TITLE :Analysis of 2nd and 3rd order Band Reject Filters

Submitted by : Yashaswa Jain IEC2013037Vartul Sharma IEC2013096Agrima Gurnani IEC2013098Raghav Garg IEC2013101

INTRODUCTION :It is sometimes desirable to have circuits capable of selectively filtering one frequency or range of frequencies out of a mix of different frequencies in a circuit. A circuit designed to perform this frequency selection is called a filter circuit, or simply a filter. A common need for filter circuits is in high-performance stereo systems, where certain ranges of audio frequencies need to be amplified or suppressed for best sound quality and power efficiency. You may be familiar with equalizers, which allow the amplitudes of several frequency ranges to be adjusted to suit the listener's taste and acoustic properties of the listening area. You may also be familiar with crossover networks, which block certain ranges of frequencies from reaching speakers. A tweeter (high-frequency speaker) is inefficient at reproducing low-frequency signals such as drum beats, so a crossover circuit is connected between the tweeter and the stereo's output terminals to block low-frequency signals, only passing high-frequency signals to the speaker's connection terminals. This gives better audio system efficiency and thus better performance. Both equalizers and crossover networks are examples of filters, designed to accomplish filtering of certain frequencies. Another practical application of filter circuits is in the conditioning of non-sinusoidal voltage waveforms in power circuits. Some electronic devices are sensitive to the presence of harