Edward Lawry Norton was an American Engineer, a scientist who graduate from MIT and Columbia. In the early 1900’s, he came up with his famous Norton’s Theorem which is the equivalent of Thévenin’s Theorem only for a current source.

Essentially, Norton’s theorem is a theoretical method to simplify down a complex circuit to a much simpler one with only 2 components: A current source and a resistor in parallel.

If the term ‘current source‘ is a bit foggy, brush up on your facts in this other article I’ve uploaded recently: Voltage and Current Source Differences

Below, on the left we have a fairly complex circuit. With a bit of circuit analysis we get to a simpler equivalent circuit which is on the right.

Example Circuit using Norton's Theorem to find the Norton Equivalent Circuit

So Why to Simplify a Circuit?

  • We connect a load (speaker, headphones, light bulb) to a source (amplifier, phone etc) which is usually this complex circuit what we are trying to simplify.
  • When we want to do any measurements on the load in the circuit, we have to consider the source too.
    • The source supplies the voltage & current for the load
  • So the source is complicated. We do not want to calculate with it every time when we really care about only the load. Thus, we do this complex calculation once and substitute the source with only a current source and the resistor in parallel. This is called a Norton Equivalent Circuit.
  • The Norton Equivalent Circuit essentially describes the ‘properties’ of the source.

Any linear circuit consisting of voltage sources, current sources and resistors can be replaced with an equivalent circuit consisting only of a single current source and a single resistor in parallel.

We refer to the single current source as Norton Current Source (In) and the single resistor as Norton resistance (Rn).

How to find In and Rn:

  • Norton Current Source (In) – To get In, we disconnect the load (if connected). Next, we short circuit (connect together) the terminals and calculate the current in this wire.
  • Norton resistance (Rn) – To get Rn, we short all voltage sources and open circuit all current sources inside the source. Next, calculate the resistance that you see when looking in at the terminals.

Thévenin to Norton and Back

  • The Thevenin and Norton circuits are interchangeable
  • There are 2 concepts to remember:
Example Circuit using Norton's Theorem to find the Norton Equivalent Circuit
Example Circuit using Norton's Theorem to find the Norton Equivalent Circuit

From Thevenin to Norton:

  • Take out load
  • Short terminals
  • In is the current through Rth:
    • In = Vth/Rth
  • Open terminals and short the voltage source
  • Rn is the resistance we measure across the terminals
    • Which is Rth
    • So Rn=Rth

From Norton to Thevenin:

  • Disconnect load and calculate voltage at the terminals
    • This will be the voltage across Rn, so
    • Vth = In x Rn
  • Next, open circuit In
  • Rth is the resistance measured across the terminals
    • which is clearly equal to Rn
    • Rth=Rn

Worked Example:

Find the Norton Equivalent of the following circuit. Next, calculate the current through the load.

Example circuit for demonstrating Norton's Theorem

 

Solution

1. Finding the Norton Equivalent Resistance:

  • STEP 1: Take out the load.
  • We do both calculations without RL.
  • For the Norton Equivalent Circuit we need to find Rn and In.
  • STEP2: Short all voltage sources and open circuit all current sources
    • We have one voltage source so let’s short it.
  • STEP3: Find the resistance you would measure at the terminals
    • since we are measuring through the terminals, R1 and R2 appear to be in parallel:
Calculating the Norton Equivalent Resistance

2. Finding the Norton Equivalent Current:

Method 1:

  • STEP 1: Short the terminals
    • Put Vs back in the circuit
    • Note, there won’t be any current flowing through R2 if we short the terminals. Why? Because a short has no resistance. The current will go the easy way.
Method 1 for calculating the Norton Equivalent Current using Norton's Theorem
  • STEP 2: Calculate the current through R1

    • We have a single resistor R1 in a closed loop.
    • In is the current in the branch where the short is. This is the same current as the current through R1.
    • Re-drawing the circuit without R2:
Method 1 for calculating the Norton Equivalent Current using Norton's Theorem step2

Method 2:

  • We know that In=Vth/Rth
  • We also know that Rth=Rn
  • All we need to do tis find Vth to get In
  • We jump back to the circuit where we took out RL but there is no short circuit.
  • The circuit what we have is a voltage divider.
    • Simply use the voltage divider equation equation to get Vth:
Method 2 for calculating the Norton Equivalent Current using Norton's Theorem step 1
  • Now use Ohm’s Law to calculate In:
Method 2 for calculating the Norton Equivalent Current using Norton's Theorem step 2

3. Calculating the current through the load:

  • Now that we know In and Rn, draw the Norton Equivalent Circuit
    • This will be a current source In with a resistance Rn in parallel.
  • In splits in two.
    • We are interested in the current I2 that goes through RL
Finding the current across the load in the Norton Equivalent Circuit

optional reading: Success in Electronics book by Tom Duncan
NEXT TOPIC: Understanding Input & Output Resistance