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Texas Instruments OA-27 Low-Sensitivity, Lowpass Filter Design (Rev. B) Application notes
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Application Report
SNOA372B – August 1996 – Revised April 2013
OA-27 Low-Sensitivity, Lowpass Filter Design
.....................................................................................................................................................
ABSTRACT
This application report covers the design of a Sallen-Key (also called KRC or VCVS [voltage-controlled,
voltage-source]) lowpass biquad with low component and op amp sensitivities. This method is valid for
either voltage-feedback or current-feedback op amps. Basic techniques for evaluating filter sensitivity
performance are included. A filter design example illustrates the method.
Contents
Introduction .................................................................................................................. 2
KRC Lowpass Biquad ...................................................................................................... 2
Design Example ............................................................................................................. 5
3.1
Overall Design ...................................................................................................... 5
3.2
Section A Design ................................................................................................... 6
3.3
Section B Design ................................................................................................... 6
4
SPICE Models ............................................................................................................... 8
5
Summary ..................................................................................................................... 8
6
References ................................................................................................................... 9
Appendix A
Transfer Function Examples .................................................................................... 10
Appendix B
Biquad Section, s Term in the Denominator That Includes K .............................................. 11
1
2
3
List of Figures
1
Lowpass Biquad ............................................................................................................. 2
2
Lowpass Filter ............................................................................................................... 5
3
Monte-Carlo Simulation Results
..........................................................................................
8
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Introduction
1
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Introduction
Changes in component values over process, environment and time affect the performance of a filter. To
achieve a greater production yield, we need to make the filter insensitive to these changes. This
application report presents a design algorithm that results in low sensitivity to component variation.
Lowpass biquad filter sections have the transfer function:
(1)
where, s=jω, Ho is the DC gain, ωp is the pole frequency, and Qp is the pole quality factor. Both ωp and Qp
affect the filter phase response, ωp the filter cutoff frequency, Qp the peaking, and Ho the gain. For these
reasons, we will minimize the sensitivities of Ho, ωp and Qp to all of the components (see Appendix A).
To achieve the best production yield, the nominal filter design must also compensate for component and
board parasitics. For information on filter component pre-distortion, see [5]. SPICE simulations, with good
component and board models, help adjust the nominal design point to compensate for parasitics.
For an overview of sensitivity analysis, with applications to filter design, see Appendix A. For useful
sensitivity properties and formulas, see Appendix B. For a more complete discussion of sensitivity
functions, their applications, and other approaches to improving the manufacturing yield of your filter, see
the references listed in Section 6.
2
KRC Lowpass Biquad
The biquad shown in Figure 1 is a Sallen-Key lowpass biquad. VIN needs to be a voltage source with low
output impedance. R1 and R2 attenuate VIN to keep the signal within the op amp’s dynamic range. The
Thevenin equivalent of VIN, R1 and R2 is a voltage source αVIN, with an output impedance of R12, where:
α = R2/(R1 + R2)
R12 = (R1 || R2)
The input impedance in the passband is:
ZIN = R1 + R2, ω << ωp
The transfer function is:
(2)
Figure 1. Lowpass Biquad
2
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KRC Lowpass Biquad
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To achieve low sensitivities, use this design algorithm:
1. Partition the gain for good Qp sensitivity and dynamic range performance:
(a) Use a low noise amplifier before this biquad if you need a large gain.
(b) Select K for good sensitivity with this empirical formula:
(3)
(c) These values also reduce the op amp bandwidth’s impact on the filter response, and increase the
bandwidth for voltage-feedback op amps. When Qp ≥ 5, the sensitivities of this biquad are very high
(d) Set α as close to 1 as possible while keeping the signal within the op amp’s dynamic range
2. Select an op amp with adequate bandwidth (f3 dB) and slew rate: (SR):
(a) f3 dB ≥ 10fc
(b) SR > 5fc Vpeak
(c) where fc is the corner frequency of the filter, and Vpeak is the largest peak voltage. Make sure the op
amp is stable at a gain of Av = K.
