Nodal Analysis

Nodal analysis is a procedure to analyze circuits. It uses node voltages as circuit variables. It is a very easy procedure for calculation as it reduces equations and makes it convenient to solve large networks. In nodal analysis we don�t take element voltages rather we take node voltages which actually reduces number of equations to be solved simultaneously. Nodal analysis is basically the implementation of KCL (Kirchoff�s Current Law).

We are interested in node voltages in nodal analysis, and after finding out node voltages we form independent circuit equations to solve the circuit. We should follow the steps below to solve a circuit using nodal analysis:

01.   First of all, select a reference node. Show it with a ground notation. You can select any node as a reference node. At that reference node, node voltage will be zero. Now assign voltages like v1, v2 �.. vn-1 etc for remaining nodes. These voltages are referenced with respect to the reference node. 

02.   Now apply KCL to all other nodes except reference node. You have to use Ohm�s law and using it express branch currents in terms of node voltages. 

03.   Finally, solve all the simultaneous equations to find the unknown node voltages.

Now let�s explain these three steps using an example. Look at the following circuit.

 

It�s a simple circuit with a voltage source of 10V and a current source of 1 mA and three different resistors. We have to find the voltage across the resistor R3. Now let us assign a reference node first. Here we can see three nodes are there. We choose the lowermost one as the reference node and assign it Vo = 0V. We also define other nodes having voltages as V1 and V2.

 

 

Now as we can see, node V1 has a voltage source of 10V and the other terminal of the source is connected to the reference node. Hence we can say that the whole 10V should appear at V1.

Therefore V1=10V.

Well, here is an important thing to mention: most of the time we assign the node which connected to a voltage source�s negative terminal as a reference node which in turn simplifies our calculation by giving the other node voltage value directly from the voltage source�s value.

 Now, let us apply KCL at node V2:

(V2 � V1) / 20 k  + V2 / 10 k � 1 m = 0

 (V2 � V1) / 20 k  + V2 / 10 k = 1 m

V2 �V1 + 2 V2 = 20  (multiplying both sides by 20k)

3 V2 � V1 = 20

3 V2 � 10 = 20 (putting the value of V1)

3 V2 = 20 + 10

3 V2 = 30

V2 = 30 / 3

V2 = 10V

Now, we have all the node voltages. We can find the voltage Vr3 easily.

Vr3 = V1 � V2 = 10 V � 10 V = 0V

We have taken arbitrary values of resistors and sources which lead the voltage across R3 to be zero.