Low-Sensitivity, Highpass Filter Design with Parasitic Compensation

Low-Sensitivity, Highpass Filter Design with Parasitic Compensation Introduction This Application Note covers the design of a Sallen-Key highpass biquad. This design gives low component and op amp sensitivities. It also shows how to compensate for the op amp s bandwidth (pre-distortion) and parasitic capacitances. A design example illustrates this method. These biquads are also called KRC or VCVS [voltage-controlled, voltage-source]. Changes in component values over process, environment and time affect the performance of a filter. To achieve a greater production yield, the filter needs to be insensitive to these changes. This Application Note presents a design algorithm that results in low sensitivity to component variation. See [6] for information on evaluating the sensitivity performance of your filter. To achieve the best production yield, the nominal filter design must also compensate for component and board parasitics. The components are pre-distorted [5] to compensate for the op amp bandwidth. This Application Note expands the pre-distortion method in [5] to include compensation for parasitic capacitances. This method is valid for either voltage-feedback or current-feedback op amps. Parasitic Compensation To pre-distort your filter components and compensate for parasitic capacitances:. Use the method in [5] to include the op amp s effect on the filter response. The result is a transfer function of the same order whose coefficients include the op amp group delay (τ oa ) evaluated at the passband edge frequency (f c ).. For all parasitic capacitances in parallel with capacitors: Add the capacitors together Simplify the resulting coefficients Use the sum of time constants form for the coefficients when possible 3. For all parasitic capacitances in parallel with resistors: Replace the resistor R x in the filter transfer function with the parallel equivalent of R x and C p. Rx Rx R C s,s j + x p Alter this impedance to a convenient form and simplify: Do not create new terms (a coefficient times a new power of s) in the transfer function after simplifying OA-9 Kumen Blake October 996 The most useful approximations are: Rx Rx RxCps ( + RxCps) ( ) RCs x p Re x These approximations are valid when: << RC x p Convert ( + R x C p s) to the exponential form (a pure time delay) when it multiplies, or divides, the entire transfer function Do not change the gain at p in allpass sections When simplifying, discard any terms that are products of the error terms (kτ oa and R x C p ); they are negligible Use the sum of time constants form for the coefficients when possible Use an op amp with adequate bandwidth (f 3dB ) and slew rate (SR): f 3dB 0f H SR > 5f H V peak where f H is the highest frequency in the passband of the filter, and V peak is the largest peak voltage. This increases the accuracy of the pre-distortion algorithm. It also reduces the filter s sensitivity to op amp performance changes over temperature and process. Make sure the op amp is stable at a gain of A V K. KRC Highpass Biquad Design The biquad shown in Figure is a Sallen-Key highpass biquad. V in needs to be a voltage source with low output impedance. The transfer function is: Vo Vin + H s p s + pq p p s Low-Sensitivity, Highpass Filter Design with Parasitic Compensation OA-9 00 Corporation AN0796 www.national.com

OA-9 KRC Highpass Biquad Design (Continued) K + R f Rg H K RC 5 RC 5 3 RC 4 3 K ( pqp) + ( ) RRCC p 4 5 3 FIGURE. Highpass Biquad To achieve low sensitivities, use this design algorithm:. Partition the gain for good Q p sensitivity and dynamic range performance: Use a low noise amplifier before this biquad if you need a large gain Select K with this empirical formula:, 0. Qp. K. Qp 0.9 Q 0.,. < Q p < 5 p + 079630 These values also reduce the op amp bandwidth s impact on the filter response. This biquad s sensitivities are too high when Q p 5. Select an op amp with adequate bandwidth (f 3dB ) and slew rate (SR): f 3dB f H f 3dB 0f c SR > 5f H V peak where f H is the highest signal frequency, f c is the corner frequency of the filter, and V peak is the largest peak voltage. Make sure the op amp is stable at a gain of A v K. 3. For current-feedback op amps, use the recommended value of R f for a gain of A V K. For voltage-feedback op amps, select R f for noise and distortion performance. Then set R g for the correct gain: Rg Rf K 4. Initialize the resistance level. R R 4 R 5 Increasing R will: Increase the output noise Reduce the distortion Improve the isolation between the op amp outputs and C and C 3 Make the parasitic capacitances a larger fraction of C and C 3 5. Initialize the capacitance level, and the component ratios C c 3 C and r 3 R5 R : 4 c ( pr) c 0.0 + + 4Q c K p ( + ) r max 0.0, Qp ( + c ) c 6. Recalculate C and initialize the capacitors: rq p c + + 4Qp K r C C c C3 cc 7. Set C and C 3 to the nearest standard values. www.national.com

