CIRCUIT ANALYSIS USING LAPLACE TRANSFORM METHODOLOGY If the

CIRCUIT ANALYSIS USING LAPLACE TRANSFORM

METHODOLOGY If the circuit is a linear circuit YES Laplace transform of the sources of excitation: s(t) S(s) Laplace transform of the all the elements in the circuit Find the output O(s) in the Laplace freq. domain Obtain the time response O(t) by taking the inverse Laplace transform NO Stop or approximate the circuit into a linear circuit and continue Examples of nonlinear circuits: logic circuits, digital circuits, or any circuits where the output is not linearly proportional to the input. Examples of linear circuits: amplifiers, lots of OPM circuits, circuits made of passive components

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

KIRCHHOFF’S VOLTAGE LAW ± Consider the KVL in time domain: ± Apply the Laplace transform:

KIRCHHOFF’S CURRENT LAW ± Consider the KCL in time domain: ± Apply the Laplace transform:

OHM’S LAW ± Consider the Ohm’s Law in time domain ± Apply the Laplace transform

INDUCTOR ± Inductor’s voltage – In the time domain: – In the s-domain:

INDUCTOR ± Inductor’s current – Rearrange VL(s) equation:

CAPACITOR ± Capacitor’s current – In the time domain: – In the s-domain:

CAPACITOR ± Capacitor’s voltage – Rearranged IC(s) equation:

RLC VOLTAGE ± The 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)

IMPEDANCE ± If we set all initial conditions to zero, the impedance is defined as: [all initial conditions=0]

IMPEDANCE & ADMITANCE ± The impedances in the s-domain are ± The admittance is defined as:

Ex. ± Find vc(t), t>0

Obtain s-Domain Circuit ± All ICs are zero since there is no source for t<0

Convert to voltage sourced s -Domain Circuit

Find I(s)

Find Capacitor’s Voltage ± The capacitor’s voltage: ± Rewritten:

Using PFE ± Expanding Vc(s) using PFE: ± Solved for K 1, K 2, and K 3:

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 Thevenin Circuit ± Using ZTH and VTH:

Obtain The Norton Circuit ± The norton current is:

Ex. ± Find v 0(t) for t>0.

s-Domain Circuit Elements Laplace transform all circuit’s elements

s-Domain Circuit

Apply Mesh-Current Analysis Loop 1 Loop 2

Substitute I 1 into eqn loop 1

Find V 0(s)

Obtain v 0(t)

CIRCUIT ANALYSIS FOR NON ZERO INITIAL CONDITION (ICs ≠ 0)

TIME DOMAIN TO s-DOMAIN CIRCUITS ± s 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 CIRCUITSRESISTOR ± In the time domain: ± In the s-domain:

TRANSFORM OF CIRCUITSINDUCTOR ± In the time domain:

TRANSFORM OF CIRCUITSINDUCTOR ± Inductor’s voltage: ± Inductor’s current:

TRANSFORM OF CIRCUITSCAPACITOR ± In the time domain:

TRANSFORM OF CIRCUITSINDUCTOR ± Capacitor’s voltage: ± Capacitor’s current:

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

s-Domain Circuit

Apply nodal analysis method

Cont’d

Using PFE ± Rewrite V 0(s) using PFE: ± Solved for K 1 and K 2:

Obtain V 0(s) and v 0(t) ± Calculate V 0(s): ± Obtain V 0(t) using look up table:

Ex. ± The input, is(t) for the circuit below is shown as in Fig. (b). Find i 0(t) is(t) 1 (a) 2 0 (b) t(s)

s-Domain Circuit

± Using current divider:

Derive Input signal, Is is 1(t) is(t) t 1 0 is 2(t) 0 2 t(sec) 0 2 t

Obtain Is(t) and Is(s) ± Expression for is(t): ± Laplace transform of is(t):

± Substitute eqn. (2) into (1):

Inverse Laplace transform