CN111669149B - Design method of self-adaptive Butterworth low-pass digital filter - Google Patents
Design method of self-adaptive Butterworth low-pass digital filter Download PDFInfo
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Abstract
The invention provides a design method of a self-adaptive Butterworth low-pass digital filter, which comprises the steps of firstly expressing coefficients of numerator polynomials and denominator polynomials of a discretized Butterworth low-pass filter transfer function as a function of a product of a sampling step length and a cut-off frequency, then converting the transfer function denominator polynomials into a first 1 form to obtain new numerator polynomial and denominator polynomial coefficients, designing a fixed proportional relation between the cut-off frequency of the Butterworth low-pass filter and the fundamental wave frequency of a detected signal, then updating the Butterworth low-pass filter transfer function according to the fundamental wave frequency and the sampling step length of the detected signal, and then filtering and outputting an input signal by adopting the updated Butterworth low-pass filter. The invention can effectively reduce the influence of harmonic wave and high-frequency interference on variable-frequency alternating-current voltage and current signals and improve the accuracy of signal detection; the gain and the phase shift of the fundamental frequency signal output by the filter are ensured to be the same when the frequency of the detected signal is rapidly changed in a large range.
Description
Technical Field
The invention belongs to the technical field of anti-aliasing filtering of a variable-frequency alternating-current power supply system of an airplane, and particularly relates to a design method of a self-adaptive Butterworth low-pass digital filter suitable for the variable-frequency alternating-current power supply system.
Background
The aircraft power supply system usually comprises a high-power diode rectifying circuit, a radar and other nonlinear loads, and harmonic pollution is caused to the alternating voltage and current of the system. In addition, high-frequency interference such as a high-intensity radiation field can also act on the control circuit, and great threat is formed to the stable operation of the system. The adoption of an anti-aliasing filter is an important means for effectively inhibiting the adverse influence of harmonic waves and high-frequency interference on the system.
The three-phase variable frequency alternating current power supply system has the characteristics of high frequency change speed (the maximum frequency change rate specified in the national standard is 250 Hz/s), large frequency change range (the normal frequency range of the currently used wide variable frequency power supply system is 360Hz-800 Hz) and the like, and brings huge challenges to the design of the anti-aliasing filter. Filter parameters are designed according to a fixed cut-off frequency, so that the phase lag is increased when the actual frequency is increased, and the stability margin of the system is reduced; when the system operates in a low frequency stage, the cut-off frequency of the filter is increased, which reduces the filtering effect and adversely affects the performance and stability of the system.
The existing filter is mostly realized by adopting a passive device, the gain in a pass band is not stable, the excessive frequency band is too wide, and the filtering effect is not good; although active low-pass filter circuits based on operational amplifiers are well developed in power systems, the corresponding structures are only suitable for applications with fixed frequency or small range variation. Another approach is to segment the frequency range and design a filter with a fixed cut-off frequency for each segment. On one hand, the method increases the complexity and cost of hardware, and simultaneously easily causes the phenomenon of discontinuous output when switching between filters with different cut-off frequencies, thereby affecting the performance and stability of the system.
Disclosure of Invention
Aiming at solving the problems in the prior art, the invention provides a design method of an adaptive Butterworth low-pass digital filter with cut-off frequency continuously adjusted along with the fundamental frequency of a detected signal for a variable-frequency alternating-current power supply system, which is used for carrying out anti-aliasing filtering on variable-frequency alternating-current voltage and current signals and improving the detection accuracy of voltage and current fundamental frequency signals, thereby improving the voltage accuracy of the variable-frequency alternating-current power supply system and enhancing the stability of the power supply system.
