ECE 110/Concept List/F22

$$\newcommand{E}[2]{#1_{\mathrm{#2}}}$$List of concepts from each lecture in ECE_110 -- this is the Fall 2022 version.

Lecture 1 - 8/29

• Main web page: http://classes.pratt.duke.edu/ECE110F22/
• Circuit terms (Element, Circuit, Path, Branch and Essential Branch, Node and Essential Node, Loop and Mesh).
• Electrical quantities (charge, current, voltage, power)

Lecture 2 - 9/2

• Passive ($$+\rightarrow -$$) Sign Convention and Active ($$-\rightarrow +$$) Sign Convention
• Circuit topology (parallel, series)
• Passive Sign Convention and Active Sign Convention and relation to calculating power absorbed and/or power delivered
• Conservation Laws (conservation of power, Kirchhoff's Voltage Law, Kirchhoff's Current Law):$$ \begin{align*} \sum_{\mbox{all elements}}\E{p}{abs}&=0 & \sum_{\mbox{closed path}}\E{v}{drop}&=0 & \sum_{\mbox{closed shape}}\E{i}{leaving}&=0 \end{align*} $$
• Accounting:
  • The number of independent KVL equations is equal to the number of meshes
  • The number of independent KCL equations is equal to the number of nodes minus one
• Example of how to find $$i$$, $$v$$, and $$p_{\mathrm{abs}}$$
• $$i$$-$$v$$ characteristics of various elements (short circuit, open circuit, switch, ideal independent voltage source, ideal independent current source, resistor)
• Resistance $$R$$ in $$\Omega$$, Conductance $$G$$ in $$\mho$$ or S.
  • For a resistor, $$v=Ri$$
  • For purely resistive elements, $$R=\frac{1}{G}$$, so $$i=Gv$$ as well!

Lecture 3 - 9/5

• Dependent sources (VCVS, VCCS, CCVS, CCCS)
• Brute Force Method and labels
• Equivalents for voltage sources in series, current sources in parallel
• Ability to rearrange items in series or parallel (no impact on element values; may impact node / mesh values)

Lecture 4 - 9/9

• How resistance is calculated $$R=\frac{\rho L}{A}$$
• Equivalent resistances; Examples/Req
• Voltage division (and redivision)

Lecture 5 - 9/12

• Current division (and redivision)
• Simple Node Voltage Method (resistors and voltage sources)

Lecture 6 - 9/16

• More Node Voltage Method
  • Examples in Resources/Examples/Methods page on Sakai

Lecture 7 - 9/19

• Mesh Current Method
  • Examples in Resources/Examples/Methods page on Sakai
• Symbolic calculations in SymPy
  • SymPy/Simultaneous Equations has some info
  • Examples in Resources/Examples/Methods page on Sakai

Lecture 8 - 9/22

• Branch Current Method
  • Examples in Resources/Examples/Methods page on Sakai
• Linearity
  • 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
  • If there are dependent sources, you must keep them activated and solve for measurements each time, and you must calculate any controlling variables each time
  • You cannot calculate power until you have the total, final currents or voltages for elements - power is nonlinear!

Lecture 9 - 9/26

• Joseph Haydn - Piano Concerto No. 11 in D major (I mean, it had to be on the board for some reason, right?
• 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 neither independent nor dependent sources, find $$R_{eq}$$.
  • 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 and find the ratio of $$v_{TEST}$$ to $$i_{TEST}$$.

