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Summary
A filter is a frequency-selective device that makes certain frequency components of a signal pass and greatly attenuates other frequency components. In the test device, the interference noise can be filtered or the spectrum analysis can be carried out by using the frequency selection function of the filter. The contents of this article belong to the scope of analog filtering. This paper will mainly introduces the principle, type, mathematical model, main parameters, RC filter design of analog filter (continuous time-invariant system). Although digital filtering technology has been widely used, analog filtering is still widely used in automatic detection, automatic control and electronic measurement instruments.
Article core | Filter Introduction | Function | A device or circuit that has a processing effect on a signal |
English name | Filter | Category | Power/Circuit Protection |
Compose | capacitance, inductance and resistance and etc. | Application | Electronic component |
Catalogs
catalogs | I.What is Filter | IV. Actual filter |
II.Classification of Filter | V.RC passive filter | |
2.1 Classification based on filter frequency selection | 4.1 First-order RC low-pass filter | |
2.2 Classification based on " Best Approximation Properties " standard | 4.2 First-order RC high-pass filter | |
2.3 Classification based on the nature of the filter components | 4.3 RC band-pass filter | |
III.Ideal Filter | VI.The application of analog filters |
Introduction
I.What is Filter
A filter is a frequency-selective device that makes certain frequency components of a signal pass and greatly attenuates other frequency components. In the test device, the interference noise can be filtered or the spectrum analysis can be carried out by using the frequency selection function of the filter.In a broad sense, any channel of information transmission (medium) can be regarded as a filter. Because the response characteristic of any device is a function of the excitation frequency, these transfer characteristics can be described in terms of frequency-domain functions. Therefore, any part of the test system, such as the mechanical system, electrical network, instrumentation and even the connecting wires and so on, will transform and process the signals that pass through in a certain frequency range according to their frequency domain characteristics.
According to the nature of the filter signal processing, filters are divided into analog filters and digital filters.
II.Classification of Filter
2.1 Classification based on filter frequency selection
⑴ Low-pass filter
From 0 to f2 , the amplitude - frequency characteristic is flat , which can make the frequency components below f2 in the signal almost unattenuated , and the frequency components above f2 are greatly attenuated .
⑵ High-pass filter
In contrast to low-pass filtering, the amplitude-frequency characteristic is flat from the frequency f _ 1 ~ ∞, which makes the frequency component above f _ 1 pass through almost without attenuation, while the frequency component below f _ 1 will be greatly attenuated.
⑶ Band-pass filter
The passband of band-pass filter is between f1 and f2 . It allows the frequency components in the signal higher than f1 and lower than f2 pass without attenuation while other components are attenuated .
⑷ Band stop filter
In contrast to band-pass filtering, the stop band of band stop filter is between frequency f1 ~ f2. It attenuates the frequency components above f1 and below f2 in the signal and passes the rest of the frequency components almost without attenuation.
Note
Low-pass filters and high-pass filters are the two most basic forms of filters. Other filters can be divided into these two types of filters. For example, the series connection of a low-pass filter and a high-pass filter is a band-pass filter; The parallel connection of a low pass filter and a high pass filter is a band-stop filter.
(Series connection of a low-pass filter and a high-pass filter)
(parallel connection of a low pass filter and a high pass filter)
2.2 Classification based on " Best Approximation Properties " standard
⑴ Butterworth filter
The requirement of Butterworth filter will be raised from the amplitude-frequency characteristic without regard of the phase-frequency characteristic. Butterworth filter has the characteristic of maximum flat amplitude.The amplitude frequency response expression is :
In is the order of the filter; wc is the cut-off frequency of the filter,
When w=wc, |H(wc)|2=1/2, so wc corresponds to
-3db point of the filter.
The Butterworth low-pass filter uses the Butterworth function as the transfer function H (s) of the filter and approximates the ideal rectangular shape of the filter in the form of the highest order Taylor series.
⑵ Chebyshev filter
The approximation requirement of Chebyshev filter is also raised from the amplitude-frequency characteristic.The amplitude frequency response expression is :
ε is the coefficient of fluctuation that determines the size of the pass-band ripple, 0 <ε <1. The ripple is due to the presence of reactance elements in the actual filter network; wc is the pass-band cut-off frequency and Tn is the n-order Chebyshev polynomial.
