Characteristics of Lead and Lag Compensators

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1 Characteritic of Lea an Lag Comenator Lea Comenator ue hae avance at the zero-croing increae high frequency gain an hift c to the right increae banwith limite amount of comenation oible Lag Comenator ue gain reuction to hift the zero-croing to the left introuce hae lag at frequencie below c reuce banwith very large amount of comenation oible Control Sytem for 3r Yr Elec. Eng. ( By Dr. M. Moutafa) Frequency Reone Deign

2 Phae (eg) Magnitue (B) Characteritic of Lea-Lag Lag Comenator Lea-Lag Lag Comenator Characteritic ' ' ' ' B Frequency (ra/ec) Control Sytem for 3r Yr Elec. Eng. ( By Dr. M. Moutafa) ' B B G m c ( ) c =.447 = 5 =.36 m = 4 m = = ' ' lea lag Frequency Reone Deign '.36 = 8 For a contraine imlementation, =

3 Lea-Lag Lag Comenator Imlementation (( = ) C (contraine) e i R R C e o E o ( ) R R R C C R C E ( ) i E o ( ) ( R C ( RC )( RC) R C R C ) R C R C ' ) ' ) Control Sytem for 3r Yr Elec. Eng. ( By Dr. M. Moutafa) ( )( ' ( )( E ( ) where... Frequency Reone Deign i

4 Lea-Lag Lag Comenator Imlementation where R C ' R C ' RC R R C C RC RC Thi reult in a contraine lea-lag lag network, =. Thee three equation are ue to etermine the circuit comonent from the ecifie,, an. An uncontraine lea-lag lag network, can be imlemente a earate lea an lag network buffere by an o-am. Control Sytem for 3r Yr Elec. Eng. ( By Dr. M. Moutafa) Frequency Reone Deign

5 Boe Plot Deign of Lea-Lag Lag Comenator G c ( ) c ' ' ' Five arameter to be choen in the eign. c : amlifier gain, low-frequency gain requirement. : the eire hae avance, m. : lea time contant, to lace m at the eign zero- croing frequency, c. : the require high frequency gain reuction. : lag time contant to lace the lag zero, one ecae below c. Control Sytem for 3r Yr Elec. Eng. ( By Dr. M. Moutafa) Frequency Reone Deign

6 Boe Plot Deign of Lea-Lag Lag. Loo gain requirement: Determine the loo gain require to meet the teay- tate error ecification.. Boe lot: Comenator: Deign Ste Plot the loo tranfer function incluing the require gain increae from te. 3. Deire Phae Avance (calculate ) Select a eirable amount of hae avance m an calculate the correoning. Control Sytem for 3r Yr Elec. Eng. ( By Dr. M. Moutafa) Frequency Reone Deign

7 Boe Plot Deign of Lea-Lag Lag Comenator: Deign Ste 4. Require Gain Reuction (calculate ) Determine the frequency, c where the eign hae margin woul be achieve, accounting for the hae avance, m an the 5 of the tail of the lag. muncom uncom mec ec m + 5 Determine the gain reuction require to make c the zero-croing frequency an calculate. gain reuction B = / B + ½ B Control Sytem for 3r Yr Elec. Eng. ( By Dr. M. Moutafa) Frequency Reone Deign

8 Boe Plot Deign of Lea-Lag Lag 5. Require an : Select to lace the m at c ; c = Select uch that the zero, / i one ecae below c ; ' ' c 6. Require c : Select c to give the require low-frequency gain increae. Imlement: Comenator: Deign Ste G c ( ) c ' ' ' Control Sytem for 3r Yr Elec. Eng. ( By Dr. M. Moutafa) Frequency Reone Deign

9 Boe Plot Deign of Lea-Lag Lag Comenator: ( = ) For the contraine eign ( = ) te 3 an 4 may require ome iteration imilar to the lea eign. In thi cae the gain reuction i / B + ½ B = ½ B c, m m ½ B Boe lot ec. + 5 in m gain reuction Control Sytem for 3r Yr Elec. Eng. ( By Dr. M. Moutafa) Frequency Reone Deign

10 Lea-Lag Lag Deign Examle The lant G ( ) Secification: teay-tate tate error <5% for a ram inut. Phae margin >45 The firt art of the eign i the ame a the hae-lea eign. Control Sytem for 3r Yr Elec. Eng. ( By Dr. M. Moutafa) Steay-tate error: E( ) G e lim E( ) 4 R( ) ( ) /.55 Draw the Boe lot of G 4 ( ) Frequency Reone Deign

