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electrical engineering Electrical engineering is an engineering discipline concerned with the study, design, and application of equipment, devices, and systems which use electricity, electronics, and electromagnetism. It emerged as an identifiable occupation in the l ...
, the maximum power transfer theorem states that, to obtain ''maximum'' external power from a power source with internal resistance, the resistance of the load must equal the resistance of the source as viewed from its output terminals.
Moritz von Jacobi Moritz Hermann or Boris Semyonovich (von) Jacobi (russian: Борис Семёнович Якоби; 21 September 1801, Potsdam – 10 March 1874, Saint Petersburg) was a Prussian and Russian Imperial engineer and physicist of Jewish descent. Ja ...
published the maximum power (transfer) theorem around 1840; it is also referred to as "Jacobi's law". The
theorem In mathematics, a theorem is a statement that has been proved, or can be proved. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of t ...
results in maximum ''power'' transfer from the power source to the load, and not maximum ''
efficiency Efficiency is the often measurable ability to avoid wasting materials, energy, efforts, money, and time in doing something or in producing a desired result. In a more general sense, it is the ability to do things well, successfully, and without ...
'' of useful power out of total power consumed. If the load resistance is made larger than the source resistance, then efficiency increases (since a higher percentage of the source power is transferred to the load), but the ''magnitude'' of the load power decreases (since the total circuit resistance increases). If the load resistance is made smaller than the source resistance, then efficiency decreases (since most of the power ends up being dissipated in the source). Although the total power dissipated increases (due to a lower total resistance), the amount dissipated in the load decreases. The theorem states how to choose (so as to maximize power transfer) the load resistance, once the source resistance is given. It is a common misconception to apply the theorem in the opposite scenario. It does ''not'' say how to choose the source resistance for a given load resistance. In fact, the source resistance that maximizes power transfer from a voltage source is always zero (the hypothetical
ideal voltage source A voltage source is a two-terminal device which can maintain a fixed voltage. An ideal voltage source can maintain the fixed voltage independent of the load resistance or the output current. However, a real-world voltage source cannot supply unl ...
), regardless of the value of the load resistance. The theorem can be extended to
alternating current Alternating current (AC) is an electric current which periodically reverses direction and changes its magnitude continuously with time in contrast to direct current (DC) which flows only in one direction. Alternating current is the form in whic ...
circuits that include reactance, and states that maximum power transfer occurs when the load impedance is equal to the
complex conjugate In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, (if a and b are real, then) the complex conjugate of a + bi is equal to a - ...
of the source impedance. The mathematics of the theorem also applies to other physical interactions, such as: * mechanical collisions between two objects, * the sharing of charge between two capacitors, * liquid flow between two cylinders, * the transmission and reflection of light at the boundary between two media.


Maximizing power transfer versus power efficiency

The theorem was originally misunderstood (notably by
Joule The joule ( , ; symbol: J) is the unit of energy in the International System of Units (SI). It is equal to the amount of work done when a force of 1 newton displaces a mass through a distance of 1 metre in the direction of the force appli ...
) to imply that a system consisting of an electric motor driven by a
battery Battery most often refers to: * Electric battery, a device that provides electrical power * Battery (crime), a crime involving unlawful physical contact Battery may also refer to: Energy source *Automotive battery, a device to provide power t ...
could not be more than 50% efficient, since the power dissipated as heat in the battery would always be equal to the power delivered to the motor when the impedances were matched. 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 "Dynamo Electric Machine" (end view, partly section, ) A dynamo is an electrical generator that creates direct current using a commutator. Dynamos were the first electrical generators capable of delivering power for industry, and the foundati ...
) 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 An electric motor is an electrical machine that converts electrical energy into mechanical energy. Most electric motors operate through the interaction between the motor's magnetic field and electric current in a wire winding to generate f ...
was a practical alternative to the
heat engine In thermodynamics and engineering, a heat engine is a system that converts heat to mechanical energy, which can then be used to do mechanical work. It does this by bringing a working substance from a higher state temperature to a lower stat ...
. The efficiency is the ratio of power dissipated by the load resistance to total power dissipated by circuit (which includes the voltage source's resistance of as well as ): \eta = \frac = \frac = \frac = \frac \, . Consider three particular cases: * If R_\mathrm/R_\mathrm \to 0, then \eta \to 0. Efficiency approaches 0% if the load resistance approaches zero (a
short circuit A short circuit (sometimes abbreviated to short or s/c) is an electrical circuit that allows a current to travel along an unintended path with no or very low electrical impedance. This results in an excessive current flowing through the circu ...
), since all power is consumed in the source and no power is consumed in the short. Note, voltage sources must have some resistance. * If R_\mathrm/R_\mathrm = 1, then \eta = \tfrac. Efficiency is only 50% if the load resistance equals the source resistance (which is the condition of maximum power transfer). * If R_\mathrm/R_\mathrm \to \infty, then \eta \to 1. Efficiency approaches 100% if the load resistance approaches infinity (though the total power level tends towards zero) or if the source resistance approaches zero. Using a large ratio is called impedance bridging.


