{"id":269,"date":"2018-11-29T06:32:00","date_gmt":"2018-11-29T06:32:00","guid":{"rendered":"http:\/\/msp04.epgpbooks.inflibnet.ac.in\/?post_type=chapter&p=269"},"modified":"2019-05-01T05:12:14","modified_gmt":"2019-05-01T05:12:14","slug":"filter-ii","status":"publish","type":"chapter","link":"https:\/\/ebooks.inflibnet.ac.in\/msp04\/chapter\/filter-ii\/","title":{"rendered":"Filter II"},"content":{"raw":"
\"epgp<\/a>\r\n<\/span><\/div>\r\n
\r\n\r\n\u00a0 \u00a0 Introduction<\/strong>\r\n\r\n \r\n\r\n \r\n

In the preceding module, we have already discussed about the various types of filters. First order and second order low-pass and high-pass filters were explained. In this module, we will discuss about the higher order filters and the detailed analysis of band-pass, band-stop and all-pass filters.<\/p>\r\n \r\n\r\nHigher order filters<\/strong>\r\n\r\n \r\n

We have already studied that in a first order filter, the gain of filter changes at a rate of 20 dB\/decade. In a second order filter, the gain changes at a rate of 40 dB\/decade. Thus, as the order of filter increases, the filter approaches towards ideal behavior i.e. change in gain in the stopband of filter becomes more steep near the cut-off frequency. Higher order filter of any order (i.e. third, fourth and so on) can be made using the first order and second order filters. For example, a third order low-pass filter can be made by integrating a first order and second order low-pass filter together in series. A fourth order filter can be made by cascading two second order low-pass filters and so on. In this way, filters of various orders can be made. However, it is important to understand that as the order of filter increases, its size also increases. Also, with an increase in the order of filter, the observed stopband response deviates increasingly from the theoretical stopband response. Thus, accuracy of the filter declines by increasing eth order of filter. Figure 8 shows the circuit diagram of third order and fourth order low-pass filters.<\/p>\r\n \r\n\r\n\"\"\r\n\r\n \r\n\r\n\"\"\r\n\r\n \r\n

Figure 8: Circuit of (a) third order and (b) fourth order low-pass filter (c) Frequency response<\/span><\/p>\r\n\r\n<\/div>\r\n

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The overall gain of the higher order filter is equal to the product of gains of the individual stages. Since the values of frequency determining resistors and capacitors (R and C) are kept to be same everywhere in third and fourth order filters shown in figure 8, therefore, the high cut-off frequency is asme for the third and fourth order low-pass filters. The cut-off frequency (fH) is given by = 2\u00a0\u00a0\u00a0 1 . The frequency response of third and fourth order low-pass filters is shown in figure 8(c). Similar to the first and second order high-pass filters, third and fourth high-pass filters can be made from third and fourth order low-pass filters by simply interchanging the positions of frequency determining resistors and capacitors in the constituting individual low-pass filters. Although, the stopband response of the higher order filter is closer to ideal case but higher order filters are more complex, occupy more space and are more expensive.<\/p>\r\n \r\n\r\nBand-pass filters<\/strong>\r\n\r\n \r\n

A band-pass filter allows only a particular band of frequencies to pass through it without attenuation and blocks others. The band of frequencies is defined by certain parameters such as lower (fL) and higher cut-off frequencies (fH), centre frequency (fC), Bandwidth (BW), passband gain and Quality factor (Q). If the lower and higher cut=off frequencies are fL and fH, then, bandwidth of the band-pass filter will be,<\/p>\r\n \r\n\r\n?? = ??<\/sub> \u2212 ??<\/sub>\r\n\r\n \r\n

Band-pass filters are basically classified into two categories as (1) Wide band-pass and (2) Narrow band-pass. It is difficult to define that whether a particular filter is wide band-pass or narrow. However, in order to differentiate the two, a parameter called figure of merit or quality factor (Q) is introduced. Q is a measure of selectivity of the filter i.e. higher the value of Q, more selective is the filter or narrower is its bandwidth. Q is related to the centre frequency of the filter and bandwidth as<\/p>\r\n\r\n<\/div>\r\n

