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In a previous lab you were directed to use node-voltage analysis to find the voltages at all nodes in an electrical circuit.
The procedure for analyzing a circuit with the node method is summarized with the following steps.
1. Clearly label all circuit parameters and distinguish the unknown parameters from the known.
2. Identify all nodes of the circuit.
3. Select a node as the reference node also called the ground and assign to it a potential of 0 Volts. All other voltages in the circuit are measured with respect to the reference node.
4. Label the voltages at all other nodes.
5. Assign and label polarities.
6. Apply KCL at each node and express the branch currents in terms of the node voltages.
7. Solve the resulting simultaneous equations for the node voltages.
8. Now that the node voltages are known, the branch currents may be obtained using Ohm’s law.
The procedure for analyzing a circuit using the mesh-method is similar to that followed in the node-method and the various steps are given here.
1. Clearly label all circuit parameters and distinguish the unknown parameters from the known.
2. Identify all meshes of the circuit.
3. Assign mesh currents and label polarities.
4. Apply KVL at each mesh and express the voltages in terms of the mesh currents.
5. Solve the resulting simultaneous equations for the mesh currents.
6. Now that the mesh currents are known, the voltages may be obtained using Ohm’s law.
Thebasicprocedureis explained using the circuitshownintheFigure1.
Figure1:Examplecircuitformesh-currentanalysis.
a) Define a mesh current for each ‘window pane’ of the circuit
InFigure1,there are two window panes and thus two mesh currents, I1 and I2. Note that the direction of each mesh-current is in the clockwise direction. This is arbitrary but it helps maintain consistency to always define the mesh-currents in a clockwise direction.
b) Assign a reference node
Note that a reference node is designated in Figure 1. Since the definition of ground potential is fundamental in understanding circuits this is a good practice regardless of the method used in analysis.
c) Write a mesh equation for each mesh using KVL
Let’s consider each mesh separately and apply KVL around the loop following the defined direction of the mesh-current. Consider mesh1 in Figure 1. For clarity mesh1 is shown separately in Figure 2. In doing this, care must be taken to carry all information of the shared branches. In Figure 2, the direction of the mesh-current I2 is included as it is essential for the writing of an accurate KVL equation. Note also that the polarity of all voltages is clearly marked with respect to the direction of mesh-current I1.
Figure2:Sub-circuit for mesh1
The following equation is obtained when applying KVL to mesh1.
In similar fashion consider mesh2 shown in Figure 3. Note again that the direction of the mesh-current I1 is included. The polarity of all voltages in mesh2 is defined with respect to the direction of I2. Applying KVL to mesh2 results in the following equation.
Figure3:Sub-circuit for mesh2
d) Solveforunknownmesh currents
In matrix form the two mesh equations become:
The resulting matrix equation is easily solved for I1 and I2 using a calculator or computer aided tool.
e) Calculatebranch currents and node voltages
Once the mesh-currents are determined, any branch current or node voltage can be determined. For example consider the branch currents i1, i2, and i3 shown in Figure 4. Note that a lower case ‘i’ is being used for the branch currents to distinguish them from the mesh-currents I1 and I2. The branch currents are determined by the following equations.
i1 = I1
i2 = I1 – I2
i3 = I2
Figure4:Branch and Mesh Currents
Now consider the node voltages v1, v2, and v3 shown in Figure 5. These are easily determined using the branch currents that were just calculated.
v1 = Vs
v2 = i2R2
v3 = i3R4
Figure5:Node Voltages
In summary, the methods of node-voltage analysis and mesh-current analysis are powerful methods that can be used to analyze many different electrical networks. Determining node-voltages allows other circuit voltages, currents and powers to be calculated. In similar fashion, determining mesh-currents allows the calculation of other circuit values of interest. So given a free choice of which method to use, one needs to consider the number of equations for each method as well as the ease with which the equations can be developed. The purpose of this lab activity is to practice developing accurate matrix node-voltage equations or matrix mesh-current equations and solving them with a computer aided tool like Matlab. This computer aided solution method will then be confirmed using Multisim to model and analyze the same circuits.
OBJECTIVES
1. Perform mesh and nodal analysis of circuits.
2. Improve skills in developing mesh and nodal equations.
3. Determine nodal voltages and mesh currents using Matlab software.
4. Confirming the Matlab solution using Multisim.
EQUIPMENT REQUIRED
Computer facilities capable of applying Matlab and Multisim software.
PROCEDURE
First an example of how Matlab can easily solve a system of simultaneous equations.
1. Consider the following set of mesh-current equations:
I1 + 2I2 + 3I3 = 4
2I1 + 3I2 + 4I3 = 5
4I1 + 2I2 + 5I3 = 1
2. Use the Matlab Editor/Debugger to enter the following program. The lines beginning with a ‘%’ sign are comments and ignored by the Matlab compiler. The purpose of the data file: ‘lab5.dat’ is to store all output from the execution of the Matlab program so it is available for processing or reporting later.
%This example program solves for mesh currents I1, I2 and I3
%open data file
diarylab5.dat
%Defines the coefficient matrix A (resistances in the case of
%mesh-current analysis) Each row of coefficients is delimited
%by a semicolon.
A=[1, 2, 3; 2, 3, 4; 4, 2, 5];
%defines the vector B (the right hand side of the matrix eq.)
B=[4; 5; 1];
%Two different ways to solve the matrix eq. are shown.
I=A^-1*B
%or
I=inv(A)*B
%close data file
diary
3. Run the program and the output will display in Matlab’s command window. The output will also be saved in the .dat file. The output should be similar to what is shown here.
I =
-1.4000
1.8000
0.6000
I =
-1.4000
1.8000
0.6000
4. Now it is your turn. Develop the matrix equation necessary to solve for the mesh currents in the circuit of Figure 6. Modify the example Matlab program to solve the matrix equation. Document your Matlab results.
Use the Matlab results to compute all branch currents(i.e., the current through each resistor) and all node voltages. These computations will be compared to Multisim analysis in step 6.
Figure 6
5. The same process can be used to determine the nodal equations for the circuit of Figure 7. Solve for V1, V2, and V3 using MATLAB software. Document your results.
Figure 7
6. Model the circuits in Figures 6 and 7 using Multisim. Perform a DC operating point analysis. Document your results showing all branch currents and all node voltages for each circuit.
7. Create two Data Tables, one for each circuit, showing a comparison of Matlab and Multisim results. They should be in agreement. If not, resolve any discrepancies.
WRITTEN REPORT
No formal report is required. Turn in the following items.
1. Cover page
2. Documented Matlab results for each circuit
3. Multisim results for each circuit presented in a neat tabular fashion.
4. The data tables from step 7.
5. A short discussion of the perceived value of tools like Matlab and Multisim.
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