SuperMeshes and SuperNodes
Questions have been asked about 'supernodes' and 'supermeshes' in sufficient number to warrant
broadcasting this note.
Two general methods of analyzing circuits, using current variables as unknowns and the other in terms
of voltage variables were describes under the names 'cutset' and 'tieset' analysis. Then two simpler
special cases, which can deal with virtually all of the circumstances likely to be encountered, were
described as 'mesh' and 'nodal' analyses. One does not do anything fundamentally different in these
cases than for the cutset and tieset procedures. It is just that details can be recognized so much more
simply that many intermediate steps in the processing can be done by inspection.
For example, in a nodal analysis the manner of choosing (node) voltage variables makes the voltage
drop from node i to node j = vi - vj; it is not necessary to look at the circuit diagram or for that matter
even to identify voltages and polarities explicitly on the circuit diagram to assert the expression. Then
the branch current from node i to node j for, say, a resistive branch is (vi-vj)/Rij. In the more general
tieset procedure it is not always clear in an involved circuit topology what the correct relationship is by
inspection although it is not difficult to do so in a step-by-step procedure.
Similarly for the use of mesh currents as variables in a mesh analysis; choosing the current variables
appropriately (by inspection) enables the current in any branch to be expressed (without even looking at
the circuit) as a difference of no more than two mesh currents (only one if the branch is not common to
two meshes).
With just a bit more thought the equations to solve for the variables even can be written neatly factored,
ready for solving.
Alas, if there are silver threads about there often is also a cloud. In our case it is the presence of a
voltage source branch in a node analysis, or a current source branch in a loop analysis. (If there is a
resistor in series with a voltage source or in parallel with a current source then a source transformation
removes the problem. As a practical matter this is always the case; entropy rules the Universe
absolutely. As an even more practical matter computer analysis programs often simply add in a resistor
whose size is chosen to have negligible influence on the numerics of the analysis; this enables source
transformations without singularities (small > 0 and large < ∞). Unfortunately ECE 211 professors
traditionally ask questions which do not allow for source transformations. What is one to do then?
Actually this question is not difficult to answer. Consider a loop analysis in which a loop (there might
be several such loops) includes a current source branch. You just have to give up a little bit of the 'by
inspection' simplicity. It is not at all necessary to go back to the full tableau layout for which KCL,
KVL, and branch VI relations all are written out explicitly. Just a little retreat is needed. The problem
is that you can't relate the source branch voltage to the source branch current by inspection as before. So
define as a new variable Vx the voltage across the current source, and use that variable wherever you
need to include the source branch voltage. Of course since you have increased the number of unknowns
you need an additional independent equation in order to solve the augmented number of equations. But
that is right in front of you; it is the known V-I relation for the source branch. That can have two forms.
If it is an independent current source the VI relation is I = constant (polarity as indicated by the source
icon). If it is an dependent current source then I = function of other branch voltages and currents.
A mesh analysis is handled similarly. Instead of defining auxiliary voltages unknowns it is current
unknowns that are needed, and the source V-I relation provides the additional equation required.
If things get as bad as they can get you will have defined every branch voltage and current as an
unknown, then written individually KVL, KCL, and branch V-I equations, and so obtained beaucoup
equations which can be solved (if you have enough strength and endurance!).
Now after all this what is, say, a supernode? If a voltage source in a node analysis is connected to the
reference node things even the limited backing up described is not needed. If you think about it the
source strength is a node voltage; ‘solve’ for that node by inspection. The current in that voltage source
Circuits Supernode/supermesh
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M H Miller
does not appear in any other node equation, so that takes care of that. If, however, the voltage source is
connected between two nodes (no reference node) then that source branch is associated with two node
equations; define the new unknown etc. and proceed as described above.
Now consider this. The source branch current appears in two node equations; in one it appears with a +
sign and in the other with a - sign ( it flows out of one node into the other node). Add the two equations
together and the current unknown will cancel out; use this as one independent equation in place of one
of the node equations. Throw the other node equation away and replace it with what the branch V-I
relation says, namely that the difference between the two node voltages at the branch ends is the source
strength. Now there is a set of N-1 equations that do not involve the added current variable. The
topographical equivalent to writing the two node equations and adding them is to write a KCL equation
for a volume of space that encloses both nodes (and therefore the voltage source branch also). Since
only branches that have one node outside the volume are involved in this equation the voltage source is
not involved. Since this process replaces two node equations with just one another independent equation
is needed. That is the equation relating the branch node voltages to the source strength.
The same sort of procedure can be applied to a supermesh.
It iss not hard to visualize, although I myself often get confused. Generally I myself introduce the
additional current variable, write the additional equation, and if I want or need to I add the equations
algebraically to eliminate the added current variable. I do similarly for a supermesh. To be really
honest, after I made sure I understood the analysis procedure and its subtleties, I began to use a
computer program to get numerical solutions.
Circuits Supernode/supermesh
2
M H Miller