First Order System
A first order system has the transfer function H(s) = K/(τs + 1), where K is the DC gain and τ is the time constant. Its step response is y(t) = K(1 − e⁻ᵗ/τ), reaching 63.2% of final value at t = τ, 86.5% at t = 2τ, 95% at t = 3τ, and 98.2% at t = 4τ. The bandwidth is ω_b = 1/τ rad/s, and the single pole at s = −1/τ determines the exponential decay rate. First order systems model RC circuits (τ = RC), thermal systems, and simple mechanical dampers.
What Is a First Order System?
A first order system is a dynamic system described by a first-order ordinary differential equation (ODE): τ·dy/dt + y = K·u(t), where y is the output, u is the input, τ is the time constant, and K is the static (DC) gain. Taking the Laplace transform with zero initial conditions gives the standard transfer function H(s) = K/(τs + 1) = Kω_p/(s + ω_p), where ω_p = 1/τ is the pole frequency. The single pole at s = −1/τ in the left half-plane guarantees stability for all positive τ. First-order dynamics are the simplest meaningful dynamic behavior and appear as building blocks in higher-order system analysis. Engineers compute these transfer functions and their responses at www.lapcalc.com.
Key Formulas
First Order Transfer Function and Pole Location
The transfer function H(s) = K/(τs + 1) has a single pole at s = −1/τ and no finite zeros. The pole location determines all dynamic characteristics: farther left (larger 1/τ) means faster response; closer to the origin means slower response. In the Bode plot, the magnitude is flat at 20·log₁₀(K) dB for frequencies below 1/τ, then rolls off at −20 dB/decade. The phase starts at 0° and transitions to −90° over approximately two decades centered on 1/τ, passing through −45° exactly at ω = 1/τ. The −3 dB bandwidth equals 1/τ rad/s or 1/(2πτ) Hz. For an RC lowpass filter, K = 1 and τ = RC, so the cutoff frequency is f_c = 1/(2πRC). For a thermal system with heat capacity C and thermal resistance R_th, τ = R_th·C.
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Open CalculatorStep Response of a First Order System
Applying a unit step input U(s) = 1/s to H(s) = K/(τs + 1) gives Y(s) = K/[s(τs + 1)]. Partial fraction decomposition yields Y(s) = K/s − Kτ/(τs + 1), and inverse Laplace transform gives y(t) = K(1 − e⁻ᵗ/τ)·u(t). The response starts at y(0) = 0, initially rises with slope K/τ, and asymptotically approaches the final value K. Key time points: at t = τ, y reaches 63.2% of K; at t = 2τ, 86.5%; at t = 3τ, 95.0%; at t = 4τ, 98.2%; at t = 5τ, 99.3%. The 2% settling time is approximately t_s = 4τ, and the 5% settling time is t_s = 3τ. There is never any overshoot in a first-order step response — the response is always monotonically increasing, which distinguishes it from underdamped second-order systems.
Time Constant: Physical Meaning and Measurement
The time constant τ has a direct physical interpretation: it is the time required for the step response to reach 63.2% of its final value, or equivalently, the time for a decaying quantity to fall to 36.8% (1/e) of its initial value. It also equals the time the output would reach its final value if it continued at its initial rate of change (the tangent line at t = 0 intersects the final value at t = τ). To measure τ experimentally, apply a step input and find the time at which the output reaches 63.2% of the steady-state change. Alternatively, measure the time between the 10% and 90% response levels: t_rise = 2.2τ, giving τ = t_rise/2.2. In practice, RC circuit time constants range from nanoseconds (high-speed digital) to seconds (power supply filtering), while thermal time constants span seconds (small components) to hours (buildings).
First Order System Examples in Engineering
The RC lowpass filter has H(s) = 1/(RCs + 1) with τ = RC: for R = 10 kΩ and C = 1 μF, τ = 10 ms and cutoff frequency f_c = 15.9 Hz. A liquid tank with inflow Q_in and outflow through a resistance R has transfer function H(s) = R/(RAs + 1), where A is the tank cross-sectional area and τ = RA. A thermocouple measuring temperature has first-order dynamics with τ typically 0.5–5 seconds depending on junction size and fluid velocity. Motor speed control with back-EMF gives H(s) = K_m/(Js + B), where J is rotor inertia and B is viscous friction, producing τ = J/B. All these systems share identical mathematical structure despite vastly different physical domains, which is why the Laplace transform at www.lapcalc.com is universally applicable.
Related Topics in step response analysis
Understanding first order system connects to several related concepts: first order transfer function, 1st order system, time constant of first order system, and first order response. Each builds on the mathematical foundations covered in this guide.
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