60dB +135 . 20dB -20dB 给●H259b5w -60dB 135 log【 log [w] Figure 11.4 Bode curves for(1)a simple pole at s=-, and( 2)a simple zero at s=-OD 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 phase curves. Figure 11.4 shows the magnitude and phase curves for a zero term of the form(s/o+ 1)and a pole term of the form 1/(s/@.+ 1). Exact plots of the magnitude and phase curves are shown as dashed lines Straight line approximations to these 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 th n, some of the curves in this and been displaced slightly to enhance clarity. The greatest error between the exact and approximate magnitude curves is +3 dB. The approximation for phase is always within 7 of the exact curve and usually much closer. The approximations for magnitude consist of two straight lines. The points of intersection between these two lines(=@, for the zero term and a=O, 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/o)2+(25/on)s+ 1]. For quadratic terms in stable, minimum phase systems, the damping ratio s reek letter zeta) is within the range 0< 5<1. Quadratic terms cannot always be adequately represented by traight line approximations. This is especially true for lightly damped systems(small 5). The traditional approach was to draw a preliminary representation of the contribution. This consists of a straight line of o dB om dc up to the breakpoint at o, 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 representation was improved based on the value of s. The phase contribution of the quadratic term was similarly constructed. Note that Fig. 11.5 presents frequency response contributions for a quadratic pair of poles. For zeros in the corresponding locations, both the magnitude and hase curves would be negated. Digital computer applications programs render this procedure unnecessary for purposes of constructing frequency response curves. Knowledge of the technique is still valuable, however, in the qualitative and quantitative interpretation of frequency response 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 ralue of 5) is a direct indication of the degree of resonance Bode diagrams are easily constructed because, with the exception of lightly damped quadratic terms, each contribution can be reasonably approximated with straight lines. Also, the overall frequency response curve is found by adding the individual contributions. Two examples follow. sEveral such standard are used. This is the one most commonly encountered in plications. e 2000 by CRC Press LLC© 2000 by CRC Press LLC 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 phase curves. Figure 11.4 shows the magnitude and phase curves for a zero term of the form (s/wz + 1) and a pole term of the form 1/(s/wp + 1). Exact plots of the magnitude and phase curves are shown as dashed lines. Straight line approximations to these 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 phase is always within 7° of the exact curve and usually much closer. The approximations for magnitude consist of two straight lines. The points of intersection between these two lines (w = wz for the zero term and w = wp 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/wn)2 + (2z/wn)s + 1].1 For quadratic terms in stable, minimum phase systems, the damping ratio z (Greek letter zeta) is within the range 0 < z < 1. Quadratic terms cannot always be adequately represented by straight line approximations. This is especially true for lightly damped systems (small z). The traditional approach was to draw a preliminary representation of the contribution. This consists of a straight line of 0 dB from dc up to the breakpoint at wn 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 representation was improved based on the value of z. The phase contribution of the quadratic term was similarly constructed. Note that Fig. 11.5 presents frequency response contributions for a quadratic pair of poles. For zeros in the corresponding locations, both the magnitude and phase curves would be negated. Digital computer applications programs render this procedure unnecessary for purposes of constructing frequency response curves. Knowledge of the technique is still valuable, however, in the qualitative and quantitative interpretation of frequency response 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. Bode diagrams are easily constructed because, with the exception of lightly damped quadratic terms, each contribution can be reasonably approximated with straight lines. Also, the overall frequency response curve is found by adding the individual contributions. Two examples follow. 1 Several such standard forms are used. This is the one most commonly encountered in controls applications. Figure 11.4 Bode curves for (1) a simple pole at s = –wp and (2) a simple zero at s = –wz