Application Note OA-29 Low-Sensitivity, Highpass Filter Design With Parasitic Compensation. Kumen Blake

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1 N Application Note OA-9 Low-Sensitivity, Highpass Filte Design With Paasitic Compensation Kumen Blake Octobe 996 Intoduction This Application Note coves the design of a Sallen-Key highpass biquad. This design gives low component and op amp sensitivities. It also shows how to compensate fo the op amp s bandwidth (pe-distotion) and paasitic capacitances. A design example illustates this method. These biquads ae also called KRC o VCVS [voltagecontolled, voltage-souce]. Changes in component values ove pocess, envionment and time affect the pefomance of a filte. To achieve a geate poduction yield, the filte needs to be insensitive to these changes. This Application Note pesents a design algoithm that esults in low sensitivity to component vaiation. See [6] fo infomation on evaluating the sensitivity pefomance of you filte. To achieve the best poduction yield, the nominal filte design must also compensate fo component and boad paasitics. The components ae pe-distoted [5] to compensate fo the op amp bandwidth. This Application Note expands the pe-distotion method in [5] to include compensation fo paasitic capacitances. This method is valid fo eithe voltage-feedback o cuent-feedback op amps. Paasitic Compensation To pe-distot you filte components and compensate fo paasitic capacitances:. Use the method in [5] to include the op amp s effect on the filte esponse. The esult is a tansfe function of the same ode whose coefficients include the op amp goup delay (τ oa ) evaluated at the passband edge fequency (f c ).. Fo all paasitic capacitances in paallel with capacitos: Add the capacitos togethe Simplify the esulting coefficients Use the sum of time constants fom fo the coefficients when possible 3. Fo all paasitic capacitances in paallel with esistos: Replace the esisto R x in the filte tansfe function with the paallel equivalent of R x and C p : Rx Rx R C s,s j ( + x p ) Alte this impedance to a convenient fom and simplify: Do not ceate new tems (a coefficient times a new powe of s) in the tansfe function afte simplifying The most useful appoximations ae: RCs x p Re x These appoximations ae valid when: Convet (+ R x C p s) to the exponential fom (a pue time delay) when it multiplies, o divides, the entie tansfe function Do not change the gain at p in allpass sections When simplifying, discad any tems that ae poducts of the eo tems (Kτ oa and R x C p ); they ae negligible Use the sum of time constants fom fo the coefficients when possible 4. Use an op amp with adequate bandwidth (f 3dB ) and slew ate (SR): f 3dB 0f H SR > 5f H V peak whee f H is the highest fequency in the passband of the filte, and V peak is the lagest peak voltage. This inceases the accuacy of the pe-distotion algoithm. It also educes the filte s sensitivity to op amp pefomance changes ove tempeatue and pocess. Make sue the op amp is stable at a gain of A v K. KRC Highpass Biquad Design The biquad shown in Figue is a Sallen-Key highpass biquad. V in needs to be a voltage souce with low output impedance. The tansfe function is: Rx Rx RxCps ( + RxCps) ( ) Vo Vin + << RC x p H s p s + pq p p s 996 National Semiconducto Copoation Pinted in the U.S.A.

2 whee: K + R f Rg H K R5C R5C3 R4C3 K ( pqp) + ( ) R R C C p Figue : Highpass Biquad To achieve low sensitivities, use this design algoithm:. Patition the gain fo good Q p sensitivity and dynamic ange pefomance: Use a low noise amplifie befoe this biquad if you need a lage gain Select K with this empiical fomula:, 0. Qp. K. Qp 0.9 Q 0.,. < Q p < 5 p + These values also educe the op amp band width s impact on the filte esponse. This biquad s sensitivities ae too high when Q p 5. Select an op amp with adequate bandwidth (f 3dB ) and slew ate (SR): f 3dB f H f 3dB 0f c SR > 5f H V peak whee f H is the highest signal fequency, fc is the cone fequency of the filte, and V peak is the lagest peak voltage. Make sue the op amp is stable at a gain of A v K. 3. Fo cuent-feedback op amps, use the ecommended value of R f fo a gain of A v K. Fo voltage-feedback op amps, select R f fo noise and distotion pefomance. Then set R g fo the coect gain: R Rg f K 4. Initialize the esistance level R R 4 R 5. Inceasing R will: Incease the output noise Reduce the distotion Impove the isolation between the op amp outputs and C and C 3 Make the paasitic capacitances a lage faction of C and C 3 5. Initialize the capacitance level ( C C C 3 ), and the component atios C c 3 C and 3 R5 R : 4 c ( pr) c Recalculate c and initialize the capacitos: 9. Calculate the esistos: + + 4Q c K p + max 0.0, Qp ( + c ) c Qp c + + 4Qp K C C c C3 cc 7. Set C and C 3 to the neaest standad values. 8. Recalculate C, c, R and : C CC 3 C c 3 C R pc 4Q c K p Qp ( + c ) c R R4 R5 R The component sensitivity fomulas ae in the table below. The sensitivities to α i K ae a measue of this biquad s sensitivity to the op amp goup delay [5]. To evaluate this biquadís sensitivity pefomance, use the method in [6].

