Maximum power transfer theorem

The theorem was originally misunderstood (notably by Joule) to imply that a system consisting of an electric motor driven by a battery could not be more than 50% efficient since, when the impedances were matched, the power lost as heat in the battery would always be equal to the power delivered to the motor.

In 1880 this assumption was shown to be false by either Edison or his colleague Francis Robbins Upton, who realized that maximum efficiency was not the same as maximum power transfer.

To achieve maximum efficiency, the resistance of the source (whether a battery or a dynamo) could be (or should be) made as close to zero as possible. Using this new understanding, they obtained an efficiency of about 90%, and proved that the electric motor was a practical alternative to the heat engine.

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, R_L, to power developed by the source, V_S, then it is straightforward to calculate from the above circuit diagram that

${\displaystyle \eta ={\frac {R_{\mathrm {L} }}{R_{\mathrm {L} }+R_{\mathrm {S} }}}={\frac {1}{1+R_{\mathrm {S} }/R_{\mathrm {L} }}}.}$

Consider three particular cases:

• If ${\displaystyle R_{\mathrm {L} }=R_{\mathrm {S} }}$, then ${\displaystyle \eta =0.5,}$
• If ${\displaystyle R_{\mathrm {L} }\to \infty }$ or ${\displaystyle R_{\mathrm {S} }=0,}$ then ${\displaystyle \eta =1,}$
• If ${\displaystyle R_{\mathrm {L} }=0}$, then ${\displaystyle \eta =0.}$

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.

Efficiency also approaches 100% if the source resistance approaches zero, and 0% if the load resistance approaches zero. In the latter case, all the power is consumed inside the source (unless the source also has no resistance), so the power dissipated in a short circuit is zero.

A related concept is reflectionless impedance matching.

In radio frequency transmission lines, and other electronics, there is often a requirement to match the source impedance (at the transmitter) to the load impedance (such as an antenna) to avoid reflections in the transmission line that could overload or damage the transmitter.

(See Cartwright[5] for a non-calculus-based proof)

Circuit diagram

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:

${\displaystyle I={\frac {V}{R_{\mathrm {S} }+R_{\mathrm {L} }}}.}$

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

${\displaystyle P_{\mathrm {L} }=I^{2}R_{\mathrm {L} }=\left({\frac {V}{R_{\mathrm {S} }+R_{\mathrm {L} }}}\right)^{2}R_{\mathrm {L} }={\frac {V^{2}}{R_{\mathrm {S} }^{2}/R_{\mathrm {L} }+2R_{\mathrm {S} }+R_{\mathrm {L} }}}.}$

The value of R_L for which this expression is a maximum could be calculated by differentiating it, but it is easier to calculate the value of R_L for which the denominator

${\displaystyle R_{\mathrm {S} }^{2}/R_{\mathrm {L} }+2R_{\mathrm {S} }+R_{\mathrm {L} }}$

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

${\displaystyle {\frac {d}{dR_{\mathrm {L} }}}\left(R_{\mathrm {S} }^{2}/R_{\mathrm {L} }+2R_{\mathrm {S} }+R_{\mathrm {L} }\right)=-R_{\mathrm {S} }^{2}/R_{\mathrm {L} }^{2}+1.}$

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

${\displaystyle R_{\mathrm {S} }^{2}/R_{\mathrm {L} }^{2}=1}$

or

${\displaystyle R_{\mathrm {L} }=\pm R_{\mathrm {S} }.}$

In practical resistive circuits, R_S and R_L are both positive, so the positive sign in the above is the correct solution.

To find out whether this solution is a minimum or a maximum, the denominator expression is differentiated again:

${\displaystyle {{d^{2}} \over {dR_{\mathrm {L} }^{2}}}\left({R_{\mathrm {S} }^{2}/R_{\mathrm {L} }+2R_{\mathrm {S} }+R_{\mathrm {L} }}\right)={2R_{\mathrm {S} }^{2}}/{R_{\mathrm {L} }^{3}}.\,\!}$

This is always positive for positive values of ${\displaystyle R_{\mathrm {S} }\,\!}$ and ${\displaystyle R_{\mathrm {L} }\,\!}$, showing that the denominator is a minimum, and the power is therefore a maximum, when

${\displaystyle R_{\mathrm {S} }=R_{\mathrm {L} }.\,\!}$

The above proof assumes fixed source resistance ${\displaystyle R_{\mathrm {S} }\,\!}$. When the source resistance can be varied, power transferred to the load can be increased by reducing ${\displaystyle R_{\textrm {S}}\,\!}$. For example, a 100 Volt source with an ${\displaystyle R_{\textrm {S}}\,\!}$ of ${\displaystyle 10~\Omega }$ will deliver 250 watts of power to a ${\displaystyle 10~\Omega }$ load; reducing ${\displaystyle R_{\textrm {S}}\,\!}$ to ${\displaystyle 0~\Omega }$ increases the power delivered to 1000 watts.

Note that this shows that maximum power transfer can also be interpreted as the load voltage being equal to one-half of the Thevenin voltage equivalent of the source.[6]

The power transfer theorem also applies when the source and/or load are not purely resistive.

