Successful application of Active Filters

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 Introduce a specific time-delay (delay equalizer) Electronic filters are applied in applications where a specific amplitude versus frequency response is required from a circuit. Other applications exist where a specific phase shift, or time delay is needs to be incorporated in a circuit and a filters may be used in those applications as well. The new FilterPro 3 has been designed to synthesize filters that purposely affect the frequency, or time characteristics of a circuit. 2

Common filter applications
Band limiting filter in a noise reduction application The range of frequencies processed by an electronic circuit may need to be limited to maximize performance and accuracy. Frequencies laying outside the desired range will add unnecessary energy to the circuit and can result in less-than-optimum performance from the circuit. Noise is an electrical signal consisting of many different frequencies some of which may be consistent, or truly random. Noise by definition is truly random in frequency and amplitude, but we often apply the term to annoying interference such as 50 or 60Hz hum in an audio system. Bandwidth limiting a system by using filtering techniques can result in the elimination of much of this unwanted noise resulting in improved system performance. 3

Anti-aliasing filter Analog filters are often employed in digital signal processing systems. This examples shows an anti-alias, or anti-aliasing, low-pass filter placed before the analog-to-digital (A/D) converter. Its purpose is to attenuate frequencies above the half-sampling (fs/2) frequency from being converted by the A/D converter. Otherwise, these frequencies will fold back into the desired frequency range producing undesirable beat frequencies (tones) in the pass-band. Anti-imaging filter In the reverse process, an anti-imaging (or reconstruction) filter is used at the output of a sampled system such as a digital to analog converter (DAC) to eliminate spectral images of the desired signal. These are generated at integer multiples of the sample rate. The ideal filter specification is the same as that for the A/D anti-aliasing filter. The anti-imaging filter’s reconstruction tasks: Remove aliases in the stair-step approximation Correct for sinc distortion 4

Delay equalization applied to a band-pass filter application The all-pass filter is a time-domain filter that is applied to produce a specific time, or phase response at a particular frequency. When signals are processed through other filters the phase shift or timing changes with frequency. The shifts in phase can result in phase distortion. Sometimes this isn’t an important consequence, but other times it is. An all-pass filter can be added to the system to shift a signal in time or phase to correct for introduced time or phase shift and restore certain attributes of the original signal. In the diagram shown the time delay, or Group Delay, for a signal is plotted across frequency after passing through a band-pass filter. It can be seen that the delay becomes long at low frequency, becomes less in the mid-band, and then lengthens again as the frequency increases. To help correct the band-pass filter’s non-linear group delay characteristic an all-pass filter with the somewhat Gaussian group delay response is added after it. Once the two filters are cascaded their total group delay becomes more constant across the frequency band of interest. 5

Filter Types Common filters employed in analog electronics Low-pass
High-pass Band-pass Band-stop, or band-reject All-pass Each filter type listed has its own set of electrical characteristics that makes it suitable for particular applications. The response for each filter type will be described in some detail in the following slides. 6

Filter Types fc fl fh Low-pass High-pass Band-pass Band-stop
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 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 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 fc fl fh Low-pass filters are the most widely applied filter type. It is designed to readily pass all frequencies extending from DC to a set cutoff frequency (fc). This region where the frequencies readily pass through the filter is called the pass-band. Once the cutoff frequency is reached the filter begins to attenuate any frequency higher than fc. The region above fc is called the stop-band. Most often the attenuation increases, or rolls off, and achieves very high levels which is limited only by the non-ideal electronics used to construct the filter. This includes the stray and lead inductance of the capacitors, non-ideal operational-amplifier limitations, and circuit parasitics associated with the physical circuit layout A low-pass filter has a single pass-band up to a cutoff frequency - fc fc is defined as the filter bandwidth The high-pass filter has a pass-band where all frequencies above the cutoff frequency (fc) pass with little to no attenuation. Below fc, within the filter’s stop-band, the signals are attenuated at ever greater levels as the frequency moves lower. A high-pass filter has a stop-band 0<f<fc, and a pass-band f>fc fc is defined as the cutoff frequency The band-pass filter has a pass-band that allows a select band of frequencies that fall within the pass-band to pass with little, or no, attenuation. An upper and lower cutoff frequency defines the bandwidth of the filter. Frequencies beyond the pass-band lie in the two stop-bands and receive greater attenuation as the frequency moves further away from the pass-band in either direction. 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 Bandwidth = fh-fL The band-stop, or band-reject filter, is sometimes referred to as a notch filter due to its notch-like gain vs. frequency stop-band characteristic. The purpose of this filter type is to attenuate, or reject, frequencies that fall within the stop-band. The filter bandwidth is defined by upper and lower cutoff frequencies like the band-pass filter. A band-stop 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 1kHz) Time-delay filter (159us) An all-pass filter is one that passes all frequencies equally well. The phase φ(ω) generally is a function of frequency. The all-pass filter function passes all frequencies equally well (or at least within the intended pass-band) with a specific phase shift or time behavior being of primary concern. The upper plots show the phase and gain responses of an all-pass filter with a specified -45º input-to-output phase-shift at 1kHz. Note that the amplitude vs. frequency plot is flat throughout its entire range. Often, a simple 1st-order all-pass filter is sufficient to achieve the required phase-shift. Keep in mind that the phase-shift will be different from -45º at other input frequencies as the phase plot indicates. Most often Bessel filters are used for all-pass filter functions because of their very linear phase vs. frequency characteristic. The lower plots are for another all-pass filter that has been optimized for a particular time delay; in this case 159usat a 1kHz input frequency. Note that the delay is precise for this 4th-order all-pass. The precise delay was accomplished using a Bessel filter. The time delay, or group delay, will be 159us for input frequencies up to the 1kHz. The phase shift for this filter is found by: φº = -td/tp (360º) = -(159us/1ms)(360º) = -57.2º Where: td = delay time and tp = period time 8

