**Introduction**

A **band pass filter** is an electronic device or circuit that allows signals between two specific frequencies to pass. That is, allowing signals in a specific frequency band to pass while shielding other frequency bands. In other words, a band-pass filter attenuates frequency components in other ranges to an extremely low level, as opposed to the concept of a band-stop filter. For example, the RLC tank is an analog band-pass filter, it is a resistor - inductor - capacitor circuit (RLC circuit). These filtering circuit can also be made by connecting low-pass filters and high-pass filters.

How to Design Band Pass Filter Circuit

**Catalog**

## Ⅰ Band Pass Filter Circuit Characteristics

An **ideal band pass filter** should have a completely flat pass band, no amplification or attenuation. And all frequencies outside the pass band will be completely attenuated. In addition, the conversion outside the pass band is completed in an extremely small frequency range. But in fact, there is no ideal band-pass filter. Because the filter cannot completely attenuate all frequencies outside the desired frequency range, especially there is an attenuated but not isolated frequency range outside the desired pass band. This is usually called the filter roll-off phenomenon, and it is expressed in dB per decade of attenuation amplitude. Generally, the filter design should ensure that the roll-off range is as narrow as possible, so that the performance of the filter is closer to the design requirement. However, as the roll-off range gets smaller and smaller, the pass band becomes no longer flat and causes ripples.

The high-pass filter has a low **cut-off frequency**, and the low-pass filter has a high cut-off frequency. When the high cut-off frequency is lower than the low cut-off frequency, combining the two circuits, and it is possible to design a band pass filter. The gain of the band pass filter is adjusted by the feedback resistor and the current limiting resistor.

Figure 1. Band Pass Filter Circuit Parts

A band pass filter with a high quality factor refers to a filter with a narrow pass band. In other words, a high-Q factor means that fewer unwanted frequency signals will pass. A low-Q factor means that the pass band is very wide, to allow a wider range of frequencies to pass through.

## Ⅱ Band Pass Filter Parameters

### 2.1 Center Frequency

It usually defined as the midpoint between the two 3dB points of a band pass filter (or a band stop filter), generally expressed by the arithmetic average of the two 3dB points. It is a frequency when the impedance of the entire circuit is a real number.

### 2.2 Cut-off Frequency

It refers to the frequency point on the right of the low-pass filter and the frequency point on the left of the high pass filter in the pass band. That is, the boundary frequency. It is usually defined as a standard by 1dB or 3dB relative loss point. The band pass filter has two cutoff frequencies, the low cutoff frequency fp1 and the high cutoff frequency fp2.

### 2.3 Bandwidth

The difference between two cut-off frequencies. The bandwidth is defined as B=fp2－fp1.

### 2.4 Quality Factor

The reciprocal of the damping coefficient is called the quality factor, which is an important indicator of the frequency selection characteristics of band pass and band stop filters. In short, it is the ratio of the center frequency to the bandwidth. What’s more, it can be used to describe the shape of the transfer function graph.

## Ⅲ Types of Band Pass Filter

### 3.1 Active Band Pass Filter

Figure 2. Active Band Pass Filter Circuit

The active band pass filter is a cascade of high-pass and low-pass filters and amplifier components. The circuit diagram of the active band pass filter consists of three parts. The first part is the high-pass filter. Then, use the op amp for amplification. The last part of the circuit is the low-pass filter.

### 3.2 Passive Band Pass Filter

Figure 3. Passive Band Pass Filter Circuit

Passive band pass filters are a combination of passive high-pass and low-pass filters. Passive filters use only passive components, such as resistors, capacitors, and inductors. Therefore, passive band pass filters are also used as passive components and do not use op amps for amplification.

## Ⅳ Band Pass Filter Equation

### 4.1 Cutoff Frequency of Band Pass Filters

The characteristic of the band pass filter is that the output signal amplitude in the pass band is independent from the frequency. When f<fp1 or f>fp2, the output signals attenuate quickly. The amplitude-frequency characteristics are shown in the figure:

Figure 4. BPF Bandwidth

(The broken line is the ideal BPF frequency characteristic, and the solid line is the actual BPF frequency characteristic)

The resonance frequency is between fp1 and fp2, where the gain of the filter is the largest, and the bandwidth of the filter is the difference between fp2 and fp1.

It can be seen from the frequency characteristics of BPF that it can be composed of LPF and HPF in series, as long as the fpL of LPF (ie, fp2 of BPF) is greater than fpH of HPF (ie, fp1 of BPF).

