First-Order System Time Response

Understanding how systems respond over time is fundamental to engineering control, signal processing, and circuit design. First-order systems are the simplest dynamic models, but their predictable exponential behavior forms the essential building block for analyzing more complex systems. By mastering their time response, you gain the tools to predict system speed, stability margins, and accuracy, which are critical for everything from thermostat design to chemical process control.

Defining a First-Order System

A first-order system is a linear, time-invariant dynamic system whose input-output relationship is described by a first-order ordinary differential equation. In the context of control theory and systems engineering, it is most commonly represented by its transfer function in the Laplace domain. The standard form of this transfer function is:

$$G(s)=\frac{K}{\tau s+1}$$

Here, $K$ represents the system gain, which determines the final, steady-state value of the output for a given input. The parameter $\tau$ (tau) is the time constant, the single most important characteristic defining the system's speed of response. This transfer function clearly shows the system has one pole located at $s=-1/\tau$. The negative, real location of this pole is what guarantees the stable, exponential nature of the system's response to a disturbance. Physically, first-order systems are ubiquitous: the voltage across a capacitor charging through a resistor (an RC circuit), the temperature of a small object in a constant-temperature bath, and the level of liquid in a tank with a constant outlet flow are all classic examples.

The Time Constant: The Heart of the Response

The time constant $\tau$ is the definitive parameter of a first-order system. It provides a precise measure of how quickly the system reacts. A larger $\tau$ indicates a slower, more sluggish response, while a smaller $\tau$ indicates a faster reaction. Conceptually, $\tau$ represents the time it would take for the system's output to reach its final value if it continued to change at its initial rate. While it doesn't actually take that time, this definition offers an intuitive grasp of its meaning.

The power of the time constant lies in its universal scaling property. For a stable first-order system ($\tau >0$), the response to any sudden change follows an exponential curve defined by $1-e^{-t/\tau }$. This means that the fraction of the total change completed at any time $t$ depends only on the ratio $t/\tau$. This leads to key, memorable benchmarks:

• After one time constant ($t=\tau$), the response reaches $1-e^{-1}\approx 0.632$, or 63.2 percent of its final value.
• After four time constants ($t=4\tau$), the response reaches $1-e^{-4}\approx 0.982$, or 98.2 percent of its final value. For most practical purposes, the system is considered to have settled at this point.

These percentages are invariant; they hold true for any first-order system regardless of its specific gain $K$ or physical origin, provided it is in the standard form.

Step Response Characteristics

The step response—the output when the input instantaneously jumps from one constant value to another—is the primary tool for analyzing transient performance. For a unit step input, the output $y(t)$ of our standard system $G(s)=K/(\tau s+1)$ is derived via inverse Laplace transform:

$$y(t)=K(1-e^{-t/\tau })$$

This equation describes a curve that starts at zero (assuming zero initial conditions) and rises exponentially to approach the final value $K$. The initial slope of this curve at $t=0$ is $K/\tau$. If the output followed this initial slope linearly, it would reach the final value $K$ exactly at time $t=\tau$, reinforcing the time constant's conceptual definition.

Analyzing this curve yields specific performance metrics that completely characterize the first-order transient:

• Rise Time ($T_r$): While definitions vary, a common and practical one for first-order systems is the time for the response to go from 10% to 90% of its final value. For a first-order system, this is approximately $T_r\approx 2.2\tau$. It quantifies how quickly the system initially reacts.
• Settling Time ($T_s$): This is the time required for the response to enter and remain within a specified error band around the final value. Using the common 2% criterion (i.e., within 98% of the final value), the settling time is four time constants: $T_s=4\tau$. This tells you how long you must wait before the output is effectively constant.
• Steady-State Error ($e_{ss}$): For a step input, the steady-state error is the difference between the desired final value (the input step magnitude) and the actual final output. For the standard transfer function, the final value is $K$ times the input step. Therefore, if the desired gain is 1, you would design $K=1$ to achieve zero steady-state error for a step input. More formally, $e_{ss}$ can be calculated using the Final Value Theorem.

Performance in Context: Gain and Design Implications

The system gain $K$ and time constant $\tau$ are independent design parameters that address different performance goals. The gain $K$ is solely responsible for the system's steady-state accuracy. For instance, if a temperature control system has a gain of 0.8, a commanded 100°C step will only result in a steady-state temperature of 80°C—a 20% steady-state error. To correct this, you would adjust the system's components to bring $K$ to 1.

Conversely, the time constant $\tau$ governs the transient dynamics—the speed of the response. You cannot change the shape of the exponential response, but you can stretch or compress it along the time axis by modifying $\tau$. In an RC circuit, $\tau =RC$; to make the circuit charge faster, you must decrease either the resistance $R$ or the capacitance $C$. In a thermal system, reducing insulation (lowering thermal resistance) would decrease $\tau$ and speed up heating or cooling. A key design insight is that improving response speed (reducing $\tau$) often requires trade-offs, such as increased power consumption, cost, or system stress.

Common Pitfalls

1. Misidentifying the Time Constant: It's easy to misread the transfer function. A system given as $G(s)=10/(5s+2)$ is not in standard form. You must rewrite it as $G(s)=(10/2)/((5/2)s+1)=5/(2.5s+1)$ to correctly identify $K=5$ and $\tau =2.5$ seconds. Always normalize the denominator so the constant term is 1.
2. Confusing Rise Time and Settling Time: These metrics describe different phases of the response. Rise time ($\approx 2.2\tau$) measures the initial "agility," while settling time ($4\tau$) measures the total time to complete the process within a tolerance. A system can have a fast rise time but a long settling time if it oscillates, but this never happens in a pure first-order system—it is strictly monotonic.
3. Applying First-Order Metrics to Higher-Order Systems: The 63.2% and 98% rules, and the simple formulas for $T_r$ and $T_s$, are exclusive to pure first-order systems. Applying them to an underdamped second-order system (which has overshoot) will yield incorrect conclusions. Always verify the system order from its transfer function first.
4. Ignoring the Steady-State Value When Calculating Percentages: When measuring that the output is at "63.2% of the final value," you must calculate this percentage from the total change. If a system starts at 10 and ends at 50, the total change is 40. 63.2% of the change is ~25.3. Therefore, the 1$\tau$ point occurs when the output reaches $10+25.3=35.3$, not when it is at 63.2% of the maximum value (50).

Summary

• A first-order system is defined by a single energy-storage element and is characterized by its transfer function $G(s)=K/(\tau s+1)$, where $K$ is the DC gain and $\tau$ is the time constant.
• The time constant $\tau$ is the definitive speed metric. The step response reaches 63.2% of its final value at $t=\tau$ and approximately 98% at $t=4\tau$.
• Transient performance is completely described by rise time (approximately $2.2\tau$ for 10%-90%), settling time (exactly $4\tau$ for a 2% criterion), and steady-state error (determined by the gain $K$).
• The gain $K$ and time constant $\tau$ are independent parameters controlling steady-state accuracy and transient speed, respectively, and are often the focus of system design trade-offs.
• The exponential step response formula $y(t)=K(1-e^{-t/\tau })$ provides the complete time-domain solution for analyzing system behavior.