11 Lea-Lag Lag Deign Examle Phae (eg) Magnitue (B) Lea-Lag Lag Deign (uncontraine) 9 B Frequency (ra/ec) Control Sytem for 3r Yr Elec. Eng. ( By Dr. M. Moutafa) Chooe m =, then =., ½ B = 3B. muncom uncom + 5 = 3 c = 3.5 ra/ Gain reuction = -9B = / B +½ B / B = -B an Frequency Reone Deign

12 Lea-Lag Lag Deign Examle Lea time contant c = 3.5 = =. Lag time contant ' c =.35 ' G c Controller gain c = 4 Final Controller ( ) Control Sytem for 3r Yr Elec. Eng. ( By Dr. M. Moutafa) Frequency Reone Deign

13 Lea-Lag Lag Deign Examle Phae (eg) Magnitue (B) m = 45 c = 3.5 ra/ comenate uncomenate Frequency (ra/ec) Control Sytem for 3r Yr Elec. Eng. ( By Dr. M. Moutafa) Frequency Reone Deign

14 Lea-Lag Lag Deign Examle Contraine ( = ) Phae (eg) Magnitue (B) Lea-Lag Lag Deign (uncontraine) 9 B Frequency (ra/ec) Control Sytem for 3r Yr Elec. Eng. ( By Dr. M. Moutafa) For m =, =. Chooe m = 7, then =.7 The gain reuction i ½ B = 4.3 B The new zero-croing i c = 4.7 ra/ The hae margin i = 47 Frequency Reone Deign

15 Lea-Lag Lag Deign Examle Contraine ( = ) Lea time contant c = 4.7 = =.9.7 Lag time contant ' c =.47 ' G c ( ) Controller gain c = 4 Final Controller Control Sytem for 3r Yr Elec. Eng. ( By Dr. M. Moutafa) Frequency Reone Deign

16 Lea-Lag Lag Deign Examle Contraine ( = ) Lea-Lag Lag Contraine Deign 6 Phae (eg) Magnitue (B) comenate uncomenate Frequency (ra/ec) Control Sytem for 3r Yr Elec. Eng. ( By Dr. M. Moutafa) Frequency Reone Deign

17 Cloe-Loo Ste Reone Ste Reone. Amlitue lea comenator lag comenator lea-lag comenator (contraine) lea-lag comenator (uncontraine) Time (ec.) Control Sytem for 3r Yr Elec. Eng. ( By Dr. M. Moutafa) Frequency Reone Deign

18 Comenator Deign Examle G The lant 5 ( ) (.)( 5 Secification: 5) teay-tate tate error <% for a te inut. Phae margin >6 e Steay-tate error: E( ) G lim ( ) E 4.9 R( ) ( ) G. Draw the Boe lot of G 45 ( ) (.)( 5 5) ( ) Control Sytem for 3r Yr Elec. Eng. ( By Dr. M. Moutafa) Frequency Reone Deign

19 Comenator Deign Examle Phae (eg) Magnitue (B) Boe Diagram for Examle 3B Frequency (ra/ec) -5 m = 7 m(uncom uncom) = 7 at c = 6.7 ra/ A lea comenator woul nee to rovie > 87 hae avance. A lag comentor nee to introuce a high frequency gain reuction of 3B. Control Sytem for 3r Yr Elec. Eng. ( By Dr. M. Moutafa) Frequency Reone Deign

20 Lag Comenator Deign Examle The gain reuction at c =.3 ra/ to make thi the zero croing i 3B. The hae at thi frequency i 5. ( ) / B = 3B = 4.6 The time contant i / = / c =.3 = 4.35 The gain i c = 4.9 The final controller i G c ( ) Control Sytem for 3r Yr Elec. Eng. ( By Dr. M. Moutafa) Frequency Reone Deign

21 Lag Comenator Deign Examle Lag Deign for Examle Phae (eg) Magnitue (B) B m = Frequency (ra/ec) Control Sytem for 3r Yr Elec. Eng. ( By Dr. M. Moutafa) Frequency Reone Deign

