Dc Circuit Analysis (complete)

Learn how SI units ensure standard measurements in engineering, identify the six basic units and the derived charge, and apply prefixes like kilo, milli, micro, nano, pico.

Learn the flow of charge, current definitions, voltage and polarity, and how power and energy relate in direct current and alternating current circuits.

Solve basic concepts problems by calculating charge from electrons and protons, and apply current to charge integration, including piecewise current functions.

Solve basic dc circuit problems by applying energy, power, and voltage concepts to compute voltage drops, bulb power, and energy use from current, charge, and time.

Identify active and passive elements, noting that active elements generate energy and passive elements do not. Differentiate independent sources from dependent sources, and explain how the diamond shape indicates dependence.

Analyze circuit elements in dc circuit analysis using power equals voltage times current and the passive sign convention. Identify delivering versus absorbing power and apply KCL with parallel voltage rules.

Differentiate charge-time functions to obtain currents, applying the product rule when needed, then integrate currents to find charge; use voltage–current relations to compute energy and verify circuit power sums.

Apply ohm's law to solve resistance, current, and voltage in circuits. Calculate current, conductance, and power for resistor circuits using voltage, current, and resistance.

Explore circuit topology by defining branches, nodes, and loops; count branches, nodes, and loops in a circuit and differentiate series and parallel connections.

Identify branches and nodes in dc circuits, count circuit elements as branches, and classify series or parallel connections by shared nodes and currents, illustrated with voltage sources and resistors.

Apply Kirchhoff's current law and Kirchhoff's voltage law by choosing a standard current sign convention, summing currents at nodes, and summing voltages around loops to solve circuits.

Solve a series of circuit problems using Kirchhoff's laws, applying KCL and KVL to find currents and voltages across resistors, and interpret polarities.

Apply Ohm's law to series resistors, derive voltage division across R1 and R2, and determine the equivalent resistance as R1 plus R2 for dc circuit analysis.

Explore parallel resistor networks, apply Ohm's law to find i1 and i2, derive the parallel equivalent using 1/r_eq = 1/r1+1/r2 and conductance, and examine open and short-circuit cases.

Solve series and parallel resistor networks to find equivalents between terminals A and B, using circuit redraws, and apply conductance rules for parallel and series combinations.

apply ohm's law to two switch positions in the end-of-unit problem: 40 v across 100 ohm yields 0.4 a, and across 250 ohm yields 0.16 a.

Count circuit elements to determine six branches, then identify four nodes by grouping points that lack elements between them.

Apply Kirchhoff's current law at node a to determine voltage and current in a circuit with 20-ohm and 10-ohm resistors, yielding VA = -15 V and I = 0.75 A.

Solve a dc circuit analysis problem by applying Ohm's law to a parallel resistor network, determine node voltage and currents, and compute total power and powers of each element.

Master a second-unit circuit problem by applying parallel reductions and current division to find branch currents, transforming the network into simpler equivalents and confirming directions at the node.

Determine the circuit’s current by calculating the equivalent resistance of parallel resistor groups to 70 ohms, then apply 140 V to get 2 A.

Learn how to perform nodal analysis by selecting a reference ground, applying KCL at non-reference nodes, and using ohms law across resistors to relate voltages and currents.

Apply nodal analysis to solve circuits by writing KCL at each node, including a dependent current source, then solve the 3x3 system with determinants to find V1, V2, and V3.

Solve the nodal analysis problem by applying KCL to non-reference nodes, form the coefficient matrix A, and use determinants to compute the node voltages.

Master nodal analysis with voltage sources. Use the reference-to-nonreference case to set node voltage, and form a super node with KCL and KVL when the source links two nonreference nodes.

Solve nodal analysis with voltage sources using a super node and Kirchhoff's current law to determine node voltages and currents, including vA, vB, and the current from B to ground.

Apply nodal analysis with voltage sources to two supernodes, using KCL and KVL, then solve with a matrix method to determine the node voltages.

