15 February, 2007

Maximum Power Transfer Theorem

This theorem was first discovered by Moritz von Jacobi which is referred to as "jacobi's law".
The theorem states that in a circuit maximum power is transfered from source to load when the resistance of the load is same as that of the source.

The theorem applies only when the source resistance is fixed. If the source resistance were variable (but the load resistance fixed), maximum power would be transferred to the load simply by setting the source resistance to zero. Raising the source impedance to match the load would, in this case, reduce power transfer. This is the case when driving a load such as a loudspeaker with a modern amplifier. In this case, the load presented by the loudspeaker is fixed (typically, 8 ohms for home audio) and maximum power occurs with an impedance bridging connection. This type of connection also serves to maximize control of the speaker cone (due to high damping factor), which serves to lower distortion.

It is imporant to note that maximum efficiency is not the same as maximum power transfer.To achieve maximum efficiency, the resistance of the source (whether a battery or a dynamo) could be made close to zero.

The condition of maximum power transfer does not result in maximum efficiency. If we define the efficiency η as the ratio of power dissipated by the load to power developed by the source, then it is straightforward to calculate from the circuit diagram that

Consider three particular cases :

  • If RLoad = RSource then η = 0.5
  • If RLoad = infinity then η = 1
  • If RLoad = 0 then η = 0

The efficiency is only 50% when maximum power transfer is achieved, but approaches 100% as the load resistance approaches infinity (though the total power level tends towards zero). When the load resistance is zero, all the power is consumed inside the source (the power dissipated in a short circuit is zero) so the efficiency is zero.
Voltage Source
When a load resistance RL is connected to a voltage source VS with series resistance RS, maximum power transfer to the load occurs when RL is equal to RS.
Under maximum power transfer conditions, the load resistance RL, load voltage VL, load current IL and load power PL are:
VL = VS / 2
IL = VL / RL = VS / 2RS
PL = VL *VL / RL = VS * VS/ 4RS
Current Source
When a load conductance GL is connected to a current source IS with shunt conductance GS, maximum power transfer to the load occurs when GL is equal to GS.
Under maximum power transfer conditions, the load conductance GL, load current IL, load voltage VL and load power PL are:
IL = IS / 2
VL = IL / GL = IS / 2GS
PL = IL*IL / GL = IS*IS / 4GS
For more details on this topic click on the below links :
http://www.ibiblio.org/kuphaldt/socratic/output/thev.pdf -- Some exercise problems
http://www.tina.com/course/30maximum/maximum.htm ---- In case of AC circuits

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