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Time Constant Calculations
52 questions By Tony R. Kuphaldt
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Question 31 of 52
Calculating variables in reactive circuits using time-constant formulae can be time consuming, due to all the keystrokes necessary on a calculator. Even worse is when a calculator is not available! You should be prepared to estimate circuit values without the benefit of a calculator to do the math, though, because a calculator may not always be available when you need one.
Note that Euler’s constant (e) is approximately equal to 3. This is not a close approximation, but close enough for “rough” estimations. If we use a value of three instead of e’s true value of 2.718281828…, we may greatly simplify the “decay” time constant formula:
Percentage of change ≈ 3−[t/(τ)] Suppose that a capacitive discharge circuit begins with a full-charge voltage of 10 volts. Calculate the capacitor’s voltage at the following times as it discharges, assuming τ = 1 second:
- t = 0 seconds ; EC =
- t = 1 second ; EC =
- t = 2 seconds ; EC =
- t = 3 seconds ; EC =
- t = 4 seconds ; EC =
- t = 5 seconds ; EC =
Without using a calculator, you should at least be able to calculate voltage values as fractions if not decimals!
Reveal answer- t = 0 seconds ; EC = 10 V
- t = 1 second ; EC = [10/3] V = 3.33 V
- t = 2 seconds ; EC = [10/9] V = 1.11 V
- t = 3 seconds ; EC = [10/27] V = 0.370 V
- t = 4 seconds ; EC = [10/81] V = 0.123 V
- t = 5 seconds ; EC = [10/243] V = 0.0412 V
Follow-up question: without using a calculator to check, determine whether these voltages are over-estimates or under-estimates.
Notes:Calculating the voltage for the first few time constants’ worth of time should be easy without a calculator. I strongly encourage your students to develop their estimation skills, so that they may solve problems without being dependent upon a calculator. Too many students depend heavily on calculators - some are even dependent on specific brands or models of calculators!
Equally important as being able to estimate is knowing whether or not your estimations are over or under the exact values. This is especially true when estimating quantities relevant to safety and/or reliability!
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Question 32 of 52
Calculating variables in reactive circuits using time-constant formulae can be time consuming, due to all the keystrokes necessary on a calculator. Even worse is when a calculator is not available! You should be prepared to estimate circuit values without the benefit of a calculator to do the math, though, because a calculator may not always be available when you need one.
Note that Euler’s constant (e) is approximately equal to 3. This is not a close approximation, but close enough for “rough” estimations. If we use a value of three instead of e’s true value of 2.718281828…, we may greatly simplify the “increasing” time constant formula:
Percentage of change ≈ 1 − 3−[t/(τ)] Suppose that a capacitive charging circuit begins fully discharged (0 volts), and charges to an ultimate value of 10 volts. Calculate the capacitor’s voltage at the following times as it discharges, assuming τ = 1 second:
- t = 0 seconds ; EC =
- t = 1 second ; EC =
- t = 2 seconds ; EC =
- t = 3 seconds ; EC =
- t = 4 seconds ; EC =
- t = 5 seconds ; EC =
Without using a calculator, you should at least be able to calculate voltage values as fractions if not decimals!
Reveal answer- t = 0 seconds ; EC = 0 V
- t = 1 second ; EC = [20/3] V = 6.67 V
- t = 2 seconds ; EC = [80/9] V = 8.89 V
- t = 3 seconds ; EC = [260/27] V = 9.63 V
- t = 4 seconds ; EC = [800/81] V = 9.88 V
- t = 5 seconds ; EC = [2420/243] V = 9.96 V
Follow-up question: without using a calculator to check, determine whether these voltages are over-estimates or under-estimates.
Notes:Calculating the voltage for the first few time constants’ worth of time should be easy without a calculator. I strongly encourage your students to develop their estimation skills, so that they may solve problems without being dependent upon a calculator. Too many students depend heavily on calculators - some are even dependent on specific brands or models of calculators!
Equally important as being able to estimate is knowing whether or not your estimations are over or under the exact values. This is especially true when estimating quantities relevant to safety and/or reliability!
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Question 33 of 52
Determine the number of time constants (τ) that 7.5 seconds is equal to in each of the following reactive circuits:
- RC circuit; R = 10 kΩ, C = 220 μF ; 7.5 sec =
- RC circuit; R = 33 kΩ, C = 470 μF ; 7.5 sec =
- RC circuit; R = 1.5 kΩ, C = 100 μF ; 7.5 sec =
- RC circuit; R = 790 Ω, C = 9240 nF ; 7.5 sec =
- RC circuit; R = 100 kΩ, C = 33 pF ; 7.5 sec =
- LR circuit; R = 100 Ω, L = 50 mH ; 7.5 sec =
- LR circuit; R = 45 Ω, L = 2.2 H ; 7.5 sec =
- LR circuit; R = 1 kΩ, L = 725 mH ; 7.5 sec =
- LR circuit; R = 4.7 kΩ, L = 325 mH ; 7.5 sec =
- LR circuit; R = 6.2 Ω, L = 25 H ; 7.5 sec =
Reveal answer- RC circuit; R = 10 kΩ, C = 220 μF ; 7.5 sec = 3.41 τ
- RC circuit; R = 33 kΩ, C = 470 μF ; 7.5 sec = 0.484 τ
- RC circuit; R = 1.5 kΩ, C = 100 μF ; 7.5 sec = 50.0 τ
- RC circuit; R = 790 Ω, C = 9240 nF ; 7.5 sec = 1027 τ
- RC circuit; R = 100 kΩ, C = 33 pF ; 7.5 sec = 2,272,727 τ
- LR circuit; R = 100 Ω, L = 50 mH ; 7.5 sec = 15,000 τ
- LR circuit; R = 45 Ω, L = 2.2 H ; 7.5 sec = 153.4 τ
- LR circuit; R = 1 kΩ, L = 725 mH ; 7.5 sec = 10,345 τ
- LR circuit; R = 4.7 kΩ, L = 325 mH ; 7.5 sec = 108,462 τ
- LR circuit; R = 6.2 Ω, L = 25 H ; 7.5 sec = 1.86 τ
Notes:An interesting thing to note here is the span of time constant values available from common capacitor/inductor/resistor sizes. As students should notice, the capacitor-resistor combinations (all very practical values, I might add) create both longer and shorter time constant values than the inductor-resistor combinations, and that is even including the 25 Henry - 6.2 Ohm combination, which would be difficult (read: expensive) to achieve in real life.
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)] ).