EC-1.1 Node and Mesh Analysis

What this is and why it exists

A circuit diagram is a picture. To predict what it does you need equations, and you need a way of writing them that does not depend on cleverness.

Two conservation laws generate every equation in this subject. Node analysis and mesh analysis are two systematic ways of applying them. The skill is choosing the one that leaves less algebra, and then being careful about signs.

The vocabulary

• Node — a point where two or more elements meet, at one common potential.
• Reference node — the node you declare to be zero volts. Every other potential is measured from it.
• Mesh — a window of a flat circuit diagram, with nothing drawn inside it.
• Mesh current — an imagined current circulating around one window.
• Supernode — two nodes joined by a voltage source and treated as one region.
• Supermesh — two windows sharing a current source and treated as one loop.
• Ideal voltage source — holds its voltage whatever current flows.
• Ideal current source — holds its current whatever voltage appears across it.

The mental model

The first law says that the currents leaving a node add up to zero. That is not an electrical fact so much as a statement that charge does not pile up at a junction. The second law says that the voltages around any closed loop add up to zero. That is a statement that potential has one value at each point, so walking round a loop must bring you back to where you started.

Everything else in this subject is bookkeeping on top of those two sentences.

Node analysis applies the first law. Choose one node as the reference and call it zero. Every other node has an unknown potential. At each of those nodes, write that the currents leaving it sum to zero, expressing each current as a potential difference divided by a resistance. You get one equation per unknown potential.

Choosing the reference well removes an unknown for nothing. Ground the node that the most elements touch, and several currents become simple.

A voltage source between two unknown nodes is the case people get wrong first. It fixes the difference between the two potentials but neither value on its own, and you cannot write a current through an ideal source. The answer is to draw a ring around both nodes and treat them as one region. Currents leaving that whole region still sum to zero, which gives one equation, and the source itself gives the constraint that supplies the second.

Mesh analysis applies the second law. Assign a circulating current to each window of the diagram. Then, for each window, write that the voltages around it sum to zero. An element shared by two windows carries the difference of their two currents.

Choose the same rotation for every window, clockwise for all of them. Then a shared element always subtracts rather than adds, and one whole family of sign errors disappears.

A current source shared between two windows is the mirror of the supernode. It fixes a current but leaves the voltage across itself unknown, so the loop law cannot be written through it. Combine the two windows into one larger loop that avoids the source, and let the source supply the missing equation.

The last habit is the cheapest one. Before writing anything, count the unknowns both ways. The same circuit is often three equations one way and six the other, and counting takes ten seconds. Solving one network by both methods and getting the same answer is also the fastest way to find your own sign mistakes.

What you should now be able to explain or do

State both conservation laws and say what each is really claiming about the physical world. Choose a reference node well and write one equation per unknown potential. Handle a voltage source between two unknown nodes with a supernode. Assign mesh currents with one consistent rotation and explain why that removes sign errors. Handle a shared current source with a supermesh, and count unknowns both ways before starting.

Check yourself