A refinement of the maximum power theorem says that any reactive components of source and load should be of equal magnitude but opposite sign. (See below for a derivation.)

• This means that the source and load impedances should be complex conjugates of each other.
• In the case of purely resistive circuits, the two concepts are identical.

Physically realizable sources and loads are not usually purely resistive, having some inductive or capacitive components, and so practical applications of this theorem, under the name of complex conjugate impedance matching, do, in fact, exist.

If the source is totally inductive (capacitive), then a totally capacitive (inductive) load, in the absence of resistive losses, would receive 100% of the energy from the source but send it back after a quarter cycle.

The resultant circuit is nothing other than a resonant LC circuit in which the energy continues to oscillate to and fro. This oscillation is called reactive power.

Power factor correction (where an inductive reactance is used to "balance out" a capacitive one), is essentially the same idea as complex conjugate impedance matching although it is done for entirely different reasons.

For a fixed reactive source, the maximum power theorem maximizes the real power (P) delivered to the load by complex conjugate matching the load to the source.

For a fixed reactive load, power factor correction minimizes the apparent power (S) (and unnecessary current) conducted by the transmission lines, while maintaining the same amount of real power transfer.

This is done by adding a reactance to the load to balance out the load's own reactance, changing the reactive load impedance into a resistive load impedance.

Proof

source and load impedance diagram

In this diagram, AC power is being transferred from the source, with phasor magnitude of voltage ${\displaystyle |V_{\text{S}}|}$ (positive peak voltage) and fixed source impedance ${\displaystyle Z_{\text{S}}}$ (S for source), to a load with impedance ${\displaystyle Z_{\text{L}}}$ (L for load), resulting in a (positive) magnitude ${\displaystyle |I|}$ of the current phasor ${\displaystyle I}$. This magnitude ${\displaystyle |I|}$ results from dividing the magnitude of the source voltage by the magnitude of the total circuit impedance:

${\displaystyle |I|={|V_{\text{S}}| \over |Z_{\text{S}}+Z_{\text{L}}|}.}$

The average power ${\displaystyle P_{\text{L}}}$ dissipated in the load is the square of the current multiplied by the resistive portion (the real part) ${\displaystyle R_{\text{L}}}$ of the load impedance ${\displaystyle Z_{\text{L}}}$:

${\displaystyle {\begin{aligned}P_{\text{L}}&=I_{\text{rms}}^{2}R_{\text{L}}={1 \over 2}|I|^{2}R_{\text{L}}\\&={1 \over 2}\left({|V_{\text{S}}| \over |Z_{\text{S}}+Z_{\text{L}}|}\right)^{2}R_{\text{L}}={1 \over 2}{|V_{\text{S}}|^{2}R_{\text{L}} \over (R_{\text{S}}+R_{\text{L}})^{2}+(X_{\text{S}}+X_{\text{L}})^{2}},\end{aligned}}}$

where ${\displaystyle R_{\text{S}}}$ and ${\displaystyle R_{\text{L}}}$ denote the resistances, that is the real parts, and ${\displaystyle X_{\text{S}}}$ and ${\displaystyle X_{\text{L}}}$ denote the reactances, that is the imaginary parts, of respectively the source and load impedances ${\displaystyle Z_{\text{S}}}$ and ${\displaystyle Z_{\text{L}}}$.

To determine, for a given source voltage ${\displaystyle V_{\text{S}}}$ and impedance ${\displaystyle Z_{\text{S}},}$ the value of the load impedance ${\displaystyle Z_{\text{L}},}$ for which this expression for the power yields a maximum, one first finds, for each fixed positive value of ${\displaystyle R_{\text{L}}}$, the value of the reactive term ${\displaystyle X_{\text{L}}}$ for which the denominator

${\displaystyle (R_{\text{S}}+R_{\text{L}})^{2}+(X_{\text{S}}+X_{\text{L}})^{2}\,}$

is a minimum. Since reactances can be negative, this is achieved by adapting the load reactance to

${\displaystyle X_{\text{L}}=-X_{\text{S}}.}$

This reduces the above equation to:

${\displaystyle P_{\text{L}}={\frac {1}{2}}{\frac {|V_{\text{S}}|^{2}R_{\text{L}}}{(R_{\text{S}}+R_{\text{L}})^{2}}}}$

and it remains to find the value of ${\displaystyle R_{\text{L}}}$ which maximizes this expression. This problem has the same form as in the purely resistive case, and the maximizing condition therefore is ${\displaystyle R_{\text{L}}=R_{\text{S}}.}$

The two maximizing conditions

• ${\displaystyle R_{\text{L}}=R_{\text{S}}}$
• ${\displaystyle X_{\text{L}}=-X_{\text{S}}}$

describe the complex conjugate of the source impedance, denoted by ${\displaystyle ^{*},}$ and thus can be concisely combined to:

${\displaystyle Z_{\text{L}}=Z_{\text{S}}^{*}.}$

• H.W. Jackson (1959) Introduction to Electronic Circuits, Prentice-Hall.

• Conjugate matching versus reflectionless matching (PDF) taken from Electromagnetic Waves and Antennas
• The Spark Transmitter. 2. Maximising Power, part 1.