LEAP FROG TECHNIQUE. Operational Simulation of LC Ladder Filters ECEN 622 (ESS) TAMU-AMSC

Transcription

1 LEAP FOG TEHNQUE Opeatnal Smulatn f L Ladde Fltes L pttype lw senstvty One fm f ths technque s called Leapf Technque Fundamental Buldn Blcks ae - nteats - Secnd-de ealzatns Fltes cnsdeed - LP - BP - HP - BE - jω Zes EEN 6 (ESS) TAMU-AMS

2 Ladde Fltes Ladde Netwks Elements cnnected n sees and n paallel Zes ae easly ecnzed: Zseesnfnty, Zshuntze Pblem: Hw t desn actve ladde fltes? Hw t desn nducts?

3 By means f a smple example t s llustated hw t mplement an actve flte based n the equatn f a passve L pttype. v nd v v Z Z v v (v ( (v ( v v )Z ) 0)Z )

4 Next we match the equatns ceffcents wth the mplementatns. -v v - Z x x dummy F dummy ne can make v v (v ( (v ( v v )Z ) 0)Z ) v x v x ( v v ) Z x / Tw key tansfmatns:. f needed cnvete cuent equatns t vltae equatns by multplyn by a dummy (atfcal) esst value f.. Expess the elatn f cuent and vltae f equatns always as nteat. The nteat s the basc buldn blck.

5 Geneal Pncples nte Ptn f a Geneal L Ladde Netwk nte cmpnents ae eactve elements nly. s 0 s 0 and L 0 (fnte) Banch vltaes ae vltae and cuent Hw t select the ppe STATE vaable? a) b) c) v c L L v L v dv dt c d dt L vc (t) dt v L v(t) dt L L dv d (t) L (t)dt v(t) dt dt dt L L

6 Systematc appach by wtn vltae (KL) and cuent (KL) equatns [ [ Z [ [ Z [ - mmttance Functns - ltae Tansfe Functns, cnvet t functns.e., k k

7 Suce a) b) Temnatns f L Ladde Fltes n n Z [ n Z [ n (n ) [ Z Z

8 Lad temnatn a) b) n Zn n n n Z ut n n n n[(n n (n Zn n n ut Z n ) / Z n ut ut ut n ut n ut n [n n n n n n ut ) ut Lssy nteat n ut

9 The appach t map a passve L pttype s t pck the state-vaables whch can be expessed as nteats, snce nteats ae the basc buldn blck. By applyn KL, KL, KL,, as many tmes as the de f the flte, ne can wte the state-equatns that can be mplemented by actve fltes. an example s shwn belw: KL whee KL KL S n A Typcal Passve L Flte L ( ) n S S, s an abtay value ( ) SL S L ( 5a) ( 5b) ( 6) ( 7)

10 S S S S 5 S S SL L S S S Snal Flw Gaph f L Pttype f F. Shwn n pevus pae.

11 Lw-Pass Ladde Fltes (Zes at nfnty) (All Ples) Example: A Ffth-Ode LP Flte 5 State aables [,,,, 5 5 State Equatns 5 n n L n n L n KL-KL-KL-KL-KL * uent Anals n ( n n [ n sl n L 5 n n n (Sequence) sl ) n ut () 0

12 0 s n [ s n 0 sl n [ sl n 0 s 5 n [ s 5 n 0 sl n n [ sl 5 6n 5n 5 6n 5 6n 5 ut [ sl 6n ut 6n 5n 6n ut But () () () (5)

13 n k Actve Buldn Blcks τ j/ j τ j/ j τ j/ j j j j Thus, eq() s mplemented as fllws n n sl [ n n n L n L n L n n n

14 ut 5 s [ n n n L L sl 5 5 n n L L 5 n 5 n / / n [ n s [ n 5 n 6n sl 5n [ / / 6n 5 6 n 6 n ut 6n

15 n n Lw-Pass 5 th -Ode Actve Leapf Flte

16 6n / n n / SL n S n 5 SL n S n SL 5n / n n n / SL / n / n S n SL n S n S / L5n / 6n 6n

