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Five Tau Rule

Five Second Rule
If you are old enough and have a kid or two, you say this rule quite often or, at least, try. If you are still young (and restless), you still remember this rule often heard too late to do anything at all: you already swallowed down the dropped Cheerios to your stomach. Standing with not much success, the "five second rule" is to keep (or eat) dropped and dirty food from baby or kid if we pick it up within 5 seconds. This says that a food piece dropped on the floor is bad (or done) after 5 seconds.

Five Tau Rule
There is, by the way, a rule called (by me) "Five Tau Rule." What it says is similar to the 5 second rule: A transient event (expressed usually in terms of current) dies down after five Tau seconds. In other words, a transition from a (steady) state to another (steady) state lasts only for five Tau seconds. For example, a switching off source from an R-C circuit and thus the change in the current amount flowing in the R-C circuit from a certain level to zero takes five Tau seconds. If Tau is 1, then the transition takes 5 seconds. It would be 5 micro-second if Tau is 1 millionth. What determines the Tau? The circuit. More specifically the values of R and C in R-C circuit, or R and L in R-L circuit.

Tau Dynamics
What the heck is Tau (t) then? Tau is "a time constant" and as the name implies it's a constant (or a fixed value) about time. It's a constant determined by the values of R and C, or R and L. And it is about how long or short the transition would be. In an R-C circuit, Tau is calculated as R*C; in R-L circuit, it is L/R. The transitional current is usually expressed by an exponential equation which has a term containing EXP[-t/t]. Now consider the numerical values of the exponent as we change the time, t. When t=0, at the exact moment of the switching off, the exponent (or the relative value of the transient current) is EXP[-0/t]=EXP[-0]=1. When t=1t, EXP[-t/t]=EXP[-t/t]=EXP[-1]=0.3678. In other words, after 1t second (whatever value the t may be) the current is reduced to 36.78 percent of the initial value. Let's go further down the road. When time elapses 2t, the current now is EXP[-2t/t]=EXP[-2]=0.1353, that is 13.53 percent of the starting value. When elapsed to t=5t, the current now lays at, EXP[-5t/t]=EXP[-5]=0.00673, that is less than 1 percent. As you see here, the value never reduces to zero, at least in theory. In practice, we have to compromise to say that a certain value is zero enough to be qualified to be zero. The consensus reaches at the value less than 1 % of the starting value, at which t=5t. The birth of the 5 Tau rule!  Summary: Transition from a state to anothe state takes about 5 tau seconds.

Practical Wisdom
As I explained above, in an R-C circuit, t= R*C. You know what? In a circuit, R is a power consumer. If R gets smaller, t becomes smaller, and the transition time to zero gets quicker. Similarly, if your capaciator is smaller, then you can reduce the transition time.  Or the value of the product of R and C determines the tau and the transition period.

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