11
Frequency Respon 11.1 Introduction 11.2 Linear Frequency Respon Plotting 11.3 Bode Diagrams 11.4 A Comparison of Methods The IEEE Standard Dictionary of Electrical and Electronics Terms defines frequency respon in stable, linear systems to be “the frequency-dependent relation in both gain and pha difference between steady-state sinu-soidal inputs and the resultant steady-state sinusoidal outputs” [IEEE, 1988]. In certain specialized applications,the term frequency respon may be ud with more restrictive meanings. However, all such us can be related back to the fundamental definition. The frequency respon 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
党员意见建议j w (j =
遨游大海
) and the system function becomes H (j w ) is a complex quantity. Its magnitude, ΈH (j w )Έ, and its argument or pha angle, arg H (j w ), relate,respectively, the amplitudes and pha 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 (w t + Q x )
y (t ) = Y cos (w t + Q y )
then the output’s amplitude Y and pha angle Q y are related to tho of the input by the two equations
Y = ΈH (j w )ΈX
Q y = arg H (j w ) + Q x文化活动策划
-1Paul Neudorfer
Seattle University
The phra frequency respon characteristics usually下载不了
implies a complete description of a system’s sinusoidal
steady-state behavior as a function of frequency. Becau
H (j w ) is complex and, therefore, two dimensional in nature,
cvt变速箱frequency respon characteristics cannot be graphically dis-
played as a single curve plotted with respect to frequency.
Instead, the magnitude and argument of H (j w ) can be p-arately plotted as functions of frequency. Often, only the magnitude curve is prented as a conci way of character-
izing the system’s behavior, but this must be viewed as an incomplete description. The most common form for such plots is the Bode diagram (developed by H.W. Bode of Bell Laboratories), which us a logarithmic scale for frequency. Other forms of frequency respon plots have also been developed. In the Nyquist plot (Harry Nyquist, also of Bell Labs), H (j w ) is displayed on the complex plane, Re[H (j w )] on the horizontal axis, and Im[H (j w )] on the vertical. Frequency is a parameter of such curves. It is sometimes numerically identified at lected points of the curve and sometimes omitted. The Nichols chart (N.B. Nichols) graphs magnitude versus pha for the system function. Frequency again is a parameter of the resultant curve,sometimes shown and sometimes not.
Frequency respon techniques are ud in many areas of engineering. They are most obviously applicable to such topics as communications and filters, where the frequency respon behaviors of systems are central to an understanding of their operations. It is, however, in the area of control systems where frequency respon techniques are most fully developed as analytical and design tools. The Nichols chart, for instance, is ud exclusively in the analysis and design of feedback cont
rol systems.
The remaining ctions of this chapter describe veral frequency respon plotting methods. Applications of the methods can be found in other chapters throughout the Handbook .
Linear frequency respon plots are prepared most directly by computing the magnitude and pha of H (j w )and graphing each as a function of frequency (either f or w ), the frequency axis being scaled linearly. As an example, consider the transfer function
Formally, the complex frequency variable s is replaced by the sinusoidal frequency j w and the magnitude and pha found.
The plots of magnitude and pha are shown in Fig. 11.2.
FIGURE 11.1 A single-input/single-output lin-右倾机会主义
ear system.
A Bode diagram consists of plots of the gain and pha of a transfer function, each with respect to logarithmically scaled frequency axes. In addition, the gain of the transfer function is scaled in decibels according to the definition
This definition relates to transfer functions which are ratios of voltages and/or currents. The decibel gain between two powers has a multiplying factor of 10 rather than 20. This method of plotting frequency respon information was popularized by H.W. Bode in the 1930s. There are two main advantages of the Bode approach.The first is that, with it, the gain and pha curves can be easily and accurately drawn. Second, once drawn,features of the curves can be identified both qualitatively and quantitatively with relative ea, even when tho features occur over a wide dynamic range. Digital computers have rendered the first advantage obsolete. Ea of interpretation, however, remains a powerful advantage, and the Bode diagram is today the most common method chon for the display of frequency respon data.
蚌壳肉怎么做好吃
A Bode diagram is drawn by applying a t of simple rules or procedures to a transfer function. The rules relate directly to the t of poles and zeros and/or time constants of the function. Before constructing a Bode diagram, the transfer function is normalized so that each pole or zero term (except tho at s = 0) has a dc gain of one. For instance:
Figure 11.2Linear frequency respon curves of H (j w
).
ΈΈ Έ
ΈH H H j dB dB ==2010log ( )
w
In the last form of the expression, t z =1/w z and t p =1/w p . t p is a time constant of the system and s = –w p is the corresponding natural frequency. Becau it is understood that Bode diagrams are limited to sinusoidal steady-state frequency respon analysis, one can work directly from the transfer function H (s ) rather than resorting to the formalism of making the substitution s = j w.
Bode frequency respon curves (gain and pha) for H (s ) are generated from the individual contributions of the four terms K ¢, s t z + 1, 1/s , and 1/(s t p + 1). As described in the following paragraph, the frequency respon effects of the individual terms are easily drawn. To obtain the overall frequency respon curves for the transfer function, the curves for the individual terms are added together.