3 H Q p αi Sα Sα Sα C 0 Q p c C3 0 Q p c R4 0 K Q c p R5 0 K Q c p Rf Rf K K K K i i 0 K Q KRC Highpass Biquad Paasitic Compensation To pe-distot this biquad, and compensate fo the [paasitic] non-inveting input capacitance of the op amp (C ni ), do the following (see Appendix A fo the deivation of the fomulas):. Stat the iteations by ignoing the paasitics: τ 0 τ 4 0. Estimate the pe-distoted values of p and Q p ( p(pd) and Q p(pd) ) that will compensate fo τ oa and C ni : whee p(nom) and Q p(nom) ae the nominal values of p and Q p 3. Recalculate the esistos and capacitos using p(pd) and Q p(pd) : The Design Example accomplishes this by ecalculating R and, then R 4 and R 5 : p + + p 0 ( K ) Q c p K 0 K Q c p p(pd) p(nom) τ4 p(nom) Qp(pd) Qp(nom) p(pd) Qp(nom) τp(pd) p(nom) c R R C C p(pd) R5C R5C3 R4C3 ( K ) p(pd) Qp(pd) + R p(pd) C + + 4Q c K p(pd) + Qp(pd) ( + c ) c R4 R R5 R 4. Calculate the esulting paasitic coection factos: 5. Calculate the esulting filte esponse paametes p and Q p : 6. Repeat steps -5 until: τ R4Cni τ4 KτoaR4C3 + R4R5 ( C + C3) Cni p p(pd) + τ4 p(pd) Qp(pd) Qp p + Qp(pd) τp p(pd) p p(nom) Qp Qp(nom) 7. Estimate the high fequency gain: H K + τ4p(pd) If this educes the gain too much, then epatition the gain. Design Example The cicuit shown in Figue is a 3d-ode Buttewoth highpass filte. Section A is a buffeed single pole section, and Section B is a highpass biquad. Use a voltage souce with low output impedance, such as the CLC buffe, fo V in. The nominal filte specifications ae: f c 50MHz (passband edge fequency) f s 0MHz (stopband edge fequency) f H 00MHz (highest signal fequency) A p 3.0dB (maximum passband ipple) A s 40dB (minimum stopband attenuation) H 0dB (passband voltage gain) 3

4 Figue : Highpass Filte The 3d-ode Buttewoth filte [-4] meets ou specifications. The pole fequencies and quality factos ae: Section A B p /π [MHz] Q p [ ].000 Oveall Design:. Restict the esisto and capacito atios to: 0. c, 0. Use % esistos (chip metal film, 06 SMD) 3. Use 5% capacitos (ceamic chip, 06 SMD) 4. Use standad esisto and capacito values Section A Design and Pe-distotion:. Use the CLC. This is a closed-loop buffe. f 3dB 800MHz > f H 00MHz f 3dB 800MHz > 0f c 500MHz SR 3500V/µs, while a 00MHz, V pp sinusoid equies moe than 000V/µs τ oa 0.8ns at 0MHz C ni().3pf (input capacitance). Select R A fo noise, distotion and to popely isolate the CLC s output and C A. The pedistoted value of R A, that also compensates fo C ni(), is [5]: RA p τ oa CA + Cni() The esults ae in the table below: The Initial Value column shows ideal values that ignoe any paasitic effects The Adjusted Value column shows the component values that compensate fo C ni() and CLCís goup delay (τ oa ) The Standad Value column shows the neaest standad % esistos and 5% capacitos Component Initial Value Adjusted Standad C A 30pF 30pF 30pF R A 06Ω 9.8Ω 93.Ω C ni().3pf.3pf Section B Design:. Since Q p.000, set K B to.00. Use the CLC446. This is a cuent-feedback op amp f 3dB 400MHz > f H 00MHz f 3dB < 0f c 500MHz; the design will be sensitive to the op amp goup delay SR 000V/µs > 000V/µs (see Item # in Section A Design ) τ oa 0.56ns at 0MHz C ni(446).0pf (input capacitance) 3. Use the CLC446 s ecommended R f at A v.0: R fb 453Ω Then leave R gb open so that K B Initialize the esisto level: R 00Ω 5. Initialize the capacito level, and the component atios: C 3.83pF π( 50.00MHz) ( 00Ω) c max 0.0, { } 6. Recalculate c and initialize the capacitos: c 0.7 C B 89.3pF C 3B.3pF 7. Set the capacitos to the neaest standad values: C B 9pF C 3B pf 8. Recalculate the capacito level and atio, and the esisto level and atio: 9. Calculate the esistos: R 4B 34Ω R 3B 3.Ω 0. The sensitivities fo this design ae: C ( 9pF) ( pf) 3.64pF pf c 0.09 ( 9pF) R π ( 50.00MHz) ( 3.64pF) 00.6Ω H αi Sα Sα i C B C 3B R 4B R 5B R fb R gb K i p Q Sα p 4

