mathematics

Norton's theorem

Norton's theorem is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Norton's theorem rather than just read about it. In short: In direct-current circuit theory, Norton's theorem, also called the Mayer–Norton theorem, is a simplification that can be applied to networks made of linear time-invariant resistances, voltage sources, and current sources. At a pair of terminals of the network, it can be replaced by a current source and a single resistor in parallel.

Key takeaways

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

Reference excerpt

In direct-current circuit theory, Norton's theorem, also called the Mayer–Norton theorem, is a simplification that can be applied to networks made of linear time-invariant resistances, voltage sources, and current sources. At a pair of terminals of the network, it can be replaced by a current source and a single resistor in parallel. For alternating current (AC) systems the theorem can be applied to reactive impedances as well as resistances. The Norton equivalent circuit is used to represent any network of linear sources and impedances at a given frequency. Norton's theorem and its dual, Thévenin's theorem, are widely used for circuit analysis simplification and to study circuit's initial-condition and steady-state response. Norton's theorem was independently derived in 1926 by Siemens & Halske researcher Hans Ferdinand Mayer and Bell Labs engineer Edward Lawry Norton. To find the Norton equivalent of a linear time-invariant circuit, the Norton current ⁠ I n o {\displaystyle I_{\mathrm {no} }} ⁠ is calculated as the current flowing at the two terminals ⁠ A {\displaystyle A} ⁠ and ⁠ B {\displaystyle B} ⁠ of the original circuit that is now short (zero impedance between the terminals). The Norton resistance ⁠ R n o {\displaystyle R_{\mathrm {no} }} ⁠ is found by calculating the output voltage ⁠ V o {\displaystyle V_{\mathrm {o} }} ⁠ produced at ⁠ A {\displaystyle A} ⁠ and ⁠ B {\displaystyle B} ⁠ with no resistance or load connected to, then ⁠ R n o = V o I n o {\displaystyle \textstyle R_{\mathrm {no} }={V_{\mathrm {o} } \over I_{\mathrm {no} }}} ⁠; equivalently, this is the resistance between the terminals with all (independent) voltage sources short-circuited and independent current sources open-circuited (i.e., each independent source is set to produce zero energy). This is equivalent to calculating the Thevenin resistance. When there are dependent sources, the more general method must be used. The voltage at the terminals is calculated for an injection of a 1 ampere test current at the terminals. This voltage divided by the 1 A current is the Norton impedance ⁠ R n o {\displaystyle R_{\mathrm {no} }} ⁠ (in ohms). This method must be used if the circuit contains dependent sources, but it can be used in all cases even when there are no dependent sources.

Example of a Norton equivalent circuit

In the example, the total current ⁠ I t o t a l {\displaystyle I_{\mathrm {total} }} ⁠ is given by:

I t o t a l = 15 V 2 k Ω + 1 k Ω ∥ ( 1 k Ω + 1 k Ω ) = 5.625 m A . {\displaystyle I_{\mathrm {total} }={15\,\mathrm {V} \over 2\,\mathrm {k} \Omega +1\,\mathrm {k} \Omega \parallel (1\,\mathrm {k} \Omega +1\,\mathrm {k} \Omega )}=5.625\,\mathrm {mA} .}

The current through the load is then, using the current divider rule:

… excerpt ends here. Continue reading the full article.

Illustrations

Norton's theorem: Edward Lawry Norton — Edward Lawry Norton

Worked examples

Example 1 — a first encounter with Norton's theorem

Start with the simplest possible case. Write down what Norton's theorem claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, 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 Norton's theorem 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 Norton's theorem 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 Norton's theorem

In research: Norton's theorem appears in mathematics 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 Norton's theorem 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: Norton's theorem is common in secondary-school and first-year university syllabi. It links to neighbouring topics Circuit theorems, Linear electronic circuits, so understanding it makes those chapters shorter.
In everyday life: Look for Norton's theorem 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.

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How to study Norton's theorem in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Norton's theorem 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 Norton's theorem out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Norton's theorem in simple terms?

Why does Norton's theorem matter?

Because it connects several mathematics 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 Norton's theorem?

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 Norton's theorem.