| Lecture | Date/Day | Topics | Lecture Video | Relevant Materials |
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| 1B | 2019/08/28 (W) | Intro: Logistics, motivations, syllabus. | Lec 1B video | |
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| 2B | 2019/09/04 (W) | EE Device Models: resistors, capacitors, inductors, indep. sources, controlled sources, memoryless devices vs non-memoryless devices, linearity, nonlinear devices, voltage, current and non-controlled elements, the ideal diode. | Lec 2B video | |
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| 3A | 2019/09/09 (M) | EE Device Models (contd.): diodes, BJTs, MOSFETs, nonlinear capacitors/inductors, continuity/smoothing, memristors. | Lec 3A video | |
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| 3B | 2019/09/11 (W) | DAE formulations of circuits and mechanical systems: numerical issues to watch out for in device/model codes; from ODEs to DAEs; circuit eqn formulations by example: STA, NA and MNA; casting into standard DAE form; Jacobian matrices. | Lec 3B video | |
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| 4A | 2019/09/16 (M) | DAE formulations of circuits and mechanical systems (contd.): STA, NA, and MNA formulations of circuit equations in general, using the incidence matrix. DAEs of mechanical systems. | Lec 4A video | |
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| 4B | 2019/09/18 (W) | Project possibilities; DC and Newton-Raphson: Discussion of possible projects. DC analysis. Iterative methods. The basic Newton-Raphson method. | Lec 4B video | |
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| 5A | 2019/09/23 (M) | DC and Newton-Raphson (contd.): NR converges in 1 iteration on linear equations. NR convergence/non-convergence demos; rapid convergence near solution. Quadratic convergence property of NR. Convergence criteria: residual, reltol+abstol, others. | Lec 5A video | |
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| 5B | 2019/09/25 (W) | DC and Newton-Raphson (contd.): NR on a simple diode-resistor circuit: how/why it fails. The concepts of initialization and limiting; application to the resistor-diode circuit and resulting success. The need for independent init/limiting of multiple devices/nonlinearities. Changing the system's DC function to take a vector of limited variables as an input; the need for new supporting functions. Trying limiting-variable based init/limiting on the diode-resistor circuit; resulting failure. | Lec 5B video | |
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| 6A | 2019/09/30 (M) | DC and Newton-Raphson (contd.): Debugging the failure of NR with limiting variables using a linear vsrc-resistor-resistor circuit: graphical analysis to understand the problem. Fixing the problem by applying a first-order (linear) correction term to the function computed using limited variables; success on the linear and resistor-diode circuits. Better limiting functions: pnjlim. pnjlim based first-order corrected init/limited NR on circuits with many diodes and their robust convergence. | Lec 6A video | |
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| 6B | 2019/10/02 (W) | Automatic Differentiation, Ckt Examples: Forward automatic differentiation of numerical code via operator overloading. Example circuit blocks and their hand analysis: MOS current mirror. | Lec 6B video | |
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| 7A | 2019/10/07 (M) | ModSpec: Developing what a device is, step by step. The ModSpec scheme for device specification. | Lec 7A video | |
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| 8A | 2019/10/14 (M) | Transient analysis: Basic test problem. From DAEs to ODEs. Existence and uniqueness of ODE solutions: need for initial condition, counter-examples. The Lipschitz condition and the Picard-Lindelof local existence theorem. Wintner's global existence theorem. Basic notions for numerical solution: time-discretization and sampling. Intuitive notions for solving ODEs. The notion of fitting piecewise-polynomial approximations to the ODE. | - | |
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| 8B | 2019/10/16 (W) | Transient analysis (contd.): Solving ODEs numerically using piecewise polynomials and evaluating the ODE at different points. The FE, BE and TRAP methods; demos and comparison. Behaviour for large timesteps: stability. FE blows up for large timesteps; simple analysis. Why large timesteps are unavoidable: vector systems of differential equations with widely separated intrinsic time constants (aka stiff systems). Demo: FE, BE and TRAP on a stiff system. | Lec 8B video | |
