### Band Pass Filter

A band-pass filter is a circuit which is designed to pass signals only in a certain band of frequencies while attenuating all signals outside this band. The parameters of importance in a bandpass filter are the high and low cut-off frequencies (f_{H} and f_{l}*), *the bandwidth (BW), the centre frequency f_{c},* *centre-frequency gain, and the selectivity or Q.

**There are basically two types of bandpass filters viz wide bandpass and narrow bandpass filter**s. Unfortunately, there is no set dividing line between the two. However, a bandpass filter is defined as a wide bandpass if its figure of merit or quality factor Q is less than 10 while the bandpass **filters with Q > 10 are called the narrow bandpass filters.** Thus Q is a measure of selectivity, meaning the higher the value of Q the more selective is the filter, or the narrower is the bandwidth (BW). The relationship between Q, 3-db bandwidth, and the centre frequency f_{c} is given by an equation

For a wide bandpass filter the centre frequency can be defined as where f_{H} and f_{L}* *are respectively the high and low cut-off frequencies in Hz.In a narrow bandpass filter, the output voltage peaks at the centre frequency f_{c}*.*

** Wide Bandpass Filter**

A wide bandpass filter can be formed by simply cascading high-pass and low-pass sections and is generally the choice for simplicity of design and performance though such a circuit can be realized by a number of possible circuits. To form a ± 20 db/ decade bandpass filter, a first-order high-pass and a first-order low-pass sections are cascaded; for a ± 40 db/decade bandpass filter, second-order high- pass filter and a second-order low-pass filter are connected in series, and so on. It means that, the order of the bandpass filter is governed by the order of the high-pass and low-pass filters it consists of.

A ± 20 db/decade wide bandpass filter composed of a first-order high-pass filter and a first-order low-pass filter, is illustrated in fig. (a). Its frequency response is illustrated in fig. *(b).*

**Narrow Bandpass Filter.**

A narrow bandpass filter employing multiple feedback is depicted in figure. This filter employs only one op-amp, as shown in the figure. In comparison to all the filters discussed so far, this filter has some unique features that are given below.

**1. It has two feedback paths, and this is the reason that it is called a multiple-feedback filter.**

**2. The op-amp is used in the inverting mode.**

The frequency response of a narrow bandpass filter is shown in fig(b).

Generally, the narrow bandpass filter is designed for specific values of centre frequency f_{c} and Q or f_{c} and BW. The circuit components are determined from the following relationships. For simplification of design calculations each of **C**_{1}** and C**_{2}** may be taken equal to C.**

**R**_{1}** = Q/2∏ f**_{c }**CA**_{f}

**R**_{2}** =Q/2∏ f**_{c}** C(2Q**^{2}**-A**_{f}**)**

**and R**_{3 }**= Q / ∏ f**_{c}** C**

**where A**_{f}**, is the gain at centre frequency and is given as**

**A**_{f }**= R**_{3 }**/ 2R**_{1}

The gain A_{f } however must satisfy the condition A_{f} < 2 Q^{2}.

The centre frequency f_{c} of the multiple feedback filter can be changed to a new frequency f_{c}‘ without changing, the gain or bandwidth. This is achieved simply by changing R_{2} to R’_{2} so that

**R’**_{2 }**= R**_{2}** [f**_{c}**/f’**_{c}**]**^{2}

## 12 Comments

thanks a lot for your post

Hi,

can you please let me know how can I calculate the bandwidth and Fr of the filter?

Hi

would you please let me know after selecting the standard resistors, how I can calculate the Fc, Q , Gain erros? (I mean the errors produced by rounding values).

Thanks

Customer required band pass filters for space application and confirm the same in your scope of supply and on on confirmation will send you details.

hmmmm….

Looks like it’s time for some algebra.