Understanding Transient Analysis with Scilab

in utopian-io •  7 years ago  (edited)

image.png

Intended Learning Objectives
At the end of the tutorial, readers should be able to learn how to:
1. determine the step response of a series RLC circuit;
2. perform the transient analysis on a series RLC using Scilab;
3. Plot the step response of a series RLC circuit using Scilab.
II. Requirements
image.png
(image credited to scilab.org)

To be able to follow the discussion and tutorial, readers should have a desktop PC or laptop ( Windows 7, 8, or 10) and an installed a latest Scilab (an open source high level numerical analysis and programing) app either 32 or 64 bit that can be downloaded at scilab.org.
III. Difficulty
Advance
IV. Introduction
In AC circuit analysis, passive component behavior can be characterized in two state: transient and steady-state. The rapid changes in the current and voltage level in each component at the moment you turn on a switch on a fully functional circuit that contains resistors, inductors, capacitors and, power sources is called the transient response. The transient response is the natural response of the circuit when a sudden inrush of current or voltage flow through out the circuit component until parameters normalize. The equilibrium state, on the other hand, is called a steady-state response , where current and voltage values in the circuit is normalize. For this tutorial, i will explain how to perform transient analysis on a series RLC to determine the step response of the circuit. The numerical analysis is carried out through writing a program in Scilab that enables us to plot the step response of a series RLC circuit. In addition, in this analysis, we would be dealing with differential equation that would represent the energy-storage elements.
V. Step Response of series RLC circuit
We consider the series RLC circuit in figure 1. The step response is observe when you suddenly turn on and off the switch in the circuit. Step response signifies the transient state of the circuit where the natural behavior can be observe by repeatedly change from t =0 to t > 0.
image.png
Figure 1: Series RLC circuit

Apply KVL (Kirchhoff's Voltage Law) around the loop for t >0. Kirchhoff's Voltage Law states that the sum of all voltage inside a closed loop is equal to zero. So, we obtain the following equation:
image.png
Figure 2: Applying KVL and

From figure2, the equation derives from the inductor is a factor of the inductors' self inductance and of the derive of current (i) with respect to time (t). So, we have the equavalent equation at inductor as L di/dt . The current passing through the circuit could be represented as the relationship with the circuit connected capacitance and voltage. The current across the circuit (i) is equal to C dv/dt. Substituting the current (i) as a factor of capacitance and voltage, we have the following equation:
image.png

Simplying the equation derive above, we will consider a two state in our analysis which is the transient and steady-state.
image.png

Transient responseimage.png is the component where the total response dies out with time. The transient response of a circuit is either overdamped, underdamned, and critically-damped which is defined by the following equations below.
image.png

To explain the different transient response of a series RLC circuit, we will use the circuit shown in figure 3, we will find v(t) and i(t) for t > 0 by considering three cases: R = 5 ohms, R = 4 ohms, and R = 1 ohms.
image.png
Figure 3: Example Circuit
VI. Transient Analysis with Scilab Implementation
We will create a Scilab program that will allow us to easily change values of R using the input('statement'); function. Then, write the a program to implement the steps in conducting transient analysis. Now, we write a code that enables to accept values of R, L, and C.
R1=input('Enter value of R1 in ohms:');
R2=input('Enter value of R2 in ohms:');
L=input('Enter value of L in Henry:');
C=input('Enter value of C in Farad:');
Vs=input('Enter value of source voltage (Vs) in volts');
In determining the circuit's transient response, we need to determine the initial values of current (Io) and voltage (Vo). Io is equal to the current experience between all connected resistor as to the voltage (Vs ) applied to the circuit, while Vo is the voltage across R2 when switch is flipped at t>0. So we write the code as:
Io = Vs/(R1+R2); //initial current value
Vo = R2(Io); //initial voltage value
Next, we will determine the transient condition of the circuit either it is overdamped, underdamped, or critically-damped. We will determine characteristic root a = image.png and w0 = image.png for t > 0. Take note that the value to be used in the equation stated earlier ar R1, L, and C. To be able to take the square root of a function or a number, use the sqrt() in Scilab. The following conditions will determine the transient response of the circuit:
1. when a > w0, the circuit is overdamped;
2. when a < w0, the circuit is underdamped; and,
3. when a = w0, the circuit is critically damped.
We start by solving the value of a and w0.
a=R1/(2*L); // coefficient a
w0=1/sqrt(L*C); //coefficient w0
Using if and elseif function, we will create a program that will implement the conditions stated earlier. Inside the if() and elseif() function, we will have the different mathematical operations and analysis for each of the condition stated. We will also use the equation derive in the the prior section of this text.
Case 1: When R = 5 ohms. For t < 0, the switch is closed for a long time by which capacitors behaves like open circuit while inductor acts like a short circuit.
For case1, we initialize a condition for overdamped response of the circuit which is characterize by a > w0. Inside the if function, we will implement the derive equation for overdamped response and the derivative of this equation.
image.png

