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Time Constant Calculations
52 questions By Tony R. Kuphaldt
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Question 25 of 52
A formula I like to use in calculating voltage and current values in either RC or LR circuits has two forms, one for voltage and one for current:
V(t) = ( Vf − V0 ) ( 1 − 1 e[t/(τ)]) + V0 (for calculating voltage) I(t) = ( If − I0 ) ( 1 − 1 e[t/(τ)]) + I0 (for calculating current) The “0” subscript represents the condition at time = 0 (V0 or I0, respectively), representing the “initial” value of that variable. The “f” subscript represents the “final” or “ultimate” value that the voltage or current would achieve if allowed to progress indefinitely. Obviously, one must know how to determine the “initial” and “final” values in order to use either of these formulae, but once you do you will be able to calculate any voltage and any current at any time in either an RC or LR circuit.
What is not so obvious to students is how each formula works. Specifically, what does each portion of it represent, in practical terms? This is your task: to describe what each term of the equation means in your own words. I will list the “voltage” formula terms individually for you to define:
V(t) = ( Vf − V0 ) = ( 1 − 1 e[t/(τ)]) = Reveal answerThe term V(t) uses symbolism common to calculus and pre-calculus, pronounced “V of t,” meaning “voltage at time t”. It means that the variable V changes as a function of time t.
( Vf − V0 ) represents the amount of change that the voltage would go through, from the start of the charge/discharge cycle until eternity. Note that the sign (positive or negative) of this term is very important!
( 1 − [1/(e[t/(τ)])] ) is the fractional value (between 0 and 1, inclusive) expressing how far the voltage has changed from its initial value to its final value.
Follow-up question: why is it important to add the final V0 term to the expression? Why not leave the formula at V(t) = ( Vf − V0 ) ( 1 − [1/(e[t/(τ)])] ) ?
Notes:This so-called “universal time-constant formula” is my own (Tony R. Kuphaldt’s) invention. A product of frustration from trying to teach students to calculate variables in RC and LR time-constant circuits using one formula for decay and another one for increasing values, this equation works for all conditions. Vitally important to this formula’s accuracy, however, is that the student correctly assesses the initial and final values. This is the biggest problem I see students having when they go to calculate voltages or currents in time-constant circuits.
In my Lessons In Electric Circuits textbook, I introduce this formula in a slightly different form:
∆V = ( Vf − V0 ) ( 1 − 1 e[t/(τ)]) This explains the purpose of my follow-up question: to challenge students who might simply read the book’s version of the formula and not consider the difference between it and what is presented here!
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Question 26 of 52
Determine the voltage across the capacitor three seconds after the switch is moved from the upper position to the lower position, assuming it had been left in the upper position for a long time:
Reveal answerEC = 3.974 V @ 3 seconds
Follow-up question: identify at least one failure in this circuit which would cause the capacitor to remain completely discharged no matter what position the switch was in.
Notes:This problem is unique in that the capacitor does not discharge all the way to 0 volts when the switch is moved to the lower position. Instead, it discharges down to a (final) value of 3 volts. Solving for the answer requires that students be a bit creative with the common time-constant equations (e−[t/(τ)] and 1 − e−[t/(τ)]).
The follow-up question is simply an exercise in troubleshooting theory.
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Question 27 of 52
Calculate the voltage across the switch contacts the exact moment they open, and 15 milliseconds after they have been opened:
Reveal answerEswitch = 40.91 V @ t = 0 seconds
Eswitch = 9.531 V @ t = 15 milliseconds
Follow-up question: predict all voltage drops in this circuit in the event that the inductor fails open (broken wire inside).
Notes:There is quite a lot to calculate in order to reach the solutions in this question. There is more than one valid way to approach it, as well. An important fact to note: the voltage across the switch contacts, in both examples, is greater than the battery voltage! Just as capacitive time-constant circuits can generate currents in excess of what their power sources can supply, inductive time-constant circuits can generate voltages in excess of what their power sources can supply.
The follow-up question is simply an exercise in troubleshooting theory.
Maybe this will help someone else. The general formulas for V(t) and I(t) in question 25 (and the x(t) versions in question s 23 and 14) contain typos (or maybe hypertext coding glitches). They should actually be V(t) = (Vf-Vo)(1-e^(-t/𝛕)) + Vo, I(t) = (If-Io)(1-e^(-t/𝛕)) + Io in question 25. Those are correct in the PDF download version. In questions 23 and 24 the equations are x = xinitial + ( xfinal − xinitial ) ( 1 − e[(−t)/(τ)] ).