Filter Order gain vs. frequency behavior for different low-pass filter orders
typically, one active filter stage is required for each 2nd-order function Pass-band Stop-band fC (-3dB) 1kHz The filter order will dictate a filter’s stop-band attenuation. Generally, the higher the order, the higher the attenuation. Increasing a filter’s order most often is accomplished by cascading first, and/or second-order filter stages. For the example shown in the slide the stop-band gain roll-off is -40dB/dec, for each corresponding 2nd-order stage. Conversely, the gain increases at a rate of +40dB/dec, for each 2nd-order high-pass stage. The benefit of higher-order filters is a more ideal response within the pass-band and a greater attenuation rate in the stop-band. But higher order filters come at the expense of greater circuit complexity and more stringent component tolerances. 9

2nd-order low-pass, high-pass and band-pass gain vs. frequency
Filter Order 2nd-order low-pass, high-pass and band-pass gain vs. frequency The plots shown are examples of responses for low-pass, high-pass and band-pass filters that have been normalized to 1kHz. An important point to note is the 2nd-order low-pass and high-pass decrease or increase at a 40 dB/dec rate, respectively, beyond fc. Observe that although the 2nd-order band-pass response decreases at rate similar to the low and high-pass filters close to fc, that it eventually achieves a lower roll-off rate of -20dB/dec; half that of the other filter types. Thus, twice as many filter band-pass filter stages are required to achieve the same eventual gain roll-off rate as the low and high-pass filters. 10

Filter Responses Response Considerations Amplitude vs. frequency
Phase vs. frequency Time delay vs. frequency (group delay) Step and impulse response characteristics Filters are employed in circuits to act upon an applied input and provide a specific response characteristic. Most often this is to attenuate some portion of the applied signal’s frequency content, but it may involve something other such as modifying the phase, timing, or impulse behavior in a very specific, controlled manner. 11

Filter Reponses Common active low-pass filters - amplitude vs
Filter Reponses Common active low-pass filters - amplitude vs. frequency Δ attenuation of nearly 30 dB at 1 decade Examples of the amplitude vs. frequency responses for several different, but common, 1kHz low-pass filters are shown. All of these filters are of the same 4th-order, but as can be seen they exhibit different responses over frequency. Different applications may call for one response, or another. Here’s some details regarding each of these responses: Butterworth – This is referred to as a maximally flat response. The roll-off in the pass and stop-bands is very predictable and a function of the filter order. The amplitude vs. frequency response is as mathematically flat as possible within the pass-band. It is the most often applied filter response in general filter and signal-processing applications. Chebyshev - Feature earlier roll-off than most other filter response types, but at the expense of pass-band ripple. They minimize the error between the idealized and the actual filter characteristic over the range of the filter. A faster initial roll-off rate is associated with higher pass-band ripple. Linear phase – The phase response is a linear function of the frequency. All frequency components have equal delay time. Therefore, the filter has a constant delay or group delay. Bessel - Has a maximally flat group delay (maximally linear phase response). They are often used in audio crossover systems. Analog Bessel filters are characterized by almost constant group delay across the entire pass-band, thus preserving the wave shape of filtered signals in the pass-band. Gaussian - A filter whose impulse response is a Gaussian function. Gaussian filters are designed to give no overshoot to a step function input while minimizing the rise and fall time. This behavior is closely connected to the fact that the Gaussian filter has the minimum possible group delay. 12

Filter Reponses – phase and time responses 1 kHz, 4th-order low-pass filter example
Phase vs. frequency Impulse response Group Delay In the upper left plot, the phase behavior is shown for the different types of 1kHz low-pass filter responses. The Bessel, Gaussian and Linear Phase filters provide a linear phase vs. frequency response in their pass-band, or across their frequency range, while the Butterworth and Chebyshev filters have increasingly non-linear phase responses, respectively. The upper right plots show the 1kHz filter responses to a 1V, 100us step impulse. Here the Bessel, Gaussian and Linear Phase filters provide a similar amplitude overshoot, with minimal undershoot. They settle out within about 1us. The Butterworth has less overshoot than the other responses, but exhibits some undershoot. It requires about 2us to settle. When a filter requires a long time to settle it is said to ring. The Chebyshev filter requires a period longer than 2.5us, more on the order of 4 or 5us to settle. Thus, the Butterworth and Chebyshev usually aren’t a good choice for filters requiring fast settling characteristics. The plot shown along the bottom provides a Group Delay comparison for each of the filter types. Clearly it can be seen that the Group Delay characteristics are heavily skewed near the cutoff frequency for the Chebyshev and Butterworth responses. This may be a point of concern for some applications.

Active filter topology and response development

Why Active Filters? A comparison of a 1kHz passive and active 2nd-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 Active filters have replaced conventional passive LC (inductor/capacitor) filters in many low-frequency filter applications. The size, weight, availability, cost and magnetic characteristics of large inductors often makes their use less convenient, or undesirable. LC filters are still the most viable, and sometimes the only choice, for applications involving radio frequencies and where power levels and frequencies exceed an active filter’s capabilities. Dynamic range limitations must be considered for all applications and that too may dictate the use of a passive filter, over an active filter. The circuit shown makes a comparison between a 1kHz passive low-pass filter and a 1kHz active low-pass filter. The LC filter requires specific source and load impedances to provide a prescribed response. Losses associated with the inductor(s) will cause the gain and phase responses to deviate from the ideal, lossless case given here. The active-filter must be driven by a very low impedance source, or the source impedance must be included in the filter design to keep from corrupting the ideal response. When these two filters are driven by an equivalent source voltage it can be seen that both the gain and phase response are identical. A different active filter topology may invert the output to input signal relationship and that needs to be considered in an application. 15

