ECE 6(ESS) Follow The Leader Architecture 6 th Order Elliptic andpa Filter A numerical example
Objective To deign a 6th order bandpa elliptic filter uing the Follow-the-Leader (FLF) architecture. The pecification are: Specification Value Order 6 Paband.9MHz to.mhz Paband Ripple 0.d Attenuation 0d at 0.6MHz
Realization of High Order Tranfer Function (N>) Cacade of nd order ection (one t order ection if N i odd) Leapfrog Follow-The-Leader Cacade FLF Leap-Frog Senitivity High Medium Low Eay to Tune Medium Eay Difficult
Primary Reonator loc -F -F V in P P P V out It provide compenatory internal interaction between the different filter ection through coupling the biquad building bloc.
Deign Quetion: How do we obtain the feedbac coefficient F and F? How do we determine the pecification for each biquadratic ection? Q ω o Gain
Deign Procedure Start with the Lowpa equivalent ytem. -F -F V in K 0 K K K V out ad New: Elliptic Filter need finite zero in their lowpa equivalent tranfer function.
Implementation of Finite Zero by the Summation Technique -F -F V in K 0 V 0 K V V V K K 0 V out
Deign Procedure Let for now K K K and () Applying Maon rule, the complete tranfer function i given by: () T ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( 0 0 F T F T T T T K V V H in out ( ) ( ) ( ) ( ) ( ) 0 0 F F K
Deign Procedure From Matlab or Fieta, we can obtain the lowpa prototype tranfer function of the deired 6 th Order Elliptic Filter: () Equating the denominator of equation () and (), we obtain the following et of equation from which we can olve for, F, and F. (4) Alo from equation () and () (5) 0 0 0 0 ) ( a a a b b b b b a m H ( ) 0 a F F a F a 0 0 0 b a K m
Deign Procedure To obtain the ummation coefficient, we equate the numerator of equation () and (). If H() i a bandpa: 0 b 0. Then, we obtain the following et of equation from which we can determine, and. (6) ( ) 0 0 0 b b b b
Deigning for Maximum Dynamic Range We need to ditribute the gain of each ection T(), i.e. K, K and K uch that we maximize the Dynamic Range. The maximum dynamic range will be obtained if the ignal pectra at the output of all ection have equal maxima, i.e. V out,max V,max V,max V,max V 0,max
Maximizing Dynamic Range To mae V,max V out,max where K0 K0 K0q (7) V q V out,max,max prior prior to to caling caling (8) We alo need to adjut the ummation coefficient to eep the overall gain: i (9) i i q If we aume a flat pectrum for the input, i.e. Vin(ω) V out, max Max H ( ω) V,max Max H( ω) (0)
Maximizing Dynamic Range Where () To obtain K, K and K : () 0 0 ) ( ) ( ) ( a a a K V V H in,,. ) ( ) ( 4 i for H Max H Max K i i i ω ω ω ω ω ω
Deign Procedure The feedbac coefficient need to be readjuted to eep the ame loop gain: () The ummation coefficient alo need to be readjuted again: (4) K K F F F K K K F F F q q K q K K
Summary of Deign Procedure Obtain from Matlab or Fieta the lowpa prototype for the deired filter. From equation (4), (5) and (6), obtain K 0, the feedbac and the ummation coefficient. To maximize dynamic range, obtain q uing equation (8). Recalculate K 0 uing equation (7). Calculate the gain of each ection, i.e. K, K and K uing equation (). Recalculate the feedbac and ummation coefficient uing equation () and (4). Finally, apply a lowpa-to-bandpa tranformation to obtain the deired bandpa filter pecification: ω 0 π Q Q0 where Q i the quality factor of the overall filter and Q 0 i that required for each biquad ection. Note: A Matlab program wa written to automate the deign procedure for an arbitrary filter pecification of order N. f L f U
Summary of Reult For the required pecification, the following value were obtained: Feedbac Coefficient F 0.657640 F 0.7545 Feedforward Coefficient 0 0 0.6979-0.657.984 Gain for the input and each biquad tage Center frequency and Q 0 of each biquad tage K0 0.604488 K.49 K.65 K f 0.9975MHz Q 0 5.7
Simulation Reult Sytem Level The complete filter wa imulated in Cadence at a ytem-level. The reult are hown below: Ripple 0.d Magnitude Repone Phae Repone