Explore nodal analysis with voltage sources by forming a super node and applying KCL. Solve for V1, V2, V3 and node currents using voltage relations.

Apply mesh analysis to planar circuits by assigning clockwise mesh currents and using a voltage law to form and solve the equations.

Explore practical mesh analysis in dc circuits using Kirchhoff's voltage law to solve for multiple mesh currents, illustrated with two example circuits and equations.

Using mesh analysis and KVL, the video guides solving a three-mesh circuit by forming equations, applying substitutions, and computing currents with determinants, yielding I sub-zero = 1.5 A.

Solve a mesh analysis problem using KVL for three meshes to determine i_subzero. Set up the coefficient matrix and compute its determinant to find i_subzero equals -4 A.

Solve a mesh analysis problem with current sources using a supermesh, apply KVL to the supermesh and KCL at nodes P and Q to find I1, I2, I3, and I4.

Solve a dc circuit using mesh analysis with current sources, employ a supermesh, apply kvl for both super and normal meshes, and determine i1, i2, i3 with a matrix method.

Apply nodal analysis to the unit three circuit, form a super node with the voltage source, and use KCL and KVL to compute node voltages and currents.

Apply nodal analysis to the unit three circuit to determine isobar zero, solving for VA and VB via KCL at A and B with a 60 V reference.

solve the end of unit three circuit problem by applying nodal analysis with a super node to determine the node voltages and the related currents.

Apply mesh analysis to the end-of-unit 3 dc circuit, assign clockwise mesh currents, use kvl to solve, and use node a kcl to obtain ib=3 amps and ic=-0.75 amps.

Apply mesh analysis to the unit three problem, form a supermesh around currents, solve for i1, i2, i3, i4, and derive i0 = 3.62 A using KVL and determinants.

Using nodal analysis and KCL on a two-node dc circuit with multiple current sources, this lecture derives VA and VB and then computes VX and i0 from the node voltages.

Explore linearity in dc circuit analysis, showing resistors as linear elements that obey homogeneity and additivity, with outputs proportional to inputs per Ohm's law.

This video demonstrates linearity in a circuit by solving mesh equations with Kvell, showing that doubling the input voltage doubles the output voltage and current.

Analyzing linearity in a circuit, the lecture shows how the subzero voltage across the eight-arm resistor and branch currents scale proportionally with input, solved via nodal analysis and KCL.

Apply linearity by assuming I0 equals 1 ampere, solve branch currents with KCL and voltages, then use ratios to conclude I0 equals 3 amperes and isobars equal 15 amperes.

Apply nodal analysis to a linearity problem in a dc circuit, deriving Sub-Zero voltages across 12‑ohm and 8‑ohm resistors and showing 16 V when the source is 40 V.

Apply the superposition principle to circuits with multiple independent sources by calculating each source's contribution and summing them, using short circuits for voltage sources and open circuits for current sources.

apply superposition by deactivating each independent source - open circuit for a current source and short for a voltage source - and compute and sum currents and voltages.

Apply the superposition principle in dc circuits by turning off independent sources and using mesh analysis to compute currents, including open and short circuit replacements.

Apply superposition to a circuit with a 5 v voltage source and a 5 a current source, using kcl at node x and a 0.1 v x dependent source.

Apply superposition to a three-source circuit, replacing voltage source with a short and current source with an open circuit, then compute current through the 3-ohm resistor, yielding 2 amperes.

Use superposition to calculate each independent source's contribution in a circuit with 8‑V, 6‑V sources and a current source. Sum these to find a total current of 0.375 A.

Transform sources by replacing a voltage source in series with a resistor with a current source in parallel with a resistor. This equivalence, grounded in ohm's law, simplifies circuit analysis.

Learn to perform source transformations between voltage sources in series with resistors and current sources in parallel with resistors, and apply current division to obtain the 3.2 volt result.

solve a circuit using source transformations, convert current sources to voltage sources, and combine parallel sources and resistors to apply current division and find the current through the seven-ohm resistor.

