Mathematical models and simulation of electrical systems

RLC circuit modeling and simulation in Scilab and Xcos

For a given resistor-inductor-capacitor (RLC) circuit with the following input parameters:

• constant input voltage: u_IN = 12 V
• equivalent resistance:  R₁ = 0.9 Ω
• equivalent inductance L₁ = 0.115 H
• equivalent capacitance C₁ = 0.02 F
• simulation time t = 1 s
• initial time t₀ = 0 s
• initial voltage across the capacitor u_C10(t₀) = 0 V
• initial voltage gradient across the capacitor du_C10/dt₀ = 0 V/s

we’ll perform the following tasks:

• find the governing differential equation
• find the transfer function
• find the state-space model
• simulate in Scilab the differential equation
• simulate in Scilab the transfer function
• simulate in Scilab the state-space model
• simulate in Xcos the differential equation
• simulate in Xcos the circuit using Electrical library components
• simulate in Xcos the transfer function
• simulate in Xcos the state-space model

for each activity a plot will be generated for the output and some of the states.

Observation: For the equations to be more readable we are going to remove the time variable (t), from the electrical parameter, for example i_L1(t) will be i_L1 and u_R1(t) will be u_R1. Also, bear in mind that lower case parameters are variable in time (e.g. u(t), i(t)), while upper case parameters are constant in time (e.g. R₁, L₁, C₁).

Differential equation of the RLC circuit

From Kirchhoff’s Current and Voltage Law (KCL and KVL) we can write the following equation:

\[u_{R_{1}} + u_{L_{1}} + u_{C_{1}} = u_{IN} \tag{1}\]

The voltage across the resistor is:

\[u_{R_{1}} = R_{1} i_{R_{1}} \tag{2}\]

The voltage across the inductor is:

\[u_{L_{1}} = L_{1} \frac{di_{L_{1}}}{dt} \tag{3}\]

The current through the capacitor is:

\[i_{C_{1}} = C_{1} \frac{du_{C_{1}}}{dt} \tag{4}\]

Also, the current is the same throughout the loop, which means:

\[i_{R_{1}} = i_{L_{1}} = i_{C_{1}} \tag{5}\]

Replacing (5), (4), (3) and (2) in (1) gives the governing differential equation of the RLC circuit:

\[\bbox[#FFFF9D]{L_{1} C_{1} \frac{d^2u_{C_{1}}}{dt^2}+R_{1} C_{1} \frac{du_{C_{1}}}{dt}+u_{C_{1}} = u_{IN}} \tag{6}\]

Transfer function of the RLC circuit

Applying Laplace’s transform to equation (6) gives:

\[ \begin{split}
\mathcal {L} \left [ L_{1} C_{1} \frac{d^2u_{C_{1}}}{dt^2} \right ] &= L_{1} C_{1} s^2 U_{C_{1}}(s) – s U_{C_{1}}(t_{0}) – \frac{dU_{C_{1}}}{dt_{0}}\\
\mathcal {L} \left [ R_{1} C_{1} \frac{du_{C_{1}}}{dt} \right ] &= R_{1} C_{1} s U_{C_{1}}(s) – U_{C_{1}}(t_{0})\\
\mathcal {L} \left [ u_{C_{1}}(t) \right ] &= U_{C_{1}}(s)\\
\mathcal {L} \left [ u_{IN}(t) \right ] &= U_{IN}(s)\\
\end{split} \]

From the initial data we also know the initial conditions for the capacitor voltage and voltage derivative (gradient) at the initial time are:

\[\begin{split}
U_{C_{1}}(t_{0}) = 0\\
\frac{dU_{C_{1}}}{dt_{0}}=0
\end{split}\]

Replacing the Laplace transforms and initial conditions in (6) gives:

\[L_{1} C_{1} s^2 U_{C_{1}}(s) + R_{1} C_{1} s U_{C_{1}}(s) +U_{C_{1}}(s) = U_{IN}(s) \tag{7}\]