Compared with the Butterworth approximation property, although this characteristic is fluctuating within the pass-band, it attenuates more steeply for the same value of n after entering the stop-band and is much closer to the ideal case. The smaller the value of ε, the smaller the fluctuation will be in the pass-band. At the same time, the decibel value of the attenuation at the cut-off frequency will also be smaller , but the attenuation characteristic after entering the stop-band will slow down. Compared with Butterworth filters, there is ripple in the Chebyshev filter pass-band , and the transitional band of Chebyshev filter is light steep straight; therefore, when there is not allowed ripple in the pass-band, Butterworth filter is much more desirable.
phase response from the point of view, Butterworth type is better than the Chebyshev type, through the above two comparison can be seen, the former phase frequency response is closer to a straight line.
From the response of phase frequency, Butterworth type is better than that of Chebyshev type, through comparison of the two figures below, it can be seen that the phase frequency response of the former one is closer to straight line.
⑶ Bessel filter
Only meet the phase frequency characteristics and do not care about the amplitude-frequency characteristic. Bessel filter, is also known as the most flat-time delay or delay filter. The phase shift is proportional to the frequency, that is, a linear relationship. However, due to its poor amplitude-frequency characteristic, its application is often limited.
2.3 Classification based on the nature of the filter components
Passive filter (R, L, C)
Active filter (including op amp)
Detail
III.Ideal Filter
The ideal filter is a filter that can make the amplitude and phase of the signal in the passband not distorted, and the frequency components in the stopband all decay to zero. There is a clear dividing line between passband and stopband. That is to say, the amplitude-frequency characteristic of the ideal filter in the passband should be constant, the slope of the phase-frequency characteristic is a constant value, and the amplitude-frequency characteristic outside the passband should be zero.
The ideal low-pass filter frequency response function is:
The amplitude-frequency and phase-frequency characteristic curve is as follows:
By analyzing the frequency characteristic of the upper expression, we can know that The pulse response function H (T) of the filter in the time domain is a sinc function, the graph is shown below. The waveform of the impulse response h (t) extends infinitely along the left and right of the abscissa. It can be seen from the figure that the filter has responded before the unit pulse is input to the filter at t = 0, ie at t <0 . Obviously, this is a non-causal relationship that can not be physically realized. This shows that the amplitude-frequency characteristic of right-angle sharpness at the cutoff frequency, or the ideal filter described by the rectangular window function in the frequency domain, is not possible. The actual filter's frequency domain graph will not be completely cut off at a certain frequency, but will gradually attenuate and extend to ∞.
IV. Actual filter
The basic parameters of the actual filter
The ideal filter does not exist. In the graph of actual filter amplitude-frequency characteristics, there should be no strict boundaries between pass-band and stop-band. There is a transition band between the pass and stop bands. The frequency component in the transition zone will not be completely suppressed, and it will only be attenuated in varying degrees. Of course, it is believed that the narrower the transition band the better, that is, the faster and more the frequency components out of the passband decay the better. Therefore, when designing an actual filter, it is always possible to approximate the ideal filter by various methods as much as possible.
Shown in the figure: the ideal bandpass (dashed line) and the actual (solid line) band-pass filter amplitude frequency characteristic.
As can be seen from the figure, the ideal filter characteristics only need to use the cut-off frequency description, and the actual filter curve without significant turning point, the two cut-off frequency amplitude-frequency characteristics are not constant, so it requires more parameters to describe .
⑴ Ripple amplitude d
In a certain frequency range, the amplitude-frequency characteristic of the actual filter may fluctuate. The smaller the amplitude d is compared with the average A0 of the amplitude-frequency characteristic, the better, and generally it should be far less than -3dB.
⑵ Cut-off frequency fc
The frequency corresponding to the amplitude-frequency characteristic value equal to 0.707A0 is called the cut-off frequency of the filter. Taking A0 as a reference value, 0.707A0 corresponds to a -3dB point, that is, attenuates 3dB with respect to A0. If the signal amplitude squared signal power, the corresponding point is exactly half-power point.
⑶ Bandwidth B and quality factor Q value
The frequency range between the upper and lower cut-off frequencies is called filter bandwidth or -3dB bandwidth in Hz. The bandwidth determines the ability of the filter to separate the adjacent frequency components in the signal frequency resolution. In Electrotechnics, Q is usually used to represent the quality factor of the resonant circuit.