Comparing 2nd-order passive RC and an active filters
Cascaded 1st-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; 100x Common filter responses often require stage Qs higher than 0.5 At fc C1 & C2 impedances are equal to R1 and R2 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 An example of similar passive and active low-pass filter topologies are shown. The two have very similar second-order transfer functions, but that of the active filter involves additional factors that allows for gains (K) greater than 1 V/V and a specific Q value. The Q often ranges from less than 1 and has a practical limit approaching 10, and will depend on the filter response required. The standard frequency domain equation for a second-order active low-pass filter is: HLP = K / [ -(f/fc)2 + jf/Qfc +1] Where fc is the corner frequency breakpoint between the pass-band and the stop-band and is not necessarily the -3 dB frequency point. When f << fc the transfer function reduces to K, the gain factor. When f = fc the transfer function reduces to –jKQ, and signals are enhanced by the Q factor. When f >> fc the transfer function reduces to –K(fc / f)2, and signals are attenuated by the square of the frequency ratio. Resource: Analysis of Sallen-Key Architecture, SLOA024B, July 1999, revised Sept 2002, by James Karki

Two popular single op-amp active filter topologies 2nd-order implementations
Multiple Feedback (MFB) low-pass supports common low-pass, high-pass and band-pass filter responses inverting configuration 5 passive components + 1 op-amp per stage low dependency on op-amp ac gain-bandwidth to assure filter response Q and fn have low sensitivity to R and C values maximum Q of 10 for moderate gains Sallen-Key (SK) low-pass supports common low-pass, high-pass and band-pass filter responses non-inverting configuration 4-6 passive components + 1 op-amp per stage high dependency on op-amp ac gain-bandwidth to assure filter response Q is sensitive to R and C values maximum Q approaches 25 for moderate gains Points about the Multiple Feedback (MFB) and Sallen-Key filter topologies: Each topology supports low, high and band-pass responses The MFB is a popular topology that features low sensitivity to component tolerances. Each 2nd-order MFB section produces an output phase inversion. Cascading two sections results in the output being in phase with the input. All Sallen-Key filter sections provide non-inverting stages If the Sallen-Key gain is +1V/V R3 is eliminated and R4 is 0Ω. The Sallen-Key does not show a high fn sensitivity to component tolerance. However, the filter “Q” is sensitive to component tolerance.

Popular Active Filter Topologies 2nd-order implementations
Component type for each filter topology Active Filters The active filter stage is comprised of one or more operational-amplifiers and associated resistors (R) and capacitors (C). The operational-amplifier’s output is fed back to one, or both inputs via the paths containing the resistors and capacitors. Imaginary poles are intentionally developed at specific frequencies to modify the amplitude vs. frequency response of the amplifier stage and produces the specific response characteristics. Without the use of the operational-amplifiers and feedback these poles could only be developed with the use of inductors and capacitors. The active filter stage gyrates the equivalent function of the inductors in the circuits; thus, replacing them. The two most commonly applied active filter topologies are the Sallen-Key, which is a non-inverting, voltage-controlled, voltage-source (VCVS) topology and the Infinite-gain, Multiple-feedback topology (MFB). The latter is an inverting topology in which the output polarity is inverted relative to the input. Pass Z1 Z2 Z3 Z4 Z5 Low R1 R2 C3 C4 na High C1 C2 R3 R4 Band C5 Pass Z1 Z2 Z3 Z4 Z5 Low R1 C2 R3 R4 C5 High C1 R2 C3 C4 R5 Band 18

Poles and Zero locations in the s-plane establish the filter gain and phase response
third-order low-pass transfer function k H(s) = s3/ω1ω22+ s2(ω1ω2+1/ω22)+s(1/ω1+1/ω2)+1 ω = 2πf, k = gain Transfer function roots plotted in s-plane σ jω All pole filter responses s = -σ ± jω Response pole σ jω Butterworth Real -1.0 Complex -0.5 0.866 Complex conj. -0.866 1 dB Chebyshev -0.455 -0.227 0.888 -0.277 -0.888 Bessel -1.346 -1.066 1.017 -1.017 Each filter type will have an exact transfer equation that establishes the amplitude and phase response for the filter. All transfer functions are the ratio of two polynomials, and these polynomials can be expressed in terms of their roots:1 H(s) = P1(s) /P2(s) = k [(S-S1 )(S-S2)…(S-Sm)] / [(S-Sa )(S-Sb)…(S-Sn) where: k is a constant S1→ Sm are roots of P1(s) Sa→ Sn are roots of P2(s) S1, S2, S3… are called zeros of the function. When S equals one of these values H(s) foes to zero. Sa, Sb, Sc… are called poles of the function. When S equals one of the these values H(s) goes to infinity. The numeric terms in the transfer equation’s numerator and denominator determine how that filter will respond at any frequency. The roots of the equation, the poles and zeros, when plotted on the s-plane reveal information about the filter’s gain and phase response. Any particular filter response requires that they be located in very exacting positions within the complex frequency plane (s-plane). Common Butterworth, Bessel and Chebyshev responses are all-pole filters having no zeros in their transfer equation. Their poles lie in the left half of the plane and follow a circle in the case of the Butterworth response, or an ellipse in the Bessel and Chebyshev responses although the pole locations are quite different for the latter two response types. 1 adapted from: LECTURE '12: THE S PLANE AND TRANSFER IMPEDANCES 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 4A-4, Pg. 152