3. Select Rf and Rg so that:
(a) K = 1 + Rf/Rg
(b) For current-feedback op amps, use the recommended value of Rf for a gain of Av = K. For voltagefeedback op amps, select Rf for noise and distortion performance.
4. Initialize the resistance level
This value is a compromise between noise performance,
distortion performance, and adequate isolation between the op amp outputs and the capacitors.
5. Initialize the capacitance level
and the capacitors:
(a) C = 1/(Rωp)
(b) r2 = 0.10
the resistor ratio (r2 = R12 / R3), the capacitor ratio (c2 = C4/C5)
(4)
(c) C4 = cC
(d) C5 = C/c
6. Set the capacitors C4 and C5 to the nearest standard values.
7. Recalculate C, c2, R and r2:
(a)
(b) c2 = C4/C5
(c) R = 1/(Cωp)
(5)
8. Calculate R12 and the resistors
(a) R12 = rR
(b) R1 = R12/α
(c) R2 = R12/(1–α)
(d) R3 = R/r
VIN can represent a source driving a transmission line, with R1 and R2 the source and terminating
resistances. For this type of application, make these modifications to the design algorithm:
• Select R1 and R2 to properly terminate the transmission line (R1 includes the source resistance)
• Calculate α and R12
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KRC Lowpass Biquad
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Adjust C and R so that R12 = rR
To evaluate the sensitivity performance of this design, follow these steps:
1. Calculate the resulting sensitivities:
Reducing
lowers the biquad’s sensitivity to the op amp bandwidth.
2. Calculate the relative standard deviations of Ho, ωp and Qp:
In this formula, use:
(a) The nominal values of Ho, ωp and Qp for X
(b) The nominal values of R1, R2, R3, Rf, Rg, C4 and C5 for αi (do not use K since it is not a component)
(c) The capacitor and resistor standard deviations for σα . For parts with a uniform probability
distribution,
i
(6)
3. If temperature performance is a concern, then estimate the change in nominal values of Ho, ωp and Qp
over the design temperature range:
In this formula, use:
(a) The nominal values, at room temperature, of Ho, ωp and Qp for X
(b) The nominal values, at room temperature, of R1, R2, R3, Rf, Rg, C4 and C5 for αi (do not use K since
it is not a component)
(c) The nominal resistor and capacitor values at temperature T for αi(T)
4. Estimate the probable ranges of values for Ho, ωp and Qp:
(a) X ≥ (1−3 • σx/X) • min(X(T))
(b) X ≤ 1+3 • σx/X) • max(X(T))
(c) where X is Ho, ωp and Qp
4
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Design Example
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3
Design Example
The circuit shown in Figure 2 is a third order Chebyshev lowpass filter. Section 3.2 is a buffered single
pole section, and Section 3.3 is a lowpass biquad. Use a voltage source with low output impedance, such
as the CLC111 buffer, for VIN.
Figure 2. Lowpass Filter
The nominal filter specifications are:
fc = 500 MHz (passband edge frequency)
fs = 100 MHz (stopband edge frequency)
Ap = 0.5 dB (maximum passband ripple)
As = 19 dB (minimum stopband attenuation)
Ho = 0 dB (DC voltage gain)
The third order Chebyshev filter meets our specifications (see References [1] through [4]). The resulting
−3 dB frequency is 58.4 MHz. The pole frequencies and quality factors are:
Section
3.1
A
B
ωp/2π [MHz]
53:45
31:30
Qp
1.706
—
[ ]