Lecture 10 - 9/30

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

Lecture 11 - 10/3

• First-order switched circuits with constant forcing functions
• Sketching basic exponential decays

Lecture 12 - 10/7

• Sinusoids and characteristics of sin waves
• Complex numbers and representations (Cartesian, Polar, Euler) Complex Numbers
• Basic mathematical operations with complex numbers

Lecture 13 - 10/14

• Test Review

Lecture 14 - 10/17

• Test 1

Lecture 15 - 10/21

• ACSS and phasors
• Solving ACSS using just trig gets complex very quickly - we will use complex analysis to simplify the process.
• Represent signal $$x(t)=X\,\cos(\omega t+\phi_x)$$ as the real part of $$Xe^{j\phi_x}e^{j\omega t}$$.
• For ACSS with a single frequency, all terms have $$e^{j\omega t}$$ part, so unique information can be stored in a complex number called a phasor that tracks magnitude and phase; $$\mathbb{X}=Xe^{j\phi_x}=X\angle \phi_x$$
• A derivative of an ACSS variable in the time domain is equal to $$j\omega$$ times the phasor in the frequency domain.
• A ratio of phasors is a transfer function
  • The magnitude of a transfer function represents the ratio of the output phasor magnitude to the input phasor magnitude
  • The phase of the transfer function represents the difference between the output phasor phase and the input phasor phase.
  • If $$\mathbb{H}(j\omega)=\frac{\mathbb{X}_{in}}{\mathbb{X}_{out}}$$, then:
    • $$X_{out}=X_{in}*|\mathbb{H}(j\omega)|$$
    • $$\phi_{out}=\phi_{in}+\angle \mathbb{H}(j\omega)$$

Lecture 16 - 10/24

• Impedance and AC Circuit Response
• 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
• Impedance: a ratio of phasors (though not a phasor itself)
  • $$\mathbb{Z}_R=R$$
  • $$\mathbb{Z}_L=j\omega L$$
  • $$\mathbb{Z}_R=\frac{1}{j\omega C}$$
  • $$\mathbb{Z}=R+jX$$ where $$\mathbb{Z}$$ is impedance, $$R$$ is resistance, and $$X$$ is reactance
  • $$\mathbb{Y}=\frac{1}{\mathbb{Z}}=G+jB$$ where $$\mathbb{Y}$$ is admittance, $$G$$ is conductance, and $$B$$ is susceptance
    • $$\frac{1}{\mathbb{Z}}=\frac{R-jX}{R^2+X^2}$$ so
      • $$G=\frac{R}{R^2+X^2}$$
      • $$B=\frac{-X}{R^2+X^2}$$
    • $$\frac{1}{\mathbb{Y}}=\frac{G-jB}{G^2+B^2}$$ so
      • $$R=\frac{G}{G^2+B^2}$$
      • $$X=\frac{-B}{G^2+B^2}$$
• Impedances add in series and admittances add in parallel
• Conservation laws (KCL, KVL), methods derived from conservation laws (NVM, MCM, BCM), and methods derived from Ohm's Law (voltage division, current division) apply in the phasor domain!

Lecture 17 - 10/28

• Mechanical Systems

Lecture 18 - 10/31

• Resonant circuits
  • In the ACSS, resonant circuits have inductors and capacitors that balance each other
  • Generally found by finding where the denominator of a transfer function is purely real or where the effective impedance is purely real.
• Ideal and practical first-order filters
  • Practical filters characterized by maximum gain (largest magnitude of transfer function) and half power frequency ($$\omega$$ where the magnitude is $$\frac{1}{\sqrt{2}}\approx 0.7071$$ of the maximum value.)
  • For a series RC circuit,
    • Voltage across the capacitor relative to total represents a low-pass filter with $$\mathbb{H}=\frac{1}{j\omega RC+1}$$, maximum gain of 1, cutoff frequency of $$\frac{1}{RC}$$; phase at cutoff is -45$$^{\circ}$$
    • Voltage across the resistor relative to total represents a high-pass filter with $$\mathbb{H}=\frac{j\omega RC}{j\omega RC+1}$$, maximum gain of 1, cutoff frequency of $$\frac{1}{RC}$$; phase at cutoff is 45$$^{\circ}$$
• Ideal filters are either wholly on or wholly off. Ideal filters have no phase shift.

Lecture 19 - 11/4

• Second-order filters
• Can be very dangerous near resonant frequency - ACSS voltage drop across inductor or capacitor can be larger than source!