Maximum Power transfer theorem

For CBCSS S5 Core physics 2018 syllabus revision, University of Kerala.

Course: PY 1543, Electronics.

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The theorem states that the maximum power will be delivered to the load when the load resistance is equal to the internal resistance of the network delivering the power.

Consider a two terminal network represented by its equivalent Thevenin's circuit as shown in figure 1.

According to Joules law, the power delivered to the load resistance is:

From equation (1b), it is clear that the power generated in the load will be maximum, when the internal resistance is minimum. When R=0, 

From equation (2), it is evident that P is smaller than P(Max). From equation (1a), it is apparent that, power in the load is zero, if the load resistance R(L) is very small. Also, from equation 1(b), it very clear that the power in the load is also zero, when R(L) is very large. 

Thus there must have an optimum value for load resistance ,R(L) for which power delivered in the load will be maximum.

Thus for finding this optimum value for maximum power transfer, we need to differentiate equation 1(b) with respect to R(L) and equate the result to zero. Thus we get:

Thus the theorem is proved. That is, for maximum power to be delivered to the load, the load resistance should match with the internal resistance of the circuit. If a graph is plotted between the load power and the R/R(L), the curve so obtained is shown in figure (2).

Efficiency

If the source power is P(s) and the maximum power delivered is P(max), then the efficiency is P(max)/P(s). It can be shown that efficiency of a network is 50%.

This means that half the power is lost with internal resistance of the network.

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