Overall Design
1. Restrict the resistor and capacitor ratios to:
(a) 0.1 ≤ r2 ≤ 10
(b) 0.1 ≤ c2 ≤ 10
2. Use 1% resistors (chip metal film, 1206 SMD, 25 ppm/°C)
3. Use 1% capacitors (ceramic chip, 1206 SMD, 100 ppm/°C)
4. Use standard resistor and capacitor values
5. The temperature range is −40°C to 85°C, and room temperature is 25°C
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Design Example
3.2
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Section A Design
1. Use the CLC111. This is a closed-loop buffer.
(a) f3 db = 800 MHz > 10 fc = 500 MHz
(b) SR = 3500V/µs, while a 50 MHz, 2Vpp sinusoid requires more than 250V/µs
(c) Cni(111) = 1.3 pF (input capacitance)
2. We selected R1A for noise, distortion and to properly isolate the CLC111’s output and C2A. The
capacitor C2A then sets the pole frequency:1/ωp = R1AC2A. The results are in the table below:
(a) The Initial Value column shows values from the calculations above
(b) The Adjusted Value column shows the component values that compensate for Cni(111) and for the
CLC111’s finite bandwidth (see Comlinear’s application report on filter component pre-distortion [5])
(c) The Standard Value column shows the nearest available standard 1% resistors and capacitors
Value
3.3
Component
Initial
Adjusted
Standard
R1A
108Ω
100Ω
100Ω
C2A
47 pF
47 pF
47 pF
Cni(111)
—
1.3 pF
1.3 pF
Section B Design
Set αB = Ho/KB = 0.667.
1. The recommended value of KB for Qp = 1.706 is:
2. Use the CLC446. This is a current-feedback op amp
(a) f3 dB = 400 MHz ≈ 10 fc = 500 MHz
(b) SR = 2000V/µs > 250V/µs (see Item #1 in “Section A Design”)
(c) Cni(446) = 1.0 pF (non-inverting input capacitance)
3. Set RfB to the CLC446’s recommended Rf at AV = +15:
(a) RfB = 348Ω
(b) Then set RgB = 696Ω so that KB = 1.50.
4. Initialize the resistor level for noise and distortion performance:
(a) R ≈ 200Ω
5. Initialize the capacitor level, resistor and capacitor ratios, and the
capacitors:
(a) r2 ≈ 0.10
(b) c2 ≈ max (0.0983, 0.10) = 0.1000
(c) C4B ≈ 4.7 pF
(d) C5B ≈ 4.7 pF
6. Set the capacitors to the nearest standard values:
(a) C4B = 4.7 pF
(b) C5B = 4.7 pF
7. Recalculate the capacitor level and ratio, and the resistor level and ratio:
6
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Design Example
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8. Calculate R12B and the resistor values:
(a) R12B = 64.0Ω
(b) R1B = 96.0Ω
(c) R2B = 192Ω
(d) R3B = 627Ω
(e) The resulting component values are:
Value
Component
Initial
Adjusted
Standard
R1B
96.0Ω
78.9Ω
78.7Ω
R2B
192Ω
158Ω
158Ω
R3B
627Ω
582Ω
576Ω
C4B
4.7 pF
3.7 pF
3.6 pF
Cni(446)
—
1.0 pF
1.0 pF
C5B
47 pF
47 pF
47 pF
RfB
348Ω
348Ω
348Ω
RgB
696Ω
696Ω
698Ω
ωp
Sαi
Qp
Sαi
9. The sensitivities for this design are:
Ho
Sαi
αi
K
1.00
0.00
2.58
R1B
−0.33
−0.33
0.79
R2B
0.33
−0.17
0.40
R3B
0.00
−0.50
−1.19
RfB
0.33
0.00
0.86
RgB
−0.33
0.00
−0.86
C4B
0.00
−0.50
−1.36
C5B
0.00
−0.50
1.36
10. The relative standard deviations of Ho, ωp and Qp are:
(a) σH /Ho ≈ 0.38%
(b) σω /ωp ≈ 0.55%
(c) σQ /Qp ≈ 1.58%
These standard deviations are based on a uniform distribution, with all resistors and capacitor
values being independent:
o
p
p
(7)
11. The nominal values of Ho, ωp and Qp over the design temperature range are:
−40
25
85
[V/V]
1.000
1.000
1.000
ωp/2π [MHz]
53.88
53.45
53.00
Qp
1.706
1.706
1.706
T
Ho
[°C]
[ ]
12. The probable ranges of values for Ho, ωp and Qp, over the design temperature range, are:
(a) 0.99 ≤ Ho ≤ 1.01
(b) 52.1 MHz ≤ (ωp/2π) ≤ 54.8 MHz
(c) 1.63 ≤ Qp ≤ 1.79
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SPICE Models
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13. Based on the results in #10 and #12, we can conclude that:
(a) The DC gain and cutoff frequency change little with component value and temperature changes
(b) Qp has the greatest sensitivity to fabrication changes
(c) The greatest filter response variation is in the peaking near the cutoff frequency
Figure 3 shows the results of a Monte-Carlo simulation at room temperature, with 100 cases simulated.
