Equivalent voltage source

Thévenin’s (voltage-source) equivalent circuit 

Here we show how to convert any circuit with a voltage source and several resistors into an equivalent voltage source with only one internal resistor.

Thevenin’s theorem states that any linear circuit, no matter how complex, can be simplified to an equivalent circuit consisting of a single voltage source with only one internal series resistance. 

Learning by doing. Here is a simple example:

Thévenin's (voltage-source) equivalent circuit – how to use it - example

What we want: To simplify the circuit in pic a and get an equivalent voltage source with only one internal resistor R_eq as shown in pic b:

Thévenin's (voltage-source) equivalent circuit – how to use it - example

Step 1: Determining the new source voltage V_eq

The open-circuit voltage is the voltage that is tapped at terminals A and B when no load current is flowing. In this case, the open-circuit voltage corresponds to the voltage V_R2, as no current flows via R₃. The open-circuit voltage can therefore be easily determined using the voltage divider formula:

Step 2: Determine the equivalent resistance R_eq

To determine the equivalent internal resistance, we eliminate the voltage source by short-circuiting it.

Thevenin’s theorem - Determine the equivalent resistance

The equivalent internal resistance is the resistance to be measured between terminals A and B. You can see that R₁ and R₂ are in parallel and also in series with R₃:

We have now determined all the parameters of the equivalent voltage source. If desired, the short-circuit current can be easily determined using Ohm's law:

I_max =   V_eq
.            R_eq

Exercises

Simplify the following real voltage source according to Thevenin's theorem.
Values: V₀ = 30 V; R₁ = 10 Ω; R₁ = R₂ = 20 Ω

Thévenin's (voltage-source) equivalent circuit – exercise

The circuit would be easier to understand if we redraw it:

Exercise Equivalent Voltage Source - Thevenin's theorem -redrawing the circuit

Now we proceed as usual:

Step 1: Determining the new source voltage V_eq

As you can see, the voltage V_AB in open circuit (i.e. without current flow) corresponds to the voltage at resistor R₂. We apply the voltage divider rule:

V_eq = V_R2 = V_R3 = V_o/2 = 15 V 

Step 2: Determine the equivalent resistor R_eq

To do this, we eliminate the voltage source by short-circuiting it:

Equivalent voltage source exercise - Theorem of Thevenin - determine the equivalent internal resistance

R_eq = R₂ II R₃ + R₁ = 20 Ω / 2 + 10 Ω = 20 Ω