Maximum Power Transfer Theorem
The Maximum Power Transfer Theorem is not so much a means of analysis as it is an aid to system design. Simply stated, the maximum amount of power will be dissipated by a load resistance when that load resistance is equal to the Thevenin/Norton resistance of the network supplying the power. If the load resistance is lower or higher than the Thevenin/Norton resistance of the source network, its dissipated power will be less than maximum.
This is essentially what is aimed for in radio transmitter design , where the antenna or transmission line “impedance” is matched to final power amplifier “impedance” for maximum radio frequency power output. Impedance, the overall opposition to AC and DC current, is very similar to resistance, and must be equal between source and load for the greatest amount of power to be transferred to the load. A load impedance that is too high will result in low power output. A load impedance that is too low will not only result in low power output, but possibly overheating of the amplifier due to the power dissipated in its internal (Thevenin or Norton) impedance.
Taking our Thevenin equivalent example circuit, the Maximum Power Transfer Theorem tells us that the load resistance resulting in greatest power dissipation is equal in value to the Thevenin resistance (in this case, 0.8 Ω):
With this value of load resistance, the dissipated power will be 39.2 watts:
If we were to try a lower value for the load resistance (0.5 Ω instead of 0.8 Ω, for example), our power dissipated by the load resistance would decrease:
Power dissipation increased for both the Thevenin resistance and the total circuit, but it decreased for the load resistor. Likewise, if we increase the load resistance (1.1 Ω instead of 0.8 Ω, for example), power dissipation will also be less than it was at 0.8 Ω exactly:
If you were designing a circuit for maximum power dissipation at the load resistance, this theorem would be very useful. Having reduced a network down to a Thevenin voltage and resistance (or Norton current and resistance), you simply set the load resistance equal to that Thevenin or Norton equivalent (or vice versa) to ensure maximum power dissipation at the load. Practical applications of this might include radio transmitter final amplifier stage design (seeking to maximize power delivered to the antenna or transmission line), a grid tied inverterloading a solar array, or electric vehicle design (seeking to maximize power delivered to drive motor).
The Maximum Power Transfer Theorem is not: Maximum power transfer does not coincide with maximum efficiency. Application of The Maximum Power Transfer theorem to AC power distribution will not result in maximum or even high efficiency. The goal of high efficiency is more important for AC power distribution, which dictates a relatively low generator impedance compared to load impedance.
Similar to AC power distribution, high fidelity audio amplifiers are designed for a relatively low output impedance and a relatively high speaker load impedance. As a ratio, "output impdance" : "load impedance" is known as damping factor, typically in the range of 100 to 1000. [rar] [dfd]
Maximum power transfer does not coincide with the goal of lowest noise. For example, the low-level radio frequency amplifier between the antenna and a radio receiver is often designed for lowest possible noise. This often requires a mismatch of the amplifier input impedance to the antenna as compared with that dictated by the maximum power transfer theorem.
Capacitor Circuits
Next let us consider a single capacitor of capacitance C, here the relationship between the current flow and applied voltage is given by
unlike the resistor, the current flow is proportional to the voltage gradient (with respect to time) and consequently this introduces a phase shift between the two. The impedance response for a circuit containing a single capacitor is shown in time, phasor and bode notations below.
| Circuit Component | Frequency Response |
 |  |
clearly the current is 90° out of phase with the voltage, the Bode plot shows that this relationship holds for all frequencies although the magnitude of the signal drops as the frequency increases. To explain this behaviour we need to understand how the capacitor resists the passage of current. A measure of this resistance to current flow is given by the capacitative reactance Xc which has units of Ohms. This quantity Xc has both magnitude and phase and calculated using
it may be predicted by invoking Ohms law which tells us that the circuit resistance (in this Xc) is equal to
This shows us that the quantity ‹Xc always has a -90° angle attached to its magnitude and it's usually written as above or in the complex form -j Xc.
Inductor Circuits
Finally we consider the response of an inductor with an inductance (L). Here the relationship between the current flow and applied voltage is given by
like the capacitor, the current can be seen to be out of phase with the voltage. The impedance response of a circuit containing a single inductor is shown below in time phasor and bode forms.
| Circuit Component | Frequency Response |
 |  |
In an analogous manner to the capacitor the 'resistance' to current flow is given by the inductive reactance Xl which has both a magnitude and phase:
it may be predicted by using Ohms law which tells us that the circuit resistance (in this case Xl) is equal to
This shows us that the inductive reactance always has a 90° angle attached to its magnitude and is usually written in complex form as jXl or in polar form
We now have the basic information required to analyse circuits containing combinations of the above components in series or parallel. As the majority of circuits of interest in electrochemical analysis are combinations of resistors and capacitors we will only consider these in the later sections, although the extension to examine inductive circuits requires no further developments.
