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Time Constant Calculations
52 questions By Tony R. Kuphaldt
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Question 28 of 52
The decay of a variable (either voltage or current) in a time-constant circuit (RC or LR) follows this mathematical expression:
e−[t/(τ)] Where,
e = Euler’s constant ( ≈ 2.718281828)
t = Time, in seconds
τ = Time constant of circuit, in seconds
Calculate the value of this expression as t increases, given a circuit time constant (τ) of 1 second. Express this value as a percentage:
- t = 1 second
- t = 2 seconds
- t = 3 seconds
- t = 4 seconds
- t = 5 seconds
- t = 6 seconds
- t = 7 seconds
- t = 8 seconds
- t = 9 seconds
- t = 10 seconds
Based on your calculations, how would you describe the change in the expression’s value over time as t increases?
Reveal answer- t = 1 second ; e−[t/(τ)] = 36.788%
- t = 2 seconds ; e−[t/(τ)] = 13.534%
- t = 3 seconds ; e−[t/(τ)] = 4.979%
- t = 4 seconds ; e−[t/(τ)] = 1.832%
- t = 5 seconds ; e−[t/(τ)] = 0.6738%
- t = 6 seconds ; e−[t/(τ)] = 0.2479%
- t = 7 seconds ; e−[t/(τ)] = 0.09119%
- t = 8 seconds ; e−[t/(τ)] = 0.03355%
- t = 9 seconds ; e−[t/(τ)] = 0.01234%
- t = 10 seconds ; e−[t/(τ)] = 0.004540%
Notes:The purpose of this question is for students to learn the significance of the expression e−[t/(τ)] by “playing” with the numbers. The negative exponent may confuse some students, so be sure to discuss its significance with all students, so that all understand what it means.
Another concept for students to grasp in this question is that of an asymptotic function: a function that approaches a final value in incrementally smaller intervals.
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Question 29 of 52
The decay of a variable (either voltage or current) in a time-constant circuit (RC or LR) follows this mathematical expression:
e−[t/(τ)] Where,
e = Euler’s constant ( ≈ 2.718281828)
t = Time, in seconds
τ = Time constant of circuit, in seconds
Calculate the value of this expression as t increases, given a circuit time constant (τ) of 2 seconds. Express this value as a percentage:
- t = 1 second
- t = 2 seconds
- t = 3 seconds
- t = 4 seconds
- t = 5 seconds
- t = 6 seconds
- t = 7 seconds
- t = 8 seconds
- t = 9 seconds
- t = 10 seconds
Also, express the percentage value of any increasing variables (either voltage or current) in an RC or LR charging circuit, for the same conditions (same times, same time constant).
Reveal answer- t = 1 second ; e−[t/(τ)] = 60.65% ; increasing variable = 39.35%
- t = 2 seconds ; e−[t/(τ)] = 36.79% ; increasing variable = 63.21%
- t = 3 seconds ; e−[t/(τ)] = 22.31% ; increasing variable = 77.69%
- t = 4 seconds ; e−[t/(τ)] = 13.53% ; increasing variable = 86.47%
- t = 5 seconds ; e−[t/(τ)] = 8.208% ; increasing variable = 91.79%
- t = 6 seconds ; e−[t/(τ)] = 4.979% ; increasing variable = 95.02%
- t = 7 seconds ; e−[t/(τ)] = 3.020% ; increasing variable = 96.98%
- t = 8 seconds ; e−[t/(τ)] = 1.832% ; increasing variable = 98.17%
- t = 9 seconds ; e−[t/(τ)] = 1.111% ; increasing variable = 98.89%
- t = 10 seconds ; e−[t/(τ)] = 0.6738% ; increasing variable = 99.33%
Notes:Do not simply tell your students how to calculate the values of the increasing variable. Based on their qualitative knowledge of time-constant circuit curves and their ability to evaluate the downward (decay) expression, they should be able to figure out how to calculate the increasing variable’s value over time as well.
Some students will insist that you give them an equation to do this. They want to be told what to do, rather than solve the problem on their own based on an observation of pattern. It is very important that students of any science learn to recognize patterns in data, and that they learn to fit that data into a mathematical equation. If nothing else, these figures given in the answer for both decreasing and increasing variables should be plain enough.
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Question 30 of 52
Write a mathematical expression for calculating the percentage value of any increasing variables (either voltage or current) in an RC or LR time-constant circuit.
Hint: the formula for calculating the percentage of any decreasing variables in an RC or LC time-constant circuit is as follows:
e−[t/(τ)] Where,
e = Euler’s constant ( ≈ 2.718281828)
t = Time, in seconds
τ = Time constant of circuit, in seconds
Here, the value of the expression starts at 1 (100%) at time = 0 and approaches 0 (0%) as time approaches ∞. What I’m asking you to derive is an equation that does just the opposite: start with a value of 0 when time = 0 and approach a value of 1 as time approaches ∞.
Reveal answer(1 − e−[t/(τ)])(100%) Notes:Being able to derive an equation from numerical data is a complex, but highly useful, skill in all the sciences. Sure, your students will be able to find this mathematical expression in virtually any basic electronics textbook, but the point of this question is to derive this expression from an examination of data (and, of course, an examination of the other time-constant expression: e−[t/(τ)]).
Be sure to challenge your students to do this, by asking how they obtained the answer to this question. Do not “settle” for students simply telling you what the equation is - ask them to explain their problem-solving techniques, being sure that all students have contributed their insights.
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)] ).