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 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,