Time constant

The time constant $\tau$ of a first-order system is the characteristic time over which the system’s response evolves — specifically, the time it takes for an exponential to decay to $1/e\approx 0.368$ of its initial value, or for a step response to rise to $1-1/e\approx 0.632$ of its final value.

For a decaying exponential $g(t)=A{e}^{-t/\tau }$:

• At $t=\tau$: $g=A/e\approx 0.37A$.
• At $t=2\tau$: $g\approx 0.135A$.
• At $t=5\tau$: $g\approx 0.007A$ — the signal has essentially settled.

The “five time constants to settle” rule of thumb shows up everywhere from circuit design to thermal analysis.

In an RC circuit

For an RC circuit, the time constant is

$\tau =RC$

with $R$ in ohms and $C$ in farads. A $1\text{k}\Omega$ resistor with a $1\mu \text{F}$ capacitor has $\tau =1\text{ms}$. The step response of the capacitor voltage is $v_C(t)=V_b(1-e^{-t/\tau })u(t)$: rises rapidly at first, then asymptotes to $V_b$ over roughly $5\tau$.

For an RL circuit, $\tau =L/R$.

Connection to poles

A first-order system with impulse response $h(t)=(1/\tau )e^{-t/\tau }u(t)$ has transfer function $H(s)=1/(\tau s+1)$ and a single pole at $s=-1/\tau$. The cutoff frequency of the associated lowpass filter is ${\omega }_c=1/\tau$ rad/s. So the time constant, the pole location, and the cutoff frequency are three views of the same thing.

A short time constant means a pole far from the imaginary axis (system reacts quickly, broad bandwidth); a long time constant means a pole close to the imaginary axis (system reacts slowly, narrow bandwidth). This is the time-bandwidth tradeoff in its simplest form.