7.3 Thévenin’s Theorem


By the end of the section, you will be able to:
  • Find the Thévenin equivalent circuit for any linear circuit
  • Calculate the maximum power that can be transferred to a load at any point in a circuit, and the value of the load resistance required to draw maximum power

Thévenin’s theorem states that any linear circuit containing several voltage sources and resistors can be simplified to a Thévenin-equivalent circuit with a single voltage source and resistance connected in series with a load. Specifically, the three components connected in series are (see Figure 7.3.1(b)):

  1. Load resistor, R_L;
  2. Thévenin voltage V_{\mathrm{th}}, found by removing R_L from the original circuit and calculating the potential difference from one load connection point to the other (e.g. from a to b in Figure 7.3.1(a), either across R_1, V_1, and R_3 or across R_2 and R_3);
  3. Thévenin resistance R_{\mathrm{th}}, found by removing R_L from the original circuit and calculating the total equivalent resistance between the two load connection points (e.g. between a and b in Figure 7.3.1(a), thus as the equivalent resistance of the parallel combination of R_1 and R_2, connected in series with R_3).

(Figure 7.3.1)   \begin{gather*}.\end{gather*}

Figure 7.3.1 (a) An example of a DC resistive circuit with load resistor R_L identified, and (b) its Thévenin equivalent. In fact, (b) shows the general form of all Thévenin-equivalent circuits.

Thévenin’s theorem is particularly useful when the load resistance in a circuit is subject to change. When the load’s resistance changes, so does the current it draws and the power transferred to it by the rest of the circuit. In fact, currents everywhere in a circuit will be subject to change whenever a single resistance changes, and the entire circuit would need to be re-analysed to find the new current through and power transferred to a load. Repeating circuit analysis to find the new current through a load every time its resistance changes would be very time-consuming. In contrast, according to Thévenin’s theorem once V_{\mathrm{th}} and R_{\mathrm{th}} are determined for the rest of the circuit, the current through the load is always simply calculated as

(7.3.1)   \begin{equation*}I_L=\frac{V_{\mathrm{th}}}{R_{\mathrm{th}}+R_L},\end{equation*}

from which the voltage drop across, and power transferred to the load are, respectively,

(7.3.2)   \begin{equation*}V_L=R_LI_L=\frac{R_LV_{\mathrm{th}}}{R_{\mathrm{th}}+R_L},\end{equation*}

(7.3.3)   \begin{equation*}P_L=I_L^2R_L=\frac{V_L^2}{R_L}=\frac{R_LV_{\mathrm{th}}^2}{(R_{\mathrm{th}}+R_L)^2.}\end{equation*}

Equations are easily applied, and the problem of repeated circuit analysis each time a load’s resistance changes is mainly reduced to the one-time problem of finding the Thévenin voltage V_{\mathrm{th}} and resistance R_{\mathrm{th}} with respect to R_L. Example 7.3.1 shows the procedure for doing this for the circuit in Figure 7.3.1(a).


Applying Thévenin’s Theorem

Find V_{\mathrm{th}} and R_{\mathrm{th}} for the circuit in Figure 7.3.1(a).


  1. Find V_{\mathrm{th}}: note that with the circuit open between a and b there is no current through, and therefore no voltage drop across R_3. Therefore, the potential difference between a and b must occur in the loop containing V_1, R_1, and R_2. We are free to choose either parallel branch of that loop, as the potential difference across R_2 must equal the potential difference across V_1 and R_1 by the loop rule. Therefore, we will first determine the current in this loop and apply Ohm’s law to find V_{\mathrm{th}}=V_{R_2}.
  2. Find R_{\mathrm{th}}: Proceeding from a to b we encounter a junction where the circuit branches in two directions, towards R_1 and R_2. V_1 is an ideal voltage source with no resistance, and can therefore be ignored when calculating equivalent resistance. We then encounter another junction where the two branches reconnect, so R_1 and R_2 are connected in parallel. Proceeding on, we encounter R_3 in series with the parallel connection of R_1 and R_2, and eventually reach b. We will add these resistances using the rules for adding series and parallel resistors.


