Successful application of Active Filters By Thomas Kuehl

Successful application of Active Filters By Thomas Kuehl Senior Applications Engineer and John Caldwell Applications Engineer Precision Analog – Linear Products Texas Instruments – Tucson, Arizona

A filter’s purpose in life is to… • Obtain desired amplitude versus frequency characteristics or • Introduce a purposeful phase-shift versus frequency response or • Introduce a specific time-delay (delay equalizer)

Common filter applications Band limiting filter in a noise reduction application

Common filter applications

Common filter applications Delay equalization applied to a band-pass filter application

Filter Types Common filters employed in analog electronics • Low-pass • High-pass • Band-stop, or band-reject • All-pass

Filter Types Low-pass High-pass fc fc A low-pass filter has a single pass-band up to a cutoff frequency, fc and the bandwidth is equal to fc A high-pass filter has a single stop-band 0<f<fc, and pass-band f >fc Band-pass fl Band-stop fh A band-pass filter has one pass-band, between two cutoff frequencies fl and fh>fl, and two stop-bands 0<f<fl and f >fh. The bandwidth = fh-fl fl fh A band-stop (band-reject) filter is one with a stop-band fl<f<fh and two pass-bands 0<f<fl and f >fh

Filter Types An all-pass filter is one that passes all frequencies equally well The phase φ(f) generally is a function of frequency Phase-shift filter (-45º at 1 k. Hz) Time-delay filter (159 us)

Filter Order gain vs. frequency behavior for different low-pass filter orders typically, one active filter stage is required for each 2 nd-order function Pass-band Stop-band f. C (-3 d. B) 1 k. Hz

Filter Order 2 nd-order low-pass, high-pass and band-pass gain vs. frequency

Filter Responses Response Considerations • Amplitude vs. frequency • Phase vs. frequency • Time delay vs. frequency (group delay) • Step and impulse response characteristics

Filter Reponses Common active low-pass filters - amplitude vs. frequency Δ attenuation of nearly 30 d. B at 1 decade

Filter Reponses – phase and time responses 1 k. Hz, 4 th-order low-pass filter example Phase vs. frequency Group Delay Impulse response

Active filter topology and response development

Why Active Filters? A comparison of a 1 k. Hz passive and active 2 nd-order, low-pass filter • Inductor size, weight and cost for low frequency LC filters are often prohibitive • Magnetic coupling by inductors can be a problem • Active filters offer small size, low cost and are comprised of op-amps, resistors and capacitors • Active filter R and C values can be scaled to meet electrical or physical size needs

Comparing 2 nd-order passive RC and an active filters • Cascaded 1 st-order low-pass RC stages • Overall circuit Q is less than 0. 5 • Q approaches 0. 5 when the impedance of the second is much larger than the first; 100 x • Common filter responses often require stage Qs higher than 0. 5 Resource: Analysis of Sallen-Key Architecture, SLOA 024 B, July 1999, revised Sept 2002, by James Karki • At fc C 1 & C 2 impedances are equal to R 1 and R 2 impedances. Positive feedback is present and Q enhancement occurs • Higher Q is attainable with the controlled positive feedback localized to the cutoff frequency • Q’s greater than 0. 5 are supported allowing for specific filter responses; Butterworth, Chebyshev, Bessel, Gaussian, etc

Two popular single op-amp active filter topologies 2 nd-order implementations Multiple Feedback (MFB) low-pass Sallen-Key (SK) low-pass • supports common low-pass, high-pass and band -pass filter responses • inverting configuration • non-inverting configuration • 5 passive components + 1 op-amp per stage • • low dependency on op-amp ac gain-bandwidth to assure filter response • high dependency on op-amp ac gain-bandwidth to assure filter response • Q and fn have low sensitivity to R and C values • Q is sensitive to R and C values • maximum Q of 10 for moderate gains • maximum Q approaches 25 for moderate gains 4 -6 passive components + 1 op-amp per stage

Popular Active Filter Topologies 2 nd-order implementations Component type for each filter topology Pass Z 1 Z 2 Z 3 Z 4 Z 5 Low R 1 R 2 C 3 C 4 na Low R 1 C 2 R 3 R 4 C 5 High C 1 C 2 R 3 R 4 na High C 1 R 2 C 3 C 4 R 5 Band R 1 C 2 R 3 R 4 C 5 Band R 1 R 2 C 3 C 4 R 5

