20 - Digital Logic 2

• Minterms and maxterms
• Digital logic gates and schematics
• Complexity
• Boolean algebra relationships and simplifications
• Manual logic function minimization
• Gray code motivation and construction
• Karnaugh maps
  • Structure
  • Use in finding minimum sum of products (MSOP) form
  • Use in finding minimum product of sums (MPOS) form