AC (Alternating Current) Circuit Analysis

Explore sinusoidal steady-state analysis for AC circuits, detailing the time-to-frequency domain transition, application of nodal and mesh methods, and back to time-domain results.

Use nodal analysis to convert the circuit to the frequency domain, assign a reference node, and solve two node equations to find i_x from v_a over the capacitor impedance.

Convert the circuit to the frequency domain and model its impedances. Apply nodal analysis with a reference node to solve for V1 and V2 using Cramer's rule.

Learn mesh analysis for alternating current circuits: identify loops, assign mesh currents, apply circuit laws to form equations, and solve for voltages or currents.

Explore mesh analysis with a super mesh in an ac circuit, applying kvl and kcl, solving i1 through i4 with matrix methods, and determining the capacitor current and voltage.

Analyze a circuit quiz to determine the current Io using supermesh and mesh analysis, applying KVL and KCL with complex impedances and solving a three-equation system.

Apply the superposition theorem to find Io by deactivating independent sources and evaluating each contribution. Compute Io prime via mesh analysis and matrices, then combine contributions for Io.

Explore applying the superposition theorem across a multi-frequency ac circuit. Analyze contributions from dc and two ac sources using phasor impedance, time-domain synthesis, and matrix methods to compute voltages.

Apply the superposition theorem to an AC circuit, remove the current source to form an open circuit, and solve using matrix determinants to compute io and vb.

Apply superposition to analyze an ac circuit with two sources of different frequencies, using impedance and voltage and current division to find v_out.

Explore source transformation in AC circuit analysis by converting a voltage source with series impedance to a parallel current source, and vice versa, using simple formulas and examples.

Learn to solve ac circuit problems with source transformation by converting a voltage source to a current source and simplifying impedances in parallel.

Learn to solve an ac circuit quiz with source transformation, converting current to voltage sources and combining impedances. Apply current division to find the current through the resistor and capacitor.

Represent a complex linear AC circuit with Thevenin or Norton equivalents, using a voltage source and impedance or a current source in parallel with impedance, and transform between them.

Apply Thevenin's theorem to circuits with a dependent source to compute the equivalent impedance and Thevenin voltage using mesh analysis.

Apply Norton’s theorem to find the Norton impedance between terminals a and b and the short-circuit current. Use current division to compute the load current after forming the Norton equivalent.

Solve quiz eight by computing z_ab for a–b, combining 6 Ω with -j4 in parallel, then in series with 10 Ω, and apply voltage division using 75 ∠20° V.

Analyze an example for an ideal operational amplifier by applying zero input currents and zero input voltage difference to solve for vout, using capacitor impedance and kcl.

Solve an op-amp circuit for AC analysis by computing capacitor impedances, applying ideal op-amp rules, and deriving Vout and Io with time-domain expressions.

Learn to apply nodal analysis for sinusoidal steady-state circuits, choosing a reference node, formulating node equations with impedance, current source, capacitor, and resistor, and solving for the node voltage.

Use the superposition principle to analyze sinusoidal steady-state ac circuits with different frequencies, compute impedances, and convert phasor results to time-domain currents.

solve end of unit 2 problems to find equivalent impedance between terminals a and b by deactivating independent sources, using a supermesh, and obtaining z eq = -8 + j6.

Use sinusoidal steady-state analysis with nodal analysis to compute the voltage across the dependent current source; obtain Vb(t) = 13.9 cos(35 t − 134.5°) volts.

Solve quiz question two to determine the impedance between terminals using a test voltage and mesh analysis, suppressing sources and handling a dependent source.

Compute the Norton equivalence for the circuit in sinusoidal steady state by deriving the open-circuit voltage, short-circuit current, and the resulting impedance, revealing a Norton current in parallel with the impedance.

Explore the basics of ac power analysis, introducing instantaneous and average power, and how power is measured and billed in 50/60 hz ac systems.

Compute instantaneous and average power in an AC circuit using the instantaneous power formula and phasor analysis, converting polar to rectangular form.

