Frequency response 11.1 Introduction 11.2 Linear Frequency Response Plotting Paul Neudorfer 11.3 Bode Diagrams 11.4 A Comparison of Methods 11.1 Introduction The IEEE Standard Dictionary of Electrical and Electronics Terms defines frequency response in stable, linear systems to be "the frequency-dependent relation in both gain and phase difference between steady-state sinu soidal inputs and the resultant steady-state sinusoidal outputs"[IEEE, 1988. In certain specialized applications, ne term frequency response may be used with more restrictive meanings. However, all such uses can be related back to the fundamental definition. The frequency response characteristics of a system can be found directly from its transfer function. A single-input/single-output linear time-invariant system is shown in Fig. 11.1 For dynamic linear systems with no time delay, the transfer function H(s)is in the form of a ratio of polynomials in the complex frequency s, where K is a frequency-independent constant. For a system in the sinusoidal steady state, s is replaced by the sinusoidal frequency jo(=v-1)and the system function becomes H(o)=k No)= h(o) jo D() ment or phase angle, arg HGjo), relat respectively, the amplitudes and phase angles of sinusoidal steady-state input and output signals. Using the terminology of Fig. 11. 1, if the and output x(t)=X cos(ot +O) y(t)=Y cos(ot +e) then the output's amplitude Y and phase angle e, are related to those of the input by the two equations H(jo)X O,=argHGjO)+ O c 2000 by CRC Press LLC© 2000 by CRC Press LLC 11 Frequency Response 11.1 Introduction 11.2 Linear Frequency Response Plotting 11.3 Bode Diagrams 11.4 A Comparison of Methods 11.1 Introduction The IEEE Standard Dictionary of Electrical and Electronics Terms defines frequency response in stable, linear systems to be “the frequency-dependent relation in both gain and phase difference between steady-state sinu￾soidal inputs and the resultant steady-state sinusoidal outputs” [IEEE, 1988]. In certain specialized applications, the term frequency response may be used with more restrictive meanings. However, all such uses can be related back to the fundamental definition. The frequency response characteristics of a system can be found directly from its transfer function. A single-input/single-output linear time-invariant system is shown in Fig. 11.1. For dynamic linear systems with no time delay, the transfer function H(s) is in the form of a ratio of polynomials in the complex frequency s, where K is a frequency-independent constant. For a system in the sinusoidal steady state, s is replaced by the sinusoidal frequency jw (j = ) and the system function becomes H(jw) is a complex quantity. Its magnitude, *H(jw)*, and its argument or phase angle, argH(jw), relate, respectively, the amplitudes and phase angles of sinusoidal steady-state input and output signals. Using the terminology of Fig. 11.1, if the input and output signals are x(t) = X cos (wt + Qx) y(t) = Y cos (wt + Qy) then the output’s amplitude Y and phase angle Qy are related to those of the input by the two equations Y = *H(jw)*X Qy = argH(jw) + Qx H s K N s D s ( ) ( ) ( ) = -1 H j K N j D j H j ej H j ( ) ( ) ( ) ( ) ( ) w w w w w = = * * arg Paul Neudorfer Seattle University