Apply source transformation to convert a dependent voltage source into a dependent current source, then use nodal analysis to solve for X, the current through the 10 ohm resistor.

Apply source transformation to convert voltage and current sources, arrange circuit elements in series, and use KVL to compute Vx in a dc circuit.

Explore the Thevenin's theorem by replacing a complex two terminal circuit with a voltage source and series resistance, analyzing independent and dependent sources to find the Thevenin voltage and resistance.

Apply the two-step method to find the Thevenin equivalent between terminals a and b: kill independent sources and replace with shorts or opens, then compute the Thevenin voltage.

Find the Thevenin equivalent by removing the independent voltage source and analyzing currents with the dependent source present; obtain Thevenin voltage about 5.33 V and Thevenin resistance about 0.44 Ω.

Find Thevenin equivalent by calculating R_eq with a test source in a circuit containing a dependent source, then determine V_th, giving R_eq = 6 ohms and V_th = 20 volts.

Replace a two-terminal linear circuit with a Norton equivalent—a current source in parallel with a resistor defined by I_N and R_N; handle independent and dependent sources when finding I_N.

Compute the Norton equivalent by deactivating independent sources, determine R_N between terminals A and B, then short the terminals and apply mesh analysis to obtain I_N.

Solve a Norton theorem problem by finding the Norton current and Norton resistance, producing a 4.5 ampere current source in parallel with a 90 ohm resistor.

RN equals 5 ohms by killing independent sources and applying a 1 V test source; IN equals 7 A, giving a 7 A Norton current in parallel with 5 ohms.

Apply Norton theorem to find the equivalent resistance and Norton current by killing independent sources, replacing the current source with an open circuit, and solving via mesh analysis with a test voltage for the dependent source. The final Norton circuit is a 10 A current source in parallel with a 1 ohm resistor.

Learn how maximum power transfer occurs when load resistance equals source resistance, maximizing power in linear circuits. The lecture uses calculus to derive this condition.

Solve a maximum power transfer problem by finding the Thevenin equivalent and load, then use mesh analysis to obtain the load voltage and the maximum power.

Derive the circuit's Thevenin equivalent and load for maximum power transfer, compute the load resistance equal to the equivalent resistance, and determine the maximum power.

Applying linearity and a 1-volt assumption to relate node voltages in unit four yields V Sub-Zero = -5 volts.

Apply superposition to compute X by turning off each independent source in turn and analyze with a mesh and a supermesh to sum the contributions.

This lecture applies saws transformation to convert a dependent current source to a voltage source, then uses mesh analysis to solve for i_x, yielding 1.6 amperes.

Introduce operational amplifiers as a voltage controlled voltage source that can amplify, integrate, or differentiate signals. Review basic types like inverting amplifier, summing amplifier, and difference amplifier.

Explore ideal op-amps with infinite open-loop gain, infinite input resistance, and zero output resistance, showing that input currents are zero and input terminals have zero voltage difference.

Analyze an inverting operational amplifier circuit with input and feedback resistors. Show that the input terminals are at zero voltage and the gain is -Rf/Ri, reversing signal polarity.

Examine examples of inverting op amps within dc circuit analysis. See common configurations and their behavior in practical circuits.

Examine examples of inverting op amps within DC circuit analysis, illustrating how this configuration operates in practice.

Explore practical examples of inverting op amps within DC circuit analysis, covering gain calculation, feedback behavior, and signal inversion for instructional mastery.

Explore practical examples of inverting op amps, illustrating dc circuit analysis concepts and amplifier configurations for robust problem solving.

Explain non-inverting amplifiers, where input connects to the non-inverting terminal and feedback yields a gain of 1 plus RF over RI, with possible voltage follower behavior.

Apply ideal op-amp rules to a non-inverting amplifier problem, enforce zero input current and equal input voltages, determine node voltages, and obtain an output of minus one volt.

In this non-inverting amplifier example, we solve an ideal op-amp circuit using the virtual short and zero input current, applying KCL. The output is 2 volts.