Complex frequency and active filters the s-plane provides the amplitude response of a filter
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/(2cos θ) The damping frequency fd is related to the un-damped natural frequency fn by fd = fn (1- ζ2 )½ = fn [1- 1/(4Q2)]½ (rect form) fd = fn sin θ (polar form) The pole locations p1, p2 = -ζ fn ± j fn (1-ζ2)½ s-plane Complex frequency plane The complex frequency plane (s-plane) provides a graphical means for plotting the roots of a filter’s transfer equation. Doing so provides a way to visualize their positions relative to the frequency axis (-jf) and the damping axis (σ). And provides insight into the filter’s frequency and Q relationship. The “Q” is a damping constant determining the selectivity of the frequency response around fn. The Q indicates energy loss relative to the amount of energy stored within the system. Thus, the higher the Q the lower the rate of energy loss and hence oscillations will reduce more slowly, i.e. they will have a low level of damping and they will ring for longer.1 The Q is related to the circuit damping factor (ζ) by, Q = 1/2ζ. The damping factor damps the natural frequency (fn) of the filter section resulting in its damping frequency (fd ). The damped frequency is that associated with the poles of the filter transfer equation. The pole locations as plotted on the s-plane graph provides the trigonometric relationship between fd and ζ. Filter’s have pole and zero locations assigned to provide specific response characteristics in both the time and frequency domains. It is most often based on a particular mathematical approximation of a maximally flat response (Butterworth), equiripple (Chebyshev), or linear phase (Bessel), or other type of response. 1 Q - Quality Factor Tutorial, from Radio-Electronics.com Adapted from High Frequency Circuit Design, James K. Hardy, Reston Publishing Company, 1979, Appendix 4A-1, s-PLANE

The damping frequency fd approaches the undamped natural frequency fn as the Q increases
-3 dB fd fn 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 Active 2nd-order filter stages may, or may not, exhibit peaking in the vicinity of their damped natural frequency, fd . The higher the “Q” the closer fd comes to approaching the undamped natural frequency fn . Different filter responses will have different natural frequencies and Qs depending on the type of response. Adapted from High Frequency Circuit Design, James K. Hardy, Reston Publishing Company, 1979, Appendix 4A-1, s-PLANE

The stage Q (1/2ζ) affect the time and phase responses of the filter
Increasing Q higher peaking High Q = longer settling time Decreasing Q - more linear phase change The Q, or Quality Factor, of each filter stage not only affects the filter’s amplitude response across frequency but the phase and time response of the filter as well. Each filter stage has a specific damping frequency fd and Q specified such that the overall product of the individual filter stages results in the desired response characteristics. It can be seen that progressively higher Q results in increased peaking around the damping frequency fd . Other effects are a more aggressive change in the phase per unit frequency in the region approaching fd and increased settling times for pulses and input step functions. These characteristics need to be considered as part of the active filter requirements. Adapted from Google Images: gnuplot demo script: multiplt.dem

Filter sections are cascaded to produce the intended response the cutoff frequency fc , gain, roll-off, etc is the product of all stages 1 dB Chebyshev, 6th-order LP filter, fc = 10 kHz, 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 kHz, with a pass-band gain of 10 V/V The 10 kHz pass-band bandwidth is defined by the 1 dB ripple stage 3 gain was manipulated to be low because of its high Q Doing so relaxes the GBW requirement – more on this later The goal of any filter design effort is achieve a specific response result . The filter cutoff frequency, roll-off rate, gain, phase, delay and impulse responses are considered; some of these may have particular importance in the intended application. Often, the final filter will involve more than one filter stage and the final filter response is achieved by cascading a required number of individual stages. The final overall response is the product of their individual and unique fn, Q, gain, etc, characteristics.

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-pass, band-reject and all-pass responses Active filter synthesis programs such as FilterPro V3.1 and Webench Active Filter Designer (beta) are available for free, from Texas Instruments All you need to provide are the filter pass-band and stop-band requirements, and gain requirements The programs automatically determine the filter order required to meet the stop-band requirements FilterPro 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, multi- amplifier topologies suitable for low sensitivity, and/or high-gain, high-Q filters Modern active filter synthesis programs such as TI’s FIlterPro lessen much the difficulties that the occasional, or novice filter designer would encounter. The designer must have some idea about the required gain, pass-band and stop-band characteristics and topology. The programs walk the user through the steps of entering the requirements. Once the selections are completed the filter is synthesized. No real knowledge about the filter fundamentals and mathematics is required. It is mostly just a matter of making the right decisions so that the needed filter response is attained.

Operational amplifier Gain-bandwidth (GBW) product

Operational amplifier gain-bandwidth product an important ac parameter for attaining accurate active filter response The Gain-Bandwidth product is a specification that allows you to calculate the closed loop bandwidth with a simple equation. The Gain- Bandwidth product is the closed-loop gain, multiplied by the bandwidth (Gain- Bandwidth = Gain x Bandwidth). This equation can be rearranged to find the bandwidth given a closed loop gain. Note that the open loop gain normally decreases by 20dB/decade over the range that the Gain-Bandwidth is specified. This is true for most operational amplifiers. Some operational amplifiers may have a Gain-Bandwidth that is specified as being different depending on the closed-loop gain. The Unity-Gain Bandwidth is the point on the open-loop gain curve where the gain passes though 0 dB (1 V/V). The Gain-Bandwidth-Product and the Unity-Gain Bandwidth are the same only when the gain/phase response is a single-pole system (at least until the open-loop gain crosses the 0dB line). However, when a second-pole effects the gain response above 0dB, the Unity-Gain and Gain-Bandwidth product are not the same.

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 op-amps required to support the filter function are selected based on the worst-case section’s GBP i.e. the section that has the highest GBP. Although individual op-amps could be selected to meet the minimum requirements of each stage that isn’t usually desired or even practical. Often, a dual or quad op-amp is used in the filter and one is selected that will meet the requirements of all the filter sections.

