www.olatus.com ACTIVE FILTERS The filters are the circuits used to separate the complex signals according to the frequency range. The filters are widely used in communication, signal processing, wave shaping and in almost all modern electronic systems. The analog filters using op-amps are called active filters. A filter is a circuit that is designed to pass a specified band of frequencies while rejecting all the signals outside that band. The filters are basically classified as – Active filters Passive filters The passive filter network use only passive elements such as resistor, inductors and capacitors. On the other hand, active filters use the active element such as op-amps, transistors along with resistors, inductors, capacitors. Modern filters do not use inductors as the inductors are bulky, heavy and non-linear. Advantages of Active Filters All the elements along with op-amps can be used in the integrated form. Hence, there is reduction in size and weight. The cost of the integrated circuit can be much lower than its equivalent passive network. The response is improved as compared to passive filters. 1 © OLatus Systems Pvt Ltd www.olatus.com Parameters of Filter Any filter has two important parameters. They are – 1. Transfer Function It is defined as the ratio of laplace transform of output to the laplace transform of input. It is denoted by T(s). Replacing variable s by jω, we get the frequency domain transfer function and is denoted by T(jω). The magnitude of transfer function, |T(jω)| is called Gain. It is denoted by A. = Gain representation in dB, A = 20 log |T| dB 2. Cut-off Frequency It indicates the frequency at which filter changes its function. This means for a Low Pass Filter, it passes all frequencies below ω0 and rejects all frequencies above ω0. The magnitude of transfer function, |T(jω)| remains constant for all frequencies below ω0 for a Low Pass Filter while the magnitude of transfer function, |T(jω)| decreases and becomes zero for frequencies greater than ω0 . The graph of magnitude of transfer function, |T(jω)| against the frequency, ω is called the Frequency Response of the filter. 2 © OLatus Systems Pvt Ltd www.olatus.com Classification of Filters Depending on frequencies, filters can be classified into five categories. They are – Low Pass Filter (LPF) High Pass Filter (HPF) Band Pass Filter (BPF) Band Stop Filter / Band Reject Filter (BRF) All Pass Filter (APF) 1. Low Pass Filter (LPF) LPF is a circuit which passes the low frequency signals from input to output, rejecting the higher frequency signals. The frequency response of the Ideal LPF and the practical LPF are shown below – Fig: Frequency Response of Ideal LPF and Practical LPF As shown in the above frequency response curve, gain is constant for ω < ω0 and is zero for ω > ω0 ideally and decreases for practical LPF. The frequency band 0 < ω < ω0 is called pass band while the frequency band for ω > ω0 is called stop band. 3 © OLatus Systems Pvt Ltd www.olatus.com 2. High Pass Filter (HPF) In HPF, the filter rejects the frequencies which are less than cut-off frequency, ω0 and it allows to pass the frequencies which are greater than ω0. The frequency response of the Ideal HPF and the practical HPF are shown below – Fig: Frequency Response of Ideal HPF and Practical HPF As shown in the above frequency response curve, gain is constant for ω > ω0 and is zero for ω < ω0 ideally and decreases for practical HPF. The frequency band 0 < ω < ω0 is called stop band while the frequency band for ω > ω0 is called pass band. 4 © OLatus Systems Pvt Ltd www.olatus.com 3. Band Pass Filter (BPF) The BPF passes the specific band of frequency and it has two stop bands and two cut-off frequencies. The frequency response of the Ideal BPF and the practical BPF are shown below – Fig: Frequency Response of Ideal BPF and Practical BPF It rejects the frequencies less than ω1 and the frequencies higher than ω2, while it passes the band of frequencies between ω1 and ω2. The frequency between ω1 and ω2 at which the gain attains maximum value is called centre frequency denoted by ω0. At ω1 and ω2 , the gain 3dB down from its maximum value hence, those frequencies are called 3dB frequencies. And the difference (ω2 - ω1) is called bandwidth of the filter. As shown in the above frequency response curve, gain is constant for and is zero for ω < ω1 and ω > ω2 ideally and decreases for practical BPF. The frequency band 0 < ω < ω1 and ω > ω2 is called stop band while the frequency band for is called pass band. 5 © OLatus Systems Pvt Ltd www.olatus.com 4. Band Reject Filter (BRF) This is also called Band Elimination Filter or Band Stop Filter or Notch Filter. Its characteristics are exactly opposite to band pass filter. There are