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Nodal Analysis

After understood the fundamental laws of circuit theory (Ohm's law and Kirchhoff's laws), we will now proceed to use these two to develop two powerful techniques for circuit analysis :
  1. Nodal analysis
  2. Mesh analysis
Nodal analysis is based on a systematic application of Kirchhoff's current law (KCL). With these technique we will be able to analyze any linear circuit.
nodal analysis

Nodal Analysis

Nodal analysis provides a technique to analyze circuits using node voltages as the circuit variables. Using node voltages instead of element voltages as circuit variables is more convenient and reduces the equations number to solve simultaneously.

To make it simpler, let us assume circuits in this part do not have voltage sources. The circuits with voltage sources will be covered in the next post. In nodal analysis, we are supposed to find node voltages. Given a circuit with n nodes without voltage sources, the nodal analysis uses following three steps.
Steps to Determine Node Voltages :
  1. Select a node as the reference node. Assign voltages v1, v2, ...., vn-1 to the remaining n - 1 nodes. The voltages are referenced with respect to the reference node.
  2. Apply KCL for each n - 1 nonreference nodes. Use Ohm's law to find the current flowing in the branches.
  3. Solve the resulting simultaneous equations to obtain the unknown node voltages.
The full explanation below.
First step is selecting a node as the reference or datum node. The reference node is usually called the ground since it is assumed to have zero potential. The symbol of reference node is shown in Figure.(1). Earth ground is shown in Figures.(1a) and (1b) and chasis ground is shown in Figure.(1c).
nodal analysis electric circuit
Figure 1. Common symbols of ground
After we get reference node, we assign voltage designations to reference nodes. Take a look at Figure.(2a) where node 0 is the reference node (v = 0), while nodes 1 and 2 are assigned with voltages v1 and v2 respectively. Remember, node voltages are defined with the respect to the reference node. As shown in Figure.(2a), each node voltage is the voltage rise from the reference node to the nonreference node or simply of the voltage node to reference node.
nodal analysis electric circuit
Figure 2. Typical circuit of nodal analysis
Second step is apply KCL to each nonreference node in the circuit. To reduce the complexity of variables, Figure.(2a) is redrawn in Figure.(2b), where we use i1, i2, and i3 as the currents flowing through resistors R1, R2, and R3 respectively.

At node 1 we apply KCL and gives
nodal analysis electric circuit
(1)
At node 2 gives
nodal analysis electric circuit
(2)
We apply Ohm's law to express the unknown value of i1, i2, and i3 in terms of node voltages. Since the resistance is passive element, using the passive sign convention, current must always flow from higher potential to a lower potential.
Current flows from a higher potential to a lower potential in a resistor.
We can use this principle as,
nodal analysis electric circuit
(3)
We obtain from Figure.(2b),
nodal analysis electric circuit
(4)
Substituting Equations.(4) in (1) and (2) results
nodal analysis electric circuit
(5)

nodal analysis electric circuit
(6)
Subsituting with conductances, Equations.(5) and (6) become
nodal analysis electric circuit
(7)

nodal analysis electric circuit
(8)
The third step is solving with node voltages. Applying KCL to n - 1 nonreference nodes, we obtain n - 1 simultaneous equation such as Equations.(5) and (6) or (7) and (8). For circuit in Figure.(2) we solve Equations.(5) and (6) or (7) and (8) to get the node voltages v1 and v2 using any standard method such as substitution method, elimination method, Cramer's rule, or matrix inversion.

For example Equations.(7) and (8) will use matrix form as
nodal analysis electric circuit
(9)
which can be solved to get v1 and v2.

This method can be modified into supernode analysis.
Nodal analysis also works for ac circuit. Make sure to read it to advance your study.

Nodal Analysis Examples

For better understanding, let us review some examples below :
1.Calculate the node voltages in the circuit in Figure.(3a)
nodal analysis electric circuit
Figure 3
Solution :
Consider Figure.(3b) where the circuit in Figure.(3a) has been prepared for nodal analysis. The currents have been selected for KCL except the branches with current sources. The labeling of the current is arbitrary but consistent. (Consistent means if , for example, i2 enters the 4 Ω resistor from left hand side, i2 must leave the resistor from the right hand side).

The reference node is selected and the node voltages v1 and v2 are now determined.
At node 1, applying KCL and Ohm's law gives
nodal analysis electric circuit
Multiplying each term in the last equation by 4, we obtain
nodal analysis electric circuit
or
nodal analysis electric circuit
(1.1)
At node 2, we do the same and get
nodal analysis electric circuit
Multiplying each term by 12 results
nodal analysis electric circuit
or
nodal analysis electric circuit
(1.2)
Now we have two simultaneous (1.1) and (1.2) and then we can solve using any method to get v1 and v2.
Method 1
Using elimination method we add (1.1) and (1.2) gives
nodal analysis electric circuit
Subsituting the result above with (1.1) gives
nodal analysis electric circuit
Method 2
Use Cramer's rule, we put (1.1) and (1.2) to matrix form as
nodal analysis electric circuit
The determinant is
nodal analysis electric circuit
We now obtain the voltages as
nodal analysis electric circuit
2.Determine the voltages at the nodes in Figure.(4a)
nodal analysis electric circuit
Solution :
In this example we will need three nonreference nodes instead of only two. We assign three nodes as can be seen in Figure.(4b)

At node 1,
nodal analysis electric circuit
Multiplying by 4 and rearranging terms, we get
nodal analysis electric circuit
(2.1)
At node 2,
nodal analysis electric circuit
Multiplying by 8 and rearranging terms,we get
nodal analysis electric circuit
(2.2)
At node 3,
nodal analysis electric circuit
Multiplying by 8, rearranging terms, and dividing by 3, we get
nodal analysis electric circuit
(2.3)
We now will use elimination method, we add (2.1) and (2.3)
nodal analysis electric circuit
(2.4)
Adding (2.2) and (2.3) gives
nodal analysis electric circuit
(2.5)
Substituting (2.5) into (2.4) we get
nodal analysis electric circuit
From (2.3), we get
nodal analysis electric circuit

Untuk Bahasa Indonesia baca Analisis Node
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