Poles and Zero locations in the s-plane establish the filter gain and phase response third-order low-pass transfer function k H(s) = s 3/ω1ω22+ s 2(ω1ω2+1/ω22)+s(1/ω1+1/ω2)+1 Transfer function roots plotted in s-plane jω ω = 2πf, k = gain All pole filter responses s = -σ ± jω Response pole σ jω Butterworth Real -1. 0 0 Complex -0. 5 0. 866 Complex conj. -0. 5 -0. 866 Real -0. 455 0 Complex -0. 227 0. 888 Complex conj. -0. 277 -0. 888 Real -1. 346 0 Complex -1. 066 1. 017 Complex conj. -1. 066 -1. 017 1 d. B Chebyshev Bessel Third-order low-pass transfer function from Burr-Brown, Simplified Design of Active Filters, Function Circuits – Design and Application, 1976, Pg. 228 Response table data from, High Frequency Circuit Design, James K. Hardy, Reston Publishing Company, 1979, table 4 A-4, Pg. 152 σ

Complex frequency and active filters the s-plane provides the amplitude response of a filter s-plane Complex frequency plane Damping factor (ζ) determines amplitude peaking around the damping frequency fd ζ = cos θ Q the peaking factor is related to ζ by Q = 1/(2ζ) = 1/(2 cos θ) The damping frequency fd is related to the un-damped natural frequency fn by fd = fn (1 - ζ 2 )½ = fn [1 - 1/(4 Q 2)]½ fd = fn sin θ (rect form) (polar form) The pole locations p 1, p 2 = -ζ fn ± j fn (1 -ζ 2)½ Adapted from High Frequency Circuit Design, James K. Hardy, Reston Publishing Company, 1979, Appendix 4 A-1, s-PLANE

The damping frequency fd approaches the undamped natural frequency fn as the Q increases fd Q ζ fd Hz 10 0. 05 998. 7 5 0. 10 995. 0 2. 5 0. 20 978. 3 1. 67 0. 30 953. 9 1. 25 0. 4 916. 5 1. 0 0. 5 866 0. 83 0. 60 800 0. 707 704. 0 -3 d. B fn Adapted from High Frequency Circuit Design, James K. Hardy, Reston Publishing Company, 1979, Appendix 4 A-1, s-PLANE

The stage Q (1/2ζ) affect the time and phase responses of the filter Increasing Q higher peaking Decreasing Q - more linear phase change Adapted from Google Images: gnuplot demo script: multiplt. dem High Q = longer settling time

Filter sections are cascaded to produce the intended response the cutoff frequency fc , gain, roll-off, etc is the product of all stages 1 d. B Chebyshev, 6 th-order LP filter, fc = 10 k. Hz, Av = 10 v/v • The overall response at Vo is the product of all filter stage responses • Each stage has unique Av, fn, Q • The resulting filter has an fc of 10 k. Hz, with a pass-band gain of 10 V/V • The 10 k. Hz pass-bandwidth is defined by the 1 d. B ripple • stage 3 gain was manipulated to be low because of its high Q • Doing so relaxes the GBW requirement – more on this later

Active filter synthesis programs to the rescue! • Modern filter synthesis programs make filter development fast and easy to use; no calculations, tables, or nomograms required • They may provide low-pass, high-pass, band-reject and all-pass responses • Active filter synthesis programs such as Filter. Pro V 3. 1 and Webench Active Filter Designer (beta) are available for free, from Texas Instruments • All you need to provide are the filter pass-band stop-band requirements, and gain requirements • The programs automatically determine the filter order required to meet the stop-band requirements • Filter. Pro provides Sallen-Key (SK), Multiple Feedback (MFB) and differential MFB topologies; the Webench program features the SK and MFB • Commercially available programs such as Filter Wiz Pro provide additional, multiamplifier topologies suitable for low sensitivity, and/or high-gain, high-Q filters

Operational amplifier Gain-bandwidth (GBW) product

Operational amplifier gain-bandwidth product an important ac parameter for attaining accurate active filter response 26

Op-amp gain-bandwidth requirements The active filter’s op-amps should: • Fully support the worst-case, highest frequency, filter section GBW requirements • Have sufficiently high open-loop gain at fn for the worst-case section

