Any circuit with a source and a load can be replaced by a single current source in parallel with a single resistance. Norton is Thevenin's current-source twin — work through the same real example below, one clear step at a time.
📐The rule: to find RN, short every voltage source (exactly like Rth). To find IN, replace the load with a short circuit and find the current flowing through that short.
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Watch it happen — one step at a time
Same circuit as the Thevenin example: a 10 V source (10 Ω) and a 20 V source (20 Ω) both feed a 40 Ω motor. Click through each step below.
Step 0 · The circuit
Two sources sharing one load
A 10 V source (through 10 Ω) and a 20 V source (through 20 Ω) both connect to a 40 Ω motor between terminals A and B.
What we want to find:The current through the motor
Step 1 · Short the terminals
Replace the motor with a plain wire
Norton's theorem removes the load and puts a short circuit across A–B instead (Thevenin leaves it open). This short pins both A and B to the exact same voltage.
With A and B at the same potential:each branch acts independently
Step 2 · Find R_N
Short the sources instead — exactly like R_th
R_N is found exactly the same way as Thevenin's R_th: short every voltage source, then find the resistance looking in from A–B. R1 and R2 both run from A to B, so they're in parallel.
Because A and B are pinned to the same voltage by the short, each source drives current through only its own resistor — R1 doesn't affect E2's current, and vice versa. Just Ohm's Law, twice.
I1 (from E1) = E1 ÷ R1 = 10 ÷ 10= 1 A
I2 (from E2) = E2 ÷ R2 = 20 ÷ 20= 1 A
I_N = I1 + I2 = 1 + 1= 2 A
Step 4 · Build the equivalent
Everything becomes ONE current source, ONE resistor
Instead of a battery in series, Norton's equivalent is a current source in parallel with R_N.
Norton equivalent:I_N = 2 A, R_N = 6.67 Ω
Step 5 · Reconnect the load
Now bring the motor back
The 2 A splits between R_N and the motor — a current divider, not a series calculation, because they're now two parallel paths for I_N to take.
A 12 V source (6 Ω) and a 24 V source (12 Ω) both feed a load between A and B. Find I_N (short-circuit current).
I1 = 12÷6 = 2 A I2 = 24÷12 = 2 A I_N = 2 + 2 = 4 A
Question 2 · Find R_N
Using the same circuit (R1=6Ω, R2=12Ω), find R_N.
R_N = (6×12) ÷ (6+12) = 72 ÷ 18 = 4 Ω
Question 3 · Reconnect the load
Using I_N = 4 A and R_N = 4 Ω from Questions 1–2, a 4 Ω load is reconnected. Find the load current.
I_load = I_N × R_N ÷ (R_N+R_L) = 4 × 4 ÷ 8 = 2 A (R_N = R_L here, so it's an equal 50/50 split)
📜 The story behind Norton's Theorem
Where the idea came from, why it works, and where engineers actually reach for it today.
🕰️In short: Norton's theorem was worked out independently by two engineers in the same year — 1926 — as a current-source companion to Thevenin's theorem, which had already existed for over 40 years.
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Where it came from
1883
Thevenin's theorem is publishedFrench telegraph engineer Léon Charles Thévenin extends Ohm's Law to complex networks, showing that any two-terminal circuit of sources and resistors can be replaced by one voltage source in series with one resistor.
1926
Two engineers reach the same idea, separately43 years later, Edward Lawry Norton — an engineer at Bell Telephone Laboratories in the USA — worked out the current-source equivalent while designing telephone and amplifier equipment. In the same year, Hans Ferdinand Mayer, a researcher at Siemens & Halske in Germany, derived the identical result independently. Neither knew about the other's work.
1926
Only one of them published itMayer published his version in a German telecommunications journal. Norton never formally published his — it stayed inside an internal Bell Labs technical report. Because of this, some textbooks call it the "Mayer–Norton theorem," though "Norton's theorem" became the name that stuck in English-language teaching.
🎓 Who was Edward Norton?
Norton (1898–1983) was an American engineer who studied electrical engineering at MIT and later earned a master's degree from Columbia University. He spent his career at Bell Labs working on network theory, acoustics, and data transmission — the same lab that, decades later, would also be central to inventing the transistor. He held nineteen patents. Norton wasn't setting out to invent a "theorem" for its own sake — he needed a faster way to design recording and amplifier equipment where currents split across several parallel paths, and the current-source view made that far easier to calculate than sticking with Thevenin's voltage-source view.
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The logic — why it works
🔄 It's Thevenin's mirror image
Thevenin and Norton describe the exact same circuit from two different angles. Thevenin asks "if I looked into these two terminals, what single voltage source and series resistor would behave identically?" Norton asks the opposite question: "what single current source and parallel resistor would behave identically?" Because both are true descriptions of the same physical circuit, you can always convert one into the other using Ohm's Law — I_N = E_th ÷ R_th, and R_N = R_th always.
The reason both views are valid comes down to linearity: as long as every source and resistor in the circuit is linear (doubling the input doubles the output), the terminals of that circuit can only ever "behave" one way from the outside — no matter how tangled the components are on the inside. Norton's theorem is really just a statement that a current-source-and-resistor pair is enough to reproduce that one behaviour.
⚡ Why short the terminals to find I_N?
Shorting the load terminals removes the load's resistance from the equation entirely, so whatever current flows through that short is the maximum current the source network can push into a zero-resistance path — which, by definition, is the Norton current. It's the current equivalent of what Thevenin does by leaving the terminals open to find the maximum voltage.
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Where it's actually used
Once a source network is boiled down to I_N and R_N, an engineer can swap in any load value and instantly see the current split — without re-solving the whole circuit each time. That's the real payoff, and it shows up across several fields:
⚡Power systems & fault analysisSimplifying large power networks so engineers can quickly estimate fault currents and size protection relays and breakers.
🔌Electronic circuit designModelling how a source (a battery, a sensor, an amplifier stage) will behave once it's connected to different loads, without redrawing the whole circuit each time.
📡TelecommunicationsThe original motivation — Bell Labs used it to design amplifier and transmission equipment where several signal paths share a common line.
🔋Power distribution networks (PDNs)In modern PCB and chip design, engineers use Norton equivalents to model how stable a chip's power supply will stay under varying current demand.
🛠️Troubleshooting & maintenanceTechnicians use the simplified equivalent to predict how a circuit should behave, then compare that to what they measure to isolate faults.
🎓Teaching circuit analysisIt's one of the first tools used to show students that a tangle of sources and resistors can be replaced with something small enough to reason about.
🟢 Key words on this page
Linear network
A circuit where doubling the input (voltage or current) always doubles the output — no diodes or transistors operating outside their linear range.
Dual
A theorem or circuit that describes the same behaviour as another, but from the opposite viewpoint (here: current source instead of voltage source).
Two-terminal network
Any circuit, however complicated inside, that connects to the rest of the world through just two wires — the pair Norton's theorem is applied across.
💡Worth remembering: Norton's theorem isn't a separate idea from Thevenin's — it's the same circuit fact viewed from the current side instead of the voltage side. Whichever one is quicker for the problem in front of you is the one to reach for.