Apply ideal op-amp rules to a summing circuit with 5k, 2.5k, and 10k resistors to compute the output voltage, which equals minus eight volts.

Apply ideal op-amp rules to a summing op-amp circuit, derive node voltages, and compute vout = -3.8 v and iout = -1.425 ma.

learn how a difference amplifier uses an ideal op-amp to amplify the difference between two inputs while rejecting common signals, and derive its output expression from the resistor network.

Design a single op-amp difference amplifier to realize output equals minus five v1 plus three v2. Set resistor ratios R2/R1 = 5 and R3/R4 = 1, with practical values like R1 = 5 kohms, R2 = 25 kohms, R3 = R4 = 10 kohms.

Design a difference amplifier with a gain of 7.5 by choosing R1, R2, R3, and R4 in a balanced op-amp circuit and applying the output voltage equation.

Explore cascaded operational amplifiers where each stage's output feeds the next to boost gain; the overall gain is the product of stage gains, with attention to load and saturation.

Calculate the output of a cascaded op-amp circuit using ideal op-amp rules; the lecture walks through solving voltages at intermediate nodes and the final output.

Explain cascaded op-amp analysis using KCL, showing A, B, and C are 5 V, vout is 25 V, and the 50 kΩ branch carries 100 µA.

Analyze cascaded op-amps with 1 volt and 2 volts inputs to obtain an 8.63-volt output, using ideal op-amp rules that currents into inputs are zero and input difference is zero.

Analyze cascaded op amps to determine each stage output using node voltages, voltage follower behavior, and Kirchhoff's current law. Conclude the final output is 35 volts.

This lecture analyzes a non-ideal op-amp circuit with 100 kΩ input resistance, 100 Ω output resistance, and loop gain of 10^5, using the equivalent circuit and KCL to find voltages.

This unit five problem analyzes an ideal op-amp circuit, applying the zero input voltage difference and zero input current rules, along with KCL, to compute the output as -2.5 volts.

Analyze a unit five operational amplifier problem by applying the ideal op-amp rules and node-based equations to find the output voltage and current X in an A, B, C circuit.

This video solves a unit five op-amp circuit, deriving the output in terms of V1 and V2 using ideal op-amp assumptions, node analysis, and equation combination.

Solve an ideal operational amplifier circuit from unit five using node analysis and a simple formula, showing two methods and yielding about 14.09 volts for the given inputs.

Solve the final problem on three ideal operational amplifiers by step-by-step application of castell at various points, using ideal op-amp properties to determine the output voltage.

Explore capacitors and inductors as storage elements that store energy, unlike resistors. Use mesh or nodal analysis to handle circuits with storage elements and build more complex designs.

Explore capacitors as passive energy storage elements that store energy in the electric field between two plates separated by an insulator, with q = c v defining capacitance.

Solve a simple capacitor problem: compute charge and energy for a 3 pF capacitor at 20 V using Q=CV and E=1/2CV^2, yielding 60 pC and 600 pJ.

Compute voltage across a four point five micro capacitor from the given charge, using C and Q, then calculate energy stored with E = 1/2 C V^2.

Calculate the current through a 5 μF capacitor by differentiating the voltage with respect to time and multiplying by the capacitance, illustrating i = C dv/dt in dc circuit analysis.

Apply I = C dv/dt to a 10 microfarad capacitor connected to a voltage source, with dv/dt = -4000 sin(280 t), giving I = -0.04 sin(280 t) A.

solve a 2 microfarad capacitor problem by determining the voltage using v(t) = (1/C) integral_0^t i(tau) dtau + v(0) with v(0)=0, given a decaying current i(t).

Compute voltage across 100 μF capacitor driven by a 50 sin(120π t) current with zero voltage at 1 and 5 ms using v = (1/C) ∫ i dt + v0.

Use i equals C dv/dt on a 200 μF capacitor, reading the voltage slope; currents: +10 mA (0–1), -10 mA (1–3), +10 mA (3–4), otherwise zero.