The operational-amplifier gain-bandwidth requirements TI’ s Filter. Pro calculates each filter section’s Gain-Bandwidth Product (GBW) from: GBWsection = G ∙ fn ∙ Q ∙ 100 where: G is the section closed-loop gain (V/V) fn is the section natural frequency Q is stage quality factor (Q = 1/2ζ) 100 (40 d. B) is a loop gain factor

The operational-amplifier gain-bandwidth requirements Op-amp closed loop gain error • The filter section’s closed-loop gain (ACL) error is a function of the open-loop gain (AOL) at any specified frequency AOL / ACL* Gain error ∆ Gain error % 104 10 -4 0. 01 103 10 -3 0. 10 102 10 -2 1. 00 101 10 -1 10. 0 * equivalent noise gain ACL • For example, select AOL to be ≥ 100∙ACL at fn for ≤ 1% gain error

The operational-amplifier gain-bandwidth requirement an example of the Filter. Pro estimation Let Filter. Pro estimate the minimum GBW for a 5 th-order, 10 k. Hz (fc) low-pass filter having a Chebyshev response, 2 V/V gain and a 3 d. B pass-band ripple Filter. Pro’s GBW estimation for the worst-case stage yields: GBW = G ∙ fn ∙ Q ∙ 100 GBW = (2 V/V)(10 k. Hz)(8. 82)(100) = 17. 64 MHz vs. 16. 94 MHz from the precise determination – see Appendix for details

Operational amplifier gain-bandwidth effects the Sallen-Key topology • The operational amplifier gainbandwidth (GBW) affects the close-in response • It also affects the ultimate attenuation at high frequency Op-amp f. H Hz d. B OPA 170 90 k -21. 8 OPA 314 110 k -23. 5 OPA 340 260 k -38. 1 OPA 140 428 k -44. 3 Filter. Pro GBW 7. 1 MHz

Operational amplifier gain-bandwidth effects the Multiple Feedback (MFB) topology • The MFB shows much less GBW dependency than the SK • Close-in response shows little effect • Insufficient GBW affects the roll-off at high frequencies • The lowest GBW device (1. 2 MHz) produces a gain deviation about 50 -60 d. B down on the response • A GBW ≥ 7 MHz for this example provides near ideal roll-off

An active filter response issue What a customer expected from their micro-power, 50 Hz active low-pass filter (SK) • A customer designed a 50 Hz low-pass filter using the Filter. Pro software: – Gain = 1 V/V – Butterworth response (Q = 0. 71) – The Sallen-Key topology was selected • The Filter. Pro and TINA-TI simulations using ideal operational amplifiers models produced ideal results • Note Filter. Pro recommended an operationalamplifier with a gain-bandwidth product (GBW) of 3. 55 k. Hz

An active filter response issue What the customer observed with a micro-power, 50 Hz active low-pass filter (SK) • Normal low-pass response below and around the 50 Hz cutoff frequency • The -40 d. B/dec roll-off fails about a decade beyond the 50 Hz cutoff frequency • The gain bottoms out at about 775 Hz and then trends back up • Note that the OPA 369 does meet the minimum GBW specified by Filter. Pro, 3. 55 k. Hz. Its GBW is about 8 to 10 k. Hz GBW ~8 k. Hz

The real operational amplifier can have a complex open-loop output impedance Zo For the OPA 369 FET Drain output stage Zo: • Changes with output current • is low, <10 Ω and resistive below 1 Hz R+j 0 • increases from tens-of-ohms to tens, or hundreds of kilohms, from 10 Hz to 10 k. Hz • Is complex, resitive plus inductive (R+j. X), from 1 Hz to 10 k. Hz R+j. X • Becomes resistive again above 10 k. Hz, the unity gain frequency R+j 0 Unity gain • The hi-Z behavior is reduced by closing the loop but Zo still alters the expected filter response

Net affect on response due to operational-amplifier complex Zo a result similar to low GBW fold-back, but with added peaking OPA 369 Zo-related altered response • Adding a load resistor may reduce the peaking but doesn’t resolve roll-off fold-back • The output offset-related current flow through RL significantly reduces Zo • If the operational amplifier has low offset the Zo can remain high and the problem remains • The added load resistor may draw more current than the op-amp defeating the purpose of using a ultra-low power op-amp

