Lecture 3: Frequecy Repoe of Secod-Order Sytem UHTXHQF\ 5HVSRQVH RI 6HFRQGUGHU 6\VWHPV A geeral ecod-order ytem ha a trafer fuctio of the form b + b + b H (. (9.4 a + a + a It ca be table, utable, caual or ot, depedig o the ig of the coefficiet ad the pecified ROC. Let' retrict our attetio to caual, table LTI ecod-order ytem of thi type. It ca be how that a eceary coditio for tability i that the coefficiet a i be all poitive, or all egative (thi i alo true for higher order ytem. Let' alo aume that b b, i.e. we baically have a lowpa ytem. Uder thee coditio the trafer fuctio ca be expreed a H ( A + ς +. (9.5 where ς i the dampig ratio i the udamped atural frequecy of the ecod-order ytem. Sytem uch a the ma-prig-damper ytem or a lowpa ecod-order filter ca be modeled by thi trafer fuctio. Example: Coider the ecod-order trafer fuctio H ( 6 9 (9.6 + 3+ 9 It udamped atural frequecy i 3, ad it dampig ratio i ς 3 3. 77 (9.7 3 Sice the dampig ratio i le tha oe, the two pole are complex. The pole are 3 3 p j j 3 3 ς + ς + + j (9.8 3 3 p ς j ς j There are three cae of iteret for the dampig ratio that lead to differet pole patter ad frequecy repoe type.
&DVH ς > I thi cae, the ytem i aid to be overdamped. The tep repoe doe't exhibit ay rigig. The two pole are real, egative ad ditict: p ς + ς, p ς ς. The ecod-order ytem ca be ee a a cacade of two tadard firt-order ytem (lag. H ( A + ς + A pp A + + + + p p p p The Bode plot of H( j A j ca the be ketched uig the techique j p + p + preeted i the previou lecture for ytem with real pole ad zero. (9.9 &DVH ς I thi cae, the ytem i aid to be critically damped. The two pole are egative ad real, but they are the ame. We ay that it a repeated pole; p ς + j ς ς p. I thi ituatio, the ecod-order ytem ca alo be ee a a cacade of two firt-order trafer fuctio havig the ame pole. &DVH ς < H ( A (9.3 ( p + I thi cae, the ytem i aid to be uderdamped. The tep repoe exhibit ome rigig, although it really become viible oly for ς <. 77. The two pole are ditict, complex cojugate of each other: p ς + j ς, p ς j ς. The magitude ad phae of A A H( j j ( j ς ( j ( ς + + + ( j + (9.3 are give by: log H( j log A log + 4ς H( j arcta % &K 'K ς 4 9 ( K *K %&' (* 4 9, (9.3. (9.33 Notice that the deomiator i Equatio (9.3 wa writte i uch a way that it dc gai i. The break frequecy i imply the udamped atural frequecy. At thi frequecy, the magitude i
log H( j log ς ;@. (9.34 For example, with ς. ad A, the magitude at i 3.98dB. Note that thi i ot the imum of the magitude a it occur at the reoat frequecy ς which i cloe to for low dampig ratio. At the reoat frequecy, the magitude of the peak reoace i give by ς log H( j log + 4ς 4 log 4ς + 4ς ( ς log 4ς ( ς log ς ς ( ( ς %&' 4 9 (* < A < A J L > C db. ad thu for our example log H( j log.. 4 6. The Bode plot for the cae ς < ca be ketched uig the aymptote, but at the price of a icreaigly large approximatio error aroud that icreae a the dampig ratio decreae. The roll-off rate pat the break frequecy i -4dB/decade. The phae tart at ad ted to π at high frequecie. log H( j (log -4-8 (rd π/ π H( j. (log 3
The Bode plot approximatio uig the aymptote doe ot covey the iformatio of the reoace i the ytem caued by the complex pole. That i, the dampig ratio wa ot ued to draw the aymptote for the magitude ad the phae plot. The peak reoace i the magitude produced by differet value of ς < are how i Figure 6.3 i the textbook. Thi figure alo how that the phae drop aroud become teeper a the dampig ratio i decreaed. 4XDOLW\ 4 I the field of commuicatio, the uderdamped ecod-order filter ha played a importat role a a imple frequecy-elective badpa filter. Whe the dampig ratio i very low, the filter become highly elective due to it high peak reoace at. The quality Q of the filter i defied a Q ς. (9.35 The higher the quality, the more elective the filter i. To upport thi claim, the 3dB badwidth (frequecy bad betwee the two frequecie where the magitude i 3dB lower tha log H( j of the badpa (ot really coidered a lowpa for high Q ecod-order filter ca be how to be -4-8 log H( j ς Q. (9.36 ς ς (log D[LPDO ODWQHVV DQG %XWWHUZRUWK LOWHUV I the traitio from a ecod-order ytem without reoace ad oe that tart to exhibit a peak that i higher tha the dc gai, there mut be a optimal dampig ratio ς for which the magitude tay flat over the widet poible badwidth before rollig off. It tur out that thi occur for ς <. 77. For thi dampig ratio, the real part ad imagiary part of the two pole all have the ame abolute value, ad the pole ca be expreed a 4
4 4 p e, p e. (9.37 3π j We recogize the pole of a lowpa ecod-order Butterworth filter with cutoff at. Thu, a Butterworth filter i optimized to be imally flat. 3π j -4 ς. 77 ς log H( j (log -8 5