Real exponential signal

A real exponential signal is

$g(t)=A{e}^{-t/\tau }.$

$A$ is the amplitude (the value at $t=0$). The new parameter $\tau$ is the time constant, with units of seconds. It controls how fast the exponential decays: at $t=\tau$ the signal has dropped to $A/e\approx 0.37A$; at $t=2\tau$ to $A/e^2\approx 0.135A$. Smaller $\tau$ means faster decay.

Geometric meaning of $\tau$

Draw the tangent line to $g(t)=A{e}^{-t/\tau }$ at $t=0$. That tangent crosses zero at $t=\tau$. So if you have a plot and want a quick visual estimate of the time constant, sketch the initial tangent and read off where it hits the axis.

Why this signal is everywhere

The voltage across a charging capacitor in an RC circuit, the current decay in an inductor, the response of any first-order system, the temperature of a cooling object — all are real exponentials (or oscillating exponentials, which are complex sinusoids). The reason is that $e^t$ is the unique function (up to scaling) whose derivative is proportional to itself, so any system governed by a first-order linear differential equation has exponential solutions.

The convolution of the unit step with an impulse response $h(t)=(1/RC)e^{-t/RC}u(t)$ gives the canonical RC step response $v_C(t)=V_b(1-e^{-t/RC})u(t)$.

Growing exponentials

With $\tau >0$, $A{e}^{t/\tau }$ grows without bound. Physically this usually signals trouble — an unstable amplifier or a runaway feedback loop. Mathematically we still have to handle growing signals: their Fourier transforms don’t exist (the integral diverges), but the Laplace transform handles them cleanly by allowing complex frequencies whose real part can compensate for the growth.

A growing exponential’s pole sits in the right half of the s-plane; a decaying one sits in the left half. The position of the pole encodes everything about the signal’s growth or decay rate.