## Band Pass Filters

### 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 (fH and fl*), *the bandwidth (BW), the centre frequency fc,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 fc is given by an equation

For a wide bandpass filter the centre frequency can be defined as where fH and fLare respectively the high and low cut-off frequencies in Hz.In a narrow bandpass filter, the output voltage peaks at the centre frequency fc*.*

** 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 fc and Q or fc 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 Af however must satisfy the condition Af < 2 Q2.

The centre frequency fc of the multiple feedback filter can be changed to a new frequency fc‘ without changing, the gain or bandwidth. This is achieved simply by changing R2 to R’2 so that

**R’****2 ****= R****2**** [f****c****/f’****c****]****2**