Active filter sensitivity to source impedance and components

The effect of source impedance on filter response 5 k. Hz Butterworth Low-pass, G = 10 V/V • Most active filter designs assume zero source impedance • Source impedance appears in series with the filter input • The impedance will affect the filter response characteristics • The multiple-feedback topology can develop gross gain, bandwidth and phase errors • The Sallen-Key maintains its pass-band gain better, but the cutoff frequency and Q can change • Actual results will vary with the RC values and pass-band characteristics • Active filters maintain their response when preceded by a low impedance source such as an op-amp amplifier

The affect of source impedance on filter response 5 k. Hz Butterworth Low-pass, G = 10 V/V Sallen-Key SK Av = 10 V/V MFB Rs = 0, Av = 10 V/V MFB Rs = 250, Av = 6. 7 V/V MFB Rs = 500, Av = 5. 1 V/V MFB Rs = 1000, Av = 3. 4 V/V MFB

Component sensitivity in active filters a vast subject of its own • Passive component variances and temperature sensitivity, and amplifier gain variance will alter a filter’s responses: fc , Q, phase, etc. • Each topology and filter BOM will exhibit different levels of sensitivity • Mathematical sensitivity analysis provides a method for predicting how sensitive the filter poles (and zeros) are to these variances • The sensitivity analysis for a filter topology is based on the classical sensitivity function • This equation provides the per unit change in y for a per unit change in x. Its accuracy decreases as the size of the change increases • An example an analysis - if the Q sensitivity relative to a particular resistor is 2, then a 1% change in R results in a 2% change in Q • The 1970’s Burr-Brown, “Operational Amplifiers” and “Function Circuits” books provide the sensitivity analysis for many MFB and SK filter types • A modern approach is to use a circuit simulator’s worst-case analysis capability and assigning component tolerances relative to projected changes • Low tolerance/ low drift resistors (1% and 0. 1%, ± 20 ppm/°C) and low tolerance/ low drift C 0 G and film capacitors (1% to 5%, ± 20 ppm/°C) will reduce sensitivity compared to other component types • Often, filters having two or three op-amp per section have low sensitivity

Component sensitivity in active filters a MFB band-pass filter component tolerance simulation Component tolerance Center Freq variance Gain variance 5% resistors 5% capacitors Δ 5 k. Hz Δ 1. 7 d. B 0. 1% resistors 2% capacitors Δ 1 k. Hz Δ 0. 3 d. B

Noise and distortion considerations in active filters

Comparison of Filter Topologies: Noise Gain • “Noise gain” is the amplification applied to the intrinsic noise sources of an amplifier • Sallen-Key and Multiple Feedback Filters have different noise gains – Different RMS noise voltages for the same filter bandwidth! • TINA-TI is a useful tool for determining the noise gain of a complex circuit. – Insert a voltage generator in series with the non-inverting input of the amplifier – Ground the filter input – Perform an AC transfer characteristic analysis Measuring the noise gain of a Sallen-Key low pass filter Measuring the noise gain of a MFB low pass filter

Noise Gain Comparison • Filter. Pro was used to design 2, 1 k. Hz Butterworth lowpass filters – 1 Sallen-Key topology – 1 Multiple Feedback topology • The signal gain of both circuits was 1 • Tina-TI was used to determine the noise gain of the circuits from 1 Hz to 1 MHz Noise Gain Comparison of 1 k. Hz Butterworth Lowpass Filters 10 9 • Within the passband, the MFB filter has 6 d. B higher noise gain • Noise gain above the corner frequency quickly decreases • The noise gain for both circuits peaks at the corner frequency of the filter 8 Noise Gain (d. B) – This is because it is an inverting topology Multiple Feedback Sallen-Key 7 6 5 4 3 2 1 0 1 10 1000 Frequency (Hz) Multiple Feedback 10000 Sallen Key 1000000