The operational-amplifier gain-bandwidth requirements
TI’ s FilterPro 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 dB) is a loop gain factor TI’s Filter Pro provides a key indicator of the bandwidth required for each filter section. It is referred to as the Gain-Bandwidth Product (GBP). It consists of the product of the pass-band gain, the section’s natural frequency and quality factor “Q.” The GBP for each section is listed with these factors for each stage in a filter solution synthesized by FilterPro. The “Q” is a damping constant determining the selectivity of the frequency response around fn. The Q indicates energy loss relative to the amount of energy stored within the system. Thus, the higher the Q the lower the rate of energy loss and hence oscillations will reduce more slowly, i.e. they will have a low level of damping and they will ring for longer.1 1 Q Quality Factor Tutorial, from Radio-Electronics.com

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 * equivalent noise gain ACL For example, select AOL to be ≥100∙ACL at fn for ≤1% gain error 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 The table shows the closed-loop gain error that results from limited open-loop gain. The particular application will dictate the acceptable maximum gain error. For example if a gain error of 1% is acceptable, then the ratio of open-loop gain (Aol) to closed-loop gain (Acl) should be no less than 100 at the worst-case fn. Higher gain accuracy will require a higher Aol to Acl ratio. Note that the Acl is the noise gain Acl value for the filter section. The noise gain Acl is the non-inverting stage Acl for a non-inverting filter section, and the gain viewed as driving the non-inverting input with an inverting filter stage. i.e. an inverting filter stage with a gain of -1 V/V has a gain when viewed from the non-inverting input as having a gain of +2 V/V. The Aol/Acl ratio must be based on the noise gain Acl.

The operational-amplifier gain-bandwidth requirement an example of the FilterPro estimation
Let FilterPro estimate the minimum GBW for a 5th-order, 10 kHz (fc) low-pass filter having a Chebyshev response, 2 V/V gain and a 3 dB pass-band ripple FilterPro’s GBW estimation for the worst-case stage yields: GBW = G ∙ fn ∙ Q ∙ 100 GBW = (2V/V)(10kHz)(8.82)(100) = 17.64MHz vs MHz from the precise determination – see Appendix for details FilterPro estimates the worst-case, or maximum stage Q for any active filter that it synthesizes. The GBW is calculated based on the filter selections entered. A more exhaustive treatment on minimum GBW determination is provided in the appendix. Despite the increased complexity in deriving the GBW using a more precise method, the results are almost identical to that using the simpler method used by FilterPro.

Operational amplifier gain-bandwidth effects the Sallen-Key topology
The operational amplifier gain- bandwidth (GBW) affects the close-in response It also affects the ultimate attenuation at high frequency Op-amp fH Hz dB OPA k OPA k OPA k OPA k The a.c. gain-bandwidth product (GBW) of an operational amplifier has a profound affect on the filter response. It is one of the most critical parameters in support of assuring correct filter response. Therefore, the gain-bandwidth must be taken into consideration when designing a high performance active filter. There are two effects that become evident when an operational amplifier having too low of gain-bandwidth is applied in a filter, 1) The response becomes distorted at and around the cutoff frequency (fC) even affecting the roll-off frequency and 2) the high frequency roll-off fails to ever continue flattening off at some frequency within decades of fC. The latter effect can severely limit the filter’s ultimate rejection outside the pass-band. FilterPro recommended a minimum GBW of 7.1 MHz for this 10kHz, 2nd-order Butterworth low-pass filter (Av = +10 V/V). The amplitude vs. frequency plot shows how the close-in response and ultimate filter attenuation are affected by the operational amplifier’s GBW. Different amplifiers having a value within the range of 1.2 to 11 MHz are provided. The close-in response distortion and ultimate attenuation improve as the GBW moves higher for the Sallen-Key topology. FilterPro 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 multiple-feedback topology is much less affected by low GBW than the Sallen-Key topology. The filter response within the pass-band and a decade beyond the cutoff frequency is surprisingly accurate . Only beyond that point are response curves beginning to show some deviation from the ideal -40 dB/dec roll-off rate. The higher the GBW amplifier the more close to ideal is the roll-off response. The lowest GBW device (1.2 MHz) produces a gain deviation about dB 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 FilterPro software: Gain = 1 V/V Butterworth response (Q = 0.71) The Sallen-Key topology was selected The FilterPro and TINA-TI simulations using ideal operational amplifiers models produced ideal results Note FilterPro recommended an operational- amplifier with a gain-bandwidth product (GBW) of kHz A customer made an inquiry to the TI Precision Amplifiers E2E forum regarding a 50 Hz, low-pass filter they had synthesized using TI’s FilterPro program. The filter requirements were simple enough requiring a 2nd-order , Butterworth low-pass response and having a gain of 1 V/V in the flat portion of the pass-band. Everything looked fine on the TINA plots, and when the filter was simulated using the ideal operational-amplifier model in TINA-TI the Bode plot results were just as expected.

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 dB/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 OPA369 does meet the minimum GBW specified by FilterPro, kHz . Its GBW is about 8 to 10 kHz This is a practical, Sallen-Key implementation of a 50 Hz, 2nd-order, low-pass filter. However, the overall low-pass response is not as planned. The response is correct in the low frequency passband, at the 50 Hz, -3 dB cutoff frequency and for about a decade out to 500 Hz. The roll-off in that region is the normal 20 dB/dec. Once the frequency moves beyond about 500 Hz the attenuation response begins to bottom out at about 775 Hz and then turns upward indicating less attenuation as the frequency is further increased. Normally, the roll-off would have continued to some high attenuation beyond -100 dB, or more until practical circuit limitations occurred. The OPA369 has sufficient GBW (~8 to 12 kHz) to support this filter’s requirements. Something else is work causing this corrupted response. GBW ~8 kHz

