RLC 11-30-2020

$I \ - \ \mathrm{Source \ Current} \ ; \ Q \ - \ \mathrm{C_1 \ charge} \ ; \ I_1 \ - \ \mathrm{R_2 \ current}$ $I_2 \ - \ \mathrm{L_2 \ current} \ ; \ I_3 \ - \ \mathrm{C_2 \ current} \ ; \ Q_3 \ - \ \mathrm{C_2 \ charge}$

$\mathrm{Circuit \ Equations:}$ $-V_S + I R_1 + L_1\dot{I} + \frac{Q}{C_1} +I_1R_2=0$ $\dot{Q} = I$ $I_1R_2 = L_2 \dot{I}_2$ $I_1R_2 = \frac{Q_3}{C_2}$ $\dot{Q}_3 = I_3$ $I = I_1+I_2+I_3$

$\mathrm{Rearranging \ equations \ to \ state \ space \ form:}$ $\left[\begin{matrix} \dot{I} \\ \dot{Q} \\ \dot{Q}_3 \\ \dot{I}_2\end{matrix}\right] = \left[\begin{matrix} -1&-1&-1&0 \\ 1&0&0&0 \\ 1&0&-1&-1 \\ 0&0&1&0 \end{matrix}\right]\left[\begin{matrix} I \\ Q\\ Q_3 \\ I_2\end{matrix}\right] + \left[\begin{matrix} 1 \\ 0\\ 0\\ 0\end{matrix}\right]V_s$ $\implies \dot{x} = Ax + B V_s$ $\implies I = \left[\begin{matrix} 1 & 0& 0& 0\end{matrix}\right]x$ $x(0) = \left[\begin{matrix} 0 \\ 0\\ 0\\ 0\end{matrix}\right]$

$\mathrm{ODE \ system \ solved \ using \ Implicit \ Euler:}$ $x(k+1) = (I - \delta t A)^{-1} (x(k) + \delta t \ BV_s)$

$\mathrm{Heat \ Dissipation:}$ $H = \int_{0}^{\infty} R_1I^2 \ dt \approx 37.5$

 1
 2
 3
 4
 5
 6
 7
 8
 9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27

clear all, clc

% Initial Condition:
x(:,1) = [0;0;0;0];

% Time initialisation:
dt     = 1e-4; tf = 50; t   = 0:dt:tf;

% System Matrices and source voltage:
A      = [-1 -1 -1 0;1 0 0 0;1 0 -1 -1;0 0 1 0]; B      = [1;0;0;0]; Vs     = 10;

% Heat dissipation initialisation:
H      = 0;

for k = 1:length(t)-1

    % Calculating I:
    I(k)     = [1 0 0 0]*x(:,k);

    % Heat dissipation integral calculation:
    H        = H + I(k)^2*dt;

    % Numerically integrating state space form - Implicit Euler:
    x(:,k+1) = (eye(4) - dt*A)\(x(:,k) + dt*B*Vs);
end

ANSWER = H