Advanced Circuit Analysis

Apply the Laplace transform to circuits, use the s-domain to solve any linear electrical system by relating input to output, and define a system as a physical process model.

Learn how to model circuit elements in the s-domain using Laplace transforms, including resistors, inductors, and capacitors with initial conditions, and handle dependent sources and op-amps.

Apply nodal analysis to a circuit with a resistor, capacitor, and inductor, transform to the s-domain under zero initial conditions, and obtain the inductor voltage via inverse Laplace transform.

Apply the Laplace transform to model the initial 5-volt capacitor voltage as a current source, then use nodal analysis to derive v(t) = 10 e^{-t} + 15 e^{-2t}.

Explain how to analyze a circuit with an inductor and switch, determine il(0)=io, and find i(t) for t>0 using mesh analysis and an initial-condition source.

Solve a Laplace-domain circuit quiz by transforming a parallel current source with a capacitor into a series voltage source, then compute the 4-ohm resistor voltage using impedance and inverse Laplace.

Solve a quiz in advanced circuit analysis using Laplace transforms to find v_a(t) across the 2-ohm resistor for a step input, yielding v_a(t) = -2 e^{-t/3} + 12 e^{-2t}.

Convert time-domain circuits to the s-domain with the Laplace transform, analyze in the frequency domain, and convert back to time domain, handling capacitor and inductor initial conditions with equivalent sources.

Explore Laplace-transform based circuit analysis with initial inductor and capacitor values, convert to s-domain, apply KCL, and obtain the capacitor voltage via inverse Laplace transform.

Solve circuit analysis with Laplace transform using superposition to compute the capacitor voltage; kill sources to obtain individual contributions, apply KCL and partial fractions, and sum to the total Va(t).

Apply Laplace transform to a circuit with a dependent source, remove the load to find v_out(s) and v(t) using initial and final value theorems.

Apply circuit transformation and Laplace analysis to compute the inductor current i_l(t) for t>0, yielding i_l(t) = -7 e^{-t} + 3 e^{-2t} + 4 u(t).

Use superposition and s-domain analysis to find the inductor current in a parallel lc circuit, summing contributions to obtain i_l(t) = -7 e^{-t} + 3 e^{-2t} + 4 u(t).

Solve a Laplace-domain circuit problem by applying initial and final value theorems to find v(t) from V(s), assuming zero initial energy in the capacitor and using mesh analysis.

This lecture introduces two-port networks, defines a port as a pair of terminals, contrasts one-port and two-port circuits, and explains V1, V2, I1, I2 as the parameters relating them.

Explore impedance parameters and the impedance matrix for a four-terminal, two-port network. Derive z11, z12, z21, z22 using open-circuit conditions, and distinguish input impedance, transfer impedance, and open-circuit concepts.

Compute impedance parameters for a two-port circuit by analyzing open-circuit and zero-input conditions to determine z11, z12, z21, and z22, producing the parameter matrix [[60, 40], [40, 70]].

apply impedance-parameter analysis to a two-port network to compute i1 and i2 from av1 = z11 i1 + z12 i2 and av2 = z21 i1 + z22 i2.

Derive z-parameters for a two-port by using open-circuit conditions to find z11, z12, z21, z22, obtaining z11 = 12 ohms, z12 = z21 = z22 = 4 ohms (matrix [[12,4],[4,4]]).

Solve an impedance transformation quiz using impedance parameters to derive currents and voltages in a two-port, yielding i1 = 0.8 ∠30°, i2 = 0.4 ∠120°, v2 = 0.

Introduces admittance parameters for a two-port network and derives y11, y12, y21, y22 by applying short-circuit conditions, explaining short circuit input admittance, transfer admittance, and short circuit output admittance.

derive the admittance parameters for a pi network by short-circuiting the output, applying current division, and computing y11, y12, y21, and y22 to form the y-parameter matrix.

Derive the admittance (Y) parameters for a circuit by applying short-circuit conditions and KCL to find y11, y12, y21, and y22, yielding the Y-parameter matrix [0.15, -0.05; -0.25, 0.25] siemens.

