*CIRCUIT ANALYSIS USING LAPLACE TRANSFORM

*METHODOLOGYExamples of nonlinear circuits:logic circuits, digital circuits,or any circuits where theoutput is not linearlyproportional to the input.

Examples of linear circuits:amplifiers, lots of OPMcircuits, circuits made ofpassive components (RLCs).

*THE s-DOMAIN CIRCUITSEquation of circuit analysis: integrodifferential equations.Convert to phasor circuits for AC steady state.Convert to s-domain using Laplace transform.KVL, KCL, Thevenin,etc.

*KIRCHHOFFS VOLTAGE LAWConsider the KVL in time domain:

Apply the Laplace transform:

*KIRCHHOFFS CURRENT LAWConsider the KCL in time domain:

Apply the Laplace transform:

*OHMS LAWConsider the Ohms Law in time domain

Apply the Laplace transform

*INDUCTORInductors voltageIn the time domain:

In the s-domain:

*INDUCTORInductors currentRearrange VL(s) equation:

*CAPACITORCapacitors currentIn the time domain:

In the s-domain:

*CAPACITORCapacitors voltageRearranged IC(s) equation:

*RLC VOLTAGEThe voltage across the RLC elements in the s-domain is the sum of a term proportional to its current I(s) and a term that depends on its initial condition.

*CIRCUIT ANALYSIS FOR ZERO INITIAL CONDITIONS (ICs = 0)

*IMPEDANCEIf we set all initial conditions to zero, the impedance is defined as:

[all initial conditions=0]

*IMPEDANCE & ADMITANCEThe impedances in the s-domain areThe admittance is defined as:

*Ex. Find vc(t), t>0

*Obtain s-Domain CircuitAll ICs are zero since there is no source for t

*Convert to voltage sourced s-Domain Circuit

*Find I(s)

*Find Capacitors VoltageThe capacitors voltage:

Rewritten:

*Using PFEExpanding Vc(s) using PFE:

Solved for K1, K2, and K3:

*Find v(t)

Using look up table:

*Ex.Find the Thevenin and Norton equivalent circuit at the terminal of the inductor.

*Obtain s-domain circuit

*Find ZTH

*Find VTH or Voc

*Draw The Thevenin CircuitUsing ZTH and VTH:

*Obtain The Norton CircuitThe norton current is:

*Ex.Find v0(t) for t>0.

*s-Domain Circuit Elements Laplace transform all circuits elements

*s-Domain Circuit

*Apply Mesh-Current AnalysisLoop 1Loop 2

*Substitute I1 into eqn loop 1

*Find V0(s)

*Obtain v0(t)

*Ex.The input, is(t) for the circuit below is shown as in Fig.(b). Find i0(t) (b)

*s-Domain Circuit

*Using current divider:

*Derive Input signal, Is

*Obtain Is(t) and Is(s)Expression for is(t):

Laplace transform of is(t):

*Substitute eqn. (2) into (1):

*Inverse Laplace transform

*CIRCUIT ANALYSIS FOR NON-ZERO INITIAL CONDITION (ICs 0)

*TIME DOMAIN TO s-DOMAIN CIRCUITSs replaced t in the unknown currents and voltages.Independent source functions are replaced by their s-domain transform pair.The initial condition serves as a second element, the initial condition generator.

*THE ELEMENTS LAW OF s-DOMAIN

*THE ELEMENTS LAW OF s-DOMAIN

*TRANSFORM OF CIRCUITS- RESISTORIn the time domain:

In the s-domain:

*TRANSFORM OF CIRCUITS- INDUCTORIn the time domain:

*TRANSFORM OF CIRCUITS- INDUCTORInductors voltage:Inductors current:

*TRANSFORM OF CIRCUITS- CAPACITORIn the time domain:

*TRANSFORM OF CIRCUITS- INDUCTORCapacitors voltage:Capacitors current:

*Ex. Find v0(t) if the initial voltage is given as v0(0-)=5 V

*s-Domain Circuit

*Apply nodal analysis method

*Contd

*Using PFERewrite V0(s) using PFE:

Solved for K1 and K2:

*Obtain V0(s) and v0(t)Calculate V0(s):

Obtain V0(t) using look up table: