5. First-order Derivative ElementA(0) = /1+0?T2L(0)= 20lg /1+ @°T2p(o) = arctgoTp() = arctgoT30Im(ap) /()70=02020dB/dec+202实际幅频特性0渐近线08Re0=0(045arctgot转折频率00.1/T1/T10/T(rad/sec)0上海交通大学SHANGHAIIIAOTONGUNIVERSITY
5. First-order Derivative Element 2 2 ( ) 1 ( ) arctg A T T = + = 2 2 ( ) 20lg 1 ( ) arctg L T T = + =
6. Second order oscillation element (Important)1G(j@) =L() = -20 lg /(1- / @,)2 + 4z2( / @,)0?0+ j252OnOnFor w<<wr, L(o)~0250/0p(の)-arctgFor 0>>n, L(o)=-40lgo/on1 -(0 / 0,(dB)j0=00040dB/dec10=0 =0.2-0.80.1/on-0.5-2040dB/dec()-10.1w/on-1.5010.51.5-0.5-18Q上海交通大学NyquistDiagramSHANGHAI IIAO TONGUNIVERSITYBodeDiagram
6. Second order oscillation element (Important) n n j G j 1 2 1 ( ) 2 2 + = -0.5 0 0.5 1 1.5 -1.5 -1 -0.5 0 j ζ=0.2—0.8 Nyquist Diagram ω=0 ω=∞ 2 2 2 2 2 ( ) 20lg (1 / ) 4 ( / ) L = n + n 2 1 ( / ) 2 / ( ) n n arctg = Bode Diagram ω/ωn 0.1 (dB) -20 40dB/dec -40dB/dec ( o ) -180 0.1 ω/ωn • For ω<<ωn , L(ω)≈0 • For ω>>ωn , L(ω)≈-40lgω/ωn
6.3.1 Partition of Open-loop System Into Typical ElementsAssume that the open-loop transfer function contains severaltypical elementsG(s) =G(s)G2(s)...G,(s)Frequency response of open-loop system is shown as follows:G(jo) =G(jo)G,(j@)..·G,(jo)A(0)ej0(0) = A(0)ej0(0) A,(0)ej92(0)...A,(0)e19.(0)Magnitude and Phase of open-loop systemA(0)= A()· A(0)..... A,(0)Φ(の)=Φ (の) +Φ,(の) +.. +Φ, (@)Log-magnitude and phase are as follows:L(0) = L,(0)+ L,(0)+...+ L, (0)上海交通大学SHANGHAIIAOTONGUNIVERSITY(の) =P()+P2(の) +..: +,()
Assume that the open-loop transfer function contains several typical elements Frequency response of open-loop system is shown as follows: 1 2 ( ) ( ) ( ) ( ) G s G s G s G s = n 1 2 ( ) ( ) ( ) ( ) G j G j G j G j = n 1 2 ( ) ( ) ( ) ( ) 1 2 ( ) ( ) ( ) ( ) n j j j j A e A e A e A e n = 6.3.1 Partition of Open-loop System Into Typical Elements Magnitude and Phase of open-loop system Log-magnitude and phase are as follows: 1 2 1 2 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) n n L L L L = + + + = + + + 1 2 1 2 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) n n A A A A = = + + +
6.3.2 Sketching Method of Nyquist DiagramRequirement of sketching :.Don't need to be accurate. Only need to determine the curve shape and keypoints (e.g. intercept with axises). It is based on the partition of open-loop system intotypical elements and relevant magnitude-phasecharacteristic diagrams上海交通大学SHANGHAIIIAOTONGUNIVERSITY
Requirement of sketching： • Don’t need to be accurate • Only need to determine the curve shape and key points (e.g. intercept with axises) • It is based on the partition of open-loop system into typical elements and relevant magnitude-phase characteristic diagrams. 6.3.2 Sketching Method of Nyquist Diagram
1. Nyquist Diagram of Typical Open-loop System(1) Open-loop transfer function without integrationand derivative elementsnKnKG(jo)=IIG(s)=IITj@+1T,s +1i=li=lW=0n=00mG(j0)=G(jo)=KZ0°0-90°0=08KReG(j0)=G(joo)=KZ0°0Z-180°n=lRemark:n=2Containsninertialelementsandn=3proportionalelement(K>0)7=4Nyquist diagram starts from positivereal axis and goes clockwise for n上海交通大学quadrantの=0→00SHANGHAI JIAO TONGUNIVERSITY
1. Nyquist Diagram of Typical Open-loop System (1) Open-loop transfer function without integration and derivative elements 1 ( ) 1 n i i K G s = T s = + n ω=0 ω=∞ G(j0)= K∠0° G(j∞)= 0∠－90° G(j0)= K∠0° G(j∞)= 0∠－180° 1 ( ) 1 n i i K G j T j = = + Remark： • Contains n inertial elements and proportional element (K>0) • Nyquist diagram starts from positive real axis and goes clockwise for n quadrantω=0→∞