Power relations in nonlinear reactors. An application of the principle of virtual work

A nonlinear reactor will produce from several input frequencies all possible harmonics and intermodulation products. If every single one of these input and output frequencies is fed through an ideal rotating electromechanical transducer a mechanism is created which has as many degrees of freedom as there are independent frequencies. Equilibrium requires that for any virtual angular displacement of the shafts the work done by the applied torques shall equal the associated change of stored energy. For noncommensurate independent frequencies the time average of this change is zero; i.e. the virtual work of the mean applied torques is zero. The Manley-Rowe relations as published are a specialised form of this statement, requiring that for each independent shaft the resultant of all mean torques (referred to it in accordance with the relevant gear ratios) should be zero. These relations cease to be generally valid if the independent frequencies are commensurate. Then there is synchronism between the harmonic of one frequency and a harmonic or fundamental of the other. The change in stored energy is a measure of the reaction produced if one attempts to change the relative angular positions of the two synchronous shafts. Its time average is not necessarily zero, and if it is not we have a mean power flow between these shafts. Exactly the same situation is encountered in the case of two synchronous generators working in parallel. Analogues can be classified according to the extent of simulation achieved: constraints only, mean energy storage or full energy storage. One of the simplest given is the horizontal lamina (?teatray?), floating on a needle point and loaded in positions specified by the harmonic index numbers with weights proportional to the associated mean torques. Its equilibrium is described by the Manley-Rowe equations. This analogue covers the commensurate case if there is an elastic tie between the principal ?synchronous point? and ?earth?. The associated diag- ram also draws attention to the existence of near-synchronous conditions, in which case pulling effects between the independent frequencies may be expected.