### 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}

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