KATHMANDU UNIVERSITY
School of Engineering Department of Electrical & Electronics Engineering
NETWORK ANALYSIS LABORATORY EXERCISE-II 2. Introduction Introduction to to MATLAB MATLAB – Simulink
SIMULINK is an interactive environment environment for modeling, analyzing, and simulating a wide variety of dynamic systems. A system in block diagram representation is built easily and the simulation results are displayed quickly. Simulink is started from the MATLAB command prompt by entering the following command: simulink. The Simulink Library Browser window should now appear on the screen. 2. 1 Buildi Building ng blocks blocks of MATLA MATLAB-Si B-Simul mulink ink
There There are are two major major clas classes ses of of elemen elements ts in Simu Simulin link: k: blocks blocks and lines. lines. Bloc Blocks ks are are used to to generate, modify, combine, output, and display signals. Lines are used to transfer signals from one block to another. Blocks The subfolders underneath the "Simulink" folder indicate the general classes of blocks available for us to use: • Continuous: Linear, continuous-time continuous-time system elements (integrators, transfer functions, statespace models, etc.) • Discrete: Linear, discrete-time discrete-time system elements (integrators, transfer functions, state space models, etc.) • Functions Functions & Tables: User -defined -defined functions and tables for interpolating function values • Math: Mathematical operators operators (sum, gain, dot product, etc.) • Nonlinear: Nonlinear operators (coulomb/viscous (coulomb/viscous friction, switches, relays, etc.) • Signals & Systems: Blocks for controlling/monitoring controlling/monitoring signal(s) and for creating subsystems • Sinks: Used to output or display signals (displays, scopes, graphs, etc.) • Sources: Used to generate various signals s ignals (step, ramp, sinusoidal, etc.)
Blocks have zero to several input terminals and zero to several output terminals. Unused input terminals are indicated by a small open triangle. Unused output terminals are indicated by a small triangular point. The block shown below has an unused input terminal on the left and an unused output terminal on the right.
Lines Lines transmit signals in the direction indicated by the arrow. Lines must always transmit signals from the output terminal of one block to the input terminal of another block.
1
A line can tap off of another line. This sends the original signal to each of two (or more) destination blocks. Lines can never inject a signal into another line; lines must be combined through the use of a block such as a summing junction. Building a System Example-2.1.1 The block diagram for a simple model consisting of a sinusoidal input multiplied by a constant gain is shown below:
5
Sine Wave
Gain Scope
This model will consist of three blocks: Sine Wave, Gain, and Scope. The Sine Wave is a Source Block from which a sinusoidal input signal originates. This signal is transferred through a line in the direction indicated by the arrow to the Gain Math Block. The Gain block modifies its input signal (multiplies it by a constant value) and outputs a new signal through a line to the Scope block. The Scope is a Sink Block used to display a signal (much like an oscilloscope). To build a system, a new model window is brought, in which to create the block diagram. This is done by clicking on the "New Model" button in the toolbar of the Simulink Library Browser. Building the system model is then accomplished through a series of steps: 1. The necessary blocks are gathered from the Library Browser and placed in the model window. 2. The parameters of the blocks are then modified to correspond with the system 3. Finally, the blocks are connected with lines to complete the model. Running Simulations To simulate the system, go to the Simulation menu and click on Start, or just click on the "Start/Pause Simulation" button in the model window toolbar.
2
Double-click the Scope block to view the output of the Gain block for the simulation as a function of time. Once the Scope window appears, click the "Autoscale" button in its toolbar (looks like a pair of binoculars) to scale the graph to better fit the window.
Example-2.1.2
Consider the first-order RL circuit as shown below. i
R
V
L
The differential equation representing the system is
=
−
Assume that: 3
L= 100H and R = 5 This system will be modeled in Simulink by using the system equation as above.
=
−
=
−5 100
To model this equation, insert a Sum block, two Gain blocks and an integrator block into a new model window. Change the parameters of the blocks as per the requirement. Connect the blocks with lines as shown in the figure below.
System Response to Step Input To simulate the system the applied input V is to be specified. Assume that a step input of V = 5V is applied at t = 0. Insert a Step block from the Sources subfolder into the model, and also add a Scope block from the Sinks subfolder to monitor the system's current, i.
The Step block must be modified to correctly represent the system. Double-click on it, and change the Step Time to 0 and the Final Value to 5. The Initial Value can be left as 0, since the V step input starts from 0 at t = 0. The Sample Time should remain 0 so that the Step block's input is monitored continuously during simulation. Next, run a simulation of the system (by clicking the "Start/Pause Simulation" button or selecting Simulation, Start). Once the simulation has finished, double-click on the Scope block to view the velocity response to the step input. Clicking on the "Autoscale" button (looks like a pair of binoculars) in the Scope window will produce the following graph -(a).
4
(a) (b) This graph (a) does not appear to show the current approaching a steady-state value, as expected for the first-order response to a step input. This result is due to the settling time of the system being greater than the 10 seconds the simulation was run. To observe the system reaching steadystate, click Simulation, Parameters in the model window, and change the Stop Time to 150 seconds. Now, re-running the simulation will result in the current graph as shown in (b). From this graph, we observe that the system has a steady-state current of about 1A, and a time constant of about 20 seconds. Let's check these results with our original equation. For a step input of V = 5V, the system equation is:
100
+5 =5
Setting di/dt = 0 gives a steady-state current of 1A, a result which agrees with the current graph above. To find the time constant of the system the characteristic equation as shown below can be used. 100s + 5 = 0 Solving this gives the characteristic root, s = -0.05, and thus the time constant is indeed 20 seconds ( = -1/s), as in the above the graph.