These simulations used the “Standard Values” of the components. The gain curves are:
• Lower 3-sigma limit (mean minus 3 times the standard deviation)
• Mean value
• Upper 3-sigma limit (mean plus 3 times the standard deviation)
Figure 3. Monte-Carlo Simulation Results
4
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.
It is recommended simulating with Comlinear’s SPICE models 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.
5
Summary
This application report contains an easy to use design algorithm for a low sensitivity, Sallen-Key lowpass
biquad, which works for Qp < 5. It also shows the basics of evaluating filter sensitivity performance.
Designing for low ωp and Qp sensitivities gives:
• Reduced filter variation over process, temperature and time
• Higher manufacturing yield
• Lower component cost
A low sensitivity design is not enough to produce high manufacturing yields. The nominal design must also
compensate for any component parasitics, board parasitics, and op amp bandwidth (see Comlinear’s
application report on filter component pre-distortion [5]). The components must also have low enough
tolerance and temperature coefficients.
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References
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6
References
1. R. Schaumann, M. Ghausi and K. Laker, Design of Analog Filters: Passive, Active RC, and Switched
Capacitor. New Jersey: Prentice Hall, 1990.
2. A. Zverev, Handbook of FILTER SYNTHESIS. John Wiley & Sons, 1967.
3. A. Willaims and F. Taylor, Electronic Filter Design Handbook, McGraw Hill, 1995.
4. S. Natarajan, Theory and Design of Linear Active Networks. Macmillan, 1987.
5. OA-21 Component Pre-Distortion for Sallen Key Filters (SNOA369)
6. K. Antreich, H. Graeb, and C. Wieser, “Circuit Analysis and Optimization Driven by Worst-Case
Distances,” IEEE Trans. Computer-Aided Design, vol. 13(1), pp. 59–71, Jan. 1994.
7. K. Krishna and S. Director, “The Linearized Performance Penalty (LPP) Method for Optimization of
Parametric Yield and Its Reliability,” IEEE Trans. Computer-Aided Design, vol. 14(12), pp. 1557–68,
Dec. 1995.
8. A. Lokanathan and J. Brockman, “Efficient Worst Case Analysis of Integrated Circuits,” IEEE 1995
Custom Integrated Circuits Conf., pp. 11.4.1–4, 1995.
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Appendix A Transfer Function Examples
A.1
Sensitivity Analysis Overview
The classic logarithmic sensitivity function is:
(8)
where, αi is a component value, and X is a filter performance measure (in the most general case, this is a
complex-value function or frequency). The sensitivity function is a dimensionless figure of merit used in
filter design.
You can approximate the relative change in X caused by the relative changes in the components αi as:
(9)
where,
(10)
The relative standard deviation of X is calculated using:
(11)
where,
• The summation is over all component values (αi) that affect X
• All component values (αi) are physically independent (no statistical correlation)
The nominal value of X is a function of temperature:
(12)
where,
• X is the nominal value of X at room temperature
• αi (T) is the nominal value of αi at temperature T
• X(T) is the nominal value of X at temperature T
To help reduce variation in filter performance:
•
•
•
10
Reduce the sensitivity function magnitudes ( ), where X is Ho, ωp and Qp, and αi is any of the
component values, the gain K, or operating conditions (such as temperature or supply voltage)
Use components with smaller tolerances
Use components with lower temperature coefficients
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Appendix B Biquad Section, s Term in the Denominator That Includes K
B.1
Handy Sensitivity Formulas
Notation:
• k, m, n = constants
• α, β = [non-zero] component parameters
• X, Y = [non-zero] performance measures
Formulas:
(13)
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