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Time Constant Calculations
52 questions By Tony R. Kuphaldt
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Question 22 of 52
Design an experiment to calculate the size of a capacitor based on its response in a time-constant circuit. Include in your design an equation that gives the value of the capacitor in farads, based on data obtained by running the experiment.
Reveal answerI recommend the following circuit for testing the capacitor:
The equation is yours to develop - I will not reveal it here. However, this does not mean there is no way to verify the accuracy of your equation, once you write it. Explain how it would be possible to prove the accuracy of your algebra, once you have written the equation.
Notes:In developing equations, students often feel “abandoned” if the instructor does not provide an answer for them. How will they ever know if their equation is correct? If the phenomenon the equation seeks to predict is well-understood, though, this is not a problem.
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Question 23 of 52
A helpful technique for analyzing RC time-constant circuits is to consider what the initial and final values for circuit variables (voltage and current) are. Consider these four RC circuits:
In each of these circuits, determine what the initial values will be for voltage across and current through both the capacitor and (labeled) resistor. These will be the voltage and current values at the very first instant the switch changes state from where it is shown in the schematic. Also, determine what the final values will be for the same variables (after a large enough time has passed that the variables are all “settled” into their ultimate values). The capacitor’s initial voltage is shown in all cases where it is arbitrary.
Reveal answerFigure 1:
VC(initial) = 0 V (given) VC(final) = 15 V IC(initial) = 1.5 mA IC(final) = 0 mA VR(initial) = 15 V VR(final) = 0 V IR(initial) = 1.5 mA IR(final) = 0 mA Figure 2:
VC(initial) = 25 V (given) VC(final) = 0 V IC(initial) = 25 mA IC(final) = 0 mA VR(initial) = 25 V VR(final) = 0 V IR(initial) = 25 mA IR(final) = 0 mA Figure 3:
VC(initial) = 4 V VC(final) = 7 V IC(initial) = 81 μA IC(final) = 0 μA VR(initial) = 3 V VR(final) = 0 V IR(initial) = 81 μA IR(final) = 0 μA Figure 4:
VC(initial) = 10 V VC(final) = 12 V IC(initial) = 606 μA IC(final) = 0 μA VR(initial) = 2 V VR(final) = 0 V IR(initial) = 606 μA IR(final) = 0 μA Follow-up question: explain why the inductor value (in Henrys) is irrelevant in determining ïnitial” and “final” values of voltage and current.
Notes:Once students grasp the concept of initial and final values in time-constant circuits, they may calculate any variable at any point in time for any RC or LR circuit (not for RLC circuits, though, as these require the solution of a second-order differential equation!). All they need is the universal time-constant equation:
x = xinitial ( xfinal − xinitial ) ( 1 − e[(−t)/(τ)] ) (x, of course, represents either voltage or current, depending on what is being calculated.)
One common mistake new students often commit is to consider “initial” values as those values of voltage and current existing in the circuit before the switch is thrown. However, “initial” refers to those values at the very first instant the switch moves to its new position, not before.
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Question 24 of 52
A helpful technique for analyzing LR time-constant circuits is to consider what the initial and final values for circuit variables (voltage and current) are. Consider these four LR circuits:
In each of these circuits, determine what the initial values will be for voltage across and current through both the inductor and (labeled) resistor. These will be the voltage and current values at the very first instant the switch changes state from where it is shown in the schematic. Also, determine what the final values will be for the same variables (after a large enough time has passed that the variables are all “settled” into their ultimate values).
Assume all inductors are ideal, possessing no coil resistance (Rcoil = 0 Ω).
Reveal answerFigure 1:
VL(initial) = 15 V VL(final) = 0 V IL(initial) = 0 mA IL(final) = 1.5 mA VR(initial) = 0 V VR(final) = 15 V IR(initial) = 0 mA IR(final) = 1.5 mA Figure 2:
VL(initial) = 50.4 V VL(final) = 0 V IL(initial) = 3.38 mA IL(final) = 825 μA VR(initial) = 74.4 V VR(final) = 18.1 V IR(initial) = 3.38 mA IR(final) = 825 μA Figure 3:
VL(initial) = 370 V VL(final) = 0 V IL(initial) = 10 mA IL(final) = 0 mA VR(initial) = 370 V VR(final) = 0 V IR(initial) = 10 mA IR(final) = 0 mA Figure 4:
VL(initial) = 6 V VL(final) = 0 V IL(initial) = 638 μA IL(final) = 1.91 mA VR(initial) = 3 V VR(final) = 9 V IR(initial) = 638 μA IR(final) = 1.91 mA Follow-up question: explain why the inductor value (in Henrys) is irrelevant in determining “initial” and “final” values of voltage and current.
Notes:Once students grasp the concept of initial and final values in time-constant circuits, they may calculate any variable at any point in time for any RC or LR circuit (not for RLC circuits, though, as these require the solution of a second-order differential equation!). All they need is the universal time-constant equation:
x = xinitial ( xfinal − xinitial ) ( 1 − e[(−t)/(τ)] ) (x, of course, represents either voltage or current, depending on what is being calculated.)
One common mistake new students often commit is to consider “initial” values as those values of voltage and current existing in the circuit before the switch is thrown. However, “initial” refers to those values at the very first instant the switch moves to its new position, not before.
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)] ).