From (7) and defining U_C1(s) as the output variable, we can extract the transfer function of the RLC circuit as:

\[\bbox[#FFFF9D]{H(s) = \frac{U_{C_{1}}(s)}{U_{IN}(s)}=\frac{1}{L_{1} C_{1} s^2 + R_{1} C_{1} s + 1}} \tag{8}\]

State-space model of the RLC circuit

We can rewrite equation (6) as:

\[\frac{d^{2}u_{C_{1}}}{dt^2} = – \frac{R_{1}}{L_{1}} \frac{du_{C_{1}}}{dt} – \frac{1}{L_{1} C_{1}} u_{C_{1}} + \frac{1}{L_{1} C_{1}} \cdot u_{IN} \tag{9}\]

Now, we define two state variables x₁ and x₂ as:

\[\left\{\begin{matrix} x_1 = u_{C_{1}} \tag{10}\\ x_2 = \dot{u_{C_{1}}} \end{matrix}\right.\]

Applying time derivation to (10) gives:

\[\left\{\begin{matrix} \dot{x_{1}} &=& \dot{u_{C_{1}}} &=& x_{2} \tag{11}\\\dot{x_{2}} &=& \ddot{u_{C_{1}}} &=& – \frac{R_{1}}{L_{1}} x_{2} – \frac{1}{L_{1} C_{1}} x_{1} + \frac{1}{L_{1} C_{1}} u_{IN} \end{matrix}\right.\]

We can write (11) in matrix multiplication form:

\[\begin{Bmatrix} \dot{x_{1}} \tag{12}\\ \dot{x_{2}} \end{Bmatrix}=\begin{bmatrix} 0 & 1\\-\frac{1}{L_{1}C_{1}} & -\frac{R_{1}}{L_{1}} \end{bmatrix} \cdot \begin{Bmatrix} x_{1}\\x_{2} \end{Bmatrix} + \begin{bmatrix} 0\\\frac{1}{L_{1}C_{1}} \end{bmatrix} \cdot \begin{Bmatrix} u_{IN} \end{Bmatrix}\]

From (12) we can extract the A and B matrices of the state-space:

\[ \bbox[#FFFF9D]{A =\begin{bmatrix} 0 & 1\\ -\frac{1}{L_{1}C_{1}} & -\frac{R_{1}}{L_{1}} \end{bmatrix}, \quad B =\begin{bmatrix} 0\\ \frac{1}{L_{1}C_{1}} \end{bmatrix}} \]

We also define the state variable u_C1(t) as being the output of the system:

\[y(t) = u_{C_{1}} \tag{13}\]

We can write (13) in matrix multiplication form:

\[\begin{Bmatrix} y \end{Bmatrix}= \begin{bmatrix} 1 & 0 \end{bmatrix} \cdot \begin{Bmatrix} x_{1} \tag{14}\\ x_{2} \end{Bmatrix} + \begin{bmatrix} 0 \end{bmatrix} \cdot \begin{Bmatrix} u_{IN} \end{Bmatrix}\]

From (14) we can extract the C and D matrices of the state-space:

\[ \bbox[#FFFF9D]{C = \begin{bmatrix} 1 & 0 \end{bmatrix}, \quad D = \begin{bmatrix} 0 \end{bmatrix}} \]

RLC circuit differential equation – Scilab simulation

One way of solving the differential equation of the RLC circuit is by using Scilab ode() function. In the Scilab instructions below we are defining the input parameters, the differential equations, initial parameters, solve the differential equation and plot the results.

// Clean figure, console and workspace variables
clf()
clc()
clear()

// Define input parameters
uIN = 12; // [V]
R1 = 0.9; // [ohm]
C1 = 0.02; // [F]
L1 = 0.115; // [H]
// Define t [s]
t0=0; tinc=0.001; tf=1; t=t0:tinc:tf;

// Define differential equation
function dx=f(t,x)
    dx(1)=x(2)
    dx(2)=(1/(L1*C1))*(uIN-R1*C1*x(2)-x(1))
endfunction
// Define initial conditions
uC10 = 0; // [V]
upC10 = 0; // [V/s]
// Solve differential equation
uC1=ode([uC10;upC10],t0,t,f);;
// Plot numeric solution
subplot(2,1,1)
plot(t,uC1(1,:),'b'), xgrid()
title("x-engineer.org","Color",'b')
ylabel('$\large{u_{C1}(t) \text{ [V]}}$','fontsize',2)
subplot(2,1,2)
plot(t,uC1(2,:),'r'), xgrid()
ylabel('$\large{\frac{du_{C1}(t)}{dt} \text{ [V/s]}}$','fontsize',2)
xlabel('$\large{t}$','fontsize',2)

After running the Scilab instructions above, we get the following plot.

