Laplace Transform and Its Application in Circuits

Abstract

Most of the circuits introduced so far have been analyzed in time domain. This means that the input to the circuit, the circuit variables, and the responses have been presented as a function of time. All the input functions such as unit step, ramp, impulse, exponential, sinusoidal, etc. have been introduced as a time-dependent variable, and their effects on circuits have been identified directly as a function of time. This required utilization of differential equations and solutions in time domain. However, high-order circuits result in high-order differential equations, which, considering the initial conditions, sometimes are hard to solve. In addition, for circuits which are exposed to a spectrum of frequencies such as filters, the time domain analysis is a limiting factor.

Laplace Transform

• 7.1 Find the Laplace transform of the following functions:

  • (a) f(t) = 10u(t)

  • (b) f(t) =  − 20δ(t)

  • (c) f(t) =  − 5tu(t)

  • (d) f(t) = 10e ^−30t u(t)

  • (e) f(t) = 3te ^−5t u(t)

  • (f) f(t) = 20 sin 75tu(t)

  • (g) f(t) = sin² ωtu(t)

  • (h) f(t) = cos² ωtu(t)

  • (i) f(t) = 10 sin (5t+30)u(t)

  • (j) $f(t)=110\sqrt{2}\cos 377t\ u(t)$

  • (k) f(t) = 12 sinh 20t u(t)

  • (l) f(t) = 10 cosh 2πt u(t)

• 7.2 Find the Laplace of the following function operations:

  • (a) f(t) = 2u(t)+5tu(t)

  • (b) f(t) = 10 sin 20t+5 cos 20t

  • (c) f(t) = 5 sin 3t −  cos 15t

  • (d) f(t) = 3e ^−10t sin 15t

  • (e) f(t) = 75e ^−100t cos 314t

  • (f) f(t) = 3e ^−10t sinh 15t

  • (g) f(t) = 75e ^−100t cosh 314t

  • (h) f(t) = 13tu(t − 5)

  • (i) f(t) = 10te ^−3t u(t − 4)

  • (j) f(t) = 5t ³ u(t)

• 7.3 Find the Laplace inverse of the following functions:

  • (a) F(s) = 1+s

  • (b) $F(s)=\frac{1}{s+5}$

  • (c) $F(s)=\frac{s}{s+5}$

  • (d) $F(s)=\frac{s-1}{s+5}$

  • (e) $F(s)=\frac{1}{s^2+25}$

  • (f) $F(s)=\frac{s+1}{s^2+25}$

  • (g) $F(s)=\frac{s^2-1}{s^2+25}$

  • (h) $F(s)=\frac{1}{s^2-25}$

  • (i) $F(s)=\frac{s+1}{s^2-25}$

  • (j) $F(s)=\frac{1}{s^2+7s+12}$

  • (k) $F(s)=\frac{s+1}{s^2+7s+12}$

  • (l) $F(s)=\frac{s^2+5s+6}{s^2+7s+12}$

  • (m) $F(s)=\frac{1}{s^2+6s+100}$

  • (n) $F(s)=\frac{s+1}{s^2+6s+100}$

  • (o) $F(s)=\frac{s^2+1}{s^2+6s+100}$

  • (p) $F(s)=\frac{s^2+s+1}{s^2+6s+100}$

  • (q) $F(s)=\frac{1}{s\left({s}^2+100\right)}$

  • (r) $F(s)=\frac{1}{s\left(s+3\right)\left(s+7\right)}$

  • (s) $F(s)=\frac{\left(s+1\right)\left(s+2\right)}{s\left(s+3\right)\left(s+7\right)}$

  • (t) $F(s)=\frac{1}{s^3{\left(s+3\right)}^2\left(s+7\right)}$

  • (u) $F(s)=\frac{1}{s}{e}^{-3s}$

  • (v) $F(s)=\frac{1}{s^2}\left({e}^{-3s}+{e}^{-5s}\right)$

  • (w) $F(s)={e}^{-10s}\frac{s}{s+5}$

  • (x) $F(s)=\frac{e^{-s}}{s^2+25}$

  • (y) $F(s)=\frac{\left(s+1\right){e}^{2s}}{s^2+25}$

  • (z) $F(s)=\frac{20}{{\left(s+3\pi \right)}^5}$

• 7.4 Prove that:

  • (a) $L\left\{\cosh at\cos at\right\}=\frac{s^3}{s^4+4{a}^4}$

  • (b) $L\left\{{t}^2\cos\ \omega t\right\}=\frac{2s\left({s}^2-3{\omega}^2\right)}{{\left({s}^2+{\omega}^2\right)}^3}$

• 7.5 Find the convolution of h(t) = f(t) ∗ g(t), wherein f(t) & g(t) are:

  • (a) f(t) = u(t) − u(t − 5), g(t) = 3u(t − 4) − 3u(t − 5).

  • (b) f(t) = u(t) − u(t − 5), g(t) = 3tu(t) − 3(t − 5)u(t − 5) − 3u(t − 5).

  • (c) f(t) = 5u(t) − u(t − 1) − u(t − 2) − 3u(t − 3), g(t) = 3tu(t) − 6(t − 1)u(t − 1)+3(t − 2)u(t − 2).

• 7.6 Do the following convolution integrals using Laplace and direct method:

  • (a) f ₁(t) ∗ f ₂(t)

• (b) f ₃(t) ∗ f ₄(t)

• (c) f ₅(t) ∗ f ₆(t)

• 7.7 Solve the following differential equations using Laplace transform:

(a)	$\dot{y}+9y=0$	y(0) = 5
(b)	$\ddot{y}+9y=0$	y(0) = 0,y ′(0) = 2
(c)	$\ddot{y}+2\dot{y}=10u(t)$	y(0) = 1,y ′(0) =  − 1
(d)	$\ddot{y}+2\dot{y}=10 tu(t)$	y(0) = 1,y ′(0) =  − 1
(e)	$\ddot{y}-2\dot{y}-3y=\dot{\delta}$	y(0) = 1,y ′(0) = 7
(f)	$4\ddot{y}+y=0$	y(0) = 1,y ′(0) =  − 2
(g)	$4\ddot{y}+y=\sin (t)$	y(0) = 0,y ′(0) =  − 1
(h)	$\ddot{y}+2\dot{y}+5y=0$	y(0) = 2,y ′(0) =  − 4
(i)	$\ddot{y}+2y=u(t)-u\left(t-1\right)$	y(0) = 0,y ′(0) = 0
(j)	$\ddot{y}+3\dot{y}+2y=\delta \left(t-a\right)$	y(0) = 0,y ′(0) = 0

• 7.8 Using Laplace transform find i ₁(t), i ₂(t), v(t).

• 7.9 Using Laplace transform find v _o(t).

• 7.10 Using Laplace transform find v _o(t).

• 7.11 Using Laplace transform find I ₁, V _o(t).

• 7.12 Using Laplace transform find i(t), v(t).

• 7.13 Using Laplace transform find i(t), v(t).

• 7.14 Using Laplace transform find v ₁(t), v ₂(t), v ₃(t).

• 7.15 Using Laplace transform find v ₁(t), v ₂(t), i ₁(t), i ₂(t).

• 7.16 Using Laplace transform find v ₁(t), v ₂(t), i ₁(t), i ₂(t).

• 7.17 Using Laplace transform find i ₁(t), i ₂(t), i ₃(t).

• 7.18 Using Laplace transform find v ₁(t), v _o(t).