From our previous discussion of Nodal Analysis we have seen, how voltage sources affect nodal analysis. We have also seen how a voltage source makes it easier for us to calculate the node voltages when connected with a reference node. But things get complicated when a voltage source cannot be referenced i.e. it comes in between two non-reference node. This voltage source along with two non-reference nodes forms a supernode. 

In summary, When a voltage source comes in between two non-reference node then these two non-reference nodes and the voltage source form a supernode and we take this supernode as a single node and apply KCL and KVL to solve the circuit.

To solve a problem of supernode follow these steps:

Let us explain this procedure with an example:

First of all we have marked a reference node V0. Then we marked all other nodes as we do normally for nodal analysis.

Now we have given a dotted circle to remind us that this is a supernode along with V1 and V2.

Now we apply KCL at the circuit:

2 = (V1 – 0) / 2 + (V2 – 0) / 4 +7

8 = 2 V1 + V2 +28 (multiplying both sides by 4)

2 V1 + V2 = -20 ………………………………………………. (a)

Now we apply KVL at the supernode loop:

-V1 - 2 +V2 = 0

V2 = V1 + 2 ……………………………………………………. (b)

Putting this value of V2 in equation (a):

2 V1 + (V1 + 2) = -20

3 V1 = -22

V1 = -22/3 V

Now from (b): V2 = -22/3 +2 = -16/3V

Note that the 10 Ohm resistor connected across the supernode does not make any difference in the calculations as it is connected across the supernode.

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:

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.