### 4.2 General Form of Second-order BPF Transfer Function

Frequency Characteristics

Where Aup is the pass-band magnification, center frequency , Q factor

Normalized frequency characteristics

Normalized amplitude - frequency characteristics

Figure 5. Amplitude - frequency Characteristics

Figure 6. Frequency Characteristics

It can be seen that the frequency characteristic of the band pass filter is completely determined by the center frequency ωo and the quality factor Q.

When f>fo, as the frequency f increases, the amplitude increases. According to the definition of cutoff frequency, the denominator of amplitude-frequency characteristic , that is (since f>fo, take a positive value)

1) Upper cutoff frequency

（take a positive value）

When f<fo, as the frequency f decreases, the output signal amplitude will decrease. According to the definition of cutoff frequency, the denominator of amplitude-frequency characteristic , that is (since f<fo, take a negative value)

2) Lower cutoff frequency

（take a negative), get the bandwidth

When the center frequency fo and bandwidth B (or Q) are known, the upper and lower cutoff frequencies fp1 and fp2 can be calculated. On the contrary, when the upper and lower cutoff frequencies fp1 and fp2 are known, the center frequency fo and bandwidth B (or Q) can be calculated.

(where ),

### 4.3 Second-order Band Pass Filters

A simple second-order band pass filter circuit is shown in the figure below, where R1 and C1 constitute a low-pass filter circuit, and C2 and R3 constitute a high-pass filter circuit.

Figure 7. Second-order Band Pass Filter Circuit

(1) Transfer Function

In order to reduce the amount of parameters matching, generally take C1=C2=C

Take , , that is

The transfer function can be obtained by using the node current method.

(2) Frequency Characteristics

where band-pass amplification (The negative sign means that the input and output are inverted. Because the filter circuit is an inverting filter.)

Center frequency (C1=C2=C), Q factor

When Aup, Q, and ωo are known, the resistance of each resistor is (R3 can be calculated with ωo/Q)

, (Aup<2Q^{2})

When the pass band amplification factor Aup is small, Q should not be too large (that is, the simple second-order BPF has poor selectivity), otherwise R2 will become very small (R2 is generally greater than 1K), which will attenuate the input signal seriously. In order to make the system stable, Aup is generally between 1 and 10, and Q can be between 1 and 20.

(3) Design Steps

Example: It is known that Aup=5, center frequency fo=450Hz, bandwidth B=200Hz (). Try to calculate the parameters of the band-pass filter and verify.**First**, according to the center frequency fo, check the parameter table and determine C1, C2, and operational amplifier parameters according to the nominal value.

fo=450Hz, take C1=C2=0.01uF(103 capacitor). Since the center frequency is not high, the requirement can be met by using LM358 operational amplifier.**Second**, calculate the resistance of each resistor. Among them, the range of R1 and R3 should be between 10K ~510K, and R2 should be between 1K ~100K, otherwise the capacitor C needs to be reselected.

Substituting the relevant parameters into the above formula, the result is R1=15.9K, R2=15.5K, R3=159K.**Third**, use simulation software to verify on the computer, and try to take the nominal value of each relevant resistance. The simulation schematic diagram and simulation results are shown in the figure below. The result values obtained from the AC small signal analysis and transmission characteristic analysis basically meet the requirements.

Figure 8. Filtering Circuit with LM358

Figure 9. Simulation Schematic Diagram

Figure 10. Voltage - Time Simulation (ui)

Figure 11. Voltage - Time Simulation (ui, uo)

The circuit requires a small number of components, and it can work with dual power supplies or with a single power supply (the non-inverting termination is connected to a 1/2Vcc bias potential). In fact, it is widely used in single power supply systems. Because the quality factor Q cannot be too high. Almost all band pass filter circuits with a larger bandwidth B (with a smaller Q value) adopt this circuit form.

### 4.4 High-Q second-order Band Pass Filters

The high-Q second-order band pass filter circuit is shown in the following figure. This circuit can work with dual power supplies or single power supplies, which is convenient to use in a single power supply system. Since the Q value can be made larger, it is particularly suitable as a band pass filter.

Figure 12. Bandpass Filter Circuit

(1) Transfer Function

In order to reduce the amount of parameters matching, generally take C1=C2=C

Where , , that is , , getting

The transfer function can be obtained by using the node current method.

(2) Frequency Characteristics

, where band-pass amplification

Center frequency , Q factor

In order to make the system stable, Aup and Q must be greater than 0, that is, 2R_{f}R_{4}-R_{F}R_{3}>0, which must be guaranteed . Adjusting can control Aup and Q. is more closer to 2, the greater the Aup and Q values. Adjust the capacitor C to select the center frequency ωo. When the values of Aup, Q and ωo are known, and the ratio between and is determined, the resistance of each resistor is

, ,

(3) Design steps

Example: It is known that Aup=5, center frequency fo=1kHz, bandwidth B=50Hz (). Try to calculate the parameters of the band-pass filter and verify.