To be able to solve the value of v(t), we need to manipulate the equations to solve for A1 and A2. However, it will become more complicated if we try to use subtitution or elimination technique in our program implementation. So, we will use the matrix functionality to solve the values for A1 and A2. So, we write the code as:
if(a>w0) then
t=0;
//Matrix operation to find A1 and A2
constV=[(Vo-Vs), Io/C];
coefA=[exp(s1*t),exp(s2*t);s1*exp(s1*t),s2*exp(s2*t)];
A=inv(coefA)*constV(:);
Next, we include a code. The program above is derive from this simplified matrix taken from the equation stated earlier.
image.png

To finalize all operation for case 1, we will need to implement a code that will allow as to plot v(t) as a function of time as 0 < t < 10 sec with plot step of 0.1.
disp(A);
t=[0,0.1:10];
Vt=Vs+ A(1)*exp(s1*t)+A(2)*exp(s2*t);
disp('V(t)=');
disp(Vt(:));
plot(t,Vt);
Case 2: When R = 4 ohms. For t < 0, the switch is closed for a long time by which capacitors behaves like open circuit while inductor acts like a short circuit. For case2, we initialize a condition for critically-damped response of the circuit which is characterize by a = w0. We will continue by using elseif() function.
//When a = w0 ,the circuit is critically-damped response
elseif(a==w0) then
t=0;
//Matrix operation to find A1 and A2
constV=[(Vo-Vs), Io/C];
coefA=[exp(-a*t),(t*exp(-a*t));(-a*exp(-a*t)),1];
A=inv(coefA)*constV(:);
image.png

This time the matrix of equation that we will be manipulating in the code is shown above. Instead of using s1 and s2, we will using the characteristic root a hence at critically-damped response, the circuit is in resonance. Next, we will plot and display the value of v(t) at a t is equal to 0 < t <10.
disp(A);
t=[0,0.05:10];
Vt=Vs+A(1)*exp(-a*t)+A(2)*t*(exp(-a*t)(:));
disp('V(t)=');
disp(Vt(:));
plot(t,Vt);
For the voltage Vt=Vs+A(1)*exp(-a*t)+A(2)*t*(exp(-a*t)(:)); plot equation, directly multiplying matrix with different number of rows and columns results to an error so you need to determine when to transpose the matrix. For instance, multiplying A(2)*t results to 1x10 matrix which could not be multiplied directly to an identical 1x10 matrix at the result of exp(-a*t). So, we need to transpose the last part of the equation: (exp(-a*t)(:).
Case 3: We will still be using the same circuit but now will consider R = 1 ohms in building our program. For case 3, we will identify the underdamped response of the circuit that will satisfy a < w0.
//When a > w0 ,the circuit is underdamped response
elseif(a<w0) then
t=0;
//Matrix operation to find A1 and A2
wd=imag(s2);
constV=[(Vo-Vs), Io/C];
coefA=[cos(wd*t)*exp(-a*t),t*sin(wd*t)*exp(-a*t);
exp(-a*t)*wd*sin(wd*t),exp(-a*t)*wd*cos(wd*t)];
A=inv(coefA)*constV(:);
image.png
Thw prior cases, only involves Vt as an exponential function of t. For this case, we will involve trigonometric functions such as sin() and cos(). In addition, since the underdamped response of a circuit involves the complex properties of the circuit response, we also use imag() to call out the imaginary properties of the characteristic root a. Then, we write a code to display and plot the underdamped response as a function Vt at t < 10.
disp(A);
t=[0,0.1:10];
Vt=Vs+A(1)*cos(wd*t)*exp(-a*t)(:)+A(2)*t*sin(wd*t)(:)*exp(-a*t);
disp('V(t)=');
disp(Vt(:));
plot(t,Vt);
end
In addition, we will also use matrix transpose to be able to perform multiplication of matrix which i have discuss earlier in case 2. Here is the code in the SciNotes.
image.png
VII. Running the code in Scilab Console
We will need to run thr code from the SciNotes to the Scilab Console by clicking the simulation button.
Case 1:(Overdamped Response) Input values of R1 = 5 ohms, R2= 1 ohms, L = 1 Henry, C = 0.25 Farad and Vs = 24 V. In this condition, the circuit components does not oscillate as it returns to steady-state. So, we can see in the resulting plot that there is an exponential decay of v(t) value as it approaches steady-state (t > 10).
image.png
After inputting all the values and Vt values are displayed, a overdamped response plot shows in the Scilab graphical window.
image.png

Case 2:(Critically-damped Response) Input values of R1 =4 ohms, R2= 1 ohms, L = 1 Henry, C = 0.25 Farad and Vs = 24 V. The plot shows a quick and rapid changes in v(t) as it approaches steady-state without oscillation.
image.png
After inputting all the values and Vt values are displayed, a critically-damped response plot shows in the Scilab graphical window.
image.png