5 Section B Pe-distotion:. The design gives these values: p(nom) π(50.00mhz) Q p(nom).000 K B.00 C B 9pF C 3B pf. Iteation shows the initial design esults. Iteations -4 pe-distot R 4B and R 5B to compensate fo the CLC446 s goup delay, and fo C ni(446) : Iteation # 3 4 p(pd) π [MHz] Q p(pd) [ ] R [Ω] [Ω] R 4B [Ω] R 5B [Ω] τ [ns] τ 4 [ns] p π [MHz] Q p [ ] The midband gain estimate is: H [V/V]. Iteation [V/V]. Iteation 4 The simulations gave a lowe value fo H. Inceasing K could help ovecome this loss, but would also incease the sensitivities. 3. The esulting components ae: Component Initial Value Adjusted Standad C B 9pF 9pF 9pF C 3B pf pf pf C ni(446).0pf.0pf R 4B 34Ω 68Ω 67Ω R 5B 3.Ω 8.5Ω 8.7Ω R fb 453Ω 453Ω 453Ω R gb Figues 3 and 4 show simulated gains. The cuve numbes ae:. Ideal (Initial Design Values, τ oa 0, C ni 0). Without pe-distotion (Initial Design Values, τ oa 0, C ni 0) 3. With pe-distotion (Pe-distoted Values, τ oa 0, C ni 0) Figue 3: Simulated Filte Magnitude Response Figue 4: Simulated Filte Magnitude Response SPICE Models SPICE models ae available fo most of Comlinea s amplifies. These models suppot nominal DC, AC, AC noise and tansient simulations at oom tempeatue. We ecommend simulating with Comlinea s SPICE models to: Pedict the op amp s influence on filte esponse Suppot quicke design cycles Include boad and component paasitic models to obtain a moe accuate pediction of the filte s esponse. To veify you simulations, we ecommend bead-boading you cicuit. Summay This Application Note contains an easy to use design algoithm fo a low sensitivity, Sallen-Key highpass biquad. Designing fo low p and Q p sensitivities gives: Reduced filte vaiation ove pocess, tempeatue and time Highe manufactuing yield Lowe component cost A low sensitivity design is not enough to poduce high manufactuing yields. This Application Note shows how to compensate fo the op amp bandwidth, and fo the [paasitic] input capacitance of the op amp. This method also applies to any othe component o boad paasitics. The components must also have low enough toleance and tempeatue coefficients. 5

6 Appendix A Deivation of Pe-distotion and Paasitic Capacitance Compensation Fomulas To pe-distot this filte, and compensate fo the [paasitic] input capacitance of the op amp (C ni ):. Use the method in [5] to include the op amp s effect on the filte esponse. The esult is: Vo Vin + H s p τ e s s + pq p p whee the op amp goup delay (τ oa ) is evaluated at the passband edge fequency (f c ), and: R5C R5C3 R4C3 K ( pqp) + ( ) R R C C K R C p τoa 4 3 K + R f Rg H K. Since C ni is in paallel with R 4, eplace R 4 with the paallel equivalent of R 4 and C ni : R4 R4 + R4Cnis oa s H R 4 C 3 R 5 C + K τoa τ oas s e V + R C s o 4 ni Vin R4C3 ( K) + + R5( C + C3) s + R4Cnis R4C3( R5C + Ktoa ) + + R4Cnis s 3. Afte simplifying, we obtain: Vo Vin + whee : t t ( pqp) + H s p τ e s s + pq p p τ R5C + R5C3 R4C3 K τ R4Cni p τ3 + τ 4 τ3 R4R5CC 3 oa τ4 KτoaR4C3 + R4R5 C + C3 Cni K + R f Rg H K ( τ3) τ3 + τ4 s Appendix B Bibliogaphy. R. Schaumann, M. Ghausi and K. Lake, Design of Analog Filtes: Passive, Active RC, and Switched Capacito. New Jesey: Pentice Hall, A. Zveev, Handbook of FILTER SYNTHESIS. John Wiley & Sons, A. Williams and F. Taylo, Electonic Filte Design Handbook. McGaw Hill, S. Nataajan, Theoy and Design of Linea Active Netwoks. Macmillan, K. Blake, Component Pe-distotion fo Sallen-Key Filtes, Comlinea Application Note, OA-, Rev. B, July K. Blake, Low-Sensitivity, Lowpass Filte Design, Comlinea Application Note, OA-7, July K. Blake, Low-Sensitivity, Bandpass Filte Design With Tuning Method, Comlinea Application Note, OA-8, Oct

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