The technical scheme of the invention is as follows:
the design method of the self-adaptive Butterworth low-pass digital filter is characterized by comprising the following steps of: the method comprises the following steps:
step 1: expressing the normalized transfer function of the Butterworth low-pass filter of the order of m as the transfer function of a complex frequency domain;
and 2, step: discretizing the complex frequency domain transfer function of the m-order Butterworth low-pass filter obtained in the step 1,and expressed as the sampling frequency T s And a cut-off frequency w c The function of the product is:
wherein n is m 、n m-1 …n 1 、n 0 And d m 、d m-1 …d 1 、d 0 Are each H a The coefficients of the numerator and denominator polynomials in (z), both being the sampling frequency T s And a cut-off frequency w c A function of the product;
and step 3: h is to be a The numerator polynomial is in the form of first 1 in (z), and new numerator and denominator polynomial coefficient n is obtained mn 、n (m-1)n …n 1n 、n 0n And d mn 、d (m-1)n …d 1n 、d 0n Wherein d is mn ＝1；
The new numerator polynomial coefficient and the denominator polynomial coefficient are respectively expressed as follows:
n kn ＝f kn (a 0 ,a 1 …a m ),k＝0,1,…m
d kn ＝f kd (a 0 ,a 1 …a m ),k＝0,1,…m
a 0 ＝T s w c ，
a 2 …a m is represented by a 0 Is a polynomial of a parameter, f kn () And f kd () Is also a polynomial function;
and 4, step 4: obtaining the sampling frequency T of the detected signal s And fundamental frequency w s Cut-off frequency w of Butterworth low-pass filter c Is taken as the fundamental frequency w of the detected signal s The proportional relationship of (A): w is a c ＝Kw s Wherein K is a set constant;
and 5: according to numerator and denominator polynomial coefficient expressions
n kn ＝f kn (a 0 ,a 1 …a m ),k＝0,1,…m
d kn ＝f kd (a 0 ,a 1 …a m ),k＝0,1,…m
And the sampling frequency T of the detected signal obtained in the step 4 s And the cut-off frequency w of the Butterworth low-pass filter c Obtaining coefficients n of numerator and denominator polynomials of transfer function of Butterworth low-pass filter kn 、d kn Therefore, the transfer function of the Butterworth low-pass filter is updated according to the detected signal, and then the updated Butterworth low-pass filter is adopted for filtering and outputting the input signal.
Further, the adaptive Butterworth low-pass digital filter is implemented in FPGA by programming in C language.
Further, the designed adaptive butterworth low-pass digital filter is a 3-order adaptive butterworth low-pass digital filter, wherein in step 2, the discretized transfer function numerator denominator coefficient is:
in the step 3, the step of the method is that,
a 0 ＝T s w c ，
a 3 ＝a 2 +4a 1 +8a 0 +8
obtaining a new numerator and denominator polynomial coefficient of
n 0n ＝a 2 /a 3 n 1n ＝3n 0 n 2n ＝3n 0 n 3n ＝n 0
d 0n ＝(a 2 -4a 1 +8a 0 -8)/a 3
d 1n ＝(3a 2 -4a 1 -8a 0 +24)/a 3
d 2n ＝(3a 2 +4a 1 -8a 0 -24)/a 3
In step 4, the cut-off frequency of the filter is kept to be 12 times the fundamental frequency w c ＝12w s 。
Advantageous effects
The adaptive Butterworth low-pass digital filter provided by the invention can effectively reduce the influence of harmonic and high-frequency interference on variable-frequency alternating-current voltage and current signals, and improve the accuracy of signal detection; meanwhile, when the frequency of the detected signal is rapidly changed in a large range, the gain and the phase shift of the fundamental wave frequency signal output by the filter are the same, on one hand, the design of an alternating voltage and current detection circuit in the variable frequency power supply system can be simplified, and on the other hand, the stability of the variable frequency alternating current power supply system can be enhanced.
Additional aspects and advantages of the invention will be set forth in part in the description which follows and, in part, will be obvious from the description, or may be learned by practice of the invention.
Drawings
The above and/or additional aspects and advantages of the present invention will become apparent and readily appreciated from the following description of the embodiments, taken in conjunction with the accompanying drawings of which:
fig. 1 is a process of operation of the adaptive butterworth low-pass digital filter.
Fig. 2-5 show the results of filtering a harmonically contaminated variable frequency ac voltage signal using an adaptive third-order butterworth low-pass digital filter designed in accordance with the present invention. The voltage signal adopts a per unit value, the amplitude of the fundamental component of the voltage signal is 1.0, 11-order harmonic waves and 13-order harmonic waves are superposed on the fundamental voltage, the amplitudes of the fundamental component are respectively 3 percent and 2 percent, the fundamental frequency respectively changes between 355Hz-365Hz and 797Hz-803Hz, and the change rate of the frequency respectively is 400Hz/s and 400Hz/s.
Figure 2 is a graph of the filtering effect of a frequency ramp from 355Hz at 400Hz/s up to 365 Hz.