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| 9A | 2019/10/21 (M) | Transient analysis (contd.): The stability plane: h\lambda for complex \lambda. Stability region of FE; comparison with stability of exact solution to test equation. Stability regions for BE and TRAP. Accuracy for small timesteps of FE, BE and TRAP: TRAP is 2nd order accurate, while FE/BE are only 1st order accurate. Consistency == 1st order accuracy. The notion of convergence of a numerical ODE method. Convergence theorem (Dahlquist): consistency + stability = convergence. | Lec 9A video | |
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| 12A | 2019/11/11 (M) | Transient analysis (contd.): p-step LMS (Linear Multi Step) methods. Exactness constraints (accuracy). Stability vs accuracy tradeoffs: A-stability, Dahlquist's 1st and 2nd barriers. Re-examining TRAP's "perfect stability": absolute stability. Trying to work around Dahlquist's 2nd barrier: desirable conditions for "stiff stability". | Lec 12A video | |
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| 12B | 2019/11/13 (W) | Transient analysis (contd.): Stiff stability conditions (contd.). BD (Backward Differentiation) (aka Gear) Formulae for stiff stablity. GEAR1==BE; GEAR2: A-stable, Absolutely stable, 2-order accuracy. Stability regions of GEAR methods. Applying LMS formulae to DAEs: s/\alpha_i x_{n-i}/\alpha_i q(x_{n-i})/g. DAEs using FE/BE/TRAP. The issue of initial condition consistency for DAEs; TRAP oscillation with inconsistent ICs. Why DAEs behave like infinitely stiff ODEs; the importance of absolute stability. Why BDF methods automatically correct inconsistent ICs. | Lec 12B video | |
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| 12C | 2019/11/15 (F) | Transient analysis (contd.), Biochemical reaction modelling: Starting a p>1 step method. Biochemical reaction systems. Law of mass action. A <-> B reaction; conservation and its relationship with singular Jacobians. Reaction rate equations for reactions with multiple reactants and stoichiometries. RREs for chains of reactions. Writing RREs using the stoichiometry matrix and the rate vector. | Lec 12C video | |
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| 13A | 2019/11/18 (M) | Biochemical reaction modelling (contd.): RRE simulation demos: A <-> B, A+B <-> C, sA A + sB B <-> sC C + sD D. Basic enzyme-catalyzed reaction and its Michaelis-Menton analysis: equivalent MM kinetics for enzyme-catalyzed reactions. Demo and comparison. SVD of the stoichiometry matrix to identify conservation. | Lec 13A video | |
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| 13B | 2019/11/20 (W) | Biochemical reaction modelling (contd.), Linear(ized) DAEs: SVD of the stoichiometry matrix to identify conservation (contd.). Linear DAEs. Discussion of necessity and sufficiency for f/q to be linear. Linearizing nonlinear DAEs. | Lec 13B video | |
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| 13C | 2019/11/22 (F) | Linearization, Laplace Transforms, Eigenanalysis: Linearizing nonlinear DAEs around a DC operating point (contd.). Uses of linear(ized) DAEs. Eigenanalysis of LTI DAEs: using LTI DAE matrices to obtain an analytical form for the transfer function H(s) (including brief tutorial on Laplace transforms). Poles of H(s), dynamical (BIBO) stability of the system, and implications on the validity of linearization. | Lec 13C video | |
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| 14A | 2019/11/25 (M) | Eigenanalysis, AC analysis: Summary and demo of eigenanalysis. Numerical phasor (aka "AC") analysis: sinusoidal steady state analysis of LTI systems. | Lec 14A video | |
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| 14B | 2019/11/26 (T) | AC analysis, DC sensitivity analysis, Simple mechanical Equation Engine: Summary and demo of AC (numerical phasor) analysis. DC parametric sensitivity analysis: direct and adjoint methods and their computational complexities. 1-d spring-mass type mechanical systems and their equation formulation: nodes, node x/v/F unknowns, velocity definition and force balance equations. Elements (anchor, spring, mass) and the unknowns and equations they contribute. Equation engine and ModSpec examination and demo: simple spring mass system netlist, DAE via equation engine, DC and transient analysis. | Lec 14B video | |
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| 15A | 2019/12/02 (M) | Random Variables, Stochastic Processes, Stationary Noise Analysis: Intuitive concepts of noise in systems — "inputs" whose values cannot be specified precisely. General methodology for understanding and working with noise: random variables and stochastic processes. RV basics: named boxes, sampling, counting samples and binning, histograms. Normalized histograms, the limit of infinite samples (probability), the limit of an infinite number of infinitesimal bins. Limit divided by bin size: probability density functions (PDFs). Basic properties of PDFs. | Lec 15A video | |