17 OTA- mplementatn sl [ n n n n j j mj mj j j Thus, EQ () s mplemented as n m m m m L L n n n 5 m m 5 EQ () m L n EQ () m5 n m EQ () m n 5 m 6 m7 5 5 EQ m6 5 m7 5 (5) L L 5n n 5n n

18 n m m m m m 5 m 6 5 m7 5 5 th -Ode OTA- Leapf mplementatn

19 ltae Scaln f OTA- All Ple Leapf Fltes. a x b m m m x m ( c d ) / s c d a x / k b m k m m / k k m m c d hann x k x wthut chann a, b,, c and d

20 n m m m6 m m m 5 5 m 7

21 n n n SL n S n SL n S n SL 5n n System Level epesentatn: 5 th -Ode LP Leapf Flte

22 n n n k k k k SL n k k SALNG MAXMUM DNAM S n k SL n k k k S n k ANGE k SL 5n 5 n n L s max max k 5n L T(s) s 5n 6n 6n ; ; max max max max k k ; max max k

23 ase n ase uent Level Scaln Technque m epesents nput esstance f a cuent-m f needs t be cnstant, then must be deceased by k. n summay k ptn ptn m m / s m s ptn k m s ptn k k m / k & & k / k / k / k epesents nput mpedance f next blck m m n m s m s

24 L Ladde Fltes wth Fnte Zes Z Z Z Z K, K, K,,,,, K,

25 Dependent Suces t eplace the Bdn Admttances n a smla f KL f nde and slvn f esults at KL [,, ) ( K ( ) (,,,, K K,, ) / ( K NODE

26 L Ladde Smulatn. Stat fm the desed L LP pttype netwk satsfyn SPES.. Elmnate shunt elements n the sees banches sees element n the shunt banches.. Nmalze and tansfm the netwk f the ealzatn s nt lw-pass.. Select the state vaables such that f (x, 5. Slve f each state vaable n tems f the the states and utput vaable ncludn the ppe scaln. 6. Synthesze each equatn wth ts cespndn buldn blck. dx dt µ s)

27 Example: Lw-Pass Wth Fnte jω Zes n L n L n K K K K n n 5n 5 5 n n n n n n n n n n 5n n n 5n n n n n n n Ln K K 5 5 n n K n L n 5n K 5 5n n n ut ut

28 5n n 5 n n n n 5 n n 5n n 5n n n n n n n n K K K K

29 n K Sn n n K n [ SLn [ SL n () () ( )

30 S S [ K55 K n [ K55 K n K 5 5 ˆ 6 K 5 () ( ) [ ut () SL n ut L L n n / /

31 ut ut 6 K5 S5n 6n (5) K ut 9 ut

32 K n ˆ n K 5 K Actve Ffth-Ode Ellptc Flte

33 OTA- Ellptc mplementatn n n K Sn n n m m m K n () n K n n m K [ () SL n m m m L n

34 [ K55 K sn () 5 m m5 m [ ut () sl n ut m6 m6 m 6 L n /

35 ut ut 5 K5 s5n n m ut (5) 5 5 m7 5 K 5 5n ut mx

36 n 5 5 ut Ffth-Ode OTA- Ellptc Flte

37 Smulatn Flatn apacts (Ellptc Fltes) v L vut (v)s (vut)s 5-5 -v - L/ v - - vut - -

38 Fnte jω Zes: An Altenatve Appach n n n n n L n n L n 5 5n n s n n sn ( ) () n n ; s n n sn ( ) ( ) n n

39 sl n s n sn ( ) sn ( 5 ) L s n 5 s 5 5n 5 sn ( 5 ) 6n () () () (5) mpnent alculatns f an OTA- mplementatn n n n m m 7 L n n m5 m m 6 n 6 Ln m7 m Nte that f 6 n then the OTA # 7 can be elmnated and the utput f OTA # 6 can be smply cnnected t ts neatve temnal. 6n