Saturday, July 30, 2011

Energetic consequences of doing things slowly

From charging capacitors to filling up water tanks.

Electrical engineers are familiar with the idea that when you try to charge a capacitor from a step voltage source, only half of the energy coming from the source gets into the capacitor while the other half is dissipated in the circuit resistances and/or lost as radiated electromagnetic energy. In other words, the efficiency is only 50%. Does it always have to happen that way? How can we do better?

The answer lies in shaping the applied voltage. If the rate of rise of the applied voltage is of the order of the time-constant of the circuit, the efficiency is much better than 50%. If it is very slow compared to the time-constant of the circuit, the efficiency approaches 100%. This is proved analytically as well as validated by circuit simulation in this article.

Even if we did not shape the voltage, if the capacitor doesn't have a linear relation between charge and voltage (in other words, it is a non-linear capacitor), the efficiency can still be very different from 50%.

The case of a step current source feeding an inductor is also analogous to the above scenario and gives similar results.

Taking a step back and applying this concept to any other domain (other than electrical engineering) can give some interesting insights. Pumping water into an overhead tank through a pipe having finite resistance (due to friction) is an example of an effort source feeding a potential energy storage element much like a voltage source charging a capacitor. Hence, it will take lesser energy if we were to pump the water slowly (trickle pumping) over several hours rather than doing it fast (gushing water) in fifteen minutes flat!

On a philosophical note, a very nice example is that of a teacher teaching a student. If the teacher goes far too fast beyond the grasping speed of the student, much of the knowledge doesn’t get transferred to the student’s brain.