6.4.Bode's Gain and Phase Relation 89 4.5 3.5 三15 0 2 -1 Figure 6.9:The Function In coth- 2 slowly near wo and small if it attenuates rapidly near wo.For example,suppose the slope dln()i.e(-20dB per decade),in the neighborhood of wo,then it dy is reasonable to expect -l×65.3,if the slope ofL=-lfor3≤“≤3 ∠L(jwo) -l×75.30,if the slope of L=-lfor吉≤÷≤5 -l×82.70,if the sope ofL=-l for≤品≤10 The behavior of L(jw)is particularly important near the crossover frequency we where L(jwe)=1 since +L(jwe)is the phase margin of the feedback system,and further the return difference is given by +L(jwe)=+L(jwe)=2 sin+L(jw) which must not be too small for good stability robustness.If +L(jwe)is forced to be very small by rapid gain attenuation,the feedback sy stem will amplify disturbances and exhibit little uncertainty tolerance at and near w.Since it is generally required that the loop transfer function L roll off as fast as possible in the high frequency range, it is reasonable to expect that L(jwe)is at most -ex 90 if the slope of L(jw)is- near we.Thus it is important to keep the slope of L near we not much smaller than -1 for a reasonably wide range of frequencies in order to guarantee some reasonable performance.The conflict between attenuation rate and loop quality near crossover is thus clearly evident. Bodes Gain and Phase Relation ￾ −3 −2 −1 0 1 2 3 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 ν ln coth |ν |/ 2 Figure  The Function ln coth jj  vs  slowly near ￾ and small if it attenuates rapidly near ￾ For example suppose the slope d ln jL jj d   ie  dB per decade in the neighborhood of ￾ then it is reasonable to expect ￾L j￾  ￾  if the slope of L  for        ￾  if the slope of L  for        ￾￾  if the slope of L  for  ￾      The behavior of ￾L j is particularly important near the crossover frequency c where jL jcj   since  ￾L jc is the phase margin of the feedback system and further the return dierence is given by j  L jcj  j  L jcj       sin  ￾L jc      which must not be too small for good stability robustness If  ￾L jc is forced to be very small by rapid gain attenuation the feedback system will amplify disturbances and exhibit little uncertainty tolerance at and near c Since it is generally required that the loop transfer function L roll o as fast as possible in the high frequency range it is reasonable to expect that ￾L jc is at most ￾ if the slope of L j is near c Thus it is important to keep the slope of L near c not much smaller than  for a reasonably wide range of frequencies in order to guarantee some reasonable performance The conict between attenuation rate and loop quality near crossover is thus clearly evident