# Norton's Theorem

**Norton's Theorem, aka the Mayer–Norton theorem, **states that any *linear circuit* can be represented by a **single current source, In, in parallel with resistance Rn**.

This is remarkably similar to **Thévenin's theorem**, the only difference is that we convert a complex network into a **current source** **in parallel** with a **resistor **instead of a *voltage source* *in series* with a *resistor*.

The circuit shown above on the left can be simplified to the circuit on the right "Norton Equivalent Circuit" using **Nortons Theorem**. If we attach a circuit to terminals X and Y, a **load**, then both circuits will drive the load identically.

## Mayer and Norton

Norton's theorem was independently derived in the same year 1926 by Siemens and Halske researcher Hans Ferdinand Mayer and Bell Labs engineer Edward Lawry Norton.

### Determining the Norton Equivalent Circuit

We can derive the Norton Equivalent Circuit using a process which can be broken down into two steps, calculating the **Norton Current** and the **Norton Parallel Resistance**.

### Norton Current

The **Norton Current** is the **short circuit current** measured through terminals X and Y which can be connected to an external load.

### Norton Resistance

The **Norton Resistance**, like the Thevenin Resistance, is the **resistance measured across terminals X and Y** when all of the sources are "zeroed".

We can think of this as being the resistance **looking back** into terminals X and Y.

Remember: To "zero" a **voltage source** we replace it with a **short circuit** and to "zero" a **current source** we replace it with an **open circuit**. Recall that the voltage across a short **is zero** and **no current** will flow through an **open circuit** hence effectively *zeroing* their effect.

### Relationship between Thevenin Voltages and Norton Currents:

The Thevenin and Norton resistances **are equivalent** and can also be determined by using **Vth** and **In** with Ohm's law, i.e.

### Examples:

**1. a)** Determine the Norton Current of the circuit below:

**Norton Current**, we need to find the

**short circuit current**which would flow between terminals X and Y. With a short through terminals X and Y we then have a simple current divider circuit with I1 being divided between branches containing R2 and R3 and the branch containing R3 is the current we need to calculate:

**b)** Determine the Norton Resistance.

**Norton Resistance**we need to "zero" sources and then redraw the circuit. In this case we simply replace the

**current source**with an

**open circuit**. The Norton Resistance is then equal to the resistance seen across terminals X and Y which is simply R2 in parallel with R3 as no current will flow through R1 with the open:

**c)** Draw the Norton Equivalent Circuit

**d)** Determine the current which would flow through a 2kΩ load

**2. a)** Determine the Norton Current of the circuit below:

**Va**: At node

**Vb**: From here we can multiply [2] by 3 and then add [1] and [2] together to eliminate Va to solve for Vb which we need to determine In: We can now use Vb and Ohm's Law to calculate In:

**b)** Determine the Norton Resistance.

**Norton Resistance**we need to "zero" sources and then redraw the circuit. In this case we simply replace the current sources with an open circuits: From here we can simplify the circuit further by removing the open circuit paths: The Norton resistance is then simply (R4+R3+R2)||R5:

**c)** Determine the current which would flow through a 1kΩ load

### Questions:

**1. a)** Determine the Norton Current of the circuit below:

#### Answer

**b)** Determine the Norton Resistance.

#### Answer

**c)** Draw the Norton Equivalent Circuit.

**d)** What current would flow through a 200Ω load?

#### Answer

**2. a)** Determine the Norton Current of the circuit below:

#### Answer

**b)** Determine the Norton Resistance.

#### Answer

**c)** Draw the Norton Equivalent Circuit.

**d)** What would the voltage drop across a 2kΩ load be?

#### Answer

**3. a)** Determine the Norton Current of the circuit below:

#### Answer

**b)** Determine the Norton Resistance.

#### Answer

**c)** Draw the Norton Equivalent Circuit.

**d)** What current would flow through a 5kΩ load?