Frequency Response Method Frequency Response Plots Frequency Response Measurements Performance Specifications in the frequency Domain

Introduction In previous lectures we examined the use of test signals such as a step and a ramp signal. In this chapter, we consider the steady-state response of a system to a sinusoidal input signal test signal. In the previous lectures the response and performance of a system have been described in terms of the complex frequency variable s and the location of the poles and zeros on the s plane. We will see that the response of a linear constant coefficient system to a sinusoidal input signal is an output sinusoidal signal at the same frequency as the input. The magnitude and phase of the output signal differ from those of the input sinusoidal signal, and the amount of difference is a function of the input frequency. In this lecture an alternative approach “frequency Response” is adopted.

Frequency Response The frequency response of a system is defined as the steady-state response of the system to a sinusoidal input signal. This sinusoid is a unique input signal, and the resulting output signal for a linear system, as well as signals throughout the system, is sinusoidal in the steady state; it differs from the input waveform only in amplitude and phase angle. One advantage of the frequency response method is the ready availability of sinusoidal test signals for various ranges of frequency and amplitudes. An other advantage of this method is that the transfer function describing the sinusoidal steady-state behaviour of a system can be obtained by replacing s with j in the system transfer function.

The Fourier and Laplace Transforms

Frequency Response Plots The transfer function of a system G(s) can be described in the frequency domain Im (G) = X () Re (G) = R ()

Bode Diagram Bode plots are a very useful way to represent the gain and phase of a system as a function of frequency. Steps to draw the Bode plot: Rewrite the transfer function in proper form. Separate the transfer function into its constituent parts. The next step is to split up the function into its constituent parts. There are seven types of parts: A constant Poles at the origin Zeros at the origin Real poles Real zeros Complex conjugate poles Draw the Bode diagram for each part.

Constant Gain Kb

Poles or Zeros at the Origin (j)

Poles or Zeros on the Real Axis Complex

Conjugate Poles or Zeros

Performance Specifications in the Frequency Domain At the resonant frequency, r, a maximum value of the frequency response Mp, is attained. The bandwidth is the frequency, B, at which the frequency response has declined 3 dB from its low-frequency value. 20 logMp -3 r B