The current through the loop with V_1, R_1 and R_2 all connected in series is


By Ohm’s law, the voltage across R_2 is therefore


By our above reasoning, we therefore have


To find R_{\mathrm{th}}, first write


Then, by our above reasoning,



The potential difference from a to b was calculated as a drop in potential across R_2 as current flows from the positive to the negative terminal of the voltage source V_1. Along the parallel branch (that is, parallel from the perspective of the load connection points a and b), potential rises at V_1, then drops across R_1, travelling in the clockwise direction. By the loop rule, there must be an overall potential rise in the clockwise direction along this branch that equals negative the potential drop in the clockwise direction across R_2. Thus, between a and b along the left branch, travelling in the counter-clockwise direction there is also a drop in potential, equal to


as required.

It is important to note that perspective matters when treating components as being connected in series or parallel. Here, when determining the current through R_2 in the open circuit, we noted that current flows through a single circuit loop with V_1, R_1, and R_2 all connected in series, and determined the current through R_2 as the potential drop across the series combination of resistors, divided by the equivalent resistance. However, when calculating R_{\mathrm{th}} we found that from the perspective of the connection points R_1 and R_2 are connected along parallel branches of the circuit.

The procedure used here to calculate V_{\mathrm{th}} and R_{\mathrm{th}} is the same as that which we apply to more complex circuits. When doing so, it is important to correctly account for voltage rises and drops across between the two load connection points, although to this end we do have freedom of choice in which branch to follow and can always choose the simplest path.


The circuit is the same as the one from Example 7.3.1, but with R_1 replaced by a short. Determine V_{\mathrm{th}} and R_{\mathrm{th}} in this case.

Maximum Power Transfer Theorem

Thévenin’s theorem finds a useful application in the maximum power transfer theorem, which states that maximum power will be transferred to a load when its resistance is equal to the Thévenin resistance of the network supplying the power. This interesting and highly useful fact is easily proven by taking the derivative of Equation 7.3.3 with respect to R_L, setting the result equal to 0, and solving for the value of R_L that maximises the function.


    \[\Rightarrow 1-\frac{2R_L}{R_{\mathrm{th}}+R_L}=0\]

    \[\Rightarrow R_L=R_{\mathrm{th}}.\]


Applying Maximum Power Transfer Theorem

What is the maximum amount of power that can be dissipated in R_L?

Figure 7.3.2 A DC resistive network.


The maximum amount of power that can be dissipated in R_L is, by the maximum power transfer theorem, the power dissipated when R_L=R_{\mathrm{th}}, for the Thévenin equivalent circuit calculated with respect to R_L. To find this, we first determine V_{\mathrm{th}} and R_{\mathrm{th}}, as follows.

With R_L replaced by an open circuit, there are two loops: one, passing through V_1, R_1, R_2, and V_2; the other, passing through V_2, R_2, and R_3. We will calculate the current through R_3 using Mesh Analysis techniques developed earlier, then determine V_{\mathrm{th}}=V_{R_3} using Ohm’s law. Note that we do not actually need to calculate any other currents, since V_{\mathrm{th}}, the potential difference between a and b, must equal V_{R_3} regardless which branch is taken.

To find R_{\mathrm{th}}, note that with respect to connection points a and b, R_1, R_2, and R_3 are all connected in parallel.

Finally, when R_L=R_{\mathrm{th}}, the current in the load is I_L=V_{\mathrm{th}}/2R_{\mathrm{th}} (see Equation 7.3.1), and the power dissipated in R_L is P_{L,\mathrm{max}}=I_L^2R_L=V_{\mathrm{th}}^2/4R_{\mathrm{th}} (cf. Equation 7.3.3).


Using the strategies developed in Mesh Analysis, we can write the matrix equations for this network as


where I_1 and I_2 are the clockwise mesh currents in the left and right loops, respectively.

To find I_2 (the actual current in R_3), we apply Cramer’s rule:


The Thévenin-equivalent voltage is therefore


The Thévenin-equivalent resistance is


Finally, the maximum power dissipated in R_L, when R_L=R_{\mathrm{th}}, is



It is important to be clear that P_{L,\mathrm{max}} is the power dissipated in R_L only when R_L=R_{\mathrm{th}}. The general expression for power dissipated in R_L is given by Equation 7.3.3.