Why can't mesh analysis be used for non-planar circuits?

Question

My book simply states this without any kind of explanation.

Answer 1

The basic pedestrian answer goes like this:

1. All you have in analyzing a network of simple components is KCL and KVL. If you write out all KCL and KVL equations, you (or a computer) can solve the circuit. (Assuming no impossible conditions, like a voltage source across a short circuit, or current source into an open circuit).

2. However, proceeding that way, with no other aid, is tedious and error prone, as it's extremely difficult to keep track of all the current and voltage directions.

3. So, as an accounting convenience, mesh analysis introduces the notion of "current loops". Each current loop is not a distinct phenomenon that can be individually observed. They are simply an "accounting breakdown", and they directly follow from KCL. But their great benefit is that they establish a rigorous convention for accounting for direction at every point in the network. Note: not the actual direction of current or voltage, but merely the direction to be counted as the "+" direction. If the actual direction turns out to be the other way, then account for it as negative.

4. But this "mesh"/"current-loops" accounting breakdown of KCL and KVL is only valid if our accounting method correctly totals the current on each wire attached to each node -- not omitting some portion of that current, and not double-counting some portion of that current. The customary way to accomplish this is to focus only on innermost loops. (An innermost loop is one with no other wiring or component drawn within it.) For example, we don't add additional loops for every possible closed path through the network! We rely on "only count all innermost loop currents" as the criterion for ensuring that we are only counting currents that exactly add up to what KCL expects.

5. But there are some networks in which we can't uniquely identify "innermost-loops". These are the so-called "non-planar" circuits, ones that can't be drawn on flat paper without crossovers. In that topology, KCL and KVL still work, of course. But for some parts of the network we find candidate loops that also have one end of an additional branch that passes through the interior of that loop. Whether we either include or exclude that loop in a loop analysis, the totals wouldn't add up properly to what KCL requires. We can therefore not use "count all innermost loops" as the basis of complying with KCL at all nodes. Consequently, the accounting simplification of (innermost) loop currents can't be used with non-planar circuits.

Answer 2

My old Uni copy of Network Analysis (Van Valkenburg) goes on for a chapter or three building the mathematical (topological) background, touching on Euler's solution of the Konigsberg bridge problem in 1735 and Kirchhoff in 1947 and Listing.

What they call the "window pane" method allows the essential meshes to be identified by inspection- loops without internal loops, and essential branches are branches that don't cross other branches.

If it's not planar, you can't draw it as so, so they suggest you should employ what they call the "tree and chord set" method to analyze the circuit.

You could probably get a rigorous mathematical proof based on graph theory, but not from moi.

Answer 3

I think that the reason is in terms of definitions, so that the method can be specified in such a way that it is easy to apply.

If the circuit is not planar, then the "3-D" branches don't have clearly definable meshes, since in 3-D you can't talk about "loops that don't have inner loops". The loop that contains the components in the 3-D branches can have many paths, and is not unambiguous like in planar circuits.

It is also not possible anymore to have each component only have 2 or 1 meshes (it can have more), and it is not possible to have an easy to follow convention for loop current direction.

All these complications I think made it worth limiting the mesh analysis to 2-D, and leave the "loop analysis" with its "loop currents" as a more general method. In that method you define the loops in a more general way, as long as every component is contained inside at least one loop. Same thing, just more difficult to keep track of, but equally valid.

Answer 4

I think the Spehro gives about as a complete answer one can hope for without a foray into the homology of CW complexes.

I just wanted to add an example so you can see how planar is subtly used in mesh analysis. Take a voltage source \$V\$ and two resistors \$R_1\$ and \$R_2\$ connected all in parallel but not in the plane but in three space.

You will notice there is a symmetry now and you naturally have three loops. One loop of current through \$R_1\$ and the voltage source \$V\$ (call it \$i_1\$), one loop through \$R_2\$ and the voltage source \$V\$ (call it \$i_2\$), and a latter between the two resistors (call it \$i_3\$).

Now there are six ways to lay this circuit out in the plane (where the voltage source is oriented positively). A mesh analysis on any one of these embeddings will use two of the three loops. Moreover, for any choice of two of the three loops, there is a circuit lay out whose mesh analysis will use those two loops.

Notice that you cannot use all three loops in a mesh analysis because you get the set of equations $$V = R_1 (i_1 - i_3)$$ $$V = R_2 (i_2 + i_3)$$ $$0 = R_1(i_1-i_3) - R_2(i_2+i_3)$$

which has many solutions, but not a unique one.

Hopefully you can at least now believe that being planar involves a choice of which loops to use.