Ⅰ Introduction
A high-pass filter (HPF), also called a low-cut filter or bass-cut filter, passes signals with a frequency higher than a certain cutoff frequency and attenuates signals with frequencies lower than the cutoff frequency. That is, unnecessary low-frequency components or low-frequency signal interference are removed. High-pass filters use the same two topologies as the low-pass filters: Sallen-Key and MFB. The only difference is that the positions of the resistors and the capacitors have changed. In other words, High-pass filters are complementary to low-pass filters.
Figure 1. High Pass RC Filter Circuit
Catalog
3.1 Passive High-pass Filter and Active High-pass Filter 3.2 First-order High-pass Filter and Second-order High-pass Filter |
Ⅱ High-pass Filter Basic
2.1 Terminology
A high-pass filter is a combination device of capacitors, inductors, and resistors that allows signal components above a certain frequency to pass through and greatly suppress signal components below that frequency. Its characteristics can be described in time domain and frequency domain by impulse response and frequency response respectively. The latter is represented by a function whose frequency is an independent variable. The latter is a function representation of frequency as an independent variable, which is generally a complex variable function with complex variable jω as an independent variable, expressed as H (jω). The H(ω) and amplitude φ(ω) are functions of angular frequency ω, which are called the "amplitude-frequency response" and "phase-frequency response" of the system, respectively. They show the signal components of different frequencies in the excitation source through the amplitude and phase changes encountered in this system. It can be proved that the "frequency response" of the system is the of the "impulse response" of the system, based on the Fourier Transform. When a linear passive system can be represented by an N-order linear differential equation, the frequency response H (jω) is a rational fraction, and its numerator and denominator correspond to the right and left sides of the differential equation, respectively.
2.2 High-pass Filter Circuit
Basic high-pass filters are constructed using resistors with capacitors or inductors. The high-pass filter composed of resistors and capacitors is called a high-pass RC filter, and the high-pass filter with resistors and inductors is called a high-pass RL filter.
Figure 2. Simple Passive High Pass RC Filter Circuit
2.2 Cutoff Frequency
Generally, the cutoff frequency of the filter refers to the right frequency point of the low-pass filter passband and the left frequency point of the high-pass filter passband, that is, the frequency response point of the filter. Usually defined by 1dB or 3dB relative loss point. For a high-pass filter, its cutoff frequency is that frequency at which the output (load) voltage equals 70.7% of the input voltage (source). The high-pass filter is based on the insertion loss at a sufficiently high -passband frequency without parasitic stopband.
Figure 3. High Pass Filter Cutoff Frequency
Ⅲ High-pass Filter Types
The following two classification methods are independent of each other. Active high-pass filters are more common, such as first-order active high-pass filters and second-order active high-pass filters.
3.1 Passive High-pass Filter and Active High-pass Filter
According to the different part devices, it can be divided into passive high-pass filter and active high-pass filter.
- Passive high-pass filter
A kind of filters composed of passive components (resistor R, inductor L, and capacitor C), which is constructed using the principle that the reactance of capacitors and inductive components changes with frequency. The advantages of this type of filter are: the circuit is relatively simple, no DC power supply is required, and the reliability is high. The disadvantages are: the signal in the passband has energy loss, the load effect is more obvious, and electromagnetic induction is easily caused when using inductive components. When L is large, the volume and weight of the filter are relatively large, and it is not suitable in the low frequency domain.
- Active high-pass filter
A filter consists of passive components (usually R and C) and active devices (such as integrated operational amplifiers). The advantage of this type of filter is that the signal in the passband has no energy loss, but also can be amplified, the load effect is not obvious, and the mutual influence is small when the multi-stage cascade is connected. It is easy to form a high-order filter using the simple method of cascade, and the filter is small, light, and does not require magnetic shielding (because no inductive components are used). The disadvantage is that the passband range is limited by the bandwidth of active devices (such as integrated op amps) and requires a DC power supply, and reliability is not as high as the passive filters, thus it is not suitable for high voltage, high frequency and high power.
Figure 4. LM741 Active High-pass Filter Circuit
3.2 First-order High-pass Filter and Second-order High-pass Filter
According to the mathematical characteristics, it is divided into a first-order high-pass filter and a second-order high-pass filter, third-order high-pass filterand so on.
Figure 5. Circuit Diagrams of High-pass Filter
- First Order High Pass Filter
The first order high-pass filter requires a capacitor with a very high capacity, which causes very high fundamental losses, and therefore, it is rarely used.
Figure 6. First-order High-pass Filter
The derivation process is as follows: CV = Q (C represents capacitance, V voltage, Q electric quantity)
Loop current
，
where
Laplace transform:
Differential form:
,
- Second-order High Pass Filter
The second-order high-pass filter has the best performance, but causes higher fundamental losses compared with the other type. The second-order filter means that the filter contains the second-order differential in the time domain expression, or the highest order of s of the transfer function denominator is 2, and the gain of the filter to DC component is 1.
Second-order High Pass filter Circuit (voltage controlled)
Figure 5. Second-order High Pass filter Circuit
Passband gain:
Transfer function:
Frequency response:
where, the expressing formula can be found
Conclusion: When f is less than f_{0}, the slope of the amplitude-frequency characteristic curve is + 40dB / dec; when Avp is greater than or equal to 3, the circuit self-excited.