In the second order oscillation link, the Q value is equal to the amplitude gain coefficient of the resonance point, Q = ½ξ (ξ - damping rate). For the band-pass filter, the ratio of the center frequency f0 () to the bandwidth B is usually referred to as the quality factor Q of the filter. For example, a center frequency of 500Hz filter, if the -3dB bandwidth of 10Hz, the Q value of 50. The larger the Q value, the higher the filter frequency resolution.
For example, if a filter with a center frequency of 500 Hz with a bandwidth of -3 dB is 10 Hz, the Q value is 50. The greater the Q is, the higher the frequency resolution of the filter is.
⑷ Octave selective W
Outside the two cut-off frequencies, the actual filter has a transitional band. The slope of the amplitude-frequency curve of this transition band indicates how fast the amplitude-frequency characteristic decays, which determines the filter's ability to reject the bandwidth-frequency components. It is usually characterized by octave selectivity. The so-called octave selectivity refers to the attenuation value of the amplitude-frequency characteristic between the upper cut-off frequency fc2 and 2fc2 or between the lower cut-off frequency fc1 and fc1/2, that is, the attenuation when the frequency changes by one octave
or
Octave attenuation is expressed in dB / oct (octave, octave). Obviously, the faster the decay (ie, the larger the W value), the better the selectivity of the filter. The attenuation rate that is far away from the cut-off frequency can also be expressed in terms of 10-fold frequency decay, ie. [dB / 10 oct].
⑸ Filter factor (or rectangular coefficient) λ
The filter factor is another representation of filter selectivity, which is a measure of filter selectivity using the ratio of the -60dB bandwidth to the -3dB bandwidth of the filter's amplitude-frequency characteristics, denoted as λ, ie,
Ideal filter λ= 1, common filter λ= 1-5, obviously, the closer to 1, the better the filter selectivity will be.
V.RC passive filter
In the test system, RC filters are commonly used . Because in this area, the signal frequency is relatively low. The RC filter has the advantages of simple circuit, strong anti-interference, better low-frequency performance and a selection of standard resistance capacitance element.
so the most commonly used filter in the field of engineering testing is RC filter.
4.1 First-order RC low-pass filter
RC low-pass filter circuit and the amplitude and phase frequency characteristics is shown below
Suppose the input voltage of the filter is ex, the output voltage is ey, the differential equation of the circuit is
This is a typical first-order system. Let λ =RC, called time constant, take Laplace transform on the Upper Formula
or
The amplitude frequency and the phase frequency characteristic formula:
Analysis shows that when f is small, A (f) = 1, the signal passes without attenuation; when f is large, A (f) = 0, the signal is completely blocked and can not pass. Low-pass filter upload stop frequency:
4.2 First-order RC high-pass filter
RC high-pass filter circuit and its amplitude and phase frequency characteristics is shown below
Suppose the input voltage of the filter is ex, the output voltage is ey, the differential equation of the circuit is
Similarly, Let λ =RC, called time constant, take Laplace transform on the Upper Formula
or
The amplitude frequency and the phase frequency characteristic formula:
Analysis shows that when f is small, A (f) = 0, the signal is completely blocked and can not pass; when f is large, A (f) = 1, the signal is not attenuated.
4.3 RC band-pass filter
Band-pass filter can be seen as the low-pass filter and high-pass filter in series, the circuit and its amplitude-frequency, phase-frequency characteristics is shown below
The amplitude frequency and the phase frequency characteristic formula:
In the formula, H1 (s) is the transfer function of the high-pass filter and H2 (s) is the transfer function of the low-pass filter.
At this moment, the extremely low and extremely high frequency components are completely blocked and can not pass; only the frequency components of the signal located in the frequency passband can pass through.
Lower cut-off frequency:
Upper cut off frequency:
It should be noticed that when two stages of high and low pass are connected in series, the interaction between the two stages should be eliminated because the latter stage becomes the "load" of the previous stage and the former stage is the signal source internal resistance of the latter stage. In fact, In fact , the two - stage common emitter follower is isolated by an operational amplifier. So the actual band-pass filter is often active. The active filter consists of an RC tuning network and an op amp. Operational amplifier can not only play the role of isolation, but also play an amplifying role of the signal amplitude.
Analysis
VI.The application of analog filters
Analog filters are a common type of conversion device used in test systems or specialized instrumentation. Such as band-pass filters used as frequency-selective devices in spectrum analyzers; low-pass filters used as anti-aliasing in digital signal analysis systems; high-pass filters used as elimination of low-frequency interference noise in acoustic emission detector; band stop filter used as eddy current vibration meter trap ...