Case 3:(Underdamped Response) Input values of R1 =1 ohms, R2= 1 ohms, L = 1 Henry, C = 0.25 Farad and Vs = 24 V. We can see in the plot a gradually decreasing amplitude of oscillation that approaches to zero.
image.png
After inputting all the values and Vt values are displayed, a critically-damped response plot shows in the Scilab graphical window.
image.png
VIII. Summary
Passive component behavior can be characterized in two state: transient and steady-state. Transient happen the moment you flip a switch on or off. The transient response (step response) of a circuit is determined by the characteristic roots a and w0 of the circuit which follows the conditions:
1. when a > w0, the circuit is overdamped;
2. when a < w0, the circuit is underdamped; and,
3. when a = w0, the circuit is critically damped.
The plot below shows the three conditional response od the circuit during the transient state.
image.png
This plot is obtain by simultaneously running all conditions in the Scilab program.
In terms of the Scilab functinality and implementation, we have discuss the usage of a matrix and numerical operations in a more advance way. We also have use input`` functionality, that allows as to repeatedly input different values of resistance, inductance and capacitance in the circuit. Furthermore, we also have discuss the usage ofimag()```. In which, it allow as to call out complex imaginary (j) in a function or equation.
IX. Scilab Code
This is the complete code that we have produce in this tutorial.
clc;
clear;
disp('Transient Analysis')
R1=input('Enter value of R1 in ohms:');
R2=input('Enter value of R2 in ohms:');
L=input('Enter value of L in Henry:');
C=input('Enter value of C in Farad:');
Vs=input('Enter value of source voltage (Vs) in volts');
Io=Vs/(R1+R2); //initial current value
Vo=R2*(Io); //initial voltage value
a=R1/(2*L); // coefficient a
w0=1/sqrt(L*C); //coefficient w0
s1=(-a)+sqrt((a^2)-(w0^2));
s2=(-a)+(-1*(sqrt((a^2)-(w0^2))));
//When a > w0 ,the circuit is overdamped response
if(a>w0) then
t=0;
//Matrix operation to find A1 and A2
constV=[(Vo-Vs), Io/C];
coefA=[exp(s1*t),exp(s2*t);s1*exp(s1*t),s2*exp(s2*t)];
A=inv(coefA)*constV(:);
disp(A);
t=[0,0.1:10];
Vt=Vs+A(1)*exp(s1*t)+A(2)*exp(s2*t);
disp('V(t)=');
disp(Vt(:));
plot(t,Vt);
//When a = w0 ,the circuit is critically-damped response
elseif(a==w0) then
t=0;
//Matrix operation to find A1 and A2
constV=[(Vo-Vs), Io/C];
coefA=[exp(-a*t),(t*exp(-a*t));
(-a*exp(-a*t)),1];
A=inv(coefA)*constV(:);
disp(A);
t=[0,0.05:10];
Vt=Vs+A(1)*exp(-a*t)+A(2)*t*(exp(-a*t)(:));
disp('V(t)=');
disp(Vt(:));
plot(t,Vt);
//When a > w0 ,the circuit is underdamped response
elseif(a<w0) then
t=0;
//Matrix operation to find A1 and A2
wd=imag(s2);
constV=[(Vo-Vs), Io/C];
coefA=[cos(wd*t)*exp(-a*t),t*sin(wd*t)*exp(-a*t);
exp(-a*t)*wd*sin(wd*t),exp(-a*t)*wd*cos(wd*t)];
A=inv(coefA)*constV(:);
disp(A);
t=[0,0.1:10];
Vt=Vs+ A(1)*cos(wd*t)*exp(-a*t)(:)+A(2)*t*sin(wd*t)(:)*exp(-a*t);
disp('V(t)=');
disp(Vt(:));
plot(t,Vt);
end
Reference:
Theoretical reference and circuit images credited to Alexander and Sadiku, Fundamentals of Electric Circuit
Curriculum
You can read my other tutorial about Circuit theorems and applications with examples using Scilab.
AC Circuit Ananlysis with Scilab


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      ·  7 years ago 

    Congratulations! This post has been upvoted by the communal account, @steemph.cebu by juecoree being run at Teenvestors Cebu (Road to Financial Freedom Channel). This service is exclusive to Steemians following the Steemph.cebu trail at Steemauto. Thank you for following Steemph.cebu curation trail!

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      Thanks for sharing your learnings... We never done that before😣 Performing such activity will help students to understand more behind its behavior. Through experiments and tests, students will broad their knowledge in a certain topic😊

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        Hi @dapmonylia ! I am glad it helps you. Yes, hand-on activities is very essential in learning basic and advance circuit theory. Appreciated! Cheers!

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            Thanks @laxam

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