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Time Constants Formula Calculation

Capacitance and resistance

The time constant of a capacitance C and a resistance R is equal to CR, and represents the time to change the voltage on the capacitance from zero to E at a constant charging current E / R (which produces a rate of change of voltage E / CR across the capacitance).

Similarly, the time constant CR represents the time to change the charge on the capacitance from zero to CE at a constant charging current E / R (which produces a rate of change of voltage E / CR across the capacitance).

If a voltage E is applied to a series circuit comprising a discharged capacitance C and a resistance R, then after time t the current i, the voltage vR across the resistance, the voltage vC across the capacitance and the charge qC on the capacitance are:

i = (E / R)e - t / CR
vR = iR = Ee - t / CR
vC = E - vR = E(1 - e - t / CR)
qC = CvC = CE(1 - e - t / CR)

If a capacitance C charged to voltage V is discharged through a resistance R, then after time t the current i, the voltage vR across the resistance, the voltage vC across the capacitance and the charge qC on the capacitance are:

i = (V / R)e - t / CR
vR = iR = Ve - t / CR
vC = vR = Ve - t / CR
qC = CvC = CVe - t / CR

Inductance and resistance

The time constant of an inductance L and a resistance R is equal to L / R, and represents the time to change the current in the inductance from zero to E / R at a constant rate of change of current E / L (which produces an induced voltage E across the inductance).

If a voltage E is applied to a series circuit comprising an inductance L and a resistance R, then after time t the current i, the voltage vR across the resistance, the voltage vL across the inductance and the magnetic linkage yL in the inductance are:

i = (E / R)(1 - e - tR / L)
vR = iR = E(1 - e - tR / L)
vL = E - vR = Ee - tR / L
yL = Li = (LE / R)(1 - e - tR / L)

If an inductance L carrying a current I is discharged through a resistance R, then after time t the current i, the voltage vR across the resistance, the voltage vL across the inductance and the magnetic linkage yL in the inductance are:

i = Ie - tR / L
vR = iR = IRe - tR / L
vL = vR = IRe - tR / L
yL = Li = LIe - tR / L

Rise Time and Fall Time

The rise time (or fall time) of a change is defined as the transition time between the 10% and 90% levels of the total change, so for an exponential rise (or fall) of time constant T, the rise time (or fall time) t10-90 is:

t10-90 = (ln0.9 - ln0.1)T 2.2T

The half time of a change is defined as the transition time between the initial and 50% levels of the total change, so for an exponential change of time constant T, the half time t50 is:

t50 = (ln1.0 - ln0.5)T 0.69T

Note that for an exponential change of time constant T:

  • over time interval T, a rise changes by a factor 1 - e -1 ( 0.63) of the remaining change.

  • over time interval T, a fall changes by a factor e -1 ( 0.37) of the remaining change.

  • after time interval 3T, less than 5% of the total change remains.

  • after time interval 5T, less than 1% of the total change remains.

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