S. Jayaram
GenE123
May 30th 2005
NODAL AND LOOP ANALYSIS
(Refer to Chapter 3 in your text)
Nodal Analysis:
Nodal analysis is a technique to solve network problems based on KCL.
•
Choose one of the nodes as a reference node. (Identify it by the
ground symbol
•
)
Mark all other nodes with some ID’s as Va, Vb,….. for the voltages at
these non-reference nodes with respect to the reference node.
•
Solve for these unknown “node” voltages.
•
Solve for branch currents using these node voltages.
Example:
Reference node, V = 0 volts.
Since one of the nodes is always identified as a reference node;
There will be (N - 1) nodes at which voltages are unknown.
By applying KCL at those (N-1) nodes, a set of independent simultaneous
equations can be set up and solved to find the unknown node voltages.
In the following sections, it is demonstrated how nodal analysis can be
used to solve circuit problems with different types of sources.
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S. Jayaram
May 30th 2005
GenE123
Circuit containing ONLY Independent Sources:
Circuits Containing Independent Current Sources:
Applying KCL @ node (1)
−i A + i1 + i 2 = 0 ;
Using Ohm’s law,
 V − 0   V1 − V2 
−i A +  1
+
=0
 R1   R2 
G=
1
R
−i A + G1 (V1 − 0 ) + G2 (V1 − V2 ) = 0
Or,
(G1 + G2 )V1 − G2V2 = i A
(1)
Similarly, applying KCL @ node (2)
−i 2 + i B + i 3 = 0 ;
Or
[i 2 − iB − i3 = 0]
Or, in terms of node voltages,
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S. Jayaram
May 30th 2005
GenE123
−
(V1 − V2 )
R2
+ iB +
(V2 − 0 )
R3
=0
−G2 (V1 − V2 ) + i B + G3V2 = 0
−G 2V 1 + (G 2 + G 3 )V 2 = −i B
(2)
From equations (1) and (2),
(G1 + G2 )V1 − G2V2 = i A
−G 2V 1 + (G 2 + G 3 )V 2 = −i B
From these two simultaneous linear
equations, we can solve for the 2
unknowns V1 and V2.
Given any circuit, we can write the nodal equations directly. To understand the
concept, consider the following circuit:
Nodal analysis:
@ node (1), all are outgoing currents,
V −V  V −V  V
∴ i1 +  1 2  +  1 2  + 1 = 0
 R4   R3  R2
(1)
@ node (2), all are incoming currents,
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S. Jayaram
May 30th 2005
GenE123
 V1 − V2

 R4
  V1 − V2 
 +i 2 = 0
+
R
 
3

(2)
From (1) and (2), the two unknowns, V1 and V2 can be found. We can
understand it better through the following example with numerical values.
Find V1, V2 and the current through the branch that contains the 8Ω resistor
1
2
First write node equations at @node (1) and @ node (2) independently. Then
solve for V1 and V2. Knowing these node voltages, the required current can be
found.
∴ Current through the 8Ω register, i 8 Ω
Because, current always flows from higher potential to lower potential,
i8Ω =
V2 − V1
8
Answer: 4 (A).
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S. Jayaram
May 30th 2005
GenE123
Circuits Containing Independent Voltage Sources:
If an independent voltage source is connected between the reference node and a
non-reference node, the non-reference node voltage is the source voltage. In
other words, the voltage at such nodes is no longer an unknown.
Example: In the following circuit, find io using nodal analysis.
First, find V0 in terms of V1 and V2 and then find io using, i 0 =
V0
3k Ω
Given: V1 = 6V and V2 = 3V with respect to the reference ground.
Answer: Vo =
9
3
(V ) and i o = (mA) .
4
4
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S. Jayaram
GenE123
May 30th 2005
Examples of circuits that contain both voltage and current
sources:
Step 1. Find the conditions @ Super Node,
Step 2. Then apply KCL @ Super Node.
Answer: V1 = 10V , V2 = 4V , i1 =
5
1
mA and i2 = mA
3
3
Numerical Problems with Dependent and Independent Sources.
Find V0 by Nodal Analysis:
Step 1. Find the conditions @ Super Node,
Step 2. Then apply KCL @ Super Node.
Answer: Vo = 9 (V)
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