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Active Filters Active Filters Introduction Filters are widely used in communication and signal processing and in electronic instruments. It is a frequency selective circuit specially designed to pass a specified band of signals while attenuating all the signals outside the band. They are classified as active and passive filters based on the components used for the purpose. Filters using resistors, capacitors and inductors are passive filters. Filters using transistors, op-amps along with the passive elements are active filters. Inductors are not preferred as they are bulky and non linear producing stray magnetic fields. They dissipate more power also. The active filters pose more advantages than passive filters. They are 1. Reduction in size and cost due to op-amps 2. Op-amps used can be easily controlled in closed loop fashion hence active filter input signals are not attenuated. 3. Active filters can be easily tuned as they have gain adjustments. 4. No loading of source or load. 5. Improved response 6. Simple design and provide voltage gains. Apart from the merits the active filters also suffer from demerits. The bandwidth is finite is the major demerit. Sensitivity also affects the performance. The active elements are sensitive to temperature and ambient changes. Hence the response deviates from the ideal one. The most commonly used Active filters are 1. Low Pass Filter (LPF) - A low pass filter allows low frequency signals to pass from input to output while it attenuates HF signals from the input. The graph of magnitude of transfer function against frequency is called Frequency response of a filter. The ideal and practical responses are shown below Ideal response Practical Response The cut off frequency fo of the filter denotes the frequency where the filter changes its function. For a LPF it passes all frequencies below that frequency and rejects all frequencies above that. 2. High Pass Filter (HPF): A high pass filter allows high frequency signals to pass from input to output while it attenuates LF signals below the cut off frequency from the input. After the cut off frequency the filter allows the frequencies to pass through rejecting below the cut off frequency Ideal response Practical Response 3. Band Pass Filter (BPF) - A BPF is combination of HPF and LPF. It has two stop bands, one pass band and two cut off frequencies fL and fH called Lower and Upper Cut off frequencies respectively. The filter rejects frequencies below the lower cut off frequency and above the upper cut off frequency, while passes the band of frequencies between them. The difference between the upper cut off frequency fH and the lower cut off frequency fL is called the bandwidth of the filter. Ideal response Practical Response 4. Band Reject Filter or Band Stop Filter or Band Elimination filter (BSF) - The characteristics of this filter are exactly the opposite of band pass filter. There are two pass bands and one stop band. The stop band is between two 3 dB frequencies. Ideal response Practical Response It can be seen from the responses of the filter that the gain decreases or increases or both in the stop band. Order of the filter decides the rate at which the gain of the filter changes. The gain rolls off at +20 dB / decade in the stop band for a first order HPF. The gain decreases at a rate of -20 dB / decade in the stop band of a LPF and so on. In case of II order filters the rate of change is 40 dB / decade and so on. The various types of filters that produce ideal responses approximately are 1. Butter worth filters and 2. Chebyshev filter Butterworth Approximation The filter in which the denominator polynomial of the transfer function is a Butterworth polynomial is called a Butterworth filter. The Butterworth polynomials of various orders are given by the table First Order Low Pass Butterworth filter The first order filter is realized by a RC network with an op-amp in non-inverting mode as shown in figure Circuit Diagram Response The resistances Rf and R1 decide the gain of the filter. This is also called one pole LPF. The impedance of the capacitor is given by Xc = 1 / 2fC. By potential divider rule, the voltage at the non-inverting input terminal A which is the voltage across the capacitor is given by, The transfer function of the filter is given by Thus for, 0 < f < fH the gain is almost constant i.e AF .At f = fH the gain reduces by 3 dB from the constant gain. As frequency increases the gain of the filter reduces at a rate of 20 dB / decade. The frequency fH is called the upper cut off frequency of the LPF. Design Steps First Order Butterworth High Pass Filter A high pass circuit attenuates all frequencies lesser than the cut off frequency fL. This filter is the complementary of a LPF. The cut off frequency is called the lower cut off frequency. All the frequencies greater than the lower cut off frequency are passed. The simple RC network works here also as that LPF but the positions of R and C are interchanged. The circuit can be analyzed for its cut off frequency. The above equation can be modified as As the op-amp is in non-inverting mode, the gain is written as AF = 1+Rf/R1 and the output is VO=AFVA. Here VA is the voltage at the non-inverting terminal. Hence The above equation shows the transfer function of the filter. In frequency domain we write the magnitude as, The circuit rejects frequencies below the lower cut off frequency and passes all frequencies from fL. Hence the gain increases till fL at a rate of +20 dB / decade. Hence the slope of the response in the stop band is +20 dB / decade for the first order high pass filters. The response is shown in figure The design steps are the same as that of first order low pass filter except the R and C positions interchanged. .
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