SUBSYSTEMS SIMULINK subsystems provide a capability within SIMULINK similar to subprograms in traditional programming languages. Example-2.1.3 To encapsulate a portion of an existing SIMULINK model into a subsystem, consider the SIMULINK model shown below and proceed as follows:
5
1. Select all the blocks and signal lines to be included in the subsystem with the bounding box as shown. 2. Choose Edit and select Create Subsystem from the model window menu bar. SIMULINK will replace the select blocks with a subsystem block that has an input port for each signal entering the new subsystem and an output port for each signal leaving the new subsystem. SIMULINK will assign default names to the input and output ports.
2.2 Time Response Analysis of first order system Example-2.2.1 Consider the first-order RL circuit as before. i
R
V
L
The differential equation representing the system is
=
−
Assume that: L= 100 H and R = 5 This system will be modeled in Simulink by using the system equation as above.
=
−
=
−5 100
The system will be modeled as under:
6
System response to Pulse Input Consider the response of the system if a pulse input is applied. To model a pulse input using SIMULINK, insert another Step block and a Sum block in the system as shown. The parameters for the original "Step" block can be left as they were before. Modify the "Step1" block parameters to the following: Step Time = 100 Initial Value = 0 Final Value = -5 These settings enable the "Step1" block to cancel out the input from the "Step" block starting at t = 100. To monitor the input of the system, V, we insert another scope into the model window as shown below:
Modify the simulation time (found by going to Simulation, then Parameters) to 200 seconds, and then run the simulation. After auto scaling the scopes recording the V and i signals, you should see graphs that look like:
7
From the output plot the time constant of the system is 20 seconds. At t = 100 seconds in the simulation, the system is within 2% of its steady-state response to the original step input of 5V. When V is instantaneously reduced to 0 at t = 100 seconds, it takes the system another 100 seconds (until t = 200 seconds) to respond to this new input. System response to Ramp Input Now apply a ramp input, V, to the system starting from inert condition. To model this, insert a Ramp block from the Sources subfolder and connect it so that it produces the system input signal, v. Also, insert Scope blocks into the model to monitor the voltage v and the current i.
Double-click on the Ramp block to modify it, and set the Slope equal to 1 (you can leave the Start Time and Initial Output as 0). These settings will steadily increase the voltage input by 1V every second, starting from V = 0 at t = 0. Also, set the simulation Stop Time to 100 seconds, and run the simulation. The simulation will result with the following input and output curves.
These plots show that if the input voltage of the circuit, v, is increased steadily, the current of the circuit, i, will continue to rise, and thus does not approach a specific steady-state value. Also note that as time passes, the current curve eventually settles into a straight line. So, the steady state response to the ramp input is linear, and it has a positive slope (i.e. the current of the system goes to infinity as time goes to infinity. Ramp Input with Saturation Apply a ramp input to the system as before, with the following changes: The input voltage v will not be allowed to exceed 50V. Thus, the system's input will appear as a ramp until its value reaches 50V. From that time forward, the saturated input will remain at 50V. To model this input in Simulink, we insert a Saturation block right after the Ramp block in the model window. The system model will be as under:
8
The Saturation block allows us to set an upper and lower limit for its input signal. If the signal to the block is between the minimum and maximum values we have set, the Saturation block passes it through unaltered. If the input signal is greater than the maximum, however, it outputs the set maximum value. Similarly, if the input signal is less than the minimum, the Saturation block simply outputs this userdefined minimum value. Double-click on the Saturation block to modify its parameters, and change its Upper Limit to 50 and its Lower Limit to 0. Now, run the simulation (change the Stop Time to 100 seconds), and view the v and i scope blocks.
Note that the input voltage, V, appears as a ramp input until it reaches 50V, and then stays at that maximum as time continues to pass. Also up to about t = 20 seconds, the current response of the circuit, i, is identical to what it was for the ramp input we analyzed previously. Beyond that time, the two plots take on very different shapes. 2.3 Time Response Analysis of second order system Example-2.3.1 Consider a second order system as shown below: 2
i
¼F
Vc
10V
4H
The governing equation of the system is
+
+
=
9
Let
=
and
=
, then
= =
1
[ −
−
]
The simulink model for the system would be as below:
Consider the following values for simulation: R =2 ; L = 4H; C = 1/4F. Also for the step input consider step time = 0, Initial value = 0, Final value =10. The response form the plot will be as under:
The output response is an under-damped case.
Assignments: 1. Make a Simulink model for the RC network, with your own values of R, C and V. Plot for the step input the output voltage across the capacitor. Comment on your plot.
2. For the RLC network make a Simulink model to simulate the network. Use these numerical values: (i) R = 8 Ω, L = 4H and C = 1/4F, V = 10 V and (ii) R = 20 Ω, L = 4H and C = 1/4F. Plot the response voltage across the capacitor in both the cases. Comment on the responses.
10
