Network Equations Using Laplace Transform

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.

Properties of Laplace Transform

• t-domain: L[K₁f₁(t) + K₂f₂(t)]
• s-domain: [K₁f₁(s) + K₂f₂(s)]

• t-domain: 

• s-domain: sⁿf(s) – s^n–1f(0) – s^n–2f′(0) – s^n–3f′(0)…..

• t-domain:

• s-domain:

• t-domain: L[f(at)]
• s-domain:

• t-domain: L[f(t – a)]
• s-domain: e^–as f(s)

• t-domain: L[e^–at f(t)]
• s-domain: f(s + a)

• t-domain: L[f₁(t) × f₂(t)]
• s-domain: f₁(s) ∙ f₂(s)

• t-domain: 

• s-domain:

• 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.

• 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 ( e^at ): 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 ( tⁿ (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.