\"\"<\/div>\r\n \r\n

A filter is said to be a wide band-pass filter if its quality factor (Q) is less than 10 whereas, if Q > 10, the filter is said to be a narrow band-pass filter. For a wide band-pass filter, the <\/span>centre frequency (??<\/sub>) can be defined as, = \u221a\u00a0 \u00a0. In a narrow band-pass filter, the output voltage peaks at the <\/span>centre frequency, fC.<\/span><\/p>\r\n\r\n

\r\n\r\n \r\n\r\nWide band-pass filter<\/strong>\r\n\r\n \r\n\r\nThe circuit of a wide band-pass filter is formed by cascading a high-pass and low-pass filter. The order of the wide band-pass filter depends on the order of its constituting low-pass and high-pass filters i.e to obtain a first order band-pass filter (gain rolls off at \u00b120 dB\/decade), first order high-pass and low-pass filters are cascaded. For a second order band-pass filter (\u00b140 dB\/decade), second-order high-pass filter is connected in series with second-order low-pass filter. The circuit diagram of first order band-pass filter is shown in figure 6(a).\r\n\r\n\"\"\r\n\r\n \r\n

Figure 6: (a) Circuit diagram of a first order wide band-pass filter (b) Frequency response<\/p>\r\n\r\n<\/div>\r\n\"\"\r\n

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Therefore, the total passband gain of the band pass filter is equal to the product of the individual passband gains of low-pass and high-pass filters.<\/p>\r\n \r\n

The frequency response of the band-pass filter is shown in figure 6(b). From eqn. (11) it can be inferred that for frequencies<\/p>\r\n\"\"\r\n\r\n<\/div>\r\n \r\n

\r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\nNarrow band-pass filter<\/strong>\r\n\r\n \r\n

The circuit diagram of a narrow band-pass filter is shown in figure 7(a). It may be noted that the filter uses only one op-amp in contrast to wide band-pass filter which uses two op-amps. Also, it has certain distinct features in comparison to all other filters:<\/p>\r\n\r\n

    \r\n \t
  1. The filter has two feedback paths because of which a narrow band-pass filter is also called as multiple feedback filter.<\/li>\r\n \t
  2. It uses op-amp in the inverting mode.<\/li>\r\n<\/ol>\r\nThe expression for the voltage gain can be obtained by applying the Kirchoff\u2019s current law at node V2.\r\n\r\n \r\n\r\n\"\"\r\n\r\n<\/div>\r\n
    \r\n\r\n\"\"\r\n\r\n\"\"\r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n<\/div>\r\n
    \r\n\r\n\"\"\r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n

    The maximum value of gain is limited by the quality factor. Gain(AF) of narrow bandpass filter must satisfy the condition: ?? < 2?2<\/sup> (Since R2 cannot be negative).<\/p>\r\n \r\n

    In order to analyze the variation of gain as a function of frequency, consider the expression for gain (equation (15) and apply some special cases:<\/p>\r\n \r\n\r\n1. When \u03c9\u2192 0:\r\n\r\n<\/div>\r\n\"\"\r\n

    \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\nFigure 7(b) shows the variation of gain of narrow bandpass filter as a function of frequency. It may be noted that the gain is maximum at a frequency \u03c9 = \u03c9o and falls on either sides.\r\n\r\n \r\n\r\nBand Reject Filter<\/strong>\r\n\r\n \r\n

    The operation of a band-reject filter is opposite to that of a bandpass filter. It is a frequency selective circuit which attenuates a particular band of frequencies while passing through it and allows the rest. Similar to the bandpass filters, band-reject filters are also classified as (1) Wide band-reject and (2) Narrow band-reject filter. Wide band-reject filter has a small value of Q ((Q < 10) whereas, narrow band-reject filter has a large value of Q (Q > 10) due to small bandwidth.<\/p>\r\n \r\n\r\nWide band-reject filter<\/strong>\r\n\r\n \r\n

    A wide band-reject filter consists of a low-pass filter, a high-pass filter and a summing amplifier. The circuit of band-reject filter and its frequency response is shown in figure 8. The band-reject filter response may be viewed as the sum of output of low pass filter and a high-pass filter. In order to realize the band-reject filter response, it is necessary that low cut-off frequency (fL) of the high-pass filter is greater than\u00a0the high cut-off frequency (<\/span>fH) of the low-pass filter. Also, the gain in the passbands of low-pass and high-pass sections must be equal. The bandwidth of <\/span>filter, BW = fH \u2013 fL.<\/span><\/p>\r\n\r\n<\/div>\r\n