KRC Highpass Biquad Design (Continued) 8. Recalculate C, C, R and r : C CC3 C c 3 C R pc 4Q c K p + + + r Qp ( + c ) c 9. Calculate the resistors: R R4 r R5 rr The component sensitivity formulas are in the table below. The sensitivities formulas are in the table below. The sensitivities to α i K are a measure of this biquad s sensitivity to the op amp group delay [5]. To evaluate this biquad is sensitivity performance, use the method in [6]. H Q p αi Sα Sα Sα i i C 0 Q r p c C3 0 Q r p c R4 0 K Q c ( ) p + r R5 0 K Q c ( ) p + r K Rf 0 c ( K ) Qp K r Rf K K 0 ( K ) Q c p r K 0 K Q c p r KRC Highpass Biquad Parasitic Compensation To pre-distort this biquad, and compensate for the [parasitic] non-inverting input capacitance of the op amp (C ni ), do the following (see Appendix A for the derivation of the formulas):. Start the iterations by ignoring the parasitics: τ 0 τ 4 0 p. Estimate the pre-distorted values of p and Q p ( p(pd) and Q p(pd) that will compensate for τ oa and C ni : p(pd) p(nom) τ4 p(nom) Qp(pd) Qp(nom) p(pd) Qp(nom) τp(pd) p(nom) Where p(nom) and Q p(nom) are the nominal values of p and Q p 3. Recalculate the resistors and capacitors using p(pd) and Q p(pd) : RRCC 4 5 3 p(pd) RC 5 RC 5 3 RC 4 3 ( K ) p(pd) Qp(pd) + The Design Example accomplishes this by recalculating R and r, then R 4 and R 5 : R p(pd) C + + 4Q c K p(pd) + r Qp(pd) ( + c ) c R4 R r R5 rr 4. Calculate the resulting parasitic correction factors: τ R 4 C ni τ 4 Kτ oa R 4 C 3 +R 4 R 5 (C +C 3 )C ni 5. Calculate the resulting filter response parameters p and Q p p p(pd) + τ4 p(pd) Qp(pd) Qp p + Qp(pd) τp p(pd) 6. Repeat steps -5 until: p p(nom) Q p Q p(nom) OA-9 3 www.national.com

OA-9 KRC Highpass Biquad Parasitic Compensation (Continued) 7. Estimate the high frequency gain: H K + τ4p(pd) If this reduces the gain too much, then repartition the gain Design Example The circuit shown in Figure is a 3rd-order Butterworth highpass filter. Section A is a buffered single pole section, and Section B is a highpass biquad. Use a voltage source with low output impedance, such as the CLC buffer, for V in : The nominal filter specifications are: f c 50MHz (passband edge frequency) f s 0MHz (stopband edge frequency) f H 00MHz (highest signal frequency) A p 3.0dB (maximum passband ripple) A s 40dB (minimum stopband attenuation) H 0dB (passband voltage gain) FIGURE. Highpass Filter 07968 The 3rd-order Butterworth filter [-4] meets our specifications. The pole frequencies and quality factors are: Section A B p /π [MHz] 50.00 50.00 Q p [ ].000 Overall Design. Restrict the resistor and capacitor ratios to:. 0. c,r 0 3. Use % resistors (chip metal film, 06 SMD) 4. Use 5% capacitors (ceramic chip, 06 SMD) 5. Use standard resistor and capacitor values Section A Design and Pre-distortion:. Use the CLC. This is a close-loop buffer. f 3dB 800MHz > f H 00MHz f 3dB 800MHz > 0f c 500MHz SR 3500V/µs, while a 00MHz, V pp sinusoid requires more than 00V/µs τ oa 0.8ns at 0MHz C ni().3pf (input capacitance). Select R A for noise, distortion and to properly isolate the CLC s output and C A. The pre-distorted value of R A, that also compensates for C ni(), is [5]: RA p τ oa CA + Cni() The results are in the table below: The Initial Value column shows ideal values that ignore any parasitic effect The Adjusted Value column shows the component values that compensate for C ni() and CLC is group delay (τ oa ) The Standard Value column shows the nearest standard % resistors and 5% capacitors Component Initial Value Standard Adjusted 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 current-feedback op amp f 3dB 400MHz > f H 00MHz f 3dB < 0f c 500MHz; the design will be sensitive to the op amp group 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 recommended R f at A v.0: R fb 453Ω Then leave R gb open so that K B.00 4. Initialize the resistor level: R 00Ω 5. Initialize the capacitor level, and the component ratios: C 3.83pF π( 50.00MHz) ( 00Ω) c 0.000 r max 0.0,0.086 0.000 { } 6. Recalculate C and initialize the capacitors: C 0.7 C B 89.3pF C 3B.3pF 7. Set the capacitors to the nearest standard values: C B 9pF C 3B pf 8. Recalculate the capacitor level and ratio, and the resistor level and ratio: www.national.com 4

