## 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 **R _{L} **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

**R**, then the derivative of this expression with respect to R

_{L}_{L}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 **R _{L}** is connected to an equivalent Thevenin circuit of original circuit(fig.(a)). The current for any value of load resistance is

I_{L = }V_{Th }/ R_{Th}+R_{L}

Then, the power delivered to the load is

The load power depends on both R_{Th} and R_{L} ; however, R_{Th} 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 R_{L} . To find the value of R_{L }that absorbs a maximum power from the Thevenin circuit, we differentiate P_{L} with respect to R_{L.}

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 ‘ R_{L} ‘matches the Thevenin’s resistance ‘ R_{Th} ‘ 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

P_{T} = I_{L} ^{2} ( R_{Th} + R_{L} ) = 2 ×I_{L} ^{2 }. R_{L}

This means that the Thevenin voltage source itself dissipates as much power in its internal resistance R_{Th} as the power absorbed by the load R_{L} . Efficiency under maximum power transfer condition is given by

For a given circuit, V_{Th} and R_{Th} are fixed. By varying the load resistance R_{L} , 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

** R _{source} = R_{ load}**

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

** R _{load}=R_{source} and X _{load} = – X _{source}**

**Remarks:** The Thevenin equivalent circuit is useful in finding the maximum power that a linear circuit can deliver to a load

♦ **THEVENIN’S THEOREM STUDY MATERIAL CLICK HERE**

♦ **NORTON’S THEOREM STUDY MATERIAL CLICK HERE**

**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**

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