Apply phasor analysis to an AC circuit to find instantaneous power, determine current from impedance, and use I conjugate times V to show the source and resistor share 2.5 watts.

Solving quiz 24 in AC circuit analysis, convert current to cosine form to derive instantaneous and average power for a passive network, with p(t)=3500+5445 cos(280t-10°) and average power 3500 W.

Solve quiz two by computing voltage from V = I Z for impedance, obtaining 1320 ∠ 8°, compute power using V times current conjugate divided by two, giving 20.19 kW.

Apply phasers to calculate the average power in a series ac circuit, determine source current, then compute the resistor and inductor powers, and confirm whether the source delivers power.

solve an example of maximum average power transfer by deriving the Thevenin equivalent, selecting the load impedance as the conjugate of the Thevenin impedance, and computing the maximum power.

Determine the load resistance for maximum average power transfer and compute the corresponding power using the equivalent impedance and voltage division.

Solve quiz one for maximum average power transfer by finding the load impedance that yields maximum power, using current division and the calculated branch voltage in the ac circuit.

Learn to compute the effective (rms) value of periodic, including sinusoidal, currents and voltages by matching the load’s average power with a DC equivalent.

Solve an rms value example for section 4.5 by analyzing i(t)=5t (0–2) and i(t)=-10 (2–4) in a 2-ohm resistor to obtain rms ≈8.17 A and average power ≈133.3 W.

Compute the rms value for a rectified sine driving a 10 ohm resistor and the corresponding average power, yielding 5 V rms and 2.5 W.

Derive the piecewise current i(t) for 0–1 and 1–2 seconds, compute irms and average power in a 9-ohm resistor; irms ≈ 9.24 A and P ≈ 768 W.

Explore apparent power and power factor in AC circuits, deriving average power from rms values and phase angles, and distinguish leading (capacitive) and lagging (inductive) responses.

Solve an AC circuit quiz by computing the load impedance through series and parallel combinations, determine the power factor, and calculate the average power from the source.

Analyze a 12 kVA load with a 0.856 lagging power factor at 120 V to determine real power, reactive power, peak current, and the load impedance using complex power concepts.

Calculate the power factor from a -75° phase angle, determine the apparent power, and relate real and reactive power using complex power for quiz 2 in AC circuit analysis.

Explore alternating-current power in parallel and series circuits, showing that power supplied equals power observed for complex, real, and reactive power, but not for apparent power.

Compute the current from the total impedance and demonstrate conservation of AC power by balancing the real and reactive powers of the source, line, and load.

solve quiz 1 for section 4.7 on conservation of ac power by finding voltages and currents in parallel branches, the source voltage, and the complex power of each branch.

Analyze two parallel loads, 3 kW at pf 0.725 leading and 6 kW at pf 0.95 lagging, to determine the combined power factor and the source complex power.

Solve a power factor correction example in ac circuit analysis, calculating reactive power and capacitor values to raise pf from 0.8 to 0.95 at 120 v, 60 hz.

Calculate the parallel capacitance to correct a 0.85 lagging load of 140 kVAr to unity power factor at 220 V, 60 Hz. Use Qc = Q1 – Q2.

Explore three-phase circuits, with three equal-amplitude sources and 120-degree phase shifts, and why three-phase power is widely used at 50 or 60 hz for constant instantaneous power and economy.

Examine balanced three-phase voltages from a 3-phase generator, with 120-degree phase differences and ABC/ACB sequences. Explore delta and wye load connections and impedance relations Z_delta and Z_y.

Solve quiz one for a positive-sequence abc. Compute van, vbn, vcn from vpn = 220∠30°, giving van = 270∠150°, vbn = 220∠30°, vcn = 220∠-90°.

Analyze the balanced three-phase wye connection, detailing line and phase voltages, line currents, and the relationships between line and phase quantities for different load and source configurations.