Noise Gain at the Filter Corner Frequency Noise Gain Comparison of 1 k. Hz Sallen-Key Lowpass Filters • The magnitude of the noise gain peak is dependant upon the Q of the filter • The peak in noise gain may significantly affect total integrated noise – This depends on how wide a bandwidth noise is integrated over • The table displays the total integrated noise of 1 k. Hz Sallen-Key lowpass filters of different Q’s – OPA 827 simulation model – 100 Ohm resistors used in all circuits (only capacitors changed) 9 8 Noise Gain (d. B) – Higher Q filters have higher peaking in their noise gain. 10 7 6 5 4 3 2 1 0 1 10 Bessel (Q: 0. 58) 1000 Frequency (Hz) Butterworth (Q: 0. 707) 100000 Chebyshev 1 d. B (Q: 0. 957) Noise Voltage Topology Q (2 k. Hz (20 k. Hz Bandwidth) Bessel 0. 58 249. 8 n. Vrms 615 n. Vrms Butterworth 0. 707 303. 1 n. Vrms 628. 4 n. Vrms Chebyshev 1 d. B 0. 957 393. 1 n. Vrms 693. 7 n. Vrms Noise Voltage (200 k. Hz Bandwidth) 1. 733 u. Vrms 1. 737 u. Vrms 1. 762 u. Vrms

Noise considerations in an active filter an OPA 376 inverting amplifier is compared in a 2 nd-order low-pass frequency Amp Filter en (n. V/√Hz) ratio 10 k. Hz 91 119 0. 76 : 1 100 k. Hz 89 15 6: 1 1 MHz 42 7. 4 5. 6 : 1

Noise considerations of an active filter an OPA 376 inverting amplifier is compared in a 2 nd-order band-pass application For Q = 10 frequency Amp Filter en (n. V/√Hz) ratio 1 k. Hz 93 23 4: 1 10 k. Hz 91 1390 1 : 15 100 k. Hz 90 16 5. 6 : 1 1 MHz 42 8. 2 5. 1 : 1

Total Harmonic Distortion and Noise Review Spectrum of a 500 Hz Sine Wave • Total Harmonic Distortion and Noise (THD+N) is a common figure of merit in many systems • Harmonics of the fundamental arise from non-linearity in the circuit’s transfer function. Fundamental -20 -40 Amplitude (d. B) – Intended to “quantify” the amount of unwanted content added to the input signal of a circuit – Consists of the sum of the amplitudes of the harmonics (integer multiples of the fundamental) and the RMS noise voltage of the circuit – Often presented as a (power or amplitude) ratio to the input signal 0 Harmonics -60 -80 -100 Noise -120 -140 -160 0 5000 10000 Frequency (Hz) – Integrated circuits AND passive components can cause this • Intrinsic noise is created in integrated circuits and resistances Vi: RMS voltage of the ith harmonic of the fundamental (i=1, 2, 3…) Vn: RMS noise voltage of the circuit Vf: RMS voltage of the fundamental 15000

Distortion from Passive Components Frequency Response and THD+N • A 1 k. Hz Sallen-Key lowpass filter was built using an OPA 1612, and replaceable passive components. • An Audio Precision SYS 2722 was used to determine the effects of capacitor type on measured THD+N – THD+N was measured from 20 Hz to 20 k. Hz – Harmonic content of a 500 Hz sine wave was also compared • THD+N is noise dominated – Increases as the filter attenuates the signal 0 -20 THD+N (d. BV) – Component values were chosen such that both C 0 G and X 7 R capacitors were available – Thin Film resistors in 1206 packages were used 20 -40 -60 -80 -100 -120 20 2000 Frequency (Hz). 1206 C 0 G Filter Response

Capacitor Dielectric Effects – Signal level was 1 Vrms – All caps are 50 V rated – Minimum of 15 d. B degradation of THD+N inside the filter’s passband – Maximum of almost 40 d. B degradation of THD+N -40 -50 -60 THD+N (d. BV) • 1206 C 0 G capacitors were replaced with 1206 X 7 R capacitors and the THD+N was re-measured Distortion Comparison of Different Capacitor Types -80 -90 -100 -110 -120 20 2000 Frequency (Hz) 1206 X 7 R 1206 C 0 G Spectrum of a 500 Hz Sine Wave • The spectrum of a 500 Hz sine wave was also compared 0 -20 Amplitude (d. B) – X 7 R shows a large number of harmonics – Odd order harmonics dominate the spectrum -70 -40 -60 -80 -100 -120 -140 -160 0 2000 4000 6000 8000 Frequency (Hz) 10000 12000 14000

