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:

b) Determine the Thevenin Resistance.
To determine the Thevenin Resistance we need to "zero" sources and then redraw the circuit. In this case we simply replace the voltage source with a short circuit.
c) Draw the Thevenin Equivalent Circuit

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

b) Determine the Thevenin Resistance.
As before we need to "zero" the sources and then redraw the circuit.

c) Draw the Thevenin Equivalent Circuit.

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


b) Determine the Thevenin Resistance.
As before we need to "zero" the sources and then redraw the circuit. We need to replace Vs1 with a short circuit and Is1 with an open circuit:
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.
