# Thévenin's Theorem

Léon Charles Thévenin, born in Meaux France, was a telegraph engineer and a talented violinist that was very interested in solving the problems of measurement in electrical circuits. He deeply studied KCL, KVL and Ohm's Law eventually developing the famous Thévenin's Theorem in 1883 which elegantly states that "any *linear circuit* can be represented by a **single voltage source, Vth, in series with resistance Rth**."

What's also fascinating to note is that the theorem was independently derived in 1853 by the German scientist Hermann von Helmholtz.

Let's decompose this theorem now, but first, what did Thévenin mean by a **linear circuit**? By *linear circuit* he simply meant that the circuit **obeys **the **superposition principle**. So what does this then mean? It means, we can simplify any network of **voltage**, **current sources** **and resistances** **connected together** to an **equivalent voltage source and resistor connected** to the outside world! It is really quite profound when you pause to think, I strongly encourage you to stop to think about this, any horrid rats nest of resistors and sources can be simplified to a voltage source in series with a resistor!

Note that a *linear circuit* contains all of the elements we have learned so far, namely, resistors, voltage sources and current sources. Later on we will encounter *non-linear* elements for which Thevenin's Theorem **does not** apply.

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

Determining the Thevenin Equivalent Circuit

We can derive the Thevenin Equivalent Circuit using a process which can be broken down into two steps, calculating the **Thevenin Voltage** and the **Thevenin Series Resistance**.

### Thevenin Voltage

The **Thevenin Voltage** is the **open circuit voltage** observed across terminals X and Y which can be connected to an external load.

### Thevenin Resistance

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.

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.

### Examples:

**1. a)** Determine the Thevenin Voltage of the circuit below:

**no current**will flow through X and Y and hence no current will

**flow through R3**which means that the voltage across R3 is

**zero**and that the

**voltage across R2**is then equal to

**Vxy**. Knowing this we then have a series circuit composed of R1 and R2 and we can apply the voltage divider rule to find Vxy:

**b)** Determine the Thevenin Resistance.

**c)** Draw the Thevenin Equivalent Circuit

**2. a)** Determine the thevenin voltage of the circuit below:

**first principles**we can determine the Thevenin Voltage MUCH easier than you think :) Firstly notice that

**Vs2**presents a voltage of 2V at the

**node**which connects to R3 and R4 which means that we have a voltage difference of 5V across R4 and R5 as

**0A flows**through the open terminals X and Y. This means that R4 and R5 acts as a simple

**voltage divider**and as R4 and R5 are the same value, it follows that

**half**of the voltage will be dropped across R4 which means that

**Vth**is simply 2.5V. It often pays to analyse a circuit from first principles first before going into a systematic technique!

**b)** Determine the Thevenin Resistance.

**short circuit**of course! Which means that we can eliminate all of the circuit elements to the left hand side of R4 for a considerably simpler circuit! From here

**Rth**is simply R4

**in parallel**with R5 which is 1kΩ!

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

**3. a)** Determine the thevenin voltage of the circuit below:

**systematic approach**is warranted here. The Thevenin voltage is equal to the voltage across X Y which appears as the second

**principal node**in the circuit. Let's employ

**Nodal Analysis**here and begin by labelling the nodes: At node

**Va**: At node

**Vb**: With two equations and two unknowns we can use the process of elimination to solve for Va and Vb. We can multiply [1] by 9 and then add to [2] to eliminate Vb and solve for Va: Substiting back into [1] to solve for Vb: Now Vb represents the open circuit voltage for the load and hence is equal to the Thevenin Voltage resulting in Vth = 14.78V.

**b)** Determine the Thevenin Resistance.

### Questions:

**1. a)** Determine the thevenin voltage of the circuit below:

#### Answer

**b)** Determine the Thevenin Resistance.

#### Answer

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

**2. a)** Determine the thevenin voltage of the circuit below:

#### Answer

**b)** Determine the Thevenin Resistance.

#### Answer

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

**3. a)** Determine the Thevenin Voltage of the circuit below:

#### Answer

**b)** Determine the Thevenin Resistance.

#### Answer

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