Compute capacitor voltage from area under the current-time graph for an uncharged capacitor, using V = (1/RC) ∫ i dt; 0.1 V at 2 ms, 0.4 V at 5 ms.

Replace capacitors with open circuits, solve for VA = 12 V and VB = 8 V, then compute energies: 4 µF = 128 µJ, 2 µF = 16 µJ.

Solve a dc circuit by replacing capacitors with open circuits, analyze a 50 V series network, and determine voltages and energy stored in 20 μF and 30 μF capacitors.

This lecture covers connecting capacitors in parallel and series, deriving equivalent capacitance: parallel sums, series reciprocal of sums, using i = C dv/dt and v = ∫ i dt.

Identify equivalent capacitance by simplifying parallel and series combinations, culminating in a 40 microfarad equivalent for the circuit.

Solve a bad connection of capacitors to determine the equivalent capacitance and the voltage across each capacitor using series and parallel combinations and charge–voltage relations.

Inductors store energy in magnetic fields and act as passive elements, while voltage equals L di/dt with L depending on turns, geometry, and core permeability.

Solve a basic inductors problem by calculating the voltage across a 0.1 h inductor and its stored energy using v = L di/dt and E = 1/2 L i^2.

Explore examples of inductors within dc circuit analysis, with a focus on solving related problems; the instructor offers to provide step-by-step explanations on request.

Compute current through a 5 H inductor with a piecewise voltage using i(t)=1/L ∫0^t v(τ)dτ, yielding i(t)=2t^3. Then find energy at 5 s as 156.25 kJ.

Compute the current in a 2-henry inductor from v(t)=10-10t with i(0)=2 A over t=0 to 4 s, yielding i(4)=-18 A and stored energy 324 J.

solve a dc circuit problem with an inductor and capacitor by treating the capacitor as open and the inductor as short, find currents, capacitor voltage, and energies stored.

Under dc, treat inductor as a short and capacitor as an open, giving 7.5 A inductor current and 15 V capacitor voltage, with energies 168.75 J and 450 J stored.

Learn how inductors connect in series or parallel to form an equivalent inductance. Use series: L_eq equals sum of L_i, and parallel: 1/L_eq equals sum of 1/L_i, following resistor analogies.

Replace capacitors with open circuits under dc conditions, then use mesh or nodal analysis to find V1 and V2; V1 = 42 V and V2 = 48 V.

Solve six capacitors problem to find C for a 20 μF equivalent between A and B. C equals 50 μF via parallel with 10 μF and series with 30 μF.

Solve unit six capacitors and inductors by computing v_c for a 3 μF capacitor from i(t) using v_c=(1/C)∫i dt+v(0), using uncharged initial condition and continuity to obtain C1=0 and C2=90.

Compute the inductor current in unit six by integrating the given voltage across a 50 mH inductor, with zero initial current, using I(t)=1/L ∫0^t v(τ) dτ.

Apply dc by replacing capacitors with open circuits and inductors with short circuits. Show inductor current 2 A; capacitor voltage 0 V; inductor energy 1 J, capacitor energy 0 J.

Calculate the equivalent inductance seen at terminals A and B by identifying parallel groups, forming X and Y in parallel, and obtaining 45.5 million henries.

Explore first-order RC and RL circuits built from resistors, capacitors, and inductors, solving via first-order differential equations using methods from resistive circuits, including source-free and independent-source cases.

Learn about source-free rc circuits where a charged capacitor discharges into a resistor, producing V(t)=V0 e^{-t/rc} with time constant rc and energy transfer to the resistor.

Analyze a source-free rc circuit to determine the initial capacitor voltage and the voltages and currents across components using series-parallel reduction and the time constant tau = rc.

Solve a source-free rc circuit to find initial capacitor voltage and time constant, compute equivalent resistance, and derive capacitor voltage and currents over time.

Solve a source-free rc circuit, find the initial capacitor voltage eight volts and energy 16/3 joules, with v(t)=8e^{-2t} and a 0.5 s time constant.