Solve quiz one on admittance parameters for a two-network T network, deriving y11, y12, y21, and y22 by applying short-circuit conditions to find input-output relationships and equivalent resistances.

Solve the final y-parameters problem by short-circuiting ports and applying Kirchhoff's current law to derive y11, y12, y21, and y22, yielding the admittance matrix [[0.3125, -0.0625], [0.1875, 0.0625]].

master hybrid parameters for two-port networks, deriving h11, h12, h21, h22 from short- and open-circuit conditions and their inverse g-parameters for the same network.

Explore how to compute hybrid parameters h11, h12, h21, and h22 for a two-network by applying short-circuit conditions and voltage division, using V1, I1, I2, and V2.

Learn to solve a hybrid-parameters example by calculating the equivalent impedance with h11, h12, h21, h22 and injecting a 1-volt source after killing independent sources to find v2 and i2.

Derive the G-parameters for a two-port in the s-domain by using open circuit and short circuit conditions and voltage division with inductor, capacitor, and resistor impedances.

Derive h11, h12, h21, and h22 for a two-port. Analyze a circuit with shorted output and 4 Ω in parallel with 6 Ω to compute the h-parameters.

Determine the input impedance of a black box circuit at the input port by applying a 1 V source, using h-parameter equations, and solving for V1 over I1.

Analyze a two-port to solve the last quiz on hybrid parameters, deriving g11, g21, g12, and g22 through KCL, open-circuit, and short-circuit analyses.

Express any periodic function as a Fourier series using sine and cosine, with coefficients a0, an, and bn in an infinite sum that represents f(x) over its period.

Advanced circuit analysis shows how to compute a0, an, and bn to express F(x) as a Fourier series of sines and cosines over a period T, using orthogonal integrals.

The lecture analyzes Fourier series form and a0 as the average value. It notes that if f is even, bn vanishes; if odd, an vanishes; symmetry about x-axis yields a0.

derive Fourier series for f(x)=a x with period T; odd symmetry makes a0 and an zero. We compute bn from integration by parts, yielding f(x)= sum_{n≥1} (-aT/(π n))(-1)^n sin(2π n x/T).

derive the Fourier series for a two-second periodic piecewise function with f(x)=x on [0,1] and 0 on [1,2], calculating a0, an, bn by integration by parts, noting no symmetry.

Explore complex Fourier series by converting cosine and sine to exponentials, derive even/odd properties of a_n and b_n, and express f(x) as a complex exponential sum using c_n.

Convert the Fourier series to a compact complex form with Euler's relation, show a_n is even and b_n is odd, and express f(x) as a full complex exponential series.

Review how sine and cosine build periodic functions, extract amplitude, frequency, and phase angles from trig and complex Fourier series, and interpret their graphical meanings.

Demonstrates that for a given function, the complex Fourier series equals the trigonometric Fourier series by deriving c_n, using Euler's formula, and transforming the expression.

Derive the Fourier transform by extending Fourier series to infinity, replacing discrete coefficients with a continuous spectrum C(k) and using complex exponentials.

Compare Fourier transform and Fourier series, showing how non periodic functions become periodic, define C(k) and angular frequency, and relate amplitude, frequency, and phase to the transform.

Understand periodic functions and odd/even properties, and learn sine and cosine forms with amplitude, frequency, phase, and how shifts affect graphs.

Explore complex numbers, their real and imaginary parts, and the polar form z = r e^{i theta}. Discover Euler's formula e^{i x} = cos x + i sin x.

Learn how orthogonal functions are defined via the dot product and integrals, and how sine and cosine form orthogonal pairs for Fourier series.

Explore orthogonal functions and key integrals used in Fourier series and transforms. Learn how sine and cosine products and exponential forms yield zero or L/2 depending on n versus m.

Explore how sine and cosine become building blocks for functions and how Fourier series express them as sums of sine and cosine terms with the fundamental period.