Experiment # 4
Nodal Analysis, Mesh Analysis, and Superposition
ƒ
To verify Nodal and Mesh equations.
ƒ
To verify superposition concept.
Nodal analysis is generally best in the case of several voltage sources. In nodal analysis, the variables (unknowns) are the "node voltages."
ƒ
Nodal Analysis Procedure:
1.
Count the number of principal nodes or junctions in the circuit. Call this number n. (A principal node or junction is a point where 3 or more branches join. We will indicate them in a circuit diagram with a red dot. Note that if a branch contains no voltage sources or loads then that entire branch can be considered to be one node.)
2.
Number the nodes N
1
, N
2
, . . . , N n
and draw them on the circuit diagram. Call the voltages at these nodes V
1
, V
2
, . . . , V n
, respectively.
3.
Choose one of the nodes to be the reference node or ground and assign it a voltage of zero.
4.
For each node except the reference node write down Kirchoff's Current Law in the form "the algebraic sum of the currents flowing out of a node equals zero". (By algebraic sum we mean that a current flowing into a node is to be considered a negative current flowing out of the node.)
For example, for the node (fig5.1) KCL yields the equation:
I a
+ I b
+ I c
= 0
Fig5.1
Express the current in each branch in terms of the nodal voltages at each end of the branch using Ohm's Law (I = V / R). Here are some examples: a.
The current downward out of node 1 (fig5.2) depends on the voltage difference V1 - V3 and the resistance in the branch.
Fig5.2
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Experiment # 4
Nodal Analysis, Mesh Analysis, and Superposition b.
In this case (fig5.3) the voltage difference across the resistance is V1 - V2 minus the voltage across the
voltage source. Thus the downward current is as shown.
Fig5.3 c.
In this case (fig5.4) the voltage difference across the resistance must be 100 volts greater than the difference V1 - V2. Thus the downward current is as shown.
Fig5.4
The result, after simplification, is a system of m linear equations in the m unknown nodal voltages (where m is one less than the number of nodes; m = n
- 1). The equations are of this form: where G
11
, G
12
, . . . , G mm
and I
1
, I
2
, . . . , I m
are constants.
Alternatively, the system of equations can be gotten (already in simplified form) by using the inspection method.
5.
Solve the system of equations for the m node voltages V
1
, V
2
, . . . , V m
using
Gaussian elimination or some other method.
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Experiment # 4
Nodal Analysis, Mesh Analysis, and Superposition
Mesh (loop) analysis is generally best in the case of several current sources. In loop analysis, the unknowns are the loop currents. Mesh analysis means that we choose loops that have no loops inside them.
ƒ
Loop Analysis Procedure:
1.
Label each of the loop/mesh currents.
Fig5.5 mesh analysis
2.
Apply KVL to loops/meshes to form equations with current variables. a.
For N independent loops, we may write N total equations using KVL around each loop. Loop currents are those currents flowing in a loop; they are used to define branch currents. b.
Current sources provide constraint equations.
3.
Solve the equations to determine the user defined loop currents.
In any linear circuit containing multiple independent sources, the current or voltage at any point in the network may be calculated as the algebraic sum of the individual contributions of each source acting alone.
ƒ
Procedure:
1.
For each independent voltage and current source (repeat the following): a.
Replace the other independent voltage sources with a short circuit
(i.e., v = 0). b.
Replace the other independent current sources with an open circuit
(i.e., i = 0).
Note: Dependent sources are not changed! c.
Calculate the contribution of this particular voltage or current source to the desired output parameter.
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Experiment # 4
Nodal Analysis, Mesh Analysis, and Superposition
2.
Algebraically sum the individual contributions (current and/or voltage) from each independent source.
=
+
Fig5.6 superposition analysis
ƒ
DC voltage source (SN.=CPS250W17978)
ƒ Digital millimeter DMM (SN.=8270010)
ƒ
Resistors
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Experiment # 4
Nodal Analysis, Mesh Analysis, and Superposition
R1 R2
A B
1k 2k 2
V1
10v R4
2k
R5
1k
5v
V2
G G
0
Fig5.7 circuit connection for nodal, mesh, superposition techniques
1.
Connect the circuit as shown in Fig5.7
2.
Measure the exact value of the resistor.
3.
Set the power supply voltages as shown in Fig5.7 Measure them using DMM.
4.
Measure the voltage at each node (A, B, C and D) with respect to the node
"G".
5.
Measure the voltage V
BC
.
1.
Measure the currents for the circuit shown in Fig5.7.
1.
Disconnect the voltage source Vs1 and replace it with a wire.
2.
Measure the voltage V'
BC
.
3.
Measure the current I'
R2
.
4.
Return the voltage source Vs1, then disconnect the voltage source Vs2 and replace it with a wire.
5.
Measure the voltage V''
BC
.
6.
Measure the current I''
R2
.
7.
V
BC
= V'
BC
+ V''
BC
.
8.
I
R2
= I'
R2
+ I''
R2
.