Package Size Effects – Signal level was 1 Vrms – All tested capacitors have a 50 V rating -50 -60 THD+N (d. BV) • 1206 X 7 R capacitors were replaced with 0603 X 7 R capacitors and distortion was measured again -70 -80 -90 -100 -110 • Distortion increases for smaller package sizes! -120 20 2000 Frequency (Hz) • The spectrum of a 500 Hz sine wave was again examined 0603 X 7 R 1206 C 0 G Spectrum of a 500 Hz Sine Wave 0 -20 -40 Amplitude (d. B) – Both odd and even order harmonics dominate – Odd order harmonics still dominate – 0603 capacitors produce harmonics above 20 k. Hz! Distortion Comparison of Different Capacitor Types -40 -60 -80 -100 -120 -140 -160 0 5000 10000 15000 Frequency (Hz) 20000 25000

Capacitor Distortion and Signal Level • As previously mentioned, capacitor distortion increases with electric field intensity THD+N of a 500 Hz Sine Wave vs. Signal Level 0 – Worse at signal levels – Worse for smaller packages -20 • Changing the signal level is a simple way to determine the source of distortion THD+N (d. BV) -40 -60 • If the circuit is noise dominated the plot will have a slope (m) of: -80 -100 -120 0. 001 0603 X 7 R 0. 1 Signal Level (Vrms) 1206 X 7 R 0805 C 0 G 1 1206 C 0 G 10 • Distortion from passive components will INCREASE with higher signal levels

Capacitor Distortion Over Frequency • Tina-TI was used to measure the voltage across each capacitor over frequency 1 Voltage (Vrms) – The sum of the two voltages is plotted in green – Diagram below indicates measurement points Capacitor Voltages in a 1 k. Hz Sallen-Key Lowpass Filter 1. 2 0. 8 0. 6 0. 4 0. 2 • The maximum voltage appears below the corner frequency 0 20 – This also correlates well to the measured peak in distortion 200 VC 2 Frequency (Hz) VC 1 2000 Combined Distortion Comparison of Different Capacitor Types -40 -50 THD+N (d. BV) -60 -70 -80 -90 -100 -110 -120 20 2000 Frequency (Hz) 0603 X 7 R 1206 C 0 G

Putting it all together

Achieving optimum active filter performance Signal source • Zs→ 0 Ω • An op-amp driver with low closed-loop gain often provides a low source impedance Resistors • Use quality, low tolerance resistors • 1 % and 0. 1% reduce filter sensitivity • Lower tolerance assures more accurate response • Low temperature coefficient reduces response change with temperature • Higher order filters require ever lower tolerances for accurate response Capacitors • Use quality C 0 G or film dielectric for low distortion • Type C 0 G has a low temperature coefficient (± 20 ppm) • Lower tolerance, 1 -2%, assures more accurate response • Higher order filters require ever lower tolerances for accurate response Operational Amplifier • Use required GBW - especially for the Sallen-Key • Be sure to consider the amplifier noise • High Zo effects can distort response • Higher amplifier current often equates to lower Zo and wider GBW • Consider dc specifications – especially bias current

Appendix

The source of the peaking amplifier complex Zo • R+j. X region of Zo exhibits Henries of inductance and kilohms of resistance • Here a higher current, lower Zo op-amp has a complex Zo added to its output path • Estimated R and L values have been taken from the OPA 369 Zo curves • Although the peaking frequency isn’t the same the mechanism is demonstrated

The operational amplifier gain-bandwidth requirement - a more precise determination For a 2 nd-order stage in an nth-order filter: GBW = 100 • ACL(fc/ai)√[(Qi 2 -0. 5)/(Qi 2 -0. 25)] Where: Example: ACL is the closed-loop gain (V/V) fc is the filter cutoff frequency ai is the filter section coefficient Qi is the filter section Q Determine the recommended GBW for a 5 th-order, 10 k. Hz (fc) low-pass filter having a Chebyshev response, 2 V/V gain and a 3 d. B pass-band ripple Source: Op-amps for everyone, chapter 16, by Thomas Kugelstadt

The operational amplifier gain-bandwidth requirements - a more precise determination Select constants from highest ‘Q’ stage

The operational amplifier gain-bandwidth required - a more precise determination ACL = 2 V/V, f. C = 10 k. Hz Coefficients from table: ai = 0. 1172, Qi = 8. 82 GBW = 100 ACL(f. C/ai)√[(Qi 2 -0. 5)/(Qi 2 -0. 25)] GBW = 100 2(10 k. Hz/0. 1172)√[(8. 822 -0. 50)/(8. 822 -0. 25)] GBW = 16. 94 MHz