The band-pass filter that is used in the spectrum analyzer can be divided into two types according to the numerical relationship between the center frequency and the bandwidth:
One is the bandwidth B does not change depending on the center frequency, known as constant bandwidth band-pass filter. As shown in the figure, the bandwidth is the same when the center frequency is at any frequency band.
The other one is the ratio of bandwidth B to the center frequency is constant, known as constant bandwidth ratio band-pass filter. As shown in the figure, the higher the center frequency, the bandwidth is wider.
Normally, in order to make the filter have good frequency resolution in any band, you can use a constant bandwidth bandpass filter (such as the radio frequency). The narrower the selected bandwidth is, the higher the frequency resolution is, but at this time, the number of filters required is large to cover the entire frequency range to be detected. Therefore, in many cases, the constant bandwidth bandpass filter does not necessarily have to be a fixed center frequency, but instead uses a reference signal that varies the center frequency of the filter by the frequency of the reference signal. In the process of signal spectrum analysis, the reference signal is supplied by a frequency-swept signal generator. This variable center frequency, constant bandwidth, band-pass filter is used in correlation filtering and in scan-tracking filtering.
The constant bandwidth ratio band-pass filter is used in the octave spectrum analyzer, which is a filter bank with different center frequencies, so that the individual band-pass filters can be combined to cover the entire frequency range of the signal to be analyzed. Its center frequency and bandwidth are configured according to certain rules.
If the lower cut-off frequency of any of the band-pass filters is fc1, the upper cut-off frequency is fc2, the relationship between fc1 and fc2:
fc1=2nfc1
In the formula, n value is called the octave range, if n = 1, it is called the octave filter; n = 1/3, then called 1/3 octave filter. The center frequency f0 of the filter is taken as the geometric mean, ie:
According to the above two formulas, we can obtain:
Filter bandwidth:
If the Q value of the filter's quality factor is expressed, then:
So in octave filter, if n = l, then Q = 1.41; if n = 1/3, then Q = 4.38; if n = 1/5, then Q = 7.2. The smaller the octave n value is, the larger the Q value will be, which indicating that the higher the filter resolution. According to the above relationship, the center frequency f0 and the bandwidth B value of the common octave filter can be determined.
In order to make the frequency components of the signal to be analyzed without lost, the center frequency of the band-pass filter bank is octave and the bandwidth is contiguous at the same time. The usual practice is to match the cut-off frequency of one 3dB on the previous filter with the lower cut-off frequency of the later one of the filter, which is shown in the figure. Such a set of filters will cover the entire frequency range, which is known as "abutting type".
The following figure shows the adjacent octave filter, the box number represents the center frequency of each bandpass filter. By the analysis of signal input, input, output band switch sequence connected with each filter, if the signal has a band-pass filter pass band frequency components, then we can show that the recording instrument observed on the frequency components.
After learning about filters, if you are still interested, then you can watch this video:
This video about introduction to Filter Design(It is a series)
Book Recommendation
Electronic Filter Simulation & Design 1st Edition
Electronic Filter Simulation and Design shows you how to apply simulation methods and commercially available software to catch errors early in the design stage and streamline your design process. Using 150 detailed illustrations, this hands-on resource examines cutting-edge simulation methods for lumped passive filters…active RC filters…low-pass and band-stop distributed filters…high-pass and band-pass distributed filters…high-frequency filters…discrete time filters…and much more. The book also contains a skills-building CD with files for major case studies covered in the text, together with demo versions of Mathcad and SIMetrix, so that you can work the examples and adapt them to their own projects.
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Electronic Filter Analysis and Synthesis (Microwave Library)
Electronic Filter Analysis and Synthesis helps you save time and effort in writing CAD and analysis programs for electronic filters, and provides explicit details on how to synthesize lowpass, bandpass, bandstop, and highpass realizations for passive, active, digital and switched capacitors.
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Electronic Filter Design Handbook, Fourth Edition (McGraw-Hill Handbooks)
Long-established as “The Bible” of practical electronic filter design, McGraw-Hill's classic Electronic Filter Design Handbook has now been completely revised and updated for a new generation of design engineers. The Fourth Edition includes the most recent advances in both analog and digital filter design_plus a new CD for simplifying the design process, ensuring accuracy of design, and saving hours of manual computation.
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