Explore phasor analysis in ac circuits, compute currents with given magnitudes and angles, and solve for I_ab and I_bc using base voltages and base currents.

Explore the balanced delta-delta connection in ac circuit analysis, examining voltages, load balance, and line relationships to show how source and load balance affect the circuit.

analyze a balanced delta-delta connection in an AC circuit, derive phasor magnitudes and phase angles, and validate both parts of example 43.

Explore solving a delta-connected load problem, determining line and phase currents and their angles in degrees. This calculation is essential for 4G, LTE, and 5G applications.

Explore the balanced delta–wye connection in ac circuit analysis, with delta-sourced and wye-loaded systems, deriving line currents via loop equations and positive-sequence relationships, and applying delta-to-wye transformations.

Transform the balanced delta-connected source to a wye, deriving phase voltages from line voltage 210 using v_ab as reference, then compute phase currents with z_y = 40 + j25.

Transform the load from y to delta in a balanced outdoor circuit; compute z delta as three times z y; derive phase and line currents from v ab.

Learn power in a balanced three-phase system with wye-connected loads, derive average and reactive power from per-phase voltages and currents, and express complex power with three-phase relations.

Solve power in a balanced ac system by computing complex, average, and reactive power at the source and load using three-phase formulas in a y-connected network.

Compute the motor power factor in a three-phase system using P = sqrt(3) V_L I_L and cos(phi) = P/(sqrt(3) V_L I_L) from the given line voltage and current.

an in-depth solution to quiz 1 on ac circuit analysis, calculating total impedance, line and load currents, and source and load complex powers for a balanced y-connected three-phase system.

Analyze three-phase motor modeled as a balanced y load to determine power factor, using P = sqrt(3) V_L I_L cos φ and 5.6 kW output; cos φ ≈ 0.807.

Compute line currents and neutral current in an unbalanced three-phase y-loaded system using ABC sequence; convert voltages to polar form and determine magnitudes and angles.

Analyze an unbalanced delta-connected three-phase load to compute line currents using Kirchhoff's laws and per-phase impedances, given 440 V line voltages and impedances 10 minus j5 ohms.

Explore magnetically coupled circuits and how magnetic coupling enables energy transfer between coils, as in transformers, before introducing mutual inductance in the next lecture.

Explore how two closely coupled coils with time-varying currents develop self and mutual inductances, inducing voltages in each other measured in henries.

Solve a mutual inductance example in a magnetically coupled two-loop ac circuit using mesh analysis, applying dot convention and complex impedance to find I1 and I2 phases.

Solve a mutual inductance circuit quiz using phasor analysis, form a coefficient matrix and determinants to find i2, then compute the output voltage as 12 at minus 45 degrees volts.

Explore energy in a magnetically coupled two-inductor circuit, derive E = 1/2 L1 I1^2, + 1/2 L2 I2^2 ± M I1 I2, discuss dot convention and the coupling coefficient k.

Compute the coupling coefficient and energy in a coupled inductor, using phasor and mesh analysis to find currents and evaluate energy at t = 0.5 s.

Learn about linear transformers with primary and secondary coils, understanding input and reflected impedance from mutual inductance, and transforming the circuit to an equivalent T or pi form.

Derive a t-equivalent circuit for a magnetically coupled transformer using L1 minus M for r_la, L2 minus M for r_lb, and M for Lc, yielding 8, 2, and 2 henries.

Solve an AC linear transformer example using a t-shaped equivalent, derive LA, LB, and LC from L and M, and analyze with two-mesh equations to find I1, I2, and V0.

Compute the pi equivalent of a linear transformer by evaluating L_a, L_b, L_c; obtain two 18 henries inductors and a 4.5 henries center inductance forming a pi network.

Convert the mutual inductance circuit to a pi equivalent, compute L_A, L_B, and L_C from L and M, then solve the three-mesh equations to find I1 and I3.

Investigate ideal transformers with perfect coupling and high-permeability cores, derive V2/V1 = N2/N1 via Faraday's law, and examine power conservation and reflected impedance.

solve an ideal transformer example to compute turns ratio, primary and secondary turns, and currents using complex power.

