Difference between revisions of "ECE 110/Concept List/F22"
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** The magnitude of a transfer function represents the ratio of the output phasor magnitude to the input phasor magnitude | ** 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. | ** The phase of the transfer function represents the difference between the output phasor phase and the input phasor phase. | ||
− | ** If $$\mathbb{H}(j\omega)=\frac{\mathbb{X}_{ | + | ** If $$\mathbb{H}(j\omega)=\frac{\mathbb{X}_{out}}{\mathbb{X}_{in}}$$, then: |
*** $$X_{out}=X_{in}*|\mathbb{H}(j\omega)|$$ | *** $$X_{out}=X_{in}*|\mathbb{H}(j\omega)|$$ | ||
*** $$\phi_{out}=\phi_{in}+\angle \mathbb{H}(j\omega)$$ | *** $$\phi_{out}=\phi_{in}+\angle \mathbb{H}(j\omega)$$ | ||
− | |||
== Lecture 16 - 10/24 == | == Lecture 16 - 10/24 == |
Revision as of 22:07, 17 November 2022
$$\newcommand{E}[2]{#1_{\mathrm{#2}}}$$List of concepts from each lecture in ECE_110 -- this is the Fall 2022 version.
Contents
- 1 Lecture 1 - 8/29
- 2 Lecture 2 - 9/2
- 3 Lecture 3 - 9/5
- 4 Lecture 4 - 9/9
- 5 Lecture 5 - 9/12
- 6 Lecture 6 - 9/16
- 7 Lecture 7 - 9/19
- 8 Lecture 8 - 9/22
- 9 Lecture 9 - 9/26
- 10 Lecture 10 - 9/30
- 11 Lecture 11 - 10/3
- 12 Lecture 12 - 10/7
- 13 Lecture 13 - 10/14
- 14 Lecture 14 - 10/17
- 15 Lecture 15 - 10/21
- 16 Lecture 16 - 10/24
- 17 Lecture 17 - 10/28
- 18 Lecture 18 - 10/31
- 19 Lecture 19 - 11/4
Lecture 1 - 8/29
- Main web page: http://classes.pratt.duke.edu/ECE110F22/
- Circuit terms (Element, Circuit, Path, Branch and Essential Branch, Node and Essential Node, Loop and Mesh).
- Electrical quantities (charge, current, voltage, power)
Lecture 2 - 9/2
- Passive ($$+\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}_{out}}{\mathbb{X}_{in}}$$, then:
- $$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}$$
- $$\frac{1}{\mathbb{Z}}=\frac{R-jX}{R^2+X^2}$$ so
- Impedances add in series and admittances add in parallel
- Conservation laws (KCL, KVL), methods derived from conservation laws (NVM, MCM, BCM), and methods derived from Ohm's Law (voltage division, current division) apply in the phasor domain!
Lecture 17 - 10/28
- Mechanical Systems
Lecture 18 - 10/31
- Resonant circuits
- In the ACSS, resonant circuits have inductors and capacitors that balance each other
- Generally found by finding where the denominator of a transfer function is purely real or where the effective impedance is purely real.
- Ideal and practical first-order filters
- Practical filters characterized by maximum gain (largest magnitude of transfer function) and half power frequency ($$\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!