5

Experiment # 4
Nodal Analysis, Mesh Analysis, and Superposition
™
R1 1 K
Ω
R2 2 K
Ω
R3 2 K
Ω
R4 2 K
Ω
R5 1 K
Ω
V
A
10 V
V
B
5.667 V
V
C
2.667 V
V
D
5 V
I
R1
4.333 mA
I
R2
1.500 mA
I
R3
-1.167 mA
ƒ
V
BC
Values
V
BC
' 1 V
V
BC
" -4 V
V
BC
-3 V
ƒ
I
R2
Values
I
R2
' -0.500 mA
I
R2
" 2.000 mA
I
R2
1.500 mA
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Experiment # 4
Nodal Analysis, Mesh Analysis, and Superposition
™
R1 0.9905 K
Ω
R2 1.9685 K
Ω
R3 1.9690 K
Ω
R4 1.9737 K
Ω
R5 0.9874 K
Ω
V
A
10.0130 V
V
B
5.6660 V
V
C
2.6676 V
V
D
4.9930 V
I
R1
4.321 mA
I
R2
1.514 mA
I
R3
1.171 mA
ƒ
V
BC
Values
V
BC
' -0.9985 V
V
BC
" 3.9990 V
V
BC
2.9996 V
ƒ
I
R2
Values
I
R2
' -0.502 mA
I
R2
" 2.021 mA
I
R2
1.516 mA
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Experiment # 4
Nodal Analysis, Mesh Analysis, and Superposition
A
R1 R2 R3
10v
10.00V
V1
2.833mA
R4
2k
2k
+
2.667V
2k
2.667mA
R5
1k
1.167mA
5.000V
V2
5v
4.333mA
1.167mA
0
=
0V
R1
1k
333.3uA
G
166.7uA
R4
2k
R2
G
333.3mV
2k
0
0V
500.0uA
1.333V
2k
G
1.333mA
R5
1k
G
R3
1.833mA
5.000V
V2
5v
1.833mA
10v
10.00V
V4
4.667mA
2.667mA
R4
2k
0
+
R1 R2 R3
A B C
2k
1.333mA
R5
1k
0V
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Experiment # 4
Nodal Analysis, Mesh Analysis, and Superposition
™
Verify Nodal Equations at nodes B and C using the measuring data in
Table 5.6 and the exact values of the resistors.
1.
At node (B):
(V
B
-V
S1
)/R
1
+ V
B
/R
4
+ (V
B
-V
C
)/R
2
= 0
(5.666-10.013)/0.9905 + (5.666/1.9737) + (5.666-2.6676)/1.9685) = 0.005248
≅
0
2.
At node (C):
(V
C
-V
B
)/ R
2
+ V
C
/R
5
+ (V
C
-V
S2
)/R
3
= 0
(2.6676-5.666)/1.9685 + (2.6676/0.9874) + (2.6676-4.993)/1.969 = -0.00256
≅
0
™
Verify mesh equations for the meshes (ABGA), (BCGGB), (CDGC) using the measured data and the exact values of the resistors.
1.
ABGA:
-10.013 + (1 x i
1
) + [ 2 x (i
1
– i
2
) ] = 0
-10.013 + (1*4.321) + (2*(4.321-1.514)) = -0.078
≅
0
2.
BCGGB:
[2 x (i
2
– i
1
)] + (2 x i
2
) + [1 x (i
2
– i
3
)] = 0
(2*(1.514-4.321)) + (2*1.514) + (1*(1.514+1.171))= 0.099
≅
0
3.
CDGC:
[1 x (i
3
– i
2
)] + (2 x i
3
) + 5 = 0
(-1.171-1.514) + (2*-1.171) + 4.993 = -0.034
≅
0
™
Compare between V
BC
in part A and V
BC in part C. Comment.
1.
Part A: V
BC
= V
B
– V
C
= 5.666 – 2.6676 = 2.9984 v
≅
3 v
2.
Part C: V
BC
= V
BC
' + V
BC
" = -0.9985 + 3.999 = 3.0005 v
≅
3 v
They have got the same value but the theoretical procedures of Superposition in part
C is much easier than Nodal Method in part A, where practical procedures of
Superposition in part C takes longer time than Nodal Method in part A .
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Experiment # 4
Nodal Analysis, Mesh Analysis, and Superposition
™
Compare between I
R2
in part B and I
R2
in part C. Comment.
1.
Part B: I
R2
= 1.514 mA
≅
1.5 mA
2.
Part C: I
R2
= 1.516 mA
≅
1.5 mA
They have got the same value but the theoretical procedures of Superposition in part
C is much easier than Mesh Method in part B, where practical procedures of
Superposition in part C takes longer time than Mesh Method in part B .
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Experiment # 4
Nodal Analysis, Mesh Analysis, and Superposition
ƒ
The node voltage method is generally best in the case of several voltage sources, and it works with both planar and nonplanar circuits. A reference node is chosen from among the essential nodes. Voltage variables are assigned at the remaining essential nodes, and kirchhoff's current law is used to write one equation per voltage variable. The number of equations is [n e
–
1], where n e
is the number of essential nodes.
ƒ
The Mesh (loop) analysis is generally best in the case of several current sources, and it works only with planar circuits. Mesh currents are assigned to each mesh, and kirchhoff's voltage law is used to write one equation per mesh. The number of equations is b - [n – 1], where b is the number of branches in which the current is unknown, and n is the number of nodes. The mesh currents are used to find the branch currents.
ƒ In a circuit with multiple independent sources, superposition allows us to activate one source at a time and sum the resulting voltages and currents to determine the voltages and currents that exist when all independent sources are active. Dependent sources are never deactivated when applying superposition.
ƒ The experiment shows that V
BC
obtained by nodal analysis is the same of V
BC obtained by superposition, while I
R2
obtained by mesh analysis is the same of
I
R2
obtained by superposition.
ƒ
Nodal Method is used to find out nodal voltages, while Mesh Method is used to find out branch's currents, while Superposition is used to find out the voltages and currents in the circuit.
11

Experiment # 4
Nodal Analysis, Mesh Analysis, and Superposition
http://www.eas.asu.edu/~holbert/ece201/recipes.htm
http://mathonweb.com/help/backgd5.htm
www.cybermike.net/.../liec_book/DC/DC_10.html
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