Explore source-free rl circuits by analyzing the inductor current after disconnecting the supply. Learn how i(t)=i0 e^{-Rt/L} and the time constant L/R govern exponential decay and energy transfer.

Solve a source-free rl circuit by determining the time constant and initial inductor current, using a test source, mesh analysis, and kvl to find the equivalent resistance.

Solve a source-free RL circuit to find the initial inductor current and the time constant, then derive i(t) = 7 e^{-2t} A and Vx = -1 V.

Solves a source-free RL circuit by finding the initial inductor current and a 0.25 s time constant, with current decaying as 6 e^{-t/0.25} A after the switch opens.

Determine the initial inductor current in a source-free RL circuit under DC steady-state, then compute the time constant and express I(t) as 2 e^{-t/0.5} A for t>0.

Explore singularity functions in dc circuit analysis, focusing on the unit step, unit impulse, and unit ramp functions, their discontinuities, and their roles in transient analysis.

Express voltage pulses as a combination of singularity functions and unit step functions, identify transitions at t = 2 and t = 5, and derive the derivative as delta functions.

Express the current pulses in terms of singularity functions and unit step functions, derive the piecewise expression and its ramp-based integral, and sketch the waveform.

Decompose the given singularity function into piecewise components—ramp, unit step, and delta functions—and express it as a sum of three terms with appropriate shifts and magnitudes.

Explore the step response of RC circuits when a DC source is applied, revealing natural, transient, and steady-state components and the complete response and the time-constant RC concept.

Explore the step response of an rc circuit by finding the initial capacitor voltage, final voltage, and time constant, then compute v(t)=v∞+(v0−v∞)e^{−t/τ} with v0=15 V, v∞=30 V, τ=2 s.

Solve the rc circuit step response by finding the initial 15 v, final 9.375 v, and a 0.5 s time constant, with v(t)=9.375+5.625 e^{-2t}.

Solves a step response problem for an RC circuit, determining the initial and final capacitor voltages, time constant, and the current through a 10-ohm resistor.

Analyzes the step response of an rc circuit, finding v_c(0−)=20 V. For t>0, v_c(t)=10+10 e^{-3t/2} V and i(t)=-2-2 e^{-3t/2} A, with i(t)=0 for t<0.

Learn the step response of rl circuits, separating transient and steady-state currents. Apply i infinity plus (i zero minus i infinity) e^(−t/τ) and note the inductor acts as a short.

Explore the step response of an rl circuit by determining i(0), i(∞), and the time constant; analyze t<0 and t>0 to show i(0)=5 A and i(∞)=2 A with τ=1/15 s.

solve the step response of an rl circuit by finding i(0), i(∞), and the time constant, then express i(t) = i∞ + (i0 − i∞) e^{−t/τ}.

Learn the step response of an RL circuit by analyzing three time intervals (t<0, 0≤t≤2, t>2) to derive the inductor current and time constants.

Analyze unit seven rc circuit problems: compute capacitor output for t<0, determine the initial capacitor voltage, and find the decay time to one third with a 60 ms time constant.

Compute the time constant of a single-inductor RL circuit by finding 24 kΩ equivalent resistance in parallel and applying τ = L/R with L = 5 H, giving 0.208 s.

Resolve a unit seven first-order circuit with a dependent source using thevenin to determine equivalent resistance, find a 5 s time constant, and show i(t)=5 e^{-t/5} from I(0)=5 A.

Analyze a unit seven first-order circuit: i(t)=5 e^{-t/4} A for t>0 after the switch moves to B, from 10 V to 20 V with a 4 s time constant.

Analyze unit seven problem five in a two-state rc circuit, determine capacitor voltage over time for open and closed switch scenarios, with time constants of 4 s and 12 s.

Analyze the final unit seven problem by evaluating the circuit before and after the switch, inductor current from 2 A to 1.6 A, and derive v_L(t) = -4 e^{-t/0.05} volts.