First, according to the center frequency fo, check the parameter table and determine C1, C2, and operational amplifier parameters according to the nominal value. (Aup: 1 ~10)

fo=1kHz, take C1=C2=0.01uF. Since the center frequency is not high, the requirement can be met by using LM358 operational amplifier.

Second, according to the value of Aup and Q, initially determine the value of and .

Since Aup and Q are large, is 1.8, is 0.5, and is 3.6.

Third, calculate the resistance of each resistor. The range of R1 and R3 should be between 10K and 510K, and R2 should be between 1K and 100K. Otherwise, the ratio of and needs to be reselected.

Substituting the relevant parameters into the above formula, the result is:

R1=229K, R3=63.7K, R2=4.18K, R4=127.4K: RF takes 36K, Rf takes 10K.**Fourth**, use simulation software to verify on the computer, and try to take the nominal value of each relevant resistance. The simulation schematic diagram and simulation results are shown in the figure below. The result values obtained from the AC small signal analysis and transmission characteristic analysis basically meet the requirements.

Figure 13. High-Q BPF Circuit

Figure 14. Voltage - Frequency Simulation

Figure 15. Voltage - Time Simulation

### 4.5 Dual-operational Amplifier BPF (High-Q)

The BPF circuit with high-Q value formed by dual operational amplifiers is shown in the figure. With fewer components, a very high-Q value can be obtained when the pass band amplification factor Aup is fixed equal to 2, so it is also a commonly used BPF circuit.

Figure 16. Dual-amp BPF

(1) Transfer function

According to the rule of futility, where

, , that is

, where

, so

(2) Frequency Characteristics

Compared with the standard form of the second-order BPF transfer function, the following parameters can be obtained:

pass-band magnification , , center frequency

When R4=R5,R2=R3=R,C1=C2=C, Aup=2, , （）

It can be seen that when Aup=2 (that is, when R4=R5), the value of Q can be very large.

(3) Design steps

According to the center frequency fo, check the parameter table to determine C. When C is determined, the resistance R is calculated from the center frequency. Meanwhile, etermine R1 based on the Q value.

### 4.6 Second-order Band Pass Filters (Voltage-controlled Type)

The second-order voltage-controlled BPF is shown in the figure. Among them, R1 and C1 constitute a low-pass filter, R2 and C2 constitute a high-pass filter. (The voltage positive feedback is introduced through the voltage R3 to form a voltage-controlled band-pass filter.)

Figure 17. Second-order Bandpass Filter (voltage controlled)

Rf/RF cannot be 3 to avoid self-excitation.

(1) Transfer Function

Where ，that is , , so

The transfer function can be obtained by using the node current method.

In order to reduce the amount of parameters matching, generally take R1=R3=R,R2=2R,C1=C2=C

Where

In order to make the system stable, the coefficient of the first term in the denominator must be larger than 0, that is, 3−Auf>0, in other words, Auf<3.

(2) Amplitude - frequency Characteristics

where band-pass amplification , center frequency , Q factor

It can be seen that the closer Auf is to 3, the larger the Q value. The narrower the pass band B, and the better the selectivity.

(3) Design Steps

According to the center frequency, look up the table and initially determine C1=C2=C

calculate resistance , that is ,

Calculate bandwidth based on upper and lower cutoff frequencies , Calculate the quality factor

Calculate by Q and determine the resistances Rf and RF.

As a special case, the center frequency fo=1KHz is known, so C1=C2=C=0.01uF

，R2=2R=31.8K

, getting Auf=2.95, that is . If Rf=10K, calculate RF=19.5K.

For high pass and band pass filters, the output of the op amp is not required to be 0 at static state. And the single power supply operating mode can be selected. In the low-pass or band-stop filter circuit, it is a DC-to-AC DC amplifier circuit, which generally requires the circuit to work in a dual power supply state.

## Ⅴ Band Pass Filter Applications

The filtering circuit has a wide range of uses.According to different frequency amplitude characteristics, filter circuits can be divided into low pass filter circuit (LPF), high-pass filter circuit (HPF), band pass filter circuit (BPF), band stop filter circuit (BEF) and all-pass filter circuit (APF) . The BPF is mainly used to highlight signals in useful frequency bands and weaken signals or interference and noise in other frequency bands to improve the signal-to-noise ratio. Therefore, band pass filters are often used in wireless receivers and transmitters to receive useful signals while preventing unwanted frequencies from passing through.

In addition to the fields of electronics and signal processing, an specific application of band pass filters is in the field of atmospheric sciences. A very common example is to use it to filter the weather data in the last 3 to 10 days, only the cyclone as a disturbance remains in the domain.

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