Transients

RC circuit
Derivation of RC transient function
RL circuit
Time constant
Application 1: relaxation oscillator
Application 2: pulse generator

Transients in DC circuits

When an RC circuit responds to a step change in input voltage, the time is takes to move 63.2% of the way to its new steady state voltage is known as the time constant, \tau . The value of \tau for an RC circuit is

\tau = RC

If the is v_C(t) = V_1 for t<0, begins changing at t=0, and v_c(t) \to V_2 as t \to \infty, the complete transient response of the capacitor voltage is described by

v_C(t) = V_2 - (V_2-V_1).\left( 1 - e^{\frac{-t}{\tau}} \right)

For example, if V_1 = 5 \mathrm{V} and V_2 = 0 \mathrm{V},

v_C(t) = 5.\left( 1 - e^{\frac{-t}{\tau}} \right)

Note that the significance of the 63.2% value is that 1 - e^{-1} = 0.6321.

For example, using the values of V_1 and V_2 specified above, at time t = \tau ,

v_C(\tau) = 5.\left( 1 - e^{-1} \right) = 0.6321 \times 5 \mathrm{V}

The concept of a time constant can be applied to many different systems or process that exhibit exponential decay, including for example heating and cooling of a building, or radioactive decay.

The time constant of an RL circuit is

\tau = \frac{L}{R}

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