two pass band and one stop band. The stop band is between 3dB frequencies ω1 and ω2. The frequency ranges 0 < ω < ω1 and ω2 > ω are the two pass band. The frequency at which the gain is minimum is called centre frequency and it is denoted by ω0. The frequency response of the Ideal BRF and the practical BRF are shown below – Fig: Frequency Response of Ideal BRF and Practical BRF 5. All Pass Filter The All Pass Filter passes all the frequencies but it produces the phase shift between input and output. The output and input voltages are equal in magnitude for all frequencies but out of phase. If input = sin ωt Then output = sin (ωt + 180°) = - sin ωt 6 © OLatus Systems Pvt Ltd www.olatus.com First Order Low Pass Butterworth Filter Fig: First Order Low Pass Butterworth Filter The voltage at node A, Since the op-amp is in non-inverting mode, the output voltage Where, = Pass Band Gain 7 © OLatus Systems Pvt Ltd www.olatus.com or, or, Putting s = jω, Putting ω = 2πf, Or, Let, fH = = High Cut-off Frequency = Gain Or, 8 © OLatus Systems Pvt Ltd www.olatus.com Case 1: f < fH, Case 2: f = fH, Case 3: f > fH, Design Steps for First Order LPF Select the cut-off frequency, fH Select the capacitor, C between 0.001F to 1µF Calculate resistance, R Select R1 = 10KΩ Calculate Rf. 9 © OLatus Systems Pvt Ltd www.olatus.com Second Order Low Pass Butterworth Filter In case of first order filter, the gain rolls off at a rate of 20 dB / decade. In case of second order filter, the gain rolls off at a rate of 40 dB / decade and the frequency response after f = fH is -40 dB / decade for a second order LPF. Fig: Second Order Low Pass Butterworth Filter The high cut-off frequency, If R2 = R3 = R and C2 = C3 = C Then, 10 © OLatus Systems Pvt Ltd www.olatus.com The gain, Where, Pass Band Gain Design Steps for Second Order LPF Select the cut-off frequency fH The design can be simplified by selecting R2 = R3 = R and C2 = C3 = C and select the value of C less than or equal to 1 µF Calculate the value of R from the equation, Select the value of R1 equal to or less than 10 KΩ Calculate the value of Rf 11 © OLatus Systems Pvt Ltd www.olatus.com First Order High Pass Butterworth Filter A high pass filter is a circuit that attenuates all the signals below a specified cut-off frequency denoted as fL. Thus, a high pass filter performs the opposite function to that of low pass filter. Hence, the high pass filter circuit can be obtained by interchanging frequency determining resistance and capacitance in low pass filter. The frequency at which the gain is 0.707 times the gain of filter in pass band is called low cutoff frequency and denoted by fL. So, all the frequencies greater than fL are allowed to pass. Fig: First Order High Pass Butterworth Filter By applying potential divider at node A, Or, 12 © OLatus Systems Pvt Ltd www.olatus.com As we know, this amplifier look like a non-inverting amplifier, hence output of non-inverting amplifier But VA = VB, So, Or, Where, Pass Band Gain or, Putting s = jω, Putting ω = 2πf, Or, Let, fL = = Low Cut-off Frequency 13 © OLatus Systems Pvt Ltd www.olatus.com The above equation represents the transfer function of first order high pass filter. The magnitude of the transfer function is = Gain Case 1: f < fL, Case 2: f = fL, = = Case 3: f > fL, Design Steps for First Order HPF Select the cut-off frequency, fL Select the capacitor, C between 0.001F to 1µF Calculate resistance, R Select R1 = 10KΩ Calculate Rf. 14 © OLatus Systems Pvt Ltd www.olatus.com Second Order High Pass Butterworth Filter The second order high pass butterworth filter produces a gain roll off at the rate of 40 dB / decade in the stop band. This filter also can be realized by interchanging the positions of resistors and capacitors in a second order low pass filter. Fig: Second Order High Pass Butterworth Filter The low cut-off frequency, If R2 = R3 = R and C2 = C3 = C Then, 15 © OLatus Systems Pvt Ltd www.olatus.com The gain, Where, Pass Band Gain Design Steps for Second Order LPF Select the cut-off frequency fL The design can be simplified by selecting R2 = R3 = R and C2 = C3 = C and select the value of C less than or equal to 1 µF Calculate the value of R from the equation, Select the value of R1 equal to or less than 10 KΩ Calculate the value of Rf 16 © OLatus Systems Pvt Ltd www.olatus.com Band Pass Filter A band pass filter is basically a frequency selector. It allows one particular band of frequencies to pass. Thus, the pass band is between the two cut-off frequencies fH and fL where fH > fL. Any frequency outside this band gets attenuated. The pass band which is between fH and fL is called bandwidth of the filter and is denoted by BW. BW = fH - fL The frequency at the centre of the pass band is called