Solve an ideal transformer problem using mesh analysis to find the current through the 10 ohm resistor and its 5.3-watt power, given a turns ratio of one over two.

Solve quiz 56 on an ideal transformer with 2200 v primary and 110 v secondary; determine the turns ratio 0.05, secondary current 500 a, and the complex power 55 kVA.

Solve quiz three for section 6.5 by applying primary–secondary relationships with N=2, form loop equations, and compute i3 to yield V0 across an eight-ohm resistor as 96 volts.

Compare a two-winding transformer with an ideal autotransformer, showing that the autotransformer delivers about 21 times more power under the same 240 V and 12 V, 4 A conditions.

Solve an example of an ideal autotransformer step-up, determining i1, i2, io, and the complex power delivered to the load using the turns ratio and Ohm's law.

Solve a quiz on auto transformers by analyzing primary and secondary powers, calculating currents from voltages, and determining the auto transformer power rating as 780 volt-amps.

Explore how keeping amplitude constant while varying frequency reveals a circuit's frequency response, describing gain and phase versus frequency and its use in filters and transfer functions.

Compute the transfer function for the surrogate circuit by using s-domain analysis, impedance, and current division to obtain i_out/i_in, then identify zeros and poles via the Delta method.

Derive the transfer function of the given ac circuit in the s-domain, then compute its zeros (-2, -5) and poles (-1.13, -8.87).

Learn the decibel scale and express gain with logarithms. Relate power gains to decibels and, with equal resistances, relate them to voltage or current gains, with examples.

Analyze Bode plots by examining constant term, poles and zeros, and simple pole or zero; identify corner frequencies and magnitude and phase slopes of 20 and -20 decibels per decade.

Construct the bode plots for the transfer function by identifying poles and zeros, rewriting factors, and plotting magnitude and phase, including corner frequencies and decibels per decade.

Analyze a transfer function to construct bode plots by converting to standard form, identifying poles and zeros, corner frequencies, and magnitude and phase.

Transform the transfer function into familiar zeros and poles, plot each factor's magnitude and phase, and combine them to obtain the overall H(ω) with its corner frequencies.

Convert the given transfer function to standard form, identify corner frequencies, and construct magnitude and phase plots to sketch the overall transfer function.

Explore series resonance in RLC circuits, where ω0 = 1/√LC yields a purely resistive impedance, minimum input impedance, and high selectivity via Q = ω0L/R and bandwidth B = ω0/Q.

Explore series resonance in an ac circuit by calculating resonance frequency, bandwidth, and quality factor, and determine current amplitudes at resonance and near half-power frequencies.

Solve for the capacitance yielding a quality factor of 50 in a series R L C circuit, then determine the resonant and half-power frequencies, bandwidth, and average power at 100-V.

Explore parallel resonance in an RLC circuit by analyzing admittance and deriving omega0 = sqrt(1/LC). At resonance, the LC acts as an open circuit, and current flows through the resistor.

Solve a parallel resonance problem to compute the resonance frequency, bandwidth, and quality factor for an R, L, and C circuit, then determine omega1 and omega2 and the power dissipated.

Analyze a parallel resonance example by solving for the circuit's resonant frequency via admittance; determine omega = 2 rad/s where the imaginary part of Y vanishes.

Solve a parallel resonant circuit quiz by calculating the resonance frequency, bandwidth, and quality factor, then determine omega1 and omega2. Use given R, L, C values to derive these parameters.

Solve quiz two for section 7.6 to find the resonant angular frequency, deriving impedance from admittance and isolating the imaginary part to obtain 173.21 radians per second.

Explore how magnitude and phase in phasor form convert sine to cosine and show how the current and voltage waveforms relate in a capacitive AC circuit.

Discover why the super node method can streamline finding node voltages when a voltage source sits between nodes, and how simple KCL can substitute full construction.