Design Example (Continued) 9. Calculate the resistors: R 4B 34Ω R 3B 3.Ω 0. The sensitivities for this design are: αi C ( 9pF) ( pf) 3.64pF pf c 0.09 ( 9pF) R π ( 50.00MHz) ( 3.64pF) 00.6Ω r 0.056 C B 0.00 0.50 0.39 C 3B 0.00 0.50 0.39 R 4B 0.00 0.50 0.50 R 5B 0.00 0.50 0.50 R fb 0.00 0.00 0.00 R gb 0.00 0.00 0.00 K.00 0.00. Section B Pre-distortion:. The design gives these values: p(nom) π(50.00mhz) Q p(nom).000 K B.00 C B 9pF C 3B pf. Iteration shows the initial design results. Iterations -4 pre-distort R 4B and R 5B to compensate for the CLC446 s group delay, and for C ni(446) : Iteration # 3 4 [MHz] p(pd) 50.00 59.73 56.8 57.54 π Q p(pd) [ ].000 0.930 0.956 0.9505 R [Ω ] 0.6 84. 88.54 87.4 r [Ω ] 0.096 0.08 0.053 0.065 R 4B [Ω ] 34.3 53.0 7.9 67.9 R 5B [Ω ] 3. 8.03 8.73 8.53 τ [ns] 0.34 0.53 0.73 0.68 τ 4 [ns].74.5.575.559 Iteration # 3 4 [MHz] p 43.87 5.96 49.5 50.3 π Q p [ ].034 0.984.003 0.999 The midband gain estimate is: H 0.770[/V/V]. Iteration 0.759 [V/V]. Iteration 4 The simulations gave a lower value for H. Increasing K could help overcome this loss, but would also increase the sensitivities. 3. The resulting components are: Component Initial Value Standard Adjusted 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 Figure 3 and Figure 4 show simulated gains. The curve numbers are:. Ideal (Initial Design Values, τ oa 0,C ni 0). Without pre-distortion (Initial Design Values, τ oa 0, C ni 0) 3. With pre-distortion (Pre-distorted Values, τ oa 0, C ni 0) 07964 FIGURE 3. Simulated Filter Magnitude Response OA-9 5 www.national.com

OA-9 Design Example (Continued) SPICE Models SPICE Models are available for most of Comlinear s amplifiers. These models support nominal DC, AC, AC noise and transient simulations at room temperature. We recommend simulating with Comlinear s SPICE model to: Predict the op amp s influence on filter response Support quicker design cycles Include board and component parasitic models to obtain a more accurate prediction of the filter s response. To verify your simulations, we recommend bread-boarding your circuit. Summary 07965 FIGURE 4. Simulated Filter Magnitude Response This application Note contains an easy to use design algorithm for a low sensitivities Sallen-Key highpass biquad. Designing for low p and Q p sensitivities gives: Reduced filter variation over process, temperature and time High manufacturing yield Lower component cost A low sensitivity design is not enough to produce high manufacturing yields. This Application Note shows how to compensate for the op amp bandwidth, and for the [parasitic] input capacitance of the op amp. This method also applies to any other component or board parasitics. The components must also have low enough tolerance and temperature coefficients. Appendix A Derivation of Pre-distortion and Parasitic Capacitance Compensation Formulas To pre-distort this filter, and compensate for the [parasitic] input capacitance of the op amp C ni ):. Use the method in [5] to include the op amp s effect on the filter response. The result is: H s Vo p τ e Vin s s + + pq p p where the op amp group delay (τ oa ) is evaluated at the passband edge frequency (f c ), and: oa s RC 5 RC 5 3 RC 4 3 K ( pqp) + ( ) RRCC K RC p 4 5 3 + τoa 4 3 K + R f Rg H K www.national.com 6

Appendix A Derivation of Pre-distortion and Parasitic Capacitance Compensation Formulas (Continued). Since C ni is in parallel with R 4, replace R 4 with the parallel equivalent of R 4 and C ni : R4 R4 + R4Cnis 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. After simplifying, we obtain: H s V p o τ oas e Vin s s + + ( pqp) p where : t t ( pqp) + τ RC 5 + RC 5 3 RC 4 3( K ) τ R4Cni p τ3 + τ 4 τ3 R4RCC 5 3 τ4 KτoaR4C3 + R4R5( C+ C3) Cni K + R f Rg H K ( τ3) ( τ3 + τ4) Appendix B Bibliography. R. Schaumann, M Ghausi and K. Laker, Design of Analog Filters: Passive, Active RD, and Switched Capacitor. New Jersey: Prentice Hall, 990.. A. Zverev, Handbook of FILTER SYNTHESIS. John Wiley & Sons, 967. 3. A. Williams and F. Taylor, Electronic Filter Design Handbook. McGraw Hill, 995. 4. S. Natarajan, Theory and Design of Linear Active Networks. Macmillan, 987. 5. K Blake, Component Pre-distortion for Sallen-Key Filters, Comlinear Application Note, OA-, Rev. B, July 996. 6. K. Blake, Low-Sensitivity, Lowpass Filter Design, Comlinear Application Note, OA-7, July 996. 7. K. Blake, Low-Sensitivity, Bandpass Filter Design With Tuning Method, Comlinear Application Note, OA-8, Oct. 996 Note: The circuits included in this application note have been tested with parts that may have been obsoleted and/or replaced with newer products. Please refer to the CLC to LMH conversion table to find the appropriate replacement part for the obsolete device. OA-9 7 www.national.com

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