FREQUENCY RESPONSE ANALYSIS

Abstract

Frequency response is the quantitative measure of the output spectrum of a system or device in response to a stimulus, and is used to characterise the dynamics of the system. It is a measure of magnitude and phase of the output as a function of frequency, in comparison to the input. In simplest terms, if a sine wave is injected into a system at a given frequency, a linear system will respond at that same frequency with a certain magnitude and a certain phase angle relative to the input. Also for a linear system, doubling the amplitude of the input will double the amplitude of the output. In addition, if the system is time-invariant, then the frequency response also will not vary with time. Two applications of frequency response analysis are related but have different objectives. For an audio system, the objective may be to reproduce the input signal with no distortion. That would require a uniform (flat) magnitude of response up to the bandwidth limitation of the system, with the signal delayed by precisely the same amount of time at all frequencies. That amount of time could be seconds, or weeks or months in the case of recorded media. In contrast, for a feedback apparatus used to control a dynamic system, the objective is to give the closed-loop system improved response as compared to the uncompensated system. The feedback generally needs to respond to system dynamics within a very small number of cycles of oscillation (usually less than one full cycle), and with a definite phase angle relative to the commanded control input. For feedback of sufficient amplification, getting the phase angle wrong can lead to instability for an open-loop stable system, or failure to stabilize a system that is open-loop unstable. Digital filters may be used for both audio systems and feedback control systems, but since the objectives are different, generally the phase characteristics of the filters will be significantly different for the two applications. The frequency response of a device or a circuit describes its operation over a specified range of signal frequencies by showing how its gain, or the amount of signal it lets through changes with frequency. Bode plots are graphical representations of the circuits frequency response characteristics and as such can be used in solving design problems. Generally, the circuits gain magnitude and phase functions are shown on separate graphs using logarithmic frequency scale along the x-axis. Bandwidth is the range of frequencies that a circuit operates at in between its upper and lower cut-off frequency points. These cut-off or corner frequency points indicate the frequencies at which the power associated with the output falls to half its maximum value. These half power points corresponds to a fall in gain of 3dB (0.7071) relative to its maximum dB value. Most amplifiers and filters have a flat frequency response characteristic in which the bandwidth or passband section of the circuit is flat and constant over a wide range of frequencies. Resonant circuits are designed to pass a range of frequencies and block others. They are constructed using resistors, inductors, and capacitors whose reactances vary with the frequency, their frequency response curves can look like a sharp rise or point as their bandwidth is affected by resonance which depends on the Q of the circuit, as a higher Q provides a narrower bandwidth.