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The maximum power transfer theorem states that to obtain maximum power transfer from a source with a finite internal resistance to a load, the internal resistance and the load's resistance need to be the same.

However, there is a difference between maximizing the power transfer and maximizing the electrical efficiency. Why?

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  • \$\begingroup\$ In short, sometimes you have to spend more to make more. \$\endgroup\$
    – user16324
    Commented Nov 13, 2021 at 14:25
  • \$\begingroup\$ What does "electrical efficiency" mean? Does it mean the ratio between the power that the load receives, and the power that the source's internal ideal voltage source puts out? \$\endgroup\$ Commented Nov 13, 2021 at 15:04
  • \$\begingroup\$ @TannerSwett In the general case, it means the ratio of useful power to total power. In this case, it means exactly what you said. \$\endgroup\$
    – David Cian
    Commented Nov 13, 2021 at 15:39

2 Answers 2

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Below is a diagram of a simple circuit illustrating the theorem.

Diagram of simple circuit with source and load

We have a voltage source generating a voltage \$V\$, a current \$I\$, an internal resistance of the source \$R_S\$ and the load which has a resistance \$R_L\$.

Let us define the two different quantities which interest us. The power transferred is \$P_L = V_L \times I = R_L I^2\$. The efficiency is the ratio of useful power to total power, where the useful power is that dissipated by the load and the total power is that output by the source: \$E = \frac{P_L}{P_S}\$.

What interests us is how these two quantities change with respect to the resistance of the load, so we play around a bit with the expressions.

$$I = \frac{V}{R_S + R_L}$$

$$P_L = (\frac{V}{R_S + R_L})^2 R_L$$

$$E = \frac{R_L I^2}{V_S I} = \frac{R_L}{R_S + R_L}$$

Now here's how the two change when the load resistance changes, for \$V = 3V\$ and \$R_S = 1\Omega\$, where the green line is the power and the red line is the efficiency.

Graph of power and efficiency

As can be seen, the maximum power is attained when the load resistance is equal to the source resistance, but it then keeps decreasing past that point. The efficiency, however, keeps increasing monotonically toward 1 (maximum efficiency) as the load resistance increases.

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? why-is-there-a-difference-between-maximizing-power-transfer-and-power-efficiency

Power has many meanings. e.g., Heat , kinetic and electromagnetic. It must be conserved in a closed system. So the Max. Transfer of Power or MPT always exists at 50% efficiency.

But this may exceed max heat dissipation. For a battery, there is Peukert's Law that defines loss of capacity or Ah or Wh capacity and thus efficiency loss in transferring power to a motor. This occurs when trying to load the cell by charge or discharge and reduce R towards the ESR of the battery. Obviously you don't want MPT from the battery for heat and safety reasons as well as 50% power loss x t = energy and thus lose energy capacity. But in theory, the matched impedance will instantly transfer max power to the load.

  • The corollary to Peukert's Law (IMHO) is that for every battery capacity and voltage or C charge rate or CCA rating or % load regulation error, you can estimate the ESR of the battery by this Thevenin voltage divider R Ratio.
  • Another corollary albeit unrelated to the question, is that from the MPT of any Diode is inversely related to its linear resistance incl. LED's you can compute the Rs or the incremental bulk resistance where the diode departs from its exponential behaviour into an almost linear mode. Originally it was Rs=1/Pmax but higher power diodes and LEDs have improved now with improved thermal. Conductance towards Rs= 25 to 50% of 1/Pmax.
  • Another trick is when putting LEDs in a parallel with high power, to avoid thermal runaway the series R can match only the difference in Rs safely and in many cases with matched Rs, this can be 0 Ohms in series to "normalize or equalize the impedance at Max power" and share power equally. " Matched LED Impedance for same Vf for all same family LEDs has a wide tolerance BUT ALL the same chemistry LEDs have the same Vf at low currents and thus impedance. They must also be thermally matched or on the same heatsink. So matching impedance of parallel loads is also important.

Again, MPT means two things. Maximum Power Transfer and also it means conjugate reactive (RLC) impedances is matched, or resistive impedance is matched.

The system MPT power efficiency is always 50% of the total power in the system for MPT but must include other forms of power such as kinetic if it exists.

For a resistance network, it always means 50% attenuation or 50% efficiency for MPT with 50% voltage and 50% of maximum current w.r.t. short circuit current.

For a reactive RLC network, the match conjugate impedance is used to transfer maximum power.

A Solar PV with MPT with active tracking results in the impedance of the source to match the load or MPPT, this only occurs when (open-cct V / sht. cct. I) = Voc/Isc of the PV = Load R but since the source is not a resistor but rather a current source with can also match the incremental change in Vo/Io= Vol/Isc the open circuit voltage over short circuit current. So when connecting a PV to a regulator to a battery KEEP THIS IN MIND of matching loads at each stage if you want MPT.

If you matched a load to the battery ESR (effective series resistance) you would get 50% of the voltage with 50% of the short circuit current and thus 50% efficiency, but the battery temperature rise would cause catastrophic failure if the Pd * Rth thermal resistance = Delta T temp rise exceeds maximum operating temp.

  • ( A Car or Lithium battery could explode, but you would get maximum power transferred out with poor (50%) efficiency.) Yet CCA is often rated at 5V below 12.5 so it not MPT but what the battery can tolerate for the maximum start friction at 0'F.

For a battery and motor load, the same fault would occur, so we never apply MPT to "Voltage Sources" that cannot handle this type of load, so we use another term. Load regulation , which may be defined as the attenuation in voltage from the load ratio RL/(RL+Rs)%100%. Thus maximum power out in a motor depends on only its characteristics of Torque * RPM= Pout to determine MPT and expect the voltage source is constant. Yet we know all conductors have some resistance, so load regulation always exists, even if we ignore it for values < x %. The same principle is applied to the grid with load regulation.

A signal Generator with 50 Ohms unloaded actually put out twice the rated voltage and half when the impedance is matched. ( Transmission Line Theory says Return Loss doubles Voltage with no load). This is always used for transmission line RF circuits for frequency response reasons, and to achieve maximum power gain when the system is all low impedance like 50 Ohms. Therefore, RF systems must factor this 50% power loss in all power transfers. This is also true for USB, Ethernet, etc. with controlled matched impedances.

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