Introduce second-order circuits with two storage elements, explain when multiple elements act as one, and survey series and parallel RLC circuits, initial values, and step and natural responses.

learn to determine initial values in a second-order circuit, including v(0), i(0), dv/dt(0), and di/dt(0), using capacitor and inductor continuity across t<0, t=0, and t>0.

Solve the initial values of a second-order dc circuit across three time intervals, determining i(0)=2 A, v(0)=4 V, and dv/dt(0)=20 V/s, with final values i(∞)=0 A and v(∞)=12 V.

Analyze t<0, t=0, and t>0 intervals to solve a dc circuit problem, replacing capacitors with open circuits and inductors with short circuits to obtain initial currents and capacitor voltages.

Learn to determine initial values in a dc circuit by analyzing three time intervals, using unit step and open/short circuit replacements to find i(0)=0 A and vc(0)=-20 V.

Analyze initial values in a dc circuit by examining t<0 and t>0, apply step function concepts, use open/short circuit replacements for capacitor and inductor, and solve for v_C and i_L.

This lecture derives the source-free series RLC circuit’s natural response, forms a second-order differential equation, and explains damping and its three cases: over-damped, under-damped, and critically damped.

Analyze a source-free series RLC circuit by computing alpha = R/(2L) and omega0 = 1/√(LC). For R=40, L=4, C=1/4, roots s1 ≈ -0.101 and s2 ≈ -9.899, overdamped.

Solves a source-free series RLC circuit problem, computes alpha and omega0, shows underdamped natural response with roots s1, s2 = -1 ± j√99, using j as the complex unit.

Solve a source-free series RLC circuit problem by determining initial capacitor voltage and inductor current, identifying the natural response type, and deriving i(t) with alpha, omega0, and damping.

Solve a source-free series RLC circuit to determine the initial current, alpha and omega0, and the underdamped i(t) for t<0, t=0, and t>0.

Derive the second-order differential equation for source-free parallel RLC circuits via KCL, then analyze overdamped, critically damped, and underdamped responses with roots s1 and s2 and cos/sin forms.

Explore a source-free parallel RLC circuit with L = 1 H and C = 10 uF, solving V(t) in overdamped, critically damped, and underdamped cases using alpha and omega zero.

Analyze a source-free parallel RLC circuit, establish critical damping, and derive V(t) = -2 e^{-20 t} using V(0)=0 and I(0)=50 mA.

Solve a source-free parallel RLC circuit by evaluating pre-, at-, and post-switch conditions; determine initial capacitor voltage and inductor current, and derive the damped exponential response.

Solves a source-free parallel RLC circuit problem, finds the initial capacitor voltage and inductor current, and derives the overdamped capacitor voltage V(t) with alpha and omega0.

Explore the step response of a series RLC circuit when a DC source is suddenly connected. Examine over-damped, critically damped, and under-damped cases and the role of the steady-state value.

Analyze the step response of a series RLC circuit by solving for V(t) and I(t) across the three time intervals t<0, t=0, and t>0.

Analyze the step response of a series RLC circuit, solving for v(t) for t>0 across time intervals using open capacitor for t<0 and KVL, revealing underdamped behavior with 15 V final value.

Analyze the step response of a parallel RLC circuit when a source is suddenly applied, covering over damped, critically damped, and under damped cases with final value and transient expressions.

Analyze the step response of a parallel RLC circuit to a unit step, solving for I(t) and I_r(t) after switching, using open/short circuit rules and deriving alpha and omega_0.

Explore the step response of a parallel RLC circuit by deriving v(t) and i(t) from initial conditions, determine damping alpha and natural frequency omega0, and obtain the underdamped current expression.

Apply four steps to general second-order circuits to determine the step response: initial values, transient from the characteristic equation, steady state, and total response X = X_ss + X_transient.

solve extra RC circuit problems by analyzing capacitor voltage for two time intervals, using open-circuit steady state and a time constant of 8 seconds.

Investigate the dc circuit in two time intervals, find the initial capacitor voltage via KCL, then determine the time constant and final capacitor voltage to obtain i(t) for t>0.