## Time costants

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 v_{R}
across the resistance, the voltage v_{C} across the capacitance and
the charge q_{C} on the capacitance are:

i = (E / R)e^{ - t / CR}

v_{R} = iR = Ee^{ - t / CR}

v_{C} = E - v_{R} = E(1 - e^{ - t / CR})

q_{C} = Cv_{C} = 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 v _{R} across the
resistance, the voltage v_{C} across the capacitance and the charge
q_{C} on the capacitance are:**

i = (V / R)e^{ - t / CR}

v_{R} = iR = Ve^{ - t / CR}

v_{C} = v_{R} = Ve^{ - t / CR}

q_{C} = Cv_{C} = 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 v_{R} across the resistance, the voltage v_{L}
across the inductance and the magnetic linkage
y_{L}
in the inductance are:

i = (E / R)(1 - e^{ - tR / L})

v_{R} = iR = E(1 - e^{ - tR / L})

v_{L} = E - v_{R} = Ee^{ - tR / L}

y_{L}
= 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 v _{R} across the resistance, the voltage v_{L}
across the inductance and the magnetic linkage
y_{L}
in the inductance are:**

i = Ie^{ - tR / L}

v_{R} = iR = IRe^{ - tR / L}

v_{L} = v_{R} = IRe^{ - tR / L}

y_{L} = 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) t_{10-90}
is:

t_{10-90} = (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 t_{50}
is :

t_{50} = (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.