22 Lea-Lag Lag Comenator Deign Examle Phae (eg) Magnitue (B) Boe Diagram for Examle 5.5B m = 7 Frequency (ra/ec) Control Sytem for 3r Yr Elec. Eng. ( By Dr. M. Moutafa) Chooe m = 65, then =.3,, an ½ B = 3.B B. muncom uncom + 5 = c = 5. ra/ Gain reuction = 5.5B = / B + ½ B / B = 8.6B an Frequency Reone Deign

23 Lea-Lag Lag Comenator Deign Examle Lea time contant c = 5. = =.435 Lag time contant ' c =.5 '.3 G c ( ) Controller gain c = 4.9 Final Controller Control Sytem for 3r Yr Elec. Eng. ( By Dr. M. Moutafa) Frequency Reone Deign

24 Lea-Lag Lag Comenator Deign Examle Phae (eg) Magnitue (B) Lea-Lag Lag Deign for Examle comenate uncomenate m = Frequency (ra/ec) Control Sytem for 3r Yr Elec. Eng. ( By Dr. M. Moutafa) Frequency Reone Deign

25 Deign Comarion Examle Boe Diagram for Examle Phae (eg) Magnitue (B) uncomenate lea-lag eign lag eign Frequency (ra/ec) Control Sytem for 3r Yr Elec. Eng. ( By Dr. M. Moutafa) Frequency Reone Deign

26 Deign Comarion Examle Ste Reone for Examle Amlitue uncomenate lea-lag eign lag eign Time (ec.) Control Sytem for 3r Yr Elec. Eng. ( By Dr. M. Moutafa) Frequency Reone Deign

27 Lea-Lea Comenation Deign One of the limitation of lea comenator i that the hae avance i limite to about 7-8. To overcome thi limit, conier cacaing two lea comenator. The eign follow a with a ingle lea excet the hae avance i ouble,, to m an the gain increae at the new zero croing i alo ouble, to B. Control Sytem for 3r Yr Elec. Eng. ( By Dr. M. Moutafa) Frequency Reone Deign

28 Lea-Lea Deign for Examle Phae (eg) Magnitue (B) Lea-Lea Deign for Examle 4-49 Frequency (ra/ec) Control Sytem for 3r Yr Elec. Eng. ( By Dr. M. Moutafa) Initial m = 7, o chooe m = =, m = 55 Then = The gain increae i B = B. The new reicte c = 4 ra/ an m(uncom) = 69 m = 69 + = 4 Frequency Reone Deign

29 Lea-Lea Deign for Examle Secon iteration: chooe m = = 4, m = 7 Then = 3 The gain increae i B =3 B. The new reicte c = ra/ an m(uncom) = 75 m = = 65 O Control Sytem for 3r Yr Elec. Eng. ( By Dr. M. Moutafa) Calculate m c 3 Calculate the gain c c The final comentor G c 4.9 Two lea network in erie. c.83 ( ) Frequency Reone Deign 7.8

30 Lea-Lea Deign for Examle Lea-Lea Deign for Examle lea-lea comenate Phae (eg) Magnitue (B) uncomenate m = 65-5 Frequency (ra/ec) Control Sytem for 3r Yr Elec. Eng. ( By Dr. M. Moutafa) Frequency Reone Deign

31 Cloe-loo loo Sytem Ste Reone for Examle Ste Reone Comarion.8 Amlitue.6.4 lag eign lea-lag eign lea-lea eign Time (ec.) Control Sytem for 3r Yr Elec. Eng. ( By Dr. M. Moutafa) Frequency Reone Deign

32 PID Control Deign In The Frequency Domain Treat the PD controller a a ecial cae of a lea comenator, the PI a a ecial cae of the lag comenator an the PID controller a a ecial cae of the lea-lag lag comenator. Deign with lightly moifie frequency reone roceure an then fine tune on- line. Control Sytem for 3r Yr Elec. Eng. ( By Dr. M. Moutafa) Frequency Reone Deign

33 PD Frequency Domain Deign PD Controller G c ( ) ( ( ) where Thi i a lea comenator with the ole at. ) The low frequency gain i. The gain an hae angle at frequency, G G c c ( j ) ( j ) ( j ) G c ( j ) tan ( ) Control Sytem for 3r Yr Elec. Eng. ( By Dr. M. Moutafa) Frequency Reone Deign