centre frequency denoted by fc. The gain is maximum at centre frequency, fc and is denoted as AfT called total pass band gain. Basically, the fc is not exactly at the centre of the pass band. Hence, it is called as resonant frequency. The gain at fH and fL is 0.707AfT. 17 © OLatus Systems Pvt Ltd www.olatus.com The relationship between Quality Factor (Q), Bandwidth (BW) and Centre Frequency (fc) – Or, Types of BPF There are two types of BPF which are classified based on the quality factor, Q. They are – Wide BPF Narrow BPF 1. Wide Band Pass Filter (Wide BPF) For Q < 10, the BPF is called wide BPF. In this type, the band pass is wide and we get large bandwidth. Fig: Frequency Response of wide band BPF 18 © OLatus Systems Pvt Ltd www.olatus.com 2. Narrow Band Pass Filter (Narrow BPF) For Q > 10, the BPF is called narrow BPF. The band pass is very narrow and bandwidth is very small. Higher the value of Q, narrower is the pass band and more selective is the filter. In the narrow BPF, the gain peaks at the centre frequency. Fig: Frequency Response of narrow band BPF 19 © OLatus Systems Pvt Ltd www.olatus.com Wide Band Pass Filter (Wide BPF) Wide BPF can be realized by simply cascading a HPF and LPF. If both HPF and LPF are of first order, the gain roll off in both the stop bands are ±20 dB/decade and wide BPF is of first order. To get gain roll off of ±40 dB/decade and second order wide BPF, both HPF and LPF must be of second order and so on. Fig: Wide Band Pass Filter Gain of HPF is, Gain of LPF is, 20 © OLatus Systems Pvt Ltd www.olatus.com As the two circuits are in cascade, the overall gain of Wide BPF is the product of the two gain expressions as Or, Where, AfT = Af1.Af2 = Total pass band gain f = input frequency fL = lower cut-off frequency fH = higher cut-off frequency Af1 = gain of high pass section Af1 = gain of high pass section 21 © OLatus Systems Pvt Ltd www.olatus.com Narrow Band Pass Filter (Narrow BPF) The narrow BPF uses only one op-amp as against two by wide BPF. It has two features It has two feedback paths The op-amp is in inverting configuration Due to feedback paths, it is called multiple feedback filter. Fig: Narrow Band Pass Filter For simplifying the calculations, choose C1 = C2 = C 22 © OLatus Systems Pvt Ltd www.olatus.com Band Stop / Band Elimination / Band Reject Filter The function of this filter is exactly opposite to the band pass filter. This filter is also classified as Wide Band Reject Filter (Q > 10) Narrow Band Reject Filter (Q < 10) Wide Band Reject Filter It consists of HPF and LPF. Additionally, it consists of a summing amplifier. Fig: Wide Band Reject Filter The low cut-off frequency, fL of HPF must be greater than the high cut-off frequency, fH of LPF. 23 © OLatus Systems Pvt Ltd www.olatus.com The pass band gain of both high pass and low pass sections must be equal. The design of the overall filter is based on the individual design of the various sections. The gain of the summing amplifier can be set to 1 for simplicity and thus R2 = R3 = R4 = R And Rcomp = R2|| R3|| R4 = R/3 Narrow Band Reject Filter (Notch Filter) The name of the filter i.e. notch filter is due to the characteristics shape of its frequency response curve, the stop band of this filter is very narrow. The typical application of the notch filter is the rejection of a single frequency. It is also used in the biomedical instrumentation and also blanking of control tones for telephone lines. Thus, particular unwanted frequency can be eliminated by using this filter. The passive circuit used to obtain the notch filter is the twin T network as shown below Fig: Twin T Network 24 © OLatus Systems Pvt Ltd www.olatus.com It consists of two T network. One consists of two resistors and a capacitor while the other consists of two capacitors and one resistor. The Notch out frequency, fN is the frequency at which maximum attenuation occurs. This is given by The circuit diagram of notch filter is shown below – Fig: Active Notch Filter 25 © OLatus Systems Pvt Ltd www.olatus.com The ideal and the practical frequency response of notch filter is shown below – Fig: Frequency Response of Ideal and Practical Notch filter Design steps for Notch Filter To design a notch filter i.e. to eliminate the specific notch frequency, fN Choose the value of capacitor, C ≤ 1 µF Calculate the value of resistance, R using frequency fN 26 © OLatus Systems Pvt Ltd www.olatus.com Second Order Notch Filter It also uses a twin T network. The circuit diagram of second order notch filter is shown below – Fig: Second Order Notch Filter The transfer function of this circuit is – Where, For the stability purpose, Af must be less than 2. 27 © OLatus Systems Pvt Ltd
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