Tranitor Level Implementation To implement each biquadratic ection, a two-integrator loop biquad OTA-C filter wa ued. Advantage with repect to Active-RC: Eay Tunability by changing the bia current of the OTA. (Active-RC need the ue of varactor). Lower Power Conumption and Smaller Area. Diadvantage with repect to Active-RC: Smaller Dynamic Range Poorer Linearity
Tranitor Level Implementation of each iquad Section Vout Vin gm - OUT gm4 - OUT gm - OUT gm - OUT C C R R 4 ) ( C C g g C g C g H m m m m
Thi image cannot currently be diplayed. Deign of Lole Integrator The lole integrator wa deigned to have unity gain at f 0.9975MHz. g m H ( ω) ω C The following pecification are needed if a 5% variation in Q i allowed: Exce Phae : g m 76.5µA φ E C 0 pf Q Q Q a V.5 0 rad 0.086 DC Q Gain : AV 60 55. 58d Q Q a
Deign of Lole Integrator Due to the relatively high DC gain required for the OTA, a foldedcacode topology wa ued: VDD M4 M M M0 Vbp M Ibia Vin M M Vin- Vbn Vout M6 M7 M9 M8 M M4 M5 VSS Tranconductance DC Gain GW ia Current Power Conumption Active Area 76.5µA/V 64.d 6MHz 40µA 79µW 77µm
Simulation Reult of the Lole Integrator The exce phae without any compenation wa.7. Paive exce phae compenation wa ued R55Ω. Magnitude Repone Phae Repone
Deign of iquadratic andpa To reue the deigned OTA: Filter g m g m 76.5 µ A V C C 0 pf C g m 4 ω0. 87 µ A Q 0 V Tranconductance g m depend on K i g K m i m4 For demontration purpoe, g m g m4, i.e. K g
Deign of iquadratic andpa Filter Due to the relatively mall tranconductance required, ource degeneration wa ued. Alo, a PMOS differential pair wa more VDD uitable. M9 M8 MA M M4 M5 R M6 Vbp M7 Vin M M Vin- Vout Vbn Ibia M0 M M4 M M VSS Tranconductance DC Gain GW ia Current Power Conumption Active Area.87µA/V 6.5d 6.MHz 4µA 77.µW 80µm
Simulation Reult of the iquadratic andpa Filter Frequency Repone Step Repone
Summation Node To complete the tranitor-level deign, we need two ummation node: -F -F V in K 0 V 0 K V V V K K 0 V out
Summation Node The ummation node can be implemented with OTA in the following configuration: Vin K0 gm0 V gm0 OUT OUT - - V - F gm0 V - gm0 OUT OUT V - OUT F gm0 - gm0 V OUT gm0 - gm0 OUT V0 - OUT Vout Summation Node for the Feedbac Path Summation Node at the Output If g m0 i choen large enough, the output reitance of each OTA doe not need to be very high. Exce Phae of OTA can be a concern.
Summation Node Due to the deired low exce phae introduced by the OTA, it i more convenient to ue a imple differential pair. VDD M M Vout Ibia Vin M M Vin- M9 M VSS Tranconductance ia Current Power Conumption Active Area 00µA/V 40µA 64µW 69µm
Simulation Reult of the Complete FLF Filter (Tranitor v. Sytem Level) Ripple ~0.d Magnitude Repone Phae Repone
Simulation Reult of the Complete FLF Filter (Tranitor v. Sytem Level) Tranient Repone to a Sinewave Step Repone
Summary of Reult Specification Paband Paband Ripple Attenuation Power Conumption Value.9MHz to.mhz ~0.d 40d at 0.6MHz 8.5mW Active Area,44µm Total Area ~,549µm
Problem to be olved Voltage Swing: The allowable input voltage wing i only 00mV. A mall voltage wing i expected, ince the OTA have a mall linear range limited by ±VDSAT of the input tranitor (in cae no linearization technique i ued, uch a ource degeneration or other). Neverthele, 00mV i too mall and i baically becaue the OTA with g m 76.5µA/V ue input tranitor with a mall VDSAT and no linearization technique i being ued. I need to redeign thee OTA to increae the linear range. ia Networ: To deign the bia networ for the folded-cacode OTA capable of effectively tracing change of VT due to proce variation. Senitivity and Tunability: To characterize the complete filter in term of enitivity and tunability. Layout
Reference [] Sedra, racett. Filter Theory and Deign: Active and Paive. Matrix Serie in Circuit and Sytem. pp. 589-659. [] Deliyanni, Sun, Fidler. Continuou-Time Active Filter Deign. CRC Pre 999. pp. 5-80. [] G. Hurtig, III. The Primary Reonator loc Technique of Filter Synthei Proc. Int. Filter Sympoium, p.84, 97. [4] arbargire. Explicit Deign of General High-Order FLF OTA-C Filter. Electronic Letter. 5th Augut 999, Vol. 5, No. 6, pp. 89-90. [5] Jie Wu, Ezz I. El-Mary. Synthei of Follow-the-Leader Feedbac Log- Domain Filter. IEEE 998. 0-780-5008-/98. pp. 8-84.