The plot represents the voltage across the capacitor u_C1(t) (blue) and the gradient of the voltage across the capacitor du_C1/dt (red).

RLC circuit transfer function – Scilab simulation

The same results we are going to have using the transfer function. In the Scilab instruction below we are defining the system (RLC circuit) as a transfer function using Scilab’s syslin() function. After, we run a simulation for a step input of u_IN and time t.

// Define transfer function
s=poly(0,'s');
Hs=[1/(L1*C1*s^2+R1*C1*s+1)];
sys=syslin('c',Hs)
// Run step response
uC1 = uIN*csim('step',t,sys);
figure(2)
plot(t,uC1), xgrid()
hf=gcf();
hf.background = -2;
title("x-engineer.org","Color",'b')
ylabel('$\large{u_{C1}(t) \text{ [V]}}$','fontsize',2)
xlabel('$\large{t} \text{ [s]}$','fontsize',2)

As expected, the results is the same as for the differential equation integration (simulation).

RLC circuit state-space – Scilab simulation

The same results we are going to have using the state-space model. In the Scilab instruction below we are defining the system (RLC circuit) as a state-space model using Scilab’s syslin() function. After, we run a simulation for a step input of u_IN and time t.

// Define state-space model
A = [0 1;-1/(L1*C1) -R1/L1];
B = [0; 1/(L1*C1)];
C = [1 0];
D = [0];
X0 = [uC10; upC10];
sys = syslin('c',A,B,C,D,X0);
// Run step response
uC1 = uIN*csim('step',t,sys);
figure(3)
plot(t,uC1,'m'), xgrid()
hf=gcf();
hf.background = -2;
title("x-engineer.org","Color",'b')
ylabel('$\large{u_{C1}(t) \text{ [V]}}$','fontsize',2)
xlabel('$\large{t} \text{ [s]}$','fontsize',2)

As expected, the results is the same as for the differential equation integration (simulation).

RLC circuit differential equation – Xcos simulation

In order to integrate the differential equation of the RLC circuit into Xcos, we are going to write equation (6) as:

\[\frac{d^{2}u_{C_{1}}}{dt^{2}}= \frac{1}{L_{1}C_{1}} \left ( u_{IN} – R_{1}C_{1} \frac{du_{C_{1}}}{dt} – u_{C_{1}} \right ) \tag{15}\]

Equation (15) is modelled into Xcos using block diagram as:

Parameters uIN, R1, L1 and C1 are loaded into the Scilab workspace from the previous examples.

The simulation is run for 1 s and outputs the following plot.

The plot represents the voltage across the capacitor u_C1(t).

RLC circuit electrical components – Xcos simulation

To verify that our differential equation, transfer function and state-space model are correct, we are going to model the RLC circuit using the Electrical library blocks from Xcos.

The parameters uIN, R1, L1 and C1 are defined in the Scilab workspace.

Running the simulation will output the same time variation for u_C1(t), which proves that the differential equation, transfer function and state-space model of the RLC circuit are correct.

RLC circuit transfer function – Xcos simulation

In this approach we are going to use the transfer function of the RC circuit and simulate it in Xcos.

The parameters uIN, R1, L1 and C1 are defined in the Scilab workspace.

Running the simulation will output the variation of the voltage across the capacitor u_C1(t), which is the same as the one in the previous examples.

RLC circuit state-space – Xcos simulation

In this approach we are going to use the state-space model of the RC circuit and simulate it in Xcos.

Parameters uIN, A, B, C, D and X0 are loaded into the Scilab workspace from the previous examples.

Running the simulation will output the variation of the voltage across the capacitor u_C1(t), which is the same as the one in the previous examples.