Figure 1
OBJECTIVES:
1. To construct a planar circuit having two voltage sources and five resistors.
2. To study node voltages and mesh currents.
3. To compare calculated and measured results using both nodal and mesh analysis.
BACKGROUND & THEORY:
SUMMARY OF NODE VOLTAGE (N-V) METHOD
1. Number of equations needed is one less than the number of essential nodes, except as noted in item 7 below.
2. Select one of the essential nodes as a reference node (the node with the most branches usually is a good choice).
3. Then assign node voltages at the other essential nodes. By definition, node voltages are a "rise" above ref. node.
4. Next, generate N-V equations by summing currents at each non-reference node (using KCL). Currents are to be considered leaving the node, unless a current source exists in the branch (then you use the direction of the arrow for determining the sign).
5. If a voltage source exists in the branch, subtract or add its voltage (depending on polarity) to the node voltage before dividing by the resistance in the branch.
6. When a dependent source exists, you must express the controlling voltage or current in terms of the assigned node voltages.
7. If a voltage source is connected directly between an essential node and the ref. node, that reduces the number of equations needed.
8. If a voltage source (independent or dependent) exists between two non-reference nodes, then you can use the supernode concept, and proceed as in 4. above to write the equations.
Note that the voltage existing in the supernode must be expressed as a function of the node voltages to obtain one equation.
SUMMARY OF MESH CURRENT (M-C) METHOD
1. Number of equations needed is equal to the number of meshes (windows) in the network, except as noted in 7. below.
2. The M-C method is used for planar networks only, where the network is drawn with no crossing branches.
3. Assign clockwise mesh current in each mesh. A mesh current exists only in the perimeter of a mesh. In some parts of the mesh, the mesh current may be the same as the branch current.
4. Next, generate M-C equations by summing voltages around each mesh (using KVL). Voltages are to be considered positive unless a voltage source exists in the mesh (then you use the polarity of the voltage to determine the sign). Where two meshes have a common branch, a net current (one mesh current minus the other) must be used to express voltage in that branch.
5. When a dependent source exists, you must express the controlling voltage or current in terms of the assigned mesh currents.
6. If a current source (independent or dependent) is common to two meshes, then you can use the supermesh concept, and proceed as in 4. above to write the equations.
Note that the "common" current source must be expressed as a function of the mesh currents to obtain one equation.
7. If a current source exists in the outer perimeter of the circuit, KVL need not be applied to that mesh (because that mesh current has to be equal to the current in that source).
NOTE: The primary advantage of both the N-V and M-C methods is that you can analyze a circuit (which has many unknowns) with a fewer number of simultaneous equations.
However - WHEN IS N-V METHOD USED INSTEAD OF M-C METHOD? - AND VICE VERSA
1. One approach: Use the one which requires the fewest number of simultaneous equations.
2. Look at location of v-sources and i-sources. The analysis may be simplified if v-sources exist between essential nodes and the reference you might select, or if an i-sources exist in the outer perimeter of meshes.
3. If a certain voltage is of primary interest, then the N-V method will probably be the best, or if your primary interest in a certain current, then the M-C method will probably be the best choice.
4. The N-V method can be applied to any circuit, whereas the M-C method requires that the circuit have a planar network.
5. When you have more v-sources than i-sources, the best selection will probably be the N-V method.
6. When you have more i-sources, the best selection will probably be the M-C method.
7. The mesh method is probably used more than it should be.
8. Other than the above, experience and intuition will probably cause the best selection.
9. The node voltage method is always used in circuit analysis programs written for computers. This because it is very easy to tell the computer which elements are connected between particular nodes in the circuit, and it is much harder to describe the loops of a circuit for a computer program.
EQUIPMENT AND PARTS LIST:
DC Power Supply.
Digital Multimeter (DMM)
Resistors one each: 1 kW, 1.5 kW,
1.8
kW, 3.3 kW, and 4.7
kW.
Breadboard
PROCEDURE:
1. Construct the circuit shown in Figure 1 using available power supply, resistors, breadboard, and connecting wires provided. R1 = 1.5 kW, R2 = 1 kW, R3 = 3.3 kW, R4 = 4.7 kW, R5 = 1.8 kW.
Figure 1
2. Set VS1 = 6 V and VS2 = 16 V.
3. Note that the reference node, nodal voltages (V1, V2, V3, & V4) and mesh currents (I1, I2, & I3) have already been designated.
4. Measure all node voltages (not Branch voltages) and the mesh currents (not the Branch currents Ia, & Ib).
5. Don't forget to measure all the resistor Values.
COMPARISONS AND QUESTIONS:
1. From your measured mesh currents, calculate the value of the branch currents ia and ib shown in Figure 1. Using top to bottom in the vertical branches and left to right in the horizontal branches, what are the currents in each of the other five branches?
2. By observation, what are the values of V1 and V4? With the given values of VS1 and VS2. Using plus reference at the top of the vertical branches and at the left for horizontal branches, what are the seven branch voltages in terms of the node voltages?
3. Node Equations:
a. Set up the node equations for the circuit, and solve for V2 and V3, using nominal values of resistances and nominal values of the voltage sources. Show all your calculations in your laboratory notebook.4. Mesh Equations:b. Compare all measured node voltages with the calculated values.
c. Repeat a & b using the measured values of resistances and measured values of the source voltages. You may use a computer or calculator to solve the equations.
a. Set up the mesh equations for the circuit, and solve for the three mesh currents, using nominal values of resistances and the nominal voltage sources. Show your calculations in your laboratory notebook.b. Compare all measured mesh currents with the calculated values.
c. Repeat a & b using the measured values of resistance and measured values of the source voltages. You may use a computer or calculator to solve the equations.
5. Calculate the power absorbed by resistors R2 and R4.
For each resistor calculate power by using three different methods: P=VI,
P=I2R, P=V2/R. Use measured resistances, measured
node voltages, and the branch currents calculated from the measured mesh
currents. Explain any differences in the power obtained by the three
methods. Are the differences small enough to be explained by the specified
meter errors? Justify your answers.
CONCLUSIONS:
Based on your experimental observations. (Can you verify or dispute any laws or principles based on the results of this experiment?)
Last updated on 02/29/2008