7.3 Time constants and step responses

Time Constants and System Response

Measuring System Response Time

When a circuit experiences a sudden change (like a switch flipping), it doesn't jump instantly to its new state. Instead, it transitions gradually. The time constant $\tau$ is the single most important number for describing how fast that transition happens.

For a first-order RC circuit, the time constant is:

$\tau = RC$

For a first-order RL circuit, it's:

$\tau = \frac{L}{R}$

After one time constant has passed, the circuit's response has reached 63.2% of the way from its initial value to its final value. Why 63.2%? Because the response follows $1 - e^{-t/\tau}$, and plugging in $t = \tau$ gives $1 - e^{-1} \approx 0.632$. A larger $\tau$ means a slower response.

Two other timing metrics show up frequently:

• Rise time $t_r$: the time for the response to go from 10% to 90% of its final value. For first-order systems, $t_r \approx 2.2\tau$.
• Settling time $t_s$: the time for the response to stay within a small band around the final value. For a 2% band, $t_s \approx 4\tau$. For a 5% band, $t_s \approx 3\tau$.

These metrics matter in practice. If you're designing a data acquisition system, you need to wait at least $4\tau$ before sampling so the signal has settled to within 2% of its true value.

Steady-State Response

The steady-state value is where the circuit ends up after the transient dies out. In a first-order RC circuit driven by a DC step, the capacitor voltage eventually equals the source voltage.

A useful rule of thumb: after $4\tau$, the response is within about 2% of steady state. After $5\tau$, it's within about 1%. For most engineering purposes, $5\tau$ is treated as "done."

Step Inputs and Response Types

Step Input Characteristics

A step input is a sudden, instantaneous change in the input signal from one constant level to another. Think of flipping a switch to connect a DC voltage source to a circuit.

Mathematically, this is represented by the unit step function $u(t)$, which equals 0 for $t < 0$ and 1 for $t \geq 0$. A step of magnitude $V_s$ is written as $V_s \cdot u(t)$.

The size of the step matters. A larger voltage step produces a proportionally larger transient response in a linear circuit. In real circuits, a very large step could push components into nonlinear regions (for example, clipping in an amplifier).

System Response Components

The complete response of a first-order circuit to a step input has two parts:

• Natural response: the part driven by energy already stored in the circuit (initial capacitor voltage or inductor current). It decays exponentially as $A \cdot e^{-t/\tau}$, where $A$ depends on initial conditions. With no external input, this is all you'd see.
• Forced response: the part driven by the external input. For a DC step input, the forced response is a constant equal to the new steady-state value.

The complete response is the sum of these two:

$v(t) = v(\infty) + [v(0) - v(\infty)] \cdot e^{-t/\tau}$

where $v(0)$ is the initial value and $v(\infty)$ is the final (steady-state) value. The exponential term is the transient part, which decays to zero over time, leaving only the steady-state value.

Initial and Final Conditions

Initial Conditions

Initial conditions describe the state of energy-storage elements at $t = 0$, the moment the step input is applied. For capacitors, this is the voltage $v(0)$. For inductors, it's the current $i(0)$.

These values come from whatever the circuit was doing before the change occurred. Two key principles help you find them:

• A capacitor's voltage cannot change instantaneously: $v_C(0^+) = v_C(0^-)$
• An inductor's current cannot change instantaneously: $i_L(0^+) = i_L(0^-)$

If the initial condition is zero (an uncharged capacitor, for example), the natural response is zero and the complete response equals just the forced response. A nonzero initial condition adds the exponential decay term.

Final Conditions

To find the final (steady-state) value $v(\infty)$ or $i(\infty)$, analyze the circuit as $t \to \infty$. At that point, all transients have died out:

• A capacitor acts like an open circuit (no current flows through it once fully charged)
• An inductor acts like a short circuit (no voltage across it once current is steady)

Replace the capacitor or inductor with the appropriate equivalent, then use standard DC circuit analysis to find the final value. Once you have $v(0)$, $v(\infty)$, and $\tau$, you can write the complete step response directly using the formula above.