The real operational amplifier can have a complex open-loop output impedance Zo
For the OPA369 FET Drain output stage Zo: Changes with output current is low, <10 Ω and resistive below 1 Hz increases from tens-of-ohms to tens, or hundreds of kilohms, from 10 Hz to 10 kHz Is complex, resitive plus inductive (R+jX), from 1 Hz to 10 kHz Becomes resistive again above 10 kHz, the unity gain frequency The hi-Z behavior is reduced by closing the loop but Zo still alters the expected filter response R+j0 R+jX The OPA369 features a P-channel/N-channel MOSFET output stage where the drains are connected together and are the output for the operational amplifier. This arrangement allows for a an output swing range that approaches the amplifier supply rail voltage levels. This is ideal for applications that require that feature, but the MOSFET configuration results in a complex, open-loop output impedance (Zo) that is a function of the output current level and applied frequency. This complex Zo may appear resistive at some frequencies and may contain a series inductive or capacitive reactance besides a resistance (R±jX) at other frequencies. This can be seen in the graph of OPA369 R+j0 Unity gain

Net affect on response due to operational-amplifier complex Zo a result similar to low GBW fold-back, but with added peaking OPA369 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 The very high and complex output impedance of the drain-drain (or collector-collector) output stages may not only exhibit the fold-back characteristic described earlier, but also an odd gain peaking characteristic in the stop-band. It can even be so bad as to produce a resonant peak where the gain exceeds that in the pass-band. The high, complex output impedance can be reduced by allowing some load current to flow through the output stage. One way is to add a load resistor from the amplifier’s output to one of the supply rails. Most often, the amplifier’s voltage and current offset will produce enough voltage offset at the output to allow some current flow through the added load resistor. The graph indicates that the lower the load resistance is for the given output offset, the more attenuated the peaking becomes. For the example shown, the peaking is gone by time the load resistance drops to 1 kΩ. There is the possibility that an operational amplifier may have very low output offset and then little to no current would flow through the load. The complex output impedance may still be high but the load resistance is still present shunting it. Do note that adding the load resistance to the output could result in the circuit consuming more current than was originally planned.

Active filter sensitivity to source impedance and components

The effect of source impedance on filter response
5 kHz 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 Most active filters designs assume the source impedance is zero. Chances are in real active filter implementations the source impedance will be something higher than zero ohms. Fortunately, most operational amplifier circuits that precede an active filter will have a very low, closed-loop output impedance where it may be fractions of an ohm or a little higher. However, if the active filter is intended to follow a sensor or transducer those may provide a high source impedance. If that is the case, that impedance will interact with the filter input components altering the intended filter response. An example provided here are 5 kHz Butterworth, low-pass filters based on both the Sallen-Key and multiple-feedback topologies. They are individually driven an input source having a non-zero impedance. The effects are those non-zero impedances are revealed in the following amplitude vs. frequency plot.

The affect of source impedance on filter response 5 kHz Butterworth Low-pass, G = 10 V/V
MFB Rs = 1000, Av = 3.4 V/V MFB Rs = 500, Av = 5.1 V/V MFB Rs = 250, Av = 6.7 V/V MFB Rs = 0, Av = 10 V/V Sallen-Key SK Av = 10 V/V MFB The amplitude vs. frequency plots for the Sallen-Key and MFB 5 kHz low-pass filters is plotted. The source impedance is varied from 0 ohms to 1 kilohm, in 250 ohm steps. The affects on the gain and cutoff characteristics are quite different for the two topologies. The Sallen-Key appears to be more tolerant of increasing source impedance than the MFB. The Sallen-Key amplitude does deviate from the Butterworth response as the source impedance increases. The cutoff frequency lowers and there is some peaking. Essentially, the Q and fn of the stage have been altered. The MFB reacts badly to the increasing source impedance resulting in high gain pass-band gain errors. Like the Sallen-Key its cutoff frequency moves but in this case higher. Again the fn and Q of the stage is being altered. The net result is the higher source impedance changes the filter response characteristics from the expected.

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 C0G 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
Center Freq variance Gain 5% resistors 5% capacitors Δ 5 kHz Δ 1.7 dB 0.1% resistors 2% capacitors Δ 1 kHz Δ 0.3 dB The affects of component tolerance on active filter performance can be evaluated using modern circuit simulation software such at TINA. Each passive component’s tolerance can be individually set to a range such as +/1% with a particular distribution such as a Gaussian distribution. Once the tolerance are set a worst-case analysis can be selected within the software. An example of a 25 kHz , MFB Bessel band-pass filter having as a Q of 10 and a gain of 5 V/V (14 dB) is shown. Two specific cases are tested with the simulator; the first, with both resistor and capacitor tolerances set to +/-5%, and a second with resistor tolerances of +/-0.1% and capacitors with +/-2% tolerances. The table summarizes the center frequency and gain variances for these loose and tight tolerance conditions. A similar simulation could be accomplished concerning the filter response characteristics over temperature. A temperature coefficient can be assigned to the passive components. The operational amplifier simulation model may have some temperature characteristics modeled, but often the important GBW parameter isn’t especially sensitive to temperature.

