MESH ANALYSIS
Dual procedures exist for analyzing circuits, one emphasizing voltage variables and the other current
variables. A special case of the former, the node-to-datum variable procedure probably is the one most
commonly used. No matter the complexity of the circuit topology it is comparatively easy to identify
the nodes, and the branches associated with each node. Writing node equations can be tedious but it is
rarely particularly complicated. There is a similar although more special case for current variables, less
frequently used but nevertheless sufficiently common and simple enough procedurally to warrant
becoming familiar with it. This is the 'mesh' analysis.
An advantage in the use of voltage variables is that a voltage is an 'across' parameter; measuring a
voltage difference between two nodes of a circuit does not require cutting into a branch in order to insert
a meter into the circuit. The conceptual and procedural advantage to this is that a simplicity in
identifying node voltage variables is retained however convoluted the circuit topology. In contrast a
current is a 'through' variable; measuring a current requires cutting into a circuit branch to insert a meter.
The conceptual consequence of this is that one has to follow some sort of closed path ('loop') to define a
current variable, and this can be a torturous undertaking for some circuit topologies. Fortunately there is
a class of topologies, of particular technological importance, for which this is a much-simplified
undertaking.
A 'planar' network is one which can be stretched out onto a plane in such a way that no branches cross.
Consider the cubic circuit topology shown below; assume the diagonal branch AB is removed. The
circuit then is 'planar', as the 'squashed' drawing to the left of the cube indicates (absent the dotted
diagonal branch). With the diagonal branch added the topology necessarily becomes non-planar.
Although the added complexity in defining and following loops is marginal in this relatively simple
illustration it does not require much additional topological complexity before a ready visualization of the
procedure is lost. A special (tieset) analysis can be performed readily for any topology, since it involves
visualizing just one loop at a time. But as it happens planar networks are not uncommon since circuits
are often conceptualized in two dimensions and then assembled on a planar circuit board. For ‘planar’
networks the special case of a 'mesh' analysis is easy to apply and warrants familiarization.
A planar network has an appearance (on a plane) similar to that of a fishnet laid out to dry. The
branches encircle a set of meshes (note that the meshes are the holes in the net!). Each branch
enclosing a mesh of a planar network has the interesting property that it either bounds just one mesh (a
mesh on the periphery of a circuit) or it is common to two, and no more than two, meshes. An example
is shown on the next page.
ECE210 Mesh Analysis
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M H Miller
Independent current variables can be defined in the
following manner. Imagine a loop current circulating
around each mesh. While it is not strictly theoretically
necessary there is a considerable simplification if the loop
currents are assigned polarities corresponding to circulation
in the same direction; either CW or CCW will do. A set of
such mesh currents is drawn in the example circuit. It
should be clear that the choice of mesh currents allows each
branch current to be specified as an algebraic sum of no
more than two mesh currents. (For example the branch
current in branch #6, flowing from node a to node e is
I1– I2.) using loop currents as described means KCL
equations at each node are inherently satisfied.
The mesh currents provide a sufficient set of current variables sufficient to describe the current in each
branch; no more are needed and fewer would mean some branch currents could not be described. A
branch current can be described using no more than two loop currents. In fact it should not be surprising
that the number of mesh currents can be shown to be N-1, where N is the number of circuit nodes. The
mesh current variables are independent since meshes can be formed one by one, and each newly formed
mesh involves a branch not involved with any mesh formed earlier. To compete an analysis write KVL
equations about each mesh loop; in general the branch volt-ampere relations can be used 'on the fly' to
replace branch voltages as the KVL equations are written.
For a specific illustrative mesh analysis consider the planar circuit below. There are three meshes, easily
identified, and the three mesh currents are defined as shown. (There is no special significance to the
order in which the subscripts are assigned.) We need not explicitly write down the expressions for each
branch current in terms of the mesh
currents; these are so readily obtained that
they can be called up by inspection as
needed.
The next formal step is to write KVL
equations, in terms of branch voltages,
circulating around each loop. But this also
need not be done explicitly; it can be
combined easily with the next step, which is
to substitute for the branch voltages from
the branch volt-ampere relations to obtain
expressions in terms of the mesh current
variables.
Thus, for example, circulating clockwise the KVL equation for the I1 loop is:
1 = (I1 – I3)*1+ (I1 - I2)*2
Note that the current in the 1Ω branch flowing in the direction of circulation around the mesh is read
directly as the superposition I1 – I3). Similarly the current in the 2Ω branch flowing in the direction of
circulation around the mesh is read directly as the superposition I1 – I2).
As is common practice (for good reason) the KVL equation has been expressed as
ECE210 Mesh Analysis
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M H Miller
Sum of source voltage rises = Sum of branch voltage drops
Similarly for the other loops write
0 = (I1 - I2) *2 + (I2 - I3)*8 + I2*5
(I2 loop)
0 = I3*4 + (I3 - I2)*8 + (I3 - I1)*1
(I3 loop)
The three independent equations in three unknowns can be solved for the variable values using any of
the familiar techniques for solving simultaneous equations.
Problem 3.57, Irwin
The novelty in this illustration is the dependent source; the control voltage is proportional to Vo, defined
as the voltage across the 4Ω resistor as shown. However it is easy enough to express Vo in terms of the
loop current variables, Vo = 4 I1. This substitution can be done in the process of writing the loop
equations, as shown to the right of the circuit diagram.
SuperMeshes
There is a dual to the supernode of a node-to-datum analysis that occurs if a branch is a current source;
the voltage across the current source is not available to include in a loop equation. Handling such cases
is very similar to the methods described for node voltage variables.
For example introduce the unknown voltage drop across a source as a variable, and then as an additional
independent equation an expression equating the (loop) current though the source to the source strength.
A supermesh can be formed by combining the two meshes for which the current source branch is the
common branch; use the supermesh used to write a KVL equation. Note that the loop current in one of
the meshes involved can be expressed in terms of the other mesh current and the source current. There
is then one less KVL equation written, and one less variable involved.
As it happens few computer circuit analysis programs use a mesh analysis; it is no simple matter to
program a computer to identify a planar circuit. Most commonly a node-voltage computer analysis
procedure is used.
ECE210 Mesh Analysis
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M H Miller