Systematic circuit analysis
A single diagonal resistor is enough for a circuit to stop being reducible by series and parallel. Nodal analysis solves it just the same, and it is exactly what any circuit simulator does on the inside.
01When series and parallel are not enough
As long as a circuit can be reduced by combining resistors in series and in parallel, that is enough. But it takes just one element connecting two branches “diagonally” for there to be no pair of resistors strictly in series or in parallel. The classic example is the Wheatstone bridge with a detector between its two midpoints: four nodes, six branches and no possible simplification. For those cases you need systematic methods, which directly set up a system of linear equations and always work, whatever the topology.
There are two, one for each Kirchhoff law: nodal analysis, which uses the current law, and mesh analysis, which uses the voltage law.
02Nodal analysis
A reference node is chosen —the ground— and the unknowns are the voltages of the remaining \( N-1 \) nodes. At each one you write that the sum of the currents leaving is zero, with each current expressed as a voltage difference across a resistor. The result arranges itself in matrix form:
The matrix \( \mathbf{G} \) of a resistor circuit is symmetric, and that regularity is what makes it possible to write it without setting up a single equation by hand: you read it off the schematic.
03Mesh analysis
The dual method: in a planar circuit a current is assigned to each mesh —each “window” of the drawing— and in each one you write that the sum of voltages around it is zero.
| Nodal | Mesh | |
|---|---|---|
| Law it uses | Kirchhoff’s current law | Kirchhoff’s voltage law |
| Unknowns | \( N-1 \) node voltages | \( B-N+1 \) mesh currents |
| In the Wheatstone bridge (N = 4, B = 6) | 2 equations (the source fixes one node) | 3 equations |
| Sources that complicate things | Floating voltage sources: supernode | Shared current sources: supermesh |
| Works for | Any circuit | Planar circuits only |
04A Wheatstone bridge with a Pt100
The Pt100 is a platinum temperature sensor: it is 100 Ω at 0 °C and, according to the IEC 60751 standard, 138.51 Ω at 100 °C. Placed in a bridge with three other 100 Ω resistors, at 0 °C the bridge is balanced and the detector reads nothing; any change in temperature unbalances it. With the source fixing the top node and ground the bottom one, two unknowns remain: the voltages of the two midpoints.
R₁ = R₂ = R₃ = 100 Ω, R₄ is the Pt100 and the source is 5 V. You move the temperature and choose the detector; the panel builds the conductance matrix, solves it and verifies the current law at each node.
05Where it shows up in electronics
Every circuit simulator —SPICE and its descendants, LTspice, ngspice, the one in any design program— solves exactly this system. It uses a variant called modified nodal analysis, which adds the currents of the voltage sources as unknowns so as not to have to think about supernodes, builds the matrix by inspection by reading the component list, and solves it. With nonlinear components it repeats that step many times, linearizing around each point; in the frequency domain it does so with complex conductances.
| Case | Why it is solved with nodal analysis |
|---|---|
| Sensor bridges: Pt100, strain gauges, load cells | The detector between the midpoints prevents series-parallel reduction |
| Transistor biasing networks | Several sources and a node shared between base and emitter |
| R-2R ladder D/A converter | Every node of the ladder has the same form, and the matrix comes out with a regular pattern |
06In the lab
Set up the bridge with the Pt100 at 100 °C and a 1 kΩ galvanometer using nodal analysis. Solve the 2×2 system with Cramer’s rule and compare with the lab panel.
Set up the three mesh equations of the same bridge, solve them and verify that the detector current matches. Count how many operations each method required.
Load the bridge into LTspice or ngspice, sweep the Pt100 resistance from 100 to 175 Ω and plot the detector voltage. Verify that it is not exactly linear.
07Common mistakes
- Mixing current directions at the same node. It is best to always write all the currents as “leaving”: the sign takes care of itself.
- Forgetting the minus sign off the diagonal. The shared conductance enters subtracted.
- Setting up meshes in a nonplanar circuit. The mesh method requires being able to draw the circuit without crossings; the nodal method does not.
- Choosing the method with more equations. Count first: \( N-1 \) against \( B-N+1 \).
- Assuming the bridge is linear. The detector voltage does not grow exactly in proportion to the change in resistance, except for small changes.
08Self-assessment
Why can a Wheatstone bridge with a detector not be reduced by series-parallel combination?
Because the detector joins the two midpoints and no pair of resistors is strictly in series or in parallel.
How is the conductance matrix built by inspection?
Diagonal: sum of the conductances connected to each node. Off the diagonal: minus the conductance joining the two nodes.
How many equations does each method give in a circuit with N nodes and B branches?
Nodal: \( N-1 \). Mesh: \( B-N+1 \).
When is the bridge balanced?
When \( R_1/R_2 = R_3/R_4 \): the two midpoints are at the same voltage and no current flows through the detector.
What method does SPICE use?
Modified nodal analysis: nodes, plus the voltage-source currents as additional unknowns.
09Further reading
- Charles K. Alexander and Matthew N. O. Sadiku. Fundamentals of Electric Circuits. 7th ed., McGraw-Hill, 2021. The chapters on nodal and mesh analysis, with supernodes, supermeshes and writing the equations by inspection.
- Chung-Wen Ho, Albert E. Ruehli and Pierce A. Brennan. The Modified Nodal Approach to Network Analysis. IEEE Transactions on Circuits and Systems, vol. 22, 1975. The paper that defined the modified nodal analysis used by all simulators.
- IEC 60751. Industrial platinum resistance thermometers and platinum temperature sensors. IEC, 2022. The standard that fixes the Pt100 resistance curve used in the lab.