IIT JEE , AIEEE Forum - Physics , Mechanics

tony

power delivered by the source in the circuit is maximum when

talker_chatter

the power delivered is maximum when the source resistance is equal to the resistance of the load applied.

it is the well known maximum power transfer theorem......................

look in your ncert book for proof if required.

edison

Well it is decided by MPT (MAXIMUM POWER TRANSFER THEOREM)

According to this theorem the maximum power is delivered when the impedance of surce matches with that of LOAD.

So SOurce resistacne (like internal resistance of battery) = Load resistance.

edison

Proof for purely resistive circuits

In the diagram opposite, power is being transferred from the source, with voltage

V

and fixed source resistance

R S

, to a load with resistance

R L

, resulting in a current

I

. By Ohm's law,

I

is simply the source voltage divided by the total circuit resistance:

$I = {V over R_mathrm{S} + R_mathrm{L}}.$

The power

P L

dissipated in the load is the square of the current multiplied by the resistance:

$P_mathrm{L} = I^2 R_mathrm{L} = {{ left( {V over {R_mathrm{S} + R_mathrm{L}}} ight) }^2} R_mathrm{L} = {{V^2} over {R_mathrm{S}^2 / R_mathrm{L} + 2R_mathrm{S} + R_mathrm{L}}}.$

We could calculate the value of

R L

for which this expression is a maximum, but it is easier to calculate the value of

R L

for which the denominator

$R_mathrm{S}^2 / R_mathrm{L} + 2R_mathrm{S} + R_mathrm{L}$

is a minimum. The result will be the same in either case. Differentiating with respect to

R L

:

${dover{dR_mathrm{L}}} left( {R_mathrm{S}^2 / R_mathrm{L} + 2R_mathrm{S} + R_mathrm{L}} ight) = -R_mathrm{S}^2 / R_mathrm{L}^2+1.$

For a maximum or minimum, the first derivative is zero, so

${R_mathrm{S}^2 / R_mathrm{L}^2} = 1$

or

$R_mathrm{L} = pm R_mathrm{S}.$

In practical resistive circuits,

R S

and

R L

are both positive. To find out whether this solution is a minimum or a maximum, we must differentiate again:

${{d^2} over {dR_mathrm{L}^2}} left( {R_mathrm{S}^2 / R_mathrm{L} + 2 R_mathrm{S} + R_mathrm{L}} ight) = {2 R_mathrm{S}^2} / {R_mathrm{L}^3}$

This is positive for positive values of

R S

and

R L

, showing that the denominator is a minimum, and the power is therefore a maximum, when

R S = R L .

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