This might help some experimenters who are pulsing inductors (e.g. motor windings) and later are trying to recover energy from them. When a rectangular pulse transitioning abruptly from 0 to some voltage V is applied to a resistor in series with an ideal inductor (e.g. a coil) by closing the switch in the diagram below, then the following sequence of events happens:  1) At the beginning (point A) no energy and no current is flowing (the switch is open).
 2) Shortly after the rising edge of the stimulating pulse (after the switch closes), the current increases linearly
 3) Some of the energy of the pulse is converted into the magnetic field in the inductor and some energy is dissipated in the resistance as heat. At this point the energy flows into the inductor faster than it is dissipated by the resistor.
 4) After the time equal to 0.69 Tau (point B) the energy flow (a.k.a. power) into the inductor reaches its peak and starts decreasing afterwards, eventually reaching zero power and magnetic energy equal to ½*L*(V/R)^{2}, at Tau >> 5
 5) However the current through the resistor keeps increasing nonlinearly but monotonically and asymptotically up to the V/R limit and the energy flow (a.k.a power), dissipated as heat in the resistor, increases similarly up to the V^{2}/R limit.
 6) After time equal to 1.15 Tau (point C), the magnetic energy accumulated in the inductor reaches the break even point with the total energy dissipated as heat in the resistor up to that point in time. Continuing beyond point C guarantees that more energy is dissipated as heat in the resistor than stored as the magnetic field of the inductor.
 7) After a very long time the current reaches the V/R limit and the magnetic energy stored in the inductor reaches ½*L*(V/R)^{2} limit but the energy dissipated in the resistor increases ad infinitum at the rate (a.k.a. power) equal to V^{2}/R.
For transformers, putting a load on the secondary winding (e.g. shorting it) has the same effect as decreasing the inductance of the primary winding (L). As a result of this, the Tau decreases and the current in the primary rises faster with time. THE POINT: If a constant and linear inductor is charged and later discharged at the same rate, then from efficiency point of view, it makes no sense to charge it longer than 0.5757 Tau (½ of the time C, see pt.6), because if you do, then the energy dissipated in the resistance will be higher than the energy recovered from the inductor during its discharge. For realistic good recovery efficiency from the above inductor, the charging time should be less than ⅛Tau. LEGEND: Tau = L/R (a time constant) V = The high level voltage of the stimulating rectangular pulse. E _{TOT} = Total energy delivered by the supply to the series RL circuit. E _{L} = Energy stored in the inductor as magnetic field E _{R} = Energy dissipated in any resistance as heat P _{L} = Instantaneous Power (energy flow) flowing into the inductor P _{R} = Instantaneous Power (energy flow) dissipation in the resistance i _{L} = The current flowing through the inductor (and resistor) ( I can post the relevant timedomain equations on request )
« Last Edit: 20211016, 13:23:09 by verpies »
