# ELECTRICAL STUDY MATERIAL | MAXIMUM POWER TRANSFER THEOREM

THEOREMS | MAXIMUM POWER TRANSFER THEOREM |MAXIMUM POWER TRANSFER THEOREM TUTORIAL | MAXIMUM POWER TRANSFER THEOREM NOTES | MAXIMUM POWER TRANSFER THEOREM STUDY MATERIAL | MAXIMUM POWER TRANSFER THEOREM MATERIAL | MAXIMUM POWER TRANSFER THEOREM PREPARATION MATERIAL|NETWORK THEOREMS

MAXIMUM POWER TRANSFER THEOREM

In an electric circuit, the load receives electric energy via the supply sources and converts that energy into a useful form. The maximum allowable power receives by the load is always limited either by the heating effect (incase of resistive load) or by the other power conversion taking place in the load. The Thevenin and Norton models imply that the internal circuits within the source will necessarily dissipate some of power generated by the source. A logical question will arise in mind, how much power can be transferred to the load from the source under the most practical conditions? In other words, what is the value of load resistance that will absorbs the maximum power from the source? This is an important issue in many practical problems and it is discussed with a suitable example. Let us consider an electric network as shown in fig.(a), the problem is to find the choice of the resistance  L R so that the network delivers maximum power to the load or in other words what value of load resistance RL will absorb the maximum amount of power from the network. This problem can be solved using nodal or mesh current analysis to obtain an expression for the power absorbed by  RL , then the derivative of this expression with respect to RL will establish the condition under what circumstances the maximum power transfer occurs. The effort required for such an approach can be quite tedious and complex. Fortunately, the network shown in fig.(a) can be represented by an equivalent Thevenin’s voltage source as shown in fig.(b). In fig.(b) a variable load resistance   RL is connected to an equivalent Thevenin circuit of original circuit(fig.(a)). The current for any value of load resistance is

IL = VTh / RTh+RL

Then, the power delivered to the load is The load power depends on both RTh and RL ; however,  RTh is constant for the equivalent Thevenin network. So power delivered by the equivalent Thevenin network to the load resistor is entirely depends on the value of   RL . To find the value of  RL that absorbs a maximum power from the Thevenin circuit, we differentiate PL with respect to   RL. For maximum power dissipation in the load, the condition given below must be satisfied This result is known as “Matching the load” or maximum power transfer occurs when the load resistance ‘ RL ‘matches the Thevenin’s resistance ‘ RTh ‘ of a given systems. Also, notice that under the condition of maximum power  transfer, the load voltage is, by voltage division, one-half of the Thevenin voltage. The expression for maximum power dissipated to the load resistance is given by The total power delivered by the source

PT = IL 2 ( RTh + RL ) = 2 ×IL 2 . RL

This means that the Thevenin voltage source itself dissipates as much power in its  internal resistance  RTh as the power absorbed by the load RL . Efficiency under maximum power transfer condition is given by For a given circuit,  VTh and RTh are fixed. By varying the load resistance RL , the power delivered to the load varies as shown in fig.(c). IMPORTANT CONDITIONS:

• When applied to a dc network maximum power transfered from source to load, when source resistance equal to the load resistance

• When applied to ac network maximum power transfered from source to load , when source and load impedance are complex conjugate. 