Nodal analysis

Key takeaways

Nodal analysis belongs to engineering; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Nodal analysis to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Nodal analysis from memory before moving on to harder problems.

Reference excerpt

In electric circuit analysis, nodal analysis (also referred to as node-voltage analysis or the branch current method) is a method of determining the voltage between nodes (points where elements or branches connect) in an electrical circuit in terms of the branch currents. Nodal analysis is essentially a systematic application of Kirchhoff's current law (KCL) for circuit analysis. Similarly, mesh analysis is a systematic application of Kirchhoff's voltage law (KVL). Nodal analysis writes an equation at each electrical node specifying that the branch currents incident at a node must sum to zero (using KCL). The branch currents are written in terms of the circuit node voltages. As a consequence, each branch constitutive relation must give current as a function of voltage; an admittance representation. For instance, for a resistor, Ibranch = Vbranch * G, where G (=1/R) is the admittance (conductance) of the resistor. Nodal analysis is possible when all the circuit elements' branch constitutive relations have an admittance representation. Nodal analysis produces a compact set of equations for the network, which can be solved by hand if small, or can be quickly solved using linear algebra by computer. Because of the compact system of equations, many circuit simulation programs (e.g., SPICE) use nodal analysis as a basis. When elements do not have admittance representations, a more general extension of nodal analysis, modified nodal analysis, can be used.

Procedure Note all connected wire segments in the circuit. These are the nodes of nodal analysis. Select one node as the ground reference. The choice does not affect the element voltages (but it does affect the nodal voltages) and is just a matter of convention. Choosing the node with the most connections can simplify the analysis. For a circuit of N nodes the number of nodal equations is N−1. Assign a variable for each node whose voltage is unknown. If the voltage is already known, it is not necessary to assign a variable. For each unknown voltage, form an equation based on Kirchhoff's Current Law (i.e. add together all currents leaving from the node and mark the sum equal to zero). The current between two nodes is equal to the voltage of the node where the current exits minus the voltage of the node where the current enters the node, both divided by the resistance between the two nodes. If there are voltage sources between two unknown voltages, join the two nodes as a supernode. The currents of the two nodes are combined in a single equation, and a new equation for the voltages is formed. Solve the system of simultaneous equations for each unknown voltage.

Examples

Basic case

The only unknown voltage in this circuit is V 1 {\displaystyle V_{1}} . There are three connections to this node and consequently three currents to consider. The direction of the currents in calculations is chosen to be away from the node.

Current through resistor R 1 {\displaystyle R_{1}} : ( V 1 − V S ) / R 1 {\displaystyle (V_{1}-V_{S})/R_{1}}

Current through resistor R 2 {\displaystyle R_{2}} : V 1 / R 2 {\displaystyle V_{1}/R_{2}}

Current through current source I S {\displaystyle I_{S}} : − I S {\displaystyle -I_{S}}

With Kirchhoff's current law, we get:

V 1 − V S R 1 + V 1 R 2 − I S = 0 {\displaystyle {\frac {V_{1}-V_{S}}{R_{1}}}+{\frac {V_{1}}{R_{2}}}-I_{S}=0}

This equation can be solved with respect to V1:

V 1 = ( V S R 1 + I S ) ( 1 R 1 + 1 R 2 ) {\displaystyle V_{1}={\frac {\left({\frac {V_{S}}{R_{1}}}+I_{S}\right)}{\left({\frac {1}{R_{1}}}+{\frac {1}{R_{2}}}\right)}}}

… excerpt ends here. Continue reading the full article.

Illustrations

Worked examples

Example 1 — a first encounter with Nodal analysis

Start with the simplest possible case. Write down what Nodal analysis claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In engineering, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Nodal analysis before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Nodal analysis ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Nodal analysis

In research: Nodal analysis appears in engineering research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Nodal analysis in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Nodal analysis is common in secondary-school and first-year university syllabi. It links to neighbouring topics Electrical engineering, Electronic circuits, so understanding it makes those chapters shorter.
In everyday life: Look for Nodal analysis outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

How to study Nodal analysis in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Nodal analysis means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Nodal analysis out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Nodal analysis in simple terms?

In electric circuit analysis, nodal analysis (also referred to as node-voltage analysis or the branch current method) is a method of determining the voltage between nodes (points where elements or branches connect) in an electrical circuit in terms of the branch currents. Nodal analysis is essentia…

Why does Nodal analysis matter?

Because it connects several engineering ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Nodal analysis?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Nodal analysis.