    \r\n\r\n\"\"\r\n\r\n \r\n

    Figure 8: (a) Circuit diagram of a wide band-reject filter and it\u2019s (b) frequency response<\/p>\r\n \r\n\r\nNarrow band-reject filter<\/strong>\r\n\r\n \r\n

    The narrow band-reject filter is commonly known as a notch filter and is used for the rejection of a single frequency such as 60-Hz power line frequency hum. The circuit of a notch filter is a twin-T network which is a passive filter composed of two T-shaped networks connected in parallel. One of the T-networks is made up of two resistors and one capacitor (R, R and 2C) whereas other T-network is made up of one resistor and two capacitors (C, C, R\/2). The upper T-network works as a low-pass filter and the lower T-network works as a high-pass filter. A twin T-network used in a notch filter is shown in figure\u00a09(a). The analysis of the notch filter can be carried out by converting the two T-networks into their equivalent -networks. The -equivalent circuit is drawn in figure 9(b) where,<\/span><\/p>\r\n\r\n<\/div>\r\n

    \r\n\r\n\"\"\r\n\r\n\"\"\r\n\r\n \r\n

    Figure 9: (a) Passive twin T-notch filter (b) Equivalent\u00a0 - network (c) Simplified circuit<\/p>\r\n\r\n<\/div>\r\nThe circuit can be reduced to that shown in figure 9(c) where,\r\n

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    A passive notch filter shown in figure 9(a) suffers from various drawbacks. Firstly, the gain of the filter at frequencies less than the notch frequency is not same as that for frequencies greater than the notch frequency. This is because of the higher voltage drop across the two resistors in the low-pass filter section in comparison to that across the capacitors in the high-pass section leading to lower gain at frequencies f<fN. Secondly, the notch filter possesses relatively low figure of merit (Q) which corresponds to a large bandwidth. Such large bandwidth is not desirable for various applications. The figure of merit (Q) can be increased by integrating the passive T-network with the voltage follower resulting in active notch filter as shown in figure 9(b). The frequency response of active notch filter is shown in figure 9(c). The circuit exhibits a flat response with a voltage gain equal to 1 over the entire frequency range except a particular frequency called notch frequency (fN). It is the frequency at which maximum attenuation occurs.<\/p>\r\n\"\"\r\n

    Thus, the name notch filter is derived by the ability of the circuit to notch out or block frequencies near fN. Notch filters are highly useful in situations where it is necessary to attenuate a single frequency which is\u00a0responsible for generating electrical noise such as that generated from inductive loads (motors) etc. Notch filters find increasing applications in communication and biomedical instruments for eliminating unwanted frequencies.<\/span><\/p>\r\n\r\n<\/div>\r\n

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    Figure 9: (a) Active notch filter and (b) Frequency response of the active notch filter<\/p>\r\n \r\n\r\nAll-pass filter<\/strong>\r\n\r\n \r\n

    An all-pass filter, in accordance to it name allows all frequency components of the signal to pass through it without attenuation but introduces predictable phase shifts for different frequencies of input. These filters are widely used as phase compensators when signals are transmitted over transmission lines such as telephone wires leading to phase change. All-pass filters are therefore, also commonly known as delay equalizers or phase correctors. The circuit of an all-pass filter is shown in figure ---. RF is set to be equal to R1 in the circuit.<\/p>\r\n\r\n<\/div>\r\n

    \"\"Figure ----: (a) Circuit of <\/span>all-pass filter and (b) Phase shift between the input and output voltages<\/span><\/p>\r\n\r\n

    \r\n\r\n\"\"\r\n\r\n \r\n\r\n<\/div>\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n\"\"\r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n

    If the parameters R and C are fixed for a filter, then the phase angle ? can be determined using the above equation. Expression ---- also reveals that as the frequency of input signal is varied from 0 to \u221e, the phase angle (?) changes from 0 to \u2212180\u00b0. Figure ---(b) shows the phase shift between the input and output voltages for an all-pass filter where Vo lags Vin by 90\u00b0. If the position of R and C are interchanges in figure ---, then the phase shift between input and output signal becomes positive i.e. output leads the input signal by angle ?.<\/p>\r\n\r\n\r\n\r\n
    you can view video on Filter II<\/strong><\/td>\r\n\"\"<\/a><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n\r\nReferences:<\/strong>\r\n