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 In order to compare the total output noise of the two filter topologies, the noise gain of the circuits must first be determined. “Noise gain” is a term used to describe the amount of amplification that will be applied to the intrinsic noise sources of the amplifier (e.g. input voltage noise and current noise). Because there are differences in the filter topologies themselves, multiple feedback and Sallen-Key filters actually have different noise gains. This fact indicates for filters that have identical transfer functions (same filter order and corner frequency) but are constructed using different topologies will produce different levels of output noise. Tina-TI is a useful tool to determine the noise gain in a particular circuit. An AC source is inserted in series with the non-inverting input and the standard input to the circuit is grounded. By plotting the voltage at the output of the circuit using an AC transfer characteristic analysis with the circuit configured in this manner we can produce a graph of the circuit’s noise gain over frequency. Measuring the noise gain of a MFB low pass filter

Noise Gain Comparison FilterPro was used to design 2, 1kHz 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 1Hz to 1MHz Within the passband, the MFB filter has 6dB higher noise gain This is because it is an inverting topology Noise gain above the corner frequency quickly decreases The noise gain for both circuits peaks at the corner frequency of the filter Sallen-Key Multiple Feedback To demonstrate this concept, Texas Instruments FilterPro was used to design two filters with identical transfer functions (Butterworth second order lowpass filters) but with alternate topologies (one Sallen-Key, one multiple feedback as is shown at the top of the slide). The noise gain of the two circuits was measured over bandwidth from 1Hz to 1MHz and is shown below the two circuits. Because the multiple feedback filter is an inverting topology, filters of this type will show a 6dB higher noise gain within their passband than Sallen-Key filters with the same signal gain. Although noise above the corner frequency is quickly attenuated it is important to notice that the noise gain of the circuit is at its highest level at the filter corner frequency and this warrants further examination.

Noise Gain at the Filter Corner Frequency
The magnitude of the noise gain peak is dependant upon the Q of the filter Higher Q filters have higher peaking in their noise gain. 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 1kHz Sallen-Key lowpass filters of different Q’s OPA827 simulation model 100 Ohm resistors used in all circuits (only capacitors changed) The amount of additional noise gain at the filters corner frequency is proportional to the Q of the filter. That is to say, higher Q filters will have a higher noise gain at their corner frequency. To illustrate this point, three Sallen-Key lowpass filters were simulated, all with a corner frequency of 1kHz but of differing Q values (Bessel, Butterworth, and Chebyshev). Notice that the Chebyshev filter, which has a Q of exhibits the highest noise gain at its corner frequency, about 9dB. Conversely, the Bessel filter, with a Q of 0.58 has less than half the noise gain at the corner frequency, around 4.3dB of gain. This peak in noise gain can significantly effect the total output noise of the filter if the measurement bandwidth (the bandwidth over which noise is integrated) is not much larger than the filter’s passband. As an example, the table shows the total integrated noise of the three Sallen-Key filters when constructed with an OPA827 and low value resistors (100 ohms). For a low bandwidth of integration (2kHz) the Chebyshev filter’s integrated noise is MUCH higher than the other two filter topologies. However, as the bandwidth of integration is increased the total noise of all three filters converge to approximately the same value. Topology Q Noise Voltage (2kHz Bandwidth) Noise Voltage (20kHz Bandwidth) Noise Voltage (200kHz Bandwidth) Bessel 0.58 249.8 nVrms 615 nVrms 1.733 uVrms Butterworth 0.707 303.1 nVrms 628.4 nVrms 1.737 uVrms Chebyshev 1dB 0.957 393.1 nVrms 693.7 nVrms 1.762 uVrms

Noise considerations in an active filter an OPA376 inverting amplifier is compared in a 2nd-order low-pass Some of the noise seen at the amplifier output is generated internally by the operational amplifier while the rest is generated by the passive components connected to it. However, all of the noise can be referred to the non-inverting input of the operational amplifier as a series of noise generators connected in series with that input. Therefore, the filter circuit acts upon the circuit-generated noise spectrum with the intended filter function. The filter response for the active filter circuit internally generated noise does show some deviation from the ideal Butterworth 2nd-order response. This is likely due to the way the energy is distributed in the noise source. frequency Amp en (nV/√Hz) Filter ratio 10 kHz 91 119 0.76 : 1 100 kHz 89 15 6 : 1 1 MHz 42 7.4 5.6 : 1

Noise considerations of an active filter an OPA376 inverting amplifier is compared in a 2nd-order band-pass application For Q = 10 frequency Amp en (nV/√Hz) Filter ratio 1 kHz 93 23 4 : 1 10 kHz 91 1390 1 :15 100 kHz 90 16 5.6 : 1 1 MHz 42 8.2 5.1 : 1

Total Harmonic Distortion and Noise Review
Total Harmonic Distortion and Noise (THD+N) is a common figure of merit in many systems 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 Harmonics of the fundamental arise from non-linearity in the circuit’s transfer function. Integrated circuits AND passive components can cause this Intrinsic noise is created in integrated circuits and resistances Fundamental Harmonics Noise Total harmonic distortion and noise (THD+N) is a figure of merit that seeks to quantify the amount of unwanted content added to a signal by a circuit’s noise and non-linearity. This quantity can be expressed as a ratio of the summation of the harmonics of the fundamental and the system RMS noise voltage to the RMS voltage of the fundamental (equation shown). Harmonics, or signals at frequencies that are integer multiples of the input signal, arise from non-linear behavior of passive components and integrated circuits. Noise is produced by the intrinsic noise of integrated circuits, the thermal noise of resistors, or may be coupled into a circuit by extrinsic sources. 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

Distortion from Passive Components
A 1kHz Sallen-Key lowpass filter was built using an OPA1612, and replaceable passive components. Component values were chosen such that both C0G and X7R capacitors were available Thin Film resistors in 1206 packages were used An Audio Precision SYS was used to determine the effects of capacitor type on measured THD+N THD+N was measured from 20Hz to 20kHz Harmonic content of a 500 Hz sine wave was also compared THD+N is noise dominated Increases as the filter attenuates the signal Active filter circuits, anti-aliasing filters for data converters, and feedback capacitors in amplifiers are examples of circuits where the use of a high-k MLCC may introduce distortion. In order to illustrate this effect, a 1kHz Butterworth active lowpass filter using the Sallen-Key topology was designed using Texas Instruments FilterPro software. Active filters are a very common application where distortion from capacitors will degrade the overall circuit performance because many designers choose low resistor values in an effort to reduce their contribution to the output noise. This increases the value of capacitors required for a certain corner frequency and high-k MLCCs may be the only capacitors available which meet the requirements for capacitance, board area, and cost. The filter circuit shown incorporates passive component values which allow capacitors C1 and C2 to be replaced with MLCCs of varying dielectric types and package sizes; facilitating direct comparison of measurements between different capacitor types. THD+N measurements were made on the filter circuit for a 1 Vrms signal using an Audio Precision SYS-2722 over the frequency range of 20Hz to 20kHz with a measurement bandwidth of 500kHz. MLCCs of the C0G dielectric type in 1206 packages delivered exceptional performance with the measured THD+N in the filter’s passband (red curve) being at the noise floor of the measurement system. C0G capacitors in 0805 packages were also tested and provided the exact same level of performance and were removed from the graphs for simplicity. The increase in THD+N above the filter’s corner frequency (filter transfer function is shown in green) is expected as the filter’s attenuation decreases the ratio of the signal’s amplitude to the noise floor.

