# Nodal analysis - Super node needed?

If using the nodal analysis for a certain circuit one usually chooses a node as reference node such that one can reach as many other nodes as possible. Unfortunately 99% of the literature I found on this subject seems to assume that one can alway find such a node that reaches every other node within a single branch.

Consider the following example: One can see that obviously it's not possible to reach every node within a single branch, even if we change the reference node.

My text book states that one would need to use a super node in such a situation, as also shown in the example with node #4.

I assume this is due to our equations not being linear independent if we would consider #4 a "normal" node.

Please provide me some insight in why we have to use a super node here and if my assumption is correct. I'd highly appreciate to see some kind of mathematical proof.

## 1 Answer

one usually chooses a node as reference node such that one can reach as many other nodes as possible.

I've never heard of this as a criterion for choosing the reference node. Usually one of the nodes connected to the power supply is the reference node. This might chance to also be connected to a large number of circuit branches.

In any case, it does not matter which node is chosen as the reference node. It won't affect the solution at all (except to offset the numerical values of the node voltages to all be relative to the chosen reference).

A super node is needed when there is an independent voltage source in the circuit.

• "A super node is needed when there is an independent voltage source in the circuit" - did you mean when there is an independent floating (one terminal is not connected to the 'ground' node) voltage source in the circuit? – Alfred Centauri Jun 24 '17 at 3:01
• @AlfredCentauri, Even with a non-floating voltage source, you'll need some special treatment for the other node that's connected to it. You won't be able to write a KCL equation at that node only in terms of node voltages. I was never taught a different name for that special treatment than "forming a supernode". If you have some different name for it, I guess that's a semantic distinction I never learned. – The Photon Jun 24 '17 at 3:28
• (1) in node voltage analysis, one solves for the voltage at a node (with respect to the datum, zero, ground node). (2) If one terminal of a voltage source is connected to the reference node, the node to which the other terminal is connected is known, i.e., no equation is need to solve for it. On the other hand, if the voltage source is connected between two nodes, neither of which is a reference node, we cannot write a KCL equation for either node (the current through the source is not constrained by, e.g., Ohm's law) but we can write a KCL equation for a 'node' that subsumes both nodes. – Alfred Centauri Jun 24 '17 at 3:34
• For example: " 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." – Alfred Centauri Jun 24 '17 at 3:36
• @AlfredCentauri, to me formally you're writing a special equation "V_1 = V_source", for example. But that may be because I think about it more in terms of how you'd program a computer to solve it rather than how you'd do it pencil-and-paper where you'd take obvious shortcuts like treat the node voltage as known. – The Photon Jun 24 '17 at 3:36