Laplace transforms: Simple Electrical Network

Problem Statement

Using the formulas

$E(t)=L\frac{d{i}_R}{dt}+R{i}_C$
$RC\frac{d{i}_R}{dt}+i_R-i_C=0$

Solve the system when V0 = 50 V, L = 0.5 h, R = 60 Ω, C = 10^-4 f, and the currents are initially zero.

Solution

Solve the system when V0 = 50 V, L = 0.5 h, R = 60 Ω, C = 10^{-4} f, and the currents are initially zero. Substituting numbers into the equations, we have

$0.5\frac{d{i}_1}{dt}+60i_2=50$

$60(10^{-4})\frac{d{i}_2}{dt}+i_2-i_1=0$

Applying the Laplace transform to each equation gives

$\frac{1}{2}(s{ℒ}\left\{i_1\right\}-i_1(0))+60{ℒ}\left\{i_2\right\}=50$

$0.006(s{ℒ}{i}_2-i_2(0))+{ℒ}\left\{i_2\right\}-{ℒ}\left\{i_1\right\}=0$

$\Rightarrow \frac{1}{2}s{ℒ}\left\{i_1\right\}+60{ℒ}\left\{i_2\right\}=\frac{50}{s}$

$-200{ℒ}\left\{i_1\right\}+(s+200){ℒ}\left\{i_2\right\}=0$

Solving for ${ℒ}\left\{i_1\right\}$ gives