Figure 3 is a filter effect with a frequency falling from 365Hz at-400 Hz/s to 355 Hz.
FIG. 4 is the effect of filtering with a frequency rising from 797Hz at 400Hz/s to 803 Hz.
Figure 5 is a graph of the filtering effect of frequency dropping from 803Hz at-400 Hz/s to 797 Hz.
Detailed Description
The invention provides a design method of an adaptive Butterworth low-pass digital filter with cut-off frequency continuously adjusted along with the fundamental frequency of a detected signal, which is used for carrying out anti-aliasing filtering on alternating voltage and current signals in a variable-frequency alternating current power supply system and has the basic principle as follows:
(1) Expressing the coefficients of the numerator, denominator polynomial of the discretized (z-domain) butterworth low-pass filter transfer function as the sampling step T s And a cut-off frequency w c A function of the product.
Assuming that the transfer function of the discretized butterworth low-pass filter is:
wherein n is m 、n m-1 …n 1 、n 0 And d m 、d m-1 …d 1 、d 0 Are each H a (z) coefficients of the numerator and denominator polynomials in (z), both T s w c And 2 to m powers thereof.
(2) H is to be a Polynomial expression of the denominator in (z)First 1 to obtain new numerator and denominator polynomial coefficient n mn 、n (m-1)n …n 1n 、n 0n And d mn 、d (m-1)n …d 1n 、d 0n Wherein d is mn ＝1。
Wherein the new numerator and denominator polynomial coefficients can be expressed as:
n kn ＝f kn (a 0 ,a 1 …a m ),k＝0,1,…m
d kn ＝f kd (a 0 ,a 1 …a m ),k＝0,1,…m
a 0 ＝T s w c ，
a 2 …a m is a to 0 Is a polynomial of a parameter, f kn () And f kd () Is also a polynomial function;
(3) Designing the cut-off frequency w of a Butterworth low-pass filter c In a fixed proportional relationship with the fundamental frequency of the detected signal, namely:
w c ＝Kw s
where K is a set constant, w s Representing the fundamental frequency of the detected signal.
(4) According to a 0 ,a 1 …a m And w c Value of (d) calculating the numerator and denominator polynomial coefficient n kn 、d kn And updating the transfer function of the Butterworth low-pass filter, and then filtering and outputting the input signal by adopting the updated Butterworth low-pass filter.
The Butterworth low-pass filter can be a digital filter realized by programming in FPGA by adopting C language, and the operation process of the self-adaptive Butterworth low-pass digital filter is shown as an attached figure 1. In theory, the invention is applicable to butterworth filters of any order. However, the implementation of the high-order filter is complicated and the phase lag increases with the increase of the order, which is disadvantageous to the stability of the system, and the low-order filter such as the second order and the third order is most widely applied in practice. The design method proposed by the present invention is illustrated below by taking a third order butterworth low pass digital filter as an example, which is exemplary and intended to be used for explaining the present invention and is not to be construed as limiting the present invention.
The method comprises the following steps: the normalized transfer function of the third order butterworth low pass filter is expressed as a transfer function of the complex frequency domain (s-domain).
The third order butterworth low pass filter normalized transfer function is:
where p represents the normalized frequency.
Let p = s/ω c Substituting for equation (1), the s-domain transfer function of the available third-order butterworth low-pass filter is:
wherein w c Representing the cut-off frequency of the filter.
Step two: discretizing the transfer function of a three-order Butterworth low-pass filter and expressing it as the sampling frequency T s And a cut-off frequency w c A function of the product.
Discretizing the formula (2) can obtain:
wherein:
step three: the denominator polynomial is converted into a first 1 form to obtain a new numerator and denominator polynomial coefficient n 3n 、n 2n 、n 1n 、n 0n And d 3n 、d 2n 、d 1n 、d 0n And so on.
Order:
a 0 ＝T s w c ，
a 3 ＝a 2 +4a 1 +8a 0 +8，
the following can be obtained:
n 0n ＝a 2 /a 3 (12)
n 1n ＝3n 0 (13)
n 2n ＝3n 0 (14)
n 3n ＝n 0 (15)
d 0n ＝(a 2 -4a 1 +8a 0 -8)/a 3 (16)
d 1n ＝(3a 2 -4a 1 -8a 0 +24)/a 3 (17)
d 2n ＝(3a 2 +4a 1 -8a 0 -24)/a 3 (18)
step four: designing the cut-off frequency w of a Butterworth low-pass filter c The relationship with the fundamental frequency of the detected signal is:
w c ＝12w s (19)
i.e. the fundamental frequency, which keeps the cut-off frequency of the filter at 12 times.
