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优化Sallen-Key滤波器组件预失真算法
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更新于2024-09-27
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本应用笔记介绍了一种针对流行Sallen-Key滤波器(也称为KRC或VCVS,即电压控制电压源滤波器)的简单组件预失真方法。这种技术特别适用于处理由运算放大器(运算放大器)和两个电阻设置的无源增益部分,其中K值通过调整电阻和电容来实现所需的传递函数。然而,由于运算放大器有限的带宽,K值在频率上会发生变化,使得Sallen-Key滤波器通常工作在低于运放3dB截止频率的频段(f << f3dB)。 "ComponentPre-distortion for Sallen-Key Filters"这一主题的核心是解决由于运放带宽受限导致的滤波器性能问题。当频率增加时,运算放大器的群延迟会逐渐影响滤波器的相位响应,可能导致信号失真。预失真方法旨在通过修改运算放大器环节中的标准电阻和电容值来补偿这个群延迟,确保在低频下保持相对稳定的性能。 该应用笔记详细阐述了设计步骤,强调了在实际设计中如何根据运算放大器的特定特性进行预失真计算,以减小频率依赖性的影响。它提供了几个示例来展示这一补偿技术的实际应用,帮助设计师优化Sallen-Key滤波器在实际信号处理系统中的性能,尤其是在需要高精度和稳定性的工作环境中。 通过理解并实施这种方法,工程师可以避免因运算放大器带宽限制而产生的潜在失真,从而提高滤波器在信号处理链中的整体性能。值得注意的是,这种预失真策略对于那些对滤波器线性度有较高要求的应用,如音频信号处理、通信系统中的滤波或信号调理等,具有重要意义。
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Component Pre-distortion
for Sallen Key Filters
Introduction
This revision obsoletes the previous revision of this Applica-
tion Note, and covers additional material.
This Application Note shows a simple component
pre-distortion method that works for many popular
Sallen-Key (also called KRC or VCVS [voltage-controlled,
voltage-source]) filter sections. This method compensates
for voltage-feedback and current-feedback op amps. Several
examples illustrate this method.
KRC active filter sections use an op amp and two resistors to
set a non-inverting gain of K. resistors and capacitors placed
around this amplifier provide the desired transfer function.
The op amp’s finite bandwidth causes K to be a function of
frequency. For this reason, KRC filters typically operate at
frequencies well below the op amp’s bandwidth (f
<<
f
3dB
).
’Pre-distortion’ compensates for the op amp’s finite band-
width by modifying the nominal resistor and capacitor val-
ues. The pre-distortion method in the Application Note com-
pensates for the op amp’s group delay which is
approximately constant when f
<<
f
3dB
.
One possible design sequence for KRC filters is:
1. Design the filter assuming an ideal op amp (K is as-
sumed constant over frequency)
•
Select components for low sensitivities
•
Do a worst case analysis
•
Do a temperature analysis
1. Pre-distort the resistors and capacitors to compensate
for the op amp’s group delay
2. Compensate for parasitic elements
Filter Component Pre-Distortion
This section outlines a simple pre-distortion method that
works for many popular Sallen-Key filters using
current-feedback or voltage-feedback op amps. Other more
general pre-distortion methods are available (see reference
[4]) which require more design effort.
To pre-distort your filter components:
1. Calculate the op amp’s delay:
where φ(f) is the op amp phase response in degrees,
and f
c
is the cutoff frequency (passband edge fre-
quency) of your filter
— Subtract the phase shift caused by your measure-
ment jig from any measured value of φ(f
c
)
— The group delay is specified at f
c
because it has the
greatest impact on the filter response near the fre-
quency.
— Other less accurate estimates of the op amp delay at
f
c
are:
— Step response propagation delay
— 1/(2πf
3dB
)
2. The time delay around the filter feedback loop (’electrical
loop delay’) adds to the op amp delay.
For this reason,
— Make the filter feedback loop as physically short as
possible.
— If you need greater accuracy in the following calcula-
tion, use the electrical loop delay (τ
eld
) instead of the
op amp delay (τ
oa
):
τ
eld
←
τ
oa
See Appendix B for information on calculating τ
eld
.
3. Replace K in the filter transfer function with a simple
approximation to the op amp’s frequency response
— Start with a simple, single pole approximation:
K
←
K/(1 + τ
oa
s),s=jω
— Alter the approximation to K and simplify:
— Do not create new terms (a coefficient times a new
power of s) in the transfer function after simplifying
— Convert (1 + τ
oa
s) to the exponential form (a pure
time delay) when it multiplies, or divides, the entire
transfer function
— Do not change the gain at ω≈ω
p
in allpass sec-
tions
— The most useful alterations to K are:
All of these approximations are valid when: ω
<<
1/τ
oa
4. Use an op amp with adequate bandwidth (f
3db
) and slew
rate (SR):
f
3db
≥ 10f
H
SR
>
5f
H
V
peak
Where f
H
is the highest frequency in the passband of the
filter, and V
peak
is the largest peak volage. This increases the
accuracy of the pre-distortion algorithm. It also reduces the
filter’s sensitivity to op amp performance changes over tem-
perature and process. Make sure the op amp is stable at the
gain of A
v
=K.
Appendix A contains examples using transfer functions. The
next section will apply the results from Appendix A.
KRC Lowpass Biquad
The biquad shown in Figure 1 is a Sallen-Key lowpass
biquad. V
in
needs to be a voltage source with low output
impedance. R
1
and R
2
attenuate V
in
to keep the signal within
the op amp’s dynamic range. Using Example 2 in Appendix
A, we can show:
National Semiconductor
OA-21
Kumen Blake
August 1996
OA-21 Component Pre-distortion for Sallen Key Filters OA-21
© 2002 National Semiconductor Corporation AN012785 www.national.com
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