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Time Constant Calculations
52 questions By Tony R. Kuphaldt
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Question 40 of 52
Calculate the voltage across a 2.5 H inductor after “charging” through a series-connected resistor with 50 Ω of resistance for 75 milliseconds, powered by a 6 volt battery. Assume that the inductor has an internal resistance of 14 Ω.
Also, express this amount of time (75 milliseconds) in terms of how many time constants have elapsed.
Hint: it would be helpful in your analysis to draw a schematic diagram of this circuit showing the inductor’s inductance and 14 ohms of resistance as two separate (idealized) components. This is a very common analysis technique in electrical engineering: to regard the parasitic characteristics of a component as a separate component in the same circuit.
Reveal answerEquivalent schematic:
EL = 2.00 V @ t = 75 milliseconds
75 ms = 1.92 time constants (1.92τ)
Notes:Although I have revealed a problem-solving technique in this question, it does not show the students exactly how to separate the inductor’s 2.5 henrys of inductance and 14 ohms of resistance into two components, nor does it give away the answer. Discuss the analytical technique of drawing idealized components (“lumped parameters”) as a problem-solving technique, and encourage students to use it whenever they are faced with analyzing a component exhibiting parasitic characteristics.
An excellent example of this technique is in “modeling” transformers. Transformers exhibit much more than just mutual inductance. They also exhibit self-inductance, leakage inductance, capacitance, resistance, and hysteretic losses. A comprehensive model for a transformer is a very complex thing, and it appears on a schematic to be a whole network of components connected together:
Each of these components is regarded as ideal (i.e., pure: possessing no parasitic characteristics), but together they “model” the behavior of a real transformer in terms readily applicable to existing mathematical techniques.
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Question 41 of 52
Calculate the amount of time it takes for a 10 μF capacitor to discharge from 18 volts to 7 volts if its ultimate (final) voltage when fully discharged will be 0 volts, and it is discharging through a 22 kΩ resistor.
Reveal answer0.208 seconds
Notes:In order for students to solve this problem, they must algebraically manipulate the “normal” time-constant formula to solve for time instead of solving for voltage.
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Question 42 of 52
Determine the amount of time needed after switch closure for the capacitor voltage (VC) to reach the specified levels:
VC Time
0 volts
-5 volts
-10 volts
-15 volts
-19 volts
Trace the direction of current in the circuit while the capacitor is charging, and be sure to denote whether you are using electron or conventional flow notation.Note: the voltages are specified as negative quantities because they are negative with respect to (positive) ground in this particular circuit.
Reveal answer
VC Time
0 volts 0 ms
-5 volts 29.75 ms
-10 volts 71.67 ms
-15 volts 143.3 ms
-19 volts 309.8 ms
While the capacitor is charging, electron flow moves clockwise and conventional flow moves counter-clockwise.Notes:Ask your students to show how they algebraically solved the standard time constant equation for t using logarithms.
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)] ).