- Multichannel feedback high-pass filter
Figure 6. Multichannel Feedback High-pass Filter
Voltage transfer function of this circuit:
Relationship between circuit parameters and components:
This circuit is designed with equal capacitance, that is, let C2 = C3 = C, C1 = | K | C, there is
Ⅳ High-pass Filter Transfer Function
- How to determine the transfer function of each high-pass filter
The high-order filter consists of a cascade of second-order filter sections and first-order filter sections, and each filter section has a specific transfer function. In addition, the cutoff frequency and quality factor Q should be determined for the second-order filter sections, and for the first order sections, only the cutoff frequency is to be determined.
- The general form of voltage transfer function of second-order high-pass filter:
Wc is the cut-off frequency, and the amplitude-frequency characteristic around the cut-off frequency is closely related to Q. K is the passband gain, which is the amplification factor when the frequency tends to infinity in the high-pass filter.
- The general form of voltage transfer function of first-order high-pass filter:
The meaning of Wc and K is the same as that of the second-order high-pass filter, but there has no Q value.
When designing and calculating, to determine these parameters, it is necessary to check the normalized pole table of the filter to complete it through a certain conversion. (Whether a low-pass, high-pass, band-pass, or band-reject filter, its frequency characteristics can be obtained by frequency coordinate transformation of the low-pass filter, so the low-pass filter also called a original filter.)
Figure 7. High-pass Filter Block Diagram
A common voltage-controlled voltage source high-pass filter circuit is used as an example. Its circuit diagram and voltage transfer function are as follows:
Figure 8. Voltage-controlled High-pass Filter Circuit
Comparing this formula with the general expression of the transfer function of the second-order high-pass filter, we can know the cutoff frequency in the voltage-controlled voltage source high-pass filter circuit:
Ⅴ High-pass Filter Order
How to determine the order of the high-pass filter
The order of Butterworth high pass filter is
Where n represents the order of filter, F_{C} is the cut-off frequency of - 3dB, F_{2} is the frequency of a specified attenuation within the transition band, and A_{2} is the attenuation at F_{2}.
The order of Chebyshev high pass filter is:
Where n is the filter order, fc is the equal ripple cutoff frequency, a1 is the gain fluctuation (dB) in the passband, f2 is the frequency of a specified attenuation in the transition band, and a2 is the attenuation at f2 Volume (dB).
The fc in the above formula represents the equal-ripple cut-off frequency, that is, the frequency of attenuation in the transition zone at a_{1}. If the -3dB cutoff frequency is used in the design, it must be converted to an equal ripple cutoff frequency to get the above formula. The following table shows the proportional relationship between the cut-off frequency such as the Chebyshev high-pass filter and the -3dB cut-off frequency.
Table 1: Ratio FC / f-3db of equal ripple bandwidth to - 3dB bandwidth of Chebyshev high pass filter
n |
2 |
3 |
4 |
5 |
6 |
a1=0.1dB |
1.9432 |
1.3690 |
1.2131 |
1.1347 |
1.0929 |
a1=0.2dB |
1.6743 |
1.2835 |
1.1564 |
1.0992 |
1.0685 |
a1=0.5dB |
1.3897 |
1.1675 |
1.0931 |
1.0593 |
1.0410 |
a1=1dB |
1.2176 |
1.0949 |
1.0530 |
1.0338 |
1.0234 |
The high-order active high-pass filter is formed by cascading several second-order high-pass filters (first-order high-pass filters should be added for odd-order ones). Each filter is called a filter section, and each has its own independent and Q value. The filter formed after the cascade can constitute different types of filters, such as Butterworth filter and Chebyshev filter.
- Calculation of first-order high-pass filter section:
- Calculation of second-order high-pass filter section:
Ⅵ Difference between High-pass Filter and Low-pass Filter
The difference between a high-pass filter and a low-pass filter is that a high-pass filter allows high-frequency or AC component signals to pass and suppresses low-frequency or DC components. A low-pass filter is a filter that allows low-frequency or DC components in a signal to pass through and suppress high-frequency components or interference and noise. In general, the low-pass filter retains signals that are smaller than the cutoff frequency, while the high-pass filter retains signals that are larger than the cutoff frequency.
Ⅶ High-pass Filter Application
1) In the power system, a high-pass filter is used to filter out harmonics of a certain order and above during harmonic compensation.
2) In audio system
The most overlooked and most useful EQ tool is the high-pass filter, which can remove unwanted low-frequency noise from the audio source. The high-pass filter can be a part on the EQ equalizer, or it can be an independent plug-in or device. Usually our speakers, mixers or microphones are equipped with high-pass filters, because the low-cutting of the recorded human voice can make it easy to distinguish the noise, although this type of noise is very low and difficult to detect. For example, in most musical instruments, the high-pass filter is used to cut off the sound lower 100Hz. You will find that the sound work is miraculously clean, but this does not apply to low-frequency instruments such as kick drums and bass.
Ⅷ Question Related to High-pass Filter and Going Further
7.1 Question
What is the bandwidth of high pass filter?
7.2 Answer
The bandwidth of the filter denotes the value of frequency from which signals are allowed to pass. For example, if the bandwidth of the high pass filter is given as 50 kHz it means that only frequencies from 50 kHz to infinity are allowed to pass.
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