34 PD Frequency Domain Deign PD Controller Reference Plot : =, = Phae (eg) Magnitue (B) 4 3 B B ieal / two ecae above / one ecae above / / / - Frequency (ra/ec) Control Sytem for 3r Yr Elec. Eng. ( By Dr. M. Moutafa) PD controller with a high frequency cut-off. G c ( ) where Frequency Reone Deign ( )

35 PD Frequency Domain Deign The eign roce i imilar to the lea comenator eign. Chooe the hae avance an etermine from tan ( ) Determine the gain increae from G c ( j ) Control Sytem for 3r Yr Elec. Eng. ( By Dr. M. Moutafa) Check the new zero croing frequency, c an the correoning hae margin (iterate). Determine from tan ( c ) Determine from the low frequency gain requirement. / = / Frequency Reone Deign

36 PI Frequency Domain Deign PI Controller G c ( ) i ( Thi i a lag comenator with the ole at = an the zero at i i i ) i i ( i ) Note that the PI comenator introuce a ole at the origin which increae the ytem tye by. The high frequency gain. Control Sytem for 3r Yr Elec. Eng. ( By Dr. M. Moutafa) Frequency Reone Deign

37 PI Frequency Domain Deign Phae (eg) Magnitue (B) PI Controller : i =, = / - - Frequency (ra/ec) B Control Sytem for 3r Yr Elec. Eng. ( By Dr. M. Moutafa) The eign i imilar to the lag comenator. Select to give the require c that will atify the m + 5. Select i uch that the zero i one ecae below c. i i c Frequency Reone Deign

38 Frequency Reone Deign Frequency Reone Deign Control Sytem for 3r Yr Elec. Eng. ( By Dr. M. Moutafa) PID Frequency Domain Deign PID Frequency Domain Deign PID Controller PID Controller ) ( ) )( )( ( i ) ( ) ) ( ) ( ( ) ( ) ( G i i i i i i c i i ) ( where, where,

39 PID Frequency Domain Deign Phae (eg) Magnitue (B) PID Controller: =, =, i = / / B Frequency (ra/ec) Control Sytem for 3r Yr Elec. Eng. ( By Dr. M. Moutafa) The PID controller i imilar to a lea-lag lag comenator, with the lag ole at zero an the lea ole at /. rereent a frequency relative to / for a hae avance of 7. Frequency Reone Deign

40 PID Frequency Domain Deign For the PID eign, firt chooe a hae avance an calculate from tan ( ) Then etermine the frequency, c where muncom uncom mec ec + 5. Calculate from c an. Select / one ecae below /. Control Sytem for 3r Yr Elec. Eng. ( By Dr. M. Moutafa) Select / two ecae above /. Plot the comenate ytem with i = an etermine the gain reuction require to make c the zero croing frequency. Frequency Reone Deign

41 Phae (eg) Magnitue (B) PID Frequency Domain Deign: Boe Diagram for Examle Examle Frequency (ra/ec) Control Sytem for 3r Yr Elec. Eng. ( By Dr. M. Moutafa) Chooe = 7. From the PID characteritic, thi occur at = tan(7 ) =.75. Fin the frequency where, muncom uncom + 5 = 5 c = 5. ra/ Fin from c =.75 =.59 Frequency Reone Deign

42 PID Frequency Domain Deign: Examle Chooe / one ecae below /. / = / = 5.9. Chooe / two ecae above /. / = / =.59. Control Sytem for 3r Yr Elec. Eng. ( By Dr. M. Moutafa) Plot the ytem with the comenator G c ( ) i ( 5.9 )(.59 ) (.5959) where i =. Frequency Reone Deign

43 Phae (eg) Magnitue (B) PID Frequency Domain Deign: PID Comenate Sytem uncomenate comenate, i = Frequency (ra/ec) Examle comenate, i = 8.5B - Control Sytem for 3r Yr Elec. Eng. ( By Dr. M. Moutafa) Check the zero croing frequency for the hae margin of 6. c = 5. ra/ Fin the require gain reuction to make thi the zero croing frequency. 8.5B Set i = the gain reuction. i =.376 Frequency Reone Deign

44 PID Frequency Domain Deign: Ste Reone for Examle.9.8 Ste Reone for PID an Lea-Lag Lag Amlitue PID Lea-Lag Time (ec.) Control Sytem for 3r Yr Elec. Eng. ( By Dr. M. Moutafa) Frequency Reone Deign

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