Frequency Response

Frequency response describes what a dynamic system does to sinusoidal inputs of different frequencies after transients have died out. If the input is a sine wave at angular frequency \omega, a linear time-invariant system produces an output at the same frequency, but generally with a different amplitude and a phase shift. The frequency response records those two quantities across frequency.

For a system with transfer function G(s), the frequency response is obtained by substituting s = j\omega:

The magnitude |G(j\omega)| shows gain or attenuation. The phase \angle G(j\omega) shows how much the output leads or lags the input. Engineers usually plot these quantities with a Bode plot, Nyquist plot, or measured frequency response function.

What the response reveals

Frequency response makes dynamic behaviour visible. A low-pass system has high gain at low frequency and attenuation at high frequency. A resonant mechanical structure shows a peak near its natural frequency. A control loop may have adequate low-frequency disturbance rejection but poor high-frequency noise rejection. An amplifier may meet gain requirements in the middle of its band but lose phase margin near crossover.

The most important quantities depend on the discipline. In control engineering, engineers look for bandwidth, crossover frequency, gain margin, phase margin, and sensitivity to disturbances. In vibration analysis, they look for resonant peaks, damping ratio, mode separation, and anti-resonances. In electronics and signal processing, they examine passband ripple, cutoff frequency, roll-off, group delay, and stopband attenuation.

Design and analysis considerations

Frequency response is valuable because it connects models, tests, and design requirements. A controller can be shaped to increase low-frequency loop gain without sacrificing stability margin. A structure can be stiffened, damped, or isolated after its resonant frequencies are identified. A filter can be specified by passband, transition band, and stopband targets. A sensor can be selected by checking whether its bandwidth and phase lag are acceptable for the intended control loop.

Measured frequency response is commonly obtained with swept-sine excitation, broadband random excitation, impulse testing, network analysis, or operational data. The measurement setup matters: excitation level, boundary conditions, sensor placement, sampling frequency, windowing, averaging, and noise all affect the result. For nonlinear systems, the apparent frequency response may change with amplitude, operating point, temperature, wear, or load.

Common mistakes

A frequent error is to treat frequency response as a complete description of a system without checking the assumptions behind it. The classical interpretation assumes linearity, time invariance, and steady-state sinusoidal excitation. Saturation, friction, backlash, dead zones, time-varying dynamics, and large transients can make a frequency-domain model misleading.

Another mistake is to read magnitude without phase. A system can have acceptable gain but unacceptable delay or phase lag, especially in feedback control. Conversely, a resonant peak that looks harmless in a lightly loaded test may become critical when the system is excited repeatedly near that frequency in service. Good engineering documentation states the measurement method, operating point, units, frequency range, uncertainty, and whether the response is open-loop, closed-loop, simulated, or measured.