## Jacobi's Theorem

Also known as the Maximum Power Transfer Theorem, misunderstanding of it retarded development of dynamos

Professor Moritz von Jacobi of St. Petersburg (1801-1874) is probably often confused with his younger brother Carl, the eminent mathematician of Königsberg (1804-1851). Among other things, Moritz experimented with electrical motors, and built an electric boat for excursions on the River Neva. In the course of these experiments, he considered how much power he could get out of a battery. A battery can be represented as an electromotive force E in series with an internal resistance R which are about constant and do not depend much on the current that is drawn. If the external load is a resistance R', then the current is I = E / (R + R'). The power dissipated in the load is I2R', while the power dissipated in the battery is I2R.

If R' = 0, there is no external power. If R' = ∞ there is also no power, since I = 0. Therefore, for some intermediate value of R' there must be a maximum power. Calculus gives the result easily, but a little reasoning also shows that maximum power is attained when R' = R (imagine interchanging R and R'). Hence the theorem: Maximum power is transferred when the internal resistance of the source equals the resistance of the load.. We should carefully note the condition that is seldom added: When the external resistance can be varied, and the internal resistance is constant.

Jacobi quite correctly concluded that electric motors were uneconomic, considering the high price of zinc and the 50% loss of energy. The concept of energy was as yet somewhat hazy, and the fact that mechanical work out was equal to the electrical work done against a counter-emf was unknown, at the time. However, it was adopted as a maxim that the internal resistance equaled the load resistance for maximum power.

When one builds a dynamo, the resistance of the armature winding is one part of the internal resistance. The parts due to iron losses and armature reaction are rather small, and were not considered in the first electrical machines. The major part of the internal resistance was simply the resistance of the winding. Even after Gramme and Siemens showed how to make efficient dynamos, the armature resistance was made high to match the loads, and this reduced the efficiency of the machines to not much more than 40%. The worst part of this was not the energy loss, but the excess heat that was produced.

When Edison was designing his lighting system in 1880, the received wisdom was to make the armature resistance equal to the resistance of the load. Either he, or Upton, his mathematical advisor, saw that this was quite incorrect. The Z dynamos and the Jumbos were made with very low armature resistance, and at one step he obtained efficiencies of 90%. He was ridiculed in the technical press by American "experts" who proved conclusively that he could not have done what he in fact did. Edison's inefficient field structures increased the weight of the dynamos, but did not affect their electrical efficiency.

Let's look at the theorem again, and ask what is the maximum power that can be obtained when the load resistance R' is fixed, and the internal resistance R is variable. For R = ∞ we have I = 0 and so zero power. For R = 0, we have I = E / R' and power E2 / R', and this is obviously the maximum value. Hence: When the load resistance is fixed, maximum power transfer occurs for zero internal resistance.

When this was understood, the "experts" blushed with shame, and the whole affair was swept under the rug. In fact, Edison is seldom credited with this basic principle of dynamo (and motor) design, that was very soon seen everywhere, making the age of electrical power possible.

In radio circuits, the internal resistance of a source of power cannot usually be reduced to a small value, so the load resistance is made equal to it for maximum power transfer. The powers involved are very small, and losses are unimportant--only the amount that gets through is important. This is a quite different case from power transmission, when losses are important.