Any source network — however complicated — can be reduced to a Thevenin equivalent: one voltage Vth in series with one resistance Rth. This theorem answers a simple, practical question: what value of load resistance RL gets the most power out of that source? Work through a real example below, one clear step at a time.
📐The rule: maximum power is transferred to the load when the load resistance equals the source's internal (Thevenin) resistance — RL = Rth. When that condition holds, the maximum power delivered is Pmax = Vth² ÷ (4Rth).
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Watch it happen — one step at a time
Circuit: E1 = 20 V (R1 = 5 Ω) and E2 = 10 V (R2 = 5 Ω) both feed a load RL. Click through each step below to find the RL that transfers the most power, and how much that power is.
Step 0 · The circuit
Two sources, one load — what value of RL wins?
E1 and E2 both feed a load RL. Whatever value we pick for RL, some power gets delivered to it — but only one value delivers the maximum possible power.
What we want to find:The R_L that gives maximum power, and that maximum power
Step 1 · Find R_th
Deactivate both sources, remove the load
To find the source network's own internal resistance, replace every voltage source with a short circuit (never remove the resistor in that branch) and take the load out. R_th is whatever resistance you'd measure looking back into the two open terminals.
Looking into terminals A–B, R1 and R2 are in parallel:
V_th is the open-circuit voltage measured across the terminals where the load used to be — with both sources back on and no load connected, no current can flow into R2's branch from the load side, so a loop current I flows around E1–R1–R2–E2.
However complicated the original network was, it behaves — as far as R_L is concerned — exactly like a single 15 V source in series with a single 2.5 Ω resistor. This is the circuit we actually apply the rule to.
Thevenin equivalent:V_th = 15 V, R_th = 2.5 Ω
Step 4 · Apply the rule
Set R_L = R_th
The rule says the load resistance should match the source's own internal resistance. R_th = 2.5 Ω, so that's the value of R_L that transfers the maximum power.
For maximum power transfer:R_L = R_th = 2.5 Ω
Step 5 · Calculate the maximum power
Work out the current, then the power
With R_L = R_th, the total resistance in the loop is R_th + R_L. Find the current, then use P = V×I — or jump straight there with the shortcut formula.
A Thevenin equivalent has V_th = 10 V and R_th = 4 Ω. What load resistance R_L gives maximum power transfer?
R_L = R_th = 4 Ω
Question 2 · Finding the maximum power
Using the same Thevenin equivalent (V_th = 10 V, R_th = 4 Ω), calculate the maximum power delivered to the load.
P_max = V_th² ÷ (4×R_th) = 100 ÷ 16 = 6.25 W
Question 3 · A full circuit
A solar cell has an open-circuit voltage of 20 V and an internal resistance of 8 Ω. What is the maximum power (in W) it can deliver to a matched load resistor?
This theorem answers a very practical 19th-century question — how do you get the most useful power out of a source that always has some internal resistance of its own? — and the answer turns out to be beautifully simple.
🕰️In short: the rule was worked out by a physicist studying electric motors, decades before it became the standard tool it is today for matching loudspeakers, aerials, and sensors to the circuits that drive them.
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Where it came from
⚙️ Born out of motor design
1840
Moritz Hermann von JacobiJacobi, a physicist and engineer working on early electric motors, showed that a motor draws the greatest mechanical power from a battery when the motor's own resistance matches the battery's internal resistance — the first clear statement of the principle.
Late 1800s
Telegraph and telephone networksAs long-distance electrical signalling grew, engineers rediscovered the same result: matching the receiving equipment's resistance to the line's resistance gave the strongest possible signal.
20th century
Radio, audio, and impedance matchingWith AC circuits and radio-frequency engineering, "resistance" widened out to "impedance" — but the same rule underpins matching an aerial to a transmitter, or a loudspeaker to an amplifier.
Today
A standard design toolMaximum power transfer now sits alongside Thevenin, Norton, and superposition as one of the core techniques for analysing — and designing — practical circuits.
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The logic — why it works
📈 A trade-off with a peak in the middle
Picture the load resistance RL sweeping from 0 Ω up to a very large value. At RL = 0 Ω the load is short-circuited — plenty of current flows, but V = IRL = 0, so no power is delivered. At very large RL, almost no current flows at all, so again the power is close to zero. Somewhere in between, power must reach a maximum.
The power delivered to the load is P = I²RL, where I = Vth ÷ (Rth + RL). Substituting and differentiating P with respect to RL, then setting the result to zero, gives exactly one solution: RL = Rth. That's the peak of the curve — push RL either side of it and the power delivered falls.
Put another way: Rth already "eats" some of the source's voltage no matter what RL is. Matching RL to Rth is the exact balance point between driving enough current and keeping enough voltage across the load to make the most of both.
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Where it's actually used
🔧 Real applications
🔊Audio amplifiers & speakersMatching a speaker's impedance to an amplifier's output impedance for the loudest, cleanest sound at a given power rating.
📡Aerials & transmission linesMatching an aerial's impedance to its feeder cable so the maximum possible radio-frequency signal actually radiates, instead of reflecting back down the line.
☀️Solar panel designSolar cells have their own internal resistance; MPPT (maximum power point tracking) electronics constantly adjust the effective load to sit at the matched point.
🔋Battery-powered sensor circuitsChoosing a sensor's input resistance to draw the most usable signal power from a small, high-resistance source like a thermocouple.
🎓A companion to TheveninBecause it's applied directly to a Thevenin equivalent, this theorem is usually the very next tool reached for once R_th and V_th are known.
🟢 Key words on this page
Load resistance (R_L)
The resistance of whatever is connected across the source's terminals to receive power — a speaker, an aerial, a sensor, a motor.
Internal / Thevenin resistance (R_th)
The resistance the source network itself presents at its terminals, found by deactivating every source and looking back in — exactly the same R_th used in Thevenin's theorem.
Impedance matching
The AC version of this same idea — choosing a load whose impedance matches the source's impedance for maximum power transfer.
Power transfer efficiency
The fraction of the total power generated that actually reaches the load, rather than being wasted inside the source's own resistance.
💡Worth remembering: matched for maximum power is not the same as matched for maximum efficiency. At R_L = R_th exactly half the total power generated is wasted inside R_th itself — efficiency is only 50%. Mains power distribution deliberately does the opposite, keeping source resistance tiny compared to the load so almost all the power (and none of the waste heat) reaches you.
SkillLondon · Maximum Power Transfer Theorem · Level 4 HN Engineering Science · skilllondon.co.uk
