Maximum Power Transfer

 

In many practical situations, a circuit is designed to provide power to a load. There are applications in areas such as communications where it is desirable to maximize the power delivered to a load. We now address the problem of delivering the maximum power to a load when given a system with known internal losses. It should be noted that this will result in significant internal losses greater than or equal to the power delivered to the load

Figure 4.48 The circuit used for maximum power transfer.

The Thevenin equivalent is useful in finding the maximum power a linear circuit can deliver to a load. We assume that we can adjust the load resistance R_L. If the entire circuit is replaced by its Thevenin equivalent except for the load, as shown in Fig. 4.48, the power delivered to the load is:

For a given circuit, V_Th and R_Th are fixed. By varying the load resistance R_L, the power delivered to the load varies as sketched in Fig. 4.49.

Figure 4.49 Power delivered to the load as a function of RL.

We notice from Fig. 4.49 that the power is small for small or large values of R_L but maximum for some value of R_L between 0 and ∞. We now want to show that this maximum power occurs when R_L is equal to R_Th. This is known as the maximum power theorem.

Maximum power is transferred to the load when the load resistance equals the Thevenin resistance as seen from the load (R_L = R_Th).

To prove the maximum power transfer theorem, we differentiate p in Eq. (4.21) with respect to R_L and set the result equal to zero. We obtain:

This implies that:

 which yields:

showing that the maximum power transfer takes place when the load resistance R_L equals the Thevenin resistance R_Th. We can readily confirm that Eq. (4.23) gives the maximum power by showing that:

The maximum power transferred is obtained by substituting Eq. (4.23) into Eq. (4.21), for 

Equation (4.24) applies only when R_L = R_Th. When R_L ≠ R_Th, we compute the power delivered to the load using Eq. (4.21).

Example:

Find the value of R_L for maximum power transfer in the circuit of Figure. 

Find the maximum power

Solution:

We need to find the Thevenin resistance R_Th and the Thevenin voltage V_Th across the terminals a-b. To get R_Th, we use the circuit : 

To get V_Th, we consider the circuit:

Applying mesh analysis gives:

Solving for i₁, we get i₁ = −2∕3. Applying KVL around the outer loop to get V_Th across terminals a-b, we obtain:

For maximum power transfer,

and the maximum power is: