Maximum power transfer theorem

MAXIMUM POWER TRANSFER THEOREM

Statement

In DC circuits, maximum power is transferred from a source to the load when the load resistance is made equal to the resistance of the network as viewed from the load terminals with load removed and all the sources are replaced by their internal resistances.

Explanation

1. Circuit is supplying power to a load R_L. The circuit enclosed in the box replaced by Thevenin's equivalent circuit consists of open circuit voltage (V_OC) in series with Thevenin's resistance (R_TH).

2. Resistance RT is the resistance seen into terminals AB with R_L removed and sources are replaced by their internal resistances.

Proof

The current supplied to R_L is given by

I_L = V_OC / R_TH + R_L

Power delivered to R_L is P_L = I²_LR_L

PL = (V_OC / R_Th + R_L )² R_L    ...(1)

For a given circuit, Voc and RTh are constant. Therefore, power delivered to load depends upon R_L. In order to find the value of R_L, for which the value of P is maximum, it is necessary to differentiate equation (1) with respect to R_L.

If the power is maximum by varying R_L we must have,

This proves the maximum power transfer theorem.

Maximum Power (P_max)

Note

Due to low efficiency and greater voltage drop during this condition, electric power systems never operate for maximum power transfer. It is because in an electric power system the ultimate goal is higher efficiency rather than the maximum power.

Application

In electronic circuits, maximum power transfer is usually desirable. For instance in a public address system, the circuit is adjusted for maximum power transfer by making load (i.e., speaker) resistance equal to source (i.e., amplifier) resistance.

 

EXAMPLE 30: Calculate the value of Ry so that maximum power is transferred from battery.

Solution:

Thevenin's resistance (R_Th)

Remove the load resistor. Short circuit the voltage source.

Open circuit V_OC

Remove the load resistor.

The maximum power is transferred to the load resistor only when the load resistance equals the Thevenin's resistance.

P_max = 31.25 W

EXAMPLE 31: (a) Find the value of R for maximum power to R. (b) Determine the maximum power to R for the given network.

Solutions : Calculate R_Th

• Short circuit the voltage source

• Remove the load resistor

• Remove the load resistor R

Thevenin's Equivalent Circuit

The maximum power is transferred to the load resistor only when the load resistance equals the Thevenin's resistance.

P_max = 3.33 W

 

EXAMPLE 32: Find the value of Ry so that maximum power is delivered to the load resistance R_L.

Solution:

Calculate R_Th

• Remove the load resistor

• Short circuit the voltage source

5 Ω and 5 Ω are connected in parallel.

The maximum power is transferred to the load resistor only when the load resistance equals the Thevenin's resistance.

R_L = R_TH = 7.69 Ω

Calculation of P_max

P_max = 30. 76 W

 

EXAMPLE 33: In figure, R absorbs maximum power. Find the value of R and the maximum power.

Solution:

Calculate R_Th

• Remove the load resistor

• Short circuit the voltage source

Calculate V_OC:

• Remove the load resistor

The maximum power is transferred to the load resistor only when the load resistance equals the Thevenin's resistance.

R_L = R_Th = 2.72 Ω

Calculation of P_max :

 

EXAMPLE 34: Find the maximum power delivered to the load in the figure shown below.

Solution:

To find R_Th:

Remove the load resistor

Short circuit the voltage source

To find V_OC:

Remove the load resistor

The maximum power is transferred to the load resistor only when the load resistance equals the Thevenin's resistance.

 

EXAMPLE 35: Find the load resistance for maximum power across it as shown in figure. What is the maximum power?

Solution:

To find R_Th:

Remove the load resistor.

Short circuit the voltage source.

4 Ω and 12 Ω are connected in parallel.

Voltage across AB, V_OC = 30 × 2 = 60 V

For maximum power transfer theorem,

R_L = R_Th = 15 Ω

P_max = 60 W

 

EXAMPLE 36: Determine the load resistance to receive maximum power from the source and also find the power deliver to the load on the circuit shown in figure.

Solution :

To find R_Th :

Remove the load resistor.

Short circuit the voltage source.