Maximum power transfer theorem

Key takeaways

Maximum power transfer 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 Maximum power transfer theorem to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Maximum power transfer theorem from memory before moving on to harder problems.

Reference excerpt

In electrical engineering, the maximum power transfer theorem states that, to obtain maximum external power from a power source with internal resistance, the resistance of the load must equal the resistance of the source as viewed from its output terminals. Moritz von Jacobi published the maximum power (transfer) theorem around 1840; it is also referred to as "Jacobi's law". The theorem results in maximum power transfer from the power source to the load, but not maximum efficiency of useful power out of total power consumed. If the load resistance is made larger than the source resistance, then efficiency increases (since a higher percentage of the source power is transferred to the load), but the magnitude of the load power decreases (since the total circuit resistance increases). If the load resistance is made smaller than the source resistance, then efficiency decreases (since most of the power ends up being dissipated in the source). Although the total power dissipated increases (due to a lower total resistance), the amount dissipated in the load decreases. The theorem states how to choose (so as to maximize power transfer) the load resistance, once the source resistance is given. It is a common misconception to apply the theorem in the opposite scenario. It does not say how to choose the source resistance for a given load resistance. In fact, the source resistance that maximizes power transfer from a voltage source is always zero (the hypothetical ideal voltage source), regardless of the value of the load resistance. The theorem can be extended to alternating current circuits that include reactance, and states that maximum power transfer occurs when the load impedance is equal to the complex conjugate of the source impedance. The mathematics of the theorem also applies to other physical interactions, such as:

mechanical collisions between two objects, the sharing of charge between two capacitors, liquid flow between two cylinders, the transmission and reflection of light at the boundary between two media.

Maximizing power transfer versus power efficiency

The theorem was originally misunderstood (notably by Joule) to imply that a system consisting of an electric motor driven by a battery could not be more than 50% efficient, since the power dissipated as heat in the battery would always be equal to the power delivered to the motor when the impedances were matched. In 1880 this assumption was shown to be false by either Edison or his colleague Francis Robbins Upton, who realized that maximum efficiency was not the same as maximum power transfer.

To achieve maximum efficiency, the resistance of the source (whether a battery or a dynamo) could be (or should be) made as close to zero as possible. Using this new understanding, they obtained an efficiency of about 90%, and proved that the electric motor was a practical alternative to the heat engine. The efficiency η is the ratio of the power dissipated by the load resistance RL to the total power dissipated by the circuit, which includes the voltage source's resistance of RS as well as RL:

η = P L P T o t a l = I 2 ⋅ R L I 2 ⋅ ( R L + R S ) = R L R L + R S = 1 1 + R S / R L . {\displaystyle \eta ={\frac {P_{\mathrm {L} }}{P_{\mathrm {Total} }}}={\frac {I^{2}\cdot R_{\mathrm {L} }}{I^{2}\cdot (R_{\mathrm {L} }+R_{\mathrm {S} })}}={\frac {R_{\mathrm {L} }}{R_{\mathrm {L} }+R_{\mathrm {S} }}}={\frac {1}{1+R_{\mathrm {S} }/R_{\mathrm {L} }}}\,.}

Consider three particular cases (note that voltage sources must have some resistance):

… excerpt ends here. Continue reading the full article.

Illustrations

Maximum power transfer theorem: source and load impedance diagram — source and load impedance diagram

Worked examples

Example 1 — a first encounter with Maximum power transfer theorem

Start with the simplest possible case. Write down what Maximum power transfer 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 Maximum power transfer 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 Maximum power transfer 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 Maximum power transfer theorem

In research: Maximum power transfer 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 Maximum power transfer 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: Maximum power transfer theorem is common in secondary-school and first-year university syllabi. It links to neighbouring topics Circuit theorems, Electrical engineering, so understanding it makes those chapters shorter.
In everyday life: Look for Maximum power transfer 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.

How to study Maximum power transfer theorem in 20 minutes

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

Frequently asked questions

What is Maximum power transfer theorem in simple terms?

In electrical engineering, the maximum power transfer theorem states that, to obtain maximum external power from a power source with internal resistance, the resistance of the load must equal the resistance of the source as viewed from its output terminals. Moritz von Jacobi published the maximum p…

Why does Maximum power transfer 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 Maximum power transfer 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 Maximum power transfer theorem.