# Network Equations Using Laplace Transform

By Mallesham Devasane|Updated : December 28th, 2016

Laplace Transform: It is an integral transformation of a function f(t) from the time domain into the complex frequency domain F(s).

Laplace transform can be used to transform the time domain circuits into S domain circuits to simplify the solution of integral differential equations to the manipulation of a set of algebraic equations.

If f(t) is the time domain function then its Laplace transform is denoted by F(s) and is defined by the equation.

where s is complex frequency variable.

s=σ + jω; where σ is attenuation constant (damping factor, and ω is angular frequency.

Circuit Analysis using Laplace and Inverse Laplace transform

Properties of Laplace Transform

Linearity
• t-domain: L[K1f1(t) + K2f2(t)]
• s-domain: [K1f1(s) + K2f2(s)]
Differentiation
• t-domain:
• s-domain: snf(s) – sn–1f(0) – sn–2f′(0) – sn–3f′(0)…..
Integration:
• t-domain:

• s-domain:

Scaling
• t-domain: L[f(at)]
• s-domain:

Time shifting
• t-domain: L[f(t – a)]
• s-domain: e–as f(s)
Shifting in s-domain
• t-domain: L[e–at f(t)]
• s-domain: f(s + a)
Convolution
• t-domain: L[f1(t) × f2(t)]
• s-domain: f1(s) ∙ f2(s)
Initial value theorem
• t-domain:
• s-domain:

Final value theorem
• t-domain:
• s-domain:

Laplace Transform of Periodic Function

where, f1(t) = Function over one time period, and f(t) = Periodic function with time period T.

Functions & Laplace transform
• Constant function (1) or Unit step function (u(t)) : Laplace Transform is 1/s
• Unit step function shifted / delayed by T ( U(t – T) ): Laplace Transform is (e–sT)/s
• Unit impulse ( δ(t) ): Laplace Transform is 1
• Exponential function ( eat ): Laplace Transform: 1/(s – a)
• Exponential function (e–at ): Laplace Transform is 1/(s + a)
• Sine function ( sin ωt ): Laplace Transform is ω/(s2 + ω2)
• Cosine function ( cos ωt ): Laplace Transform is s/(s2 + ω2)
• Damped Sine  ( e–at sin ωt ): Laplace Transform is ω/((s+a)2 + ω2)
• Damped Cosine  ( e–at sin ωt ): Laplace Transform is (s+a) /((s+a)2 + ω2)
• Ramp function ( t(n = 1, 2, 3,….) ): Laplace Transform is
• Unit ramp function ( t ): Laplace Transform is 1/s2
• Damped sine function ( e–at sin ωt ): Laplace Transform is
• Damped cosine function ( e–at cos ωt ): Laplace Transform is

• Damped cosine function ( e–at tn ): Laplace Transform is

• Differentiation theorem ( ) Laplace Transform is sF(s) – f(0 –)
• Integration theorem () Laplace Transform is
• Hyperbolic sine function ( sin h ωt ): Laplace Transform
• Hyperbolic cosine function ( cos h ωt ): Laplace Transform is
• Damped hyperbolic sine function ( e–at sin h ωt ): Laplace Transform is

• Damped hyperbolic cosine function ( e–at cos h ωt ): Laplace Transform is

• Initial value theorem: Laplace Transform is

• Final value theorem: Laplace Transform is

• Shifting theorem f(t ± a): Laplace Transform is e±as F(s)

Laplace Transform of a Periodic Function

f(t + nt) = f(t), where n is positive or negative integer.

where, F1(s) is the Laplace transform of the first cycle of the periodic function.

Analysis of Basic elements (R,L and C): Let I = I(s) be the Laplace transform of i= i (t).

Capacitor:

Inductor:

Resistor:

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write a comment
How see see all the subject notes released like this. Have you released notes of signals and systems like this.

Satish RastogiJun 3, 2016

Excellent notes sir. Thank a lot

Amruta WandreJul 14, 2016

good notes😊

Anil Kumar MuralaNov 26, 2016

Thank u

Stephen SafrinaMay 11, 2017

mm

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