The terms ud as the basis for drawing Bode diagrams are found from factoring N (s ) and D (s ), the numerator and denominator polynomials of the transfer function. The factorization results in four standard forms. The are (1) a constant K; (2) a simple s term corresponding to either a zero (if in the numerator) or a pole (if in the denominator) at the origin; (3) a term such as (s t + 1) corresponding to a real valued (nonzero) pole or zero; and (4) a quadratic term with a possible standard form of [(s/w n )2 + (2z /w n )s + 1] corresponding to a pair of complex conjugate poles or zeros. The Bode magnitude and pha curves for the possibilities are displayed in Figs. 11.3–11.5. Note that both decibel magnitude and pha are plotted milogarithmically. The frequency axis is logarithmically scaled so that every tenfold, or decade , change in frequency occurs over an equal distance. The magnitude axis is given in decibels. Customarily, this axis is marked in 20-dB increments.Positive decibel magnitudes correspond to amplifications between input and output that are greater than one (output amplitude larger than input). Negative decibel gains correspond to attenuation between input and output.
Figure 11.3 shows three parate magnitude functions. Curve 1 is trivial; the Bode magnitude of a constant K is simply the decibel-scaled constant 20 log 10 K , shown for an arbitrary value of K = 5 (20 log 10 5 = 13.98).Pha is not shown. However, a constant of K > 0 has a pha contribution of
0° for all frequencies. For K <0, the contribution would be ±180° (Recall that –cos q = cos (q ± 180°). Curve 2 shows the magnitude frequency respon curve for a pole at the origin (1/s ). It is a straight line with a slope of –20 dB/decade. The line pass through 0 dB at w = 0 rad/s. The pha contribution of a simple pole at the origin is a constant –90°, independent of frequency. The effect of a zero at the origin (s ) is shown in Curve 3. It is again a straight line that pass through 0 dB at w = 0 rad/s; however, the slope is +20 dB/decade. The pha contribution of a simple zero at s = 0 is +90°, independent of frequency.
Figure 11.3Bode magnitude functions for (1) K = 5, (2) 1/s , and (3) s
.
Note from Fig. 11.3 and the foregoing discussion that in Bode diagrams the effect of a pole term at a given location is simply the negative of that of a zero term at the same location. This is true for both magnitude and pha curves.
Figure 11.4 shows the magnitude and pha curves for a zero term of the form (s /w z + 1) and a pole term of the form 1/(s /w p + 1). Exact plots of the magnitude and pha curves are shown as dashed lines. Straight line approximations to the curves are shown as solid lines. Note that the straight line approximations are so good that they obscure the exact curves at most frequencies. For this reason, some of the curves in this and later figures have been displaced slightly to enhance clarity. The greatest error between the exact and approximate magnitude curves is ±3 dB. The approximation for pha is always within 7° of the exact curve and usually much clor. The approximations for magnitude consist of two straight lines. The points of interction between the two lines (w = w z for the zero term and w = w p for the pole) are breakpoints of the curves. Breakpoints of Bode gain curves always correspond to locations of poles or zeros in the transfer function.
In Bode analysis complex conjugate poles or zeros are always treated as pairs in the corresponding quadratic form [(s /w n )2 + (2z /w n )s + 1].1 For quadratic terms in stable, minimum pha systems, the damping ratio z (Greek letter zeta) is within the range 0 < z < 1. Quadratic terms cannot always be adequately reprented by straight line approximations. This is especially true for lightly damped systems (small z ). The traditional approach was to draw a preliminary reprentation of the contribution. This consists of a straight line of 0 dB from dc up to the breakpoint at w n followed by a straight line of slope ±40 dB/decade beyond the breakpoint,depending on whether the plot refers to a pair of poles or a pair of zeros. Then, referring to a family of curves as shown in Fig. 11.5, the preliminary reprentation was improved bad on the value of z . The pha contribution of the quadratic term was similarly constructed. Note that Fig. 11.5 prents frequency respon contributions for a quadratic pair of poles. For zeros in the corresponding locations, both the magnitude and pha curves would be negated. Digital computer applications programs render this procedure unnecessary for purpos of constructing frequency respon curves. Knowledge of the technique is still valuable, however, in the qualitative and quantitative interpretation of frequency respon curves. Localized peaking in the gain curve is a reflection of the existence of resonance in a system. The height of such a peak (and the corresponding value of z ) is a direct indication of the degree of resonance.
三七粉最佳吃法Bode diagrams are easily constructed becau, with the exception of lightly damped quadratic terms, each contribution can be reasonably approximated with straight lines. Also, the overall frequency respon curve is found by adding the individual contributions. Two examples follow.
1Several such standard forms are ud. This is the one most commonly encountered in controls applications.
Figure 11.4Bode curves for (1) a simple pole at s = –w p and (2) a simple zero at s = –w z
.