Capacitor Dielectric Effects
1206 C0G capacitors were replaced with 1206 X7R capacitors and the THD+N was re-measured Signal level was 1Vrms All caps are 50V rated Minimum of 15dB degradation of THD+N inside the filter’s passband Maximum of almost 40dB degradation of THD+N The spectrum of a 500Hz sine wave was also compared X7R shows a large number of harmonics Odd order harmonics dominate the spectrum By changing the capacitors to X7R types in 1206 packages, an immediate degradation of the circuit’s performance is observed. The THD+N has been increased by a minimum of 15 dB at 20Hz, and peaks in the region between 400 and 800 Hz where a 35dB increase in THD+N can be seen. Examining the spectrum of a 500Hz sine wave shows a large number of harmonics of the fundamental with odd order harmonics being dominant (the red curve on the spectrum is C0G, Blue is X7R).

Package Size Effects 1206 X7R capacitors were replaced with 0603 X7R capacitors and distortion was measured again Signal level was 1Vrms All tested capacitors have a 50V rating Distortion increases for smaller package sizes! The spectrum of a 500Hz sine wave was again examined Both odd and even order harmonics dominate Odd order harmonics still dominate 0603 capacitors produce harmonics above 20kHz! These effects are further increased by an additional 10dB over much of the spectrum by switching C1 and C2 to X7R capacitors in 0603 packages. Again reviewing the spectrum for a 500Hz sine wave we can see that even more harmonic content has been added to the signal. It may be surprising that when the circuit was constructed with 0603 X7R capacitors, harmonics above 20kHz were observed for a 500Hz input signal. This demonstration confirms that distortion from capacitors increases for smaller package sizes.

Capacitor Distortion and Signal Level
As previously mentioned, capacitor distortion increases with electric field intensity Worse at signal levels Worse for smaller packages Changing the signal level is a simple way to determine the source of distortion If the circuit is noise dominated the plot will have a slope (m) of: Distortion from passive components will INCREASE with higher signal levels Faced with the task of tracking down the source of high levels of distortion, it may not be immediately intuitive to an engineer whether an integrated circuit or passive components are at fault. One method to determine the dominant source of distortion is to measure the THD+N of the circuit over a wide range of signal levels as is shown. Here the THD+N of the Sallen-Key filter in Figure 1 is measured for a 500Hz fundamental with a signal level from 1mVrms to 10Vrms. When the circuit was constructed with C0G capacitors, the THD+N decreased for increasing signal levels, eventually reaching the noise floor of the measurement system at a signal level of 2Vrms. The negative slope of the line indicates that the noise of the circuit, due to the operational amplifier and resistors, is the dominant factor in the THD+N calculation. The measured THD+N will decrease with increasing signal levels because the ratio of the signal voltage to the noise voltage is improved. Conversely, non-linearities in passive components are exacerbated at higher signal levels and should cause a rising trend in distortion for increasing signal levels [3]. This was confirmed when the capacitors in the filter circuit were replaced with those of the X7R type. X7R capacitors in 0603 packages show rising distortion beginning at signal amplitudes of 20 mVrms. X7R capacitors in 1206 packages exhibited similar behavior with the rising trend in distortion beginning at 40 mVrms.

Capacitor Distortion Over Frequency
Tina-TI was used to measure the voltage across each capacitor over frequency The sum of the two voltages is plotted in green Diagram below indicates measurement points The maximum voltage appears below the corner frequency This also correlates well to the measured peak in distortion Relating the voltage drop across a passive component to the measured distortion over frequency can determine the specific passive components that are introducing distortion. The AC transfer characteristic analysis in Tina-TI can be used to plot the voltage across a component in a circuit as a function of frequency. The upper graph shows the voltage across capacitors C1 and C2 over the frequency range from 20Hz to 20kHz. By combining the voltages using a root sum of squares method, it becomes apparent that the combined voltage across the capacitors is at a maximum at approximately 600Hz. This peak in capacitor voltage correlates very well to the maximum point of distortion in the lower figure. If the region of maximum distortion did not coincide with maximum voltages on the capacitors the analysis could be repeated for other components in the circuit.

Putting it all together

Achieving optimum active filter performance
Capacitors Use quality C0G or film dielectric for low distortion Type C0G 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 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 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+jX 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 OPA369 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 2nd-order stage in an nth-order filter: GBW = 100•ACL(fc/ai)√[(Qi2-0.5)/(Qi2-0.25)] Where: 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 Example: Determine the recommended GBW for a 5th-order, 10 kHz (fc) low-pass filter having a Chebyshev response, 2 V/V gain and a 3 dB 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, fC = 10 kHz Coefficients from table: ai = , Qi = 8.82 GBW = 100 ACL(fC/ai)√[(Qi2-0.5)/(Qi2-0.25)] GBW = 100 2(10kHz/0.1172)√[( )/( )] GBW = 16.94MHz

Successful application of Active Filters

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