Impedance matching

A related concept is reflectionless
impedance matching In electronics, impedance matching is the practice of designing or adjusting the input impedance or output impedance of an electrical device for a desired value. Often, the desired value is selected to maximize power transfer or minimize si ...
. In
radio Radio is the technology of signaling and communicating using radio waves. Radio waves are electromagnetic waves of frequency between 30  hertz (Hz) and 300  gigahertz (GHz). They are generated by an electronic device called a tr ...
frequency
transmission line In electrical engineering, a transmission line is a specialized cable or other structure designed to conduct electromagnetic waves in a contained manner. The term applies when the conductors are long enough that the wave nature of the transmi ...
s, and other
electronics The field of electronics is a branch of physics and electrical engineering that deals with the emission, behaviour and effects of electrons using electronic devices. Electronics uses active devices to control electron flow by amplification ...
, there is often a requirement to match the source impedance (at the transmitter) to the load impedance (such as an
antenna Antenna ( antennas or antennae) may refer to: Science and engineering * Antenna (radio), also known as an aerial, a transducer designed to transmit or receive electromagnetic (e.g., TV or radio) waves * Antennae Galaxies, the name of two collid ...
) to avoid reflections in the
transmission line In electrical engineering, a transmission line is a specialized cable or other structure designed to conduct electromagnetic waves in a contained manner. The term applies when the conductors are long enough that the wave nature of the transmi ...
that could overload or damage the transmitter.


Calculus-based proof for purely resistive circuits

In the simplified model of powering a load with resistance by a source with voltage and source resistance , then by
Ohm's law Ohm's law states that the current through a conductor between two points is directly proportional to the voltage across the two points. Introducing the constant of proportionality, the resistance, one arrives at the usual mathematical equa ...
the resulting current is simply the source voltage divided by the total circuit resistance: I = \frac. The power dissipated in the load is the square of the current multiplied by the resistance: P_\mathrm = I^2 R_\mathrm = \left(\frac\right)^2 R_\mathrm = \frac. The value of for which this expression is a maximum could be calculated by differentiating it, but it is easier to calculate the value of for which the denominator R_\mathrm^2 / R_\mathrm + 2 R_\mathrm + R_\mathrm is a minimum. The result will be the same in either case. Differentiating the denominator with respect to : \frac \left(R_\mathrm^2 / R_\mathrm + 2R_\mathrm + R_\mathrm\right) = -R_\mathrm^2 / R_\mathrm^2 + 1. For a maximum or minimum, the first derivative is zero, so R_\mathrm^2 / R_\mathrm^2 = 1 or R_\mathrm = \pm R_\mathrm. In practical resistive circuits, and 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: \frac \left( \right) = / . This is always positive for positive values of R_\mathrm and R_\mathrm , showing that the denominator is a minimum, and the power is therefore a maximum, when R_\mathrm = R_\mathrm. The above proof assumes fixed source resistance R_\mathrm . When the source resistance can be varied, power transferred to the load can be increased by reducing R_\textrm. For example, a 100 Volt source with an R_\textrm of 10\,\Omega will deliver 250 watts of power to a 10\,\Omega load; reducing R_\textrm to 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.


In reactive circuits

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 conjugate In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, (if a and b are real, then) the complex conjugate of a + bi is equal to a - ...
s'' 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 An LC circuit, also called a resonant circuit, tank circuit, or tuned circuit, is an electric circuit consisting of an inductor, represented by the letter L, and a capacitor, represented by the letter C, connected together. The circuit can a ...
in which the energy continues to oscillate to and fro. This oscillation is called
reactive power Reactive may refer to: *Generally, capable of having a reaction (disambiguation) *An adjective abbreviation denoting a bowling ball coverstock made of reactive resin *Reactivity (chemistry) *Reactive mind *Reactive programming See also *Reactanc ...
. 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

In this diagram, AC power is being transferred from the source, with
phasor In physics and engineering, a phasor (a portmanteau of phase vector) is a complex number representing a sinusoidal function whose amplitude (''A''), angular frequency (''ω''), and initial phase (''θ'') are time-invariant. It is related to ...
magnitude of voltage , V_\text, (positive peak voltage) and fixed source impedance Z_\text (S for source), to a load with impedance Z_\text (L for load), resulting in a (positive) magnitude , I, of the current phasor I. This magnitude , I, results from dividing the magnitude of the source voltage by the magnitude of the total circuit impedance: , I, = . The average power P_\text dissipated in the load is the square of the current multiplied by the resistive portion (the real part) R_\text of the load impedance Z_\text: \begin P_\text & = I_\text^2 R_\text = , I, ^2 R_\text\\ & = \left( \right)^2 R_\text = , \end where R_\text and R_\text denote the resistances, that is the real parts, and X_\text and X_\text denote the reactances, that is the imaginary parts, of respectively the source and load impedances Z_\text and Z_\text. To determine, for a given source voltage V_\text and impedance Z_\text, the value of the load impedance Z_\text, for which this expression for the power yields a maximum, one first finds, for each fixed positive value of R_\text, the value of the reactive term X_\text for which the denominator (R_\text + R_\text)^2 + (X_\text + X_\text)^2 is a minimum. Since reactances can be negative, this is achieved by adapting the load reactance to X_\text = -X_\text. This reduces the above equation to: P_\text = \frac 1 2 \frac and it remains to find the value of R_\text which maximizes this expression. This problem has the same form as in the purely resistive case, and the maximizing condition therefore is R_\text = R_\text. The two maximizing conditions *R_\text = R_\text *X_\text = -X_\text describe the
complex conjugate In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, (if a and b are real, then) the complex conjugate of a + bi is equal to a - ...
of the source impedance, denoted by ^*, and thus can be concisely combined to: Z_\text = Z_\text{S}^*.


Notes


References

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


External links


''Conjugate matching versus reflectionless matching''
(
PDF Portable Document Format (PDF), standardized as ISO 32000, is a file format developed by Adobe in 1992 to present documents, including text formatting and images, in a manner independent of application software, hardware, and operating systems. ...
) taken fro
''Electromagnetic Waves and Antennas''


Circuit theorems Electrical engineering