Maximum power transfer theorem or in short you can call it the MPT theorem is known for analyzing maximum power transferred at the load. This theorem is commonly used in electrical and electronic engineering where at load the power required is high. For example in electronics engineering and communication engineering maximum power transfer theorem is mostly applicable.

MPT applies to both AC and DC circuits but in AC circuits we can represent it with load impedance (Z_{L}) and for DC we can represent it with load resistance (R_{L}).

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## Maximum power transfer theorem statement:

**Maximum power transfer theorem:** It states that for maximum power transfer the load resistance should be equal to Thevenin equivalent resistance (or Norton equivalent resistance).

## Explanation of MPT theorem

The above circuit shows Thevenin’s equivalent circuit in series with load resistor R_{L}.

Load current in the circuit

I = V_{th }/ R_{th}+R_{L } …Eq1

The power generated by the circuit

P = I^{2 }* R_{L} = (V_{th}/R_{th}+R_{L})^{2 }* R_{L } …Eq2

For finding mximum value of power transferred differntiate Eq2 with respest to R_{L.}

dP/dR_{L} = 0 = V_{th} {(R_{th}+R_{L}) – R_{L} * 2(R_{L}+R_{th})} / (R_{th}+R_{L})^{4}

(R_{th}+R_{L})(R_{th}+R_{L}-2R_{L}) = 0

R_{th} = R_{L}

The above expression shows that for maximum power generation load resistance should be equal to Thevenin’s equivalent resistance.

The value of maximum power generated will be :

P_{max }= (V_{th}/R_{th}+R_{L})^{2 }* R_{L}

As R_{L} = R_{th}

P_{max }= (V_{th}/2R_{L})^{2 }* R_{L}

P_{max }= V_{th}^{2}/4R_{L}

The efficiency of any system at maximum power will be

ɳ = output/input

ɳ = (V_{th}^{2}/4R_{L}) / (V_{th}^{2}/4R_{L}) + (V_{th}^{2}/4R_{L})

ɳ = 50%

This shows that at maximum power the efficiency of any system will be 50%

### Steps to solve MPT theorem question.

- Remove the load resistance
- Find the Thevenin’s equivalent resistance R
_{th} - As per MPT R
_{th }= R_{L}, put the value of R_{L }is equal to R_{th.} - At last, find Maximum power transferred by eq. given below

P_{max }= V_{th}^{2}/4R_{L}

## Application of Maximum power transfer theorem

MPT is applied in those fields of engineering (communication, electronics, etc.) where at the output end (load) maximum power is required otherwise the system will not function properly.

- Television receiver
- An electronic amplifier in Radio
- Loudspeaker etc.

Q. Find the value of load resistance R_{L} and determine maximum power transferred.

Equivalent resistance of the circuit.

R_{th} = 8 + 4||4

R_{th} = 8 + (16/8)

R_{th} = 10

As per MPT theorem Rth=RL

Therefore, R_{L }= 10 ohm

Now, V_{th} = voltage across 4 ohm resistor

V_{th} = 10 * (4/8)

V_{th} = 5 V

Maximum power transferred,

P_{max }= V_{th}^{2}/4R_{L}

P_{max }= 5^{2}/ (4*10)

P_{max }= 5/8 or 0.625 watts

Hence maximum power transferred is 0.625 watts.

That’s all about Maximum power transfer theorem.