Step five: according to a 0 ,a 1 ,a 2 ,a 3 And the equations (12) to (19) calculate n 0n 、n 1n 、n 2n 、n 3n And d 0n 、d 1n 、d 2n 、d 3n Thereby updating the butterworth low-pass filter transfer function, and then filtering the input signal with the updated butterworth low-pass filter output based on the output result of the previous step and the system state.
Although embodiments of the present invention have been shown and described above, it is understood that the above embodiments are exemplary and should not be construed as limiting the present invention, and that variations, modifications, substitutions and alterations can be made in the above embodiments by those of ordinary skill in the art without departing from the principle and spirit of the present invention.
Claims (3)
1. A design method of an adaptive Butterworth low-pass digital filter is characterized by comprising the following steps: the method comprises the following steps:
step 1: expressing the normalized transfer function of the Butterworth low-pass filter of the order of m as the transfer function of a complex frequency domain;
step 2: the battevo of m-order obtained in the step 1The complex frequency domain transfer function of the slowpass filter is discretized and expressed as a sampling frequency T s And a cut-off frequency w c The function of the product is:
wherein n is m 、n m-1 …n 1 、n 0 And d m 、d m-1 …d 1 、d 0 Are each H a The coefficients of the numerator and denominator polynomials in (z), both being the sampling frequency T s And a cut-off frequency w c A function of the product;
and step 3: h is to be a The numerator polynomial in (z) is in the form of first 1 to obtain new numerator and denominator polynomial coefficient n mn 、n (m-1)n …n 1n 、n 0n And d mn 、d (m-1)n …d 1n 、d 0n Wherein d is mn ＝1；
The new numerator polynomial coefficient and the denominator polynomial coefficient are respectively expressed as follows:
n kn ＝f kn (a 0 ,a 1 …a m ),k＝0,1,…m
d kn ＝f kd (a 0 ,a 1 …a m ),k＝0,1,…m
a 0 ＝T s w c ，
a 2 …a m is a to 0 Is a polynomial of a parameter, f kn () And f kd () Is also a polynomial function;
and 4, step 4: obtaining the sampling frequency T of the detected signal s And fundamental frequency w s The cut-off frequency w of the Butterworth low-pass filter c Is taken as the fundamental frequency w of the detected signal s The proportional relationship of (A): w is a c ＝Kw s Wherein K is a set constant;
and 5: according to the coefficient expression of numerator and denominator polynomials
n kn ＝f kn (a 0 ,a 1 …a m ),k＝0,1,…m
d kn ＝f kd (a 0 ,a 1 …a m ),k＝0,1,…m
And the sampling frequency T of the detected signal obtained in the step 4 s And the cut-off frequency w of the Butterworth low-pass filter c Obtaining coefficients n of transfer function numerator and denominator polynomial of the Butterworth low-pass filter kn 、d kn Therefore, the transfer function of the Butterworth low-pass filter is updated according to the detected signal, and then the updated Butterworth low-pass filter is adopted for filtering and outputting the input signal.
2. The adaptive butterworth low-pass digital filter design method of claim 1, wherein: the self-adaptive Butterworth low-pass digital filter is realized in an FPGA by programming in a C language.
3. The adaptive butterworth low-pass digital filter design method of claim 1, wherein: the designed adaptive butterworth low-pass digital filter is a 3-order adaptive butterworth low-pass digital filter, wherein in the step 2, the discretized transfer function numerator denominator coefficient is as follows:
in the step 3, the step of the method is that,
a 0 ＝T s w c ，
a 3 ＝a 2 +4a 1 +8a 0 +8
obtain a new numerator and denominator polynomial coefficient of
n 0n ＝a 2 /a 3 n 1n ＝3n 0 n 2n ＝3n 0 n 3n ＝n 0
d 0n ＝(a 2 -4a 1 +8a 0 -8)/a 3
d 1n ＝(3a 2 -4a 1 -8a 0 +24)/a 3
d 2n ＝(3a 2 +4a 1 -8a 0 -24)/a 3
In step 4, the cut-off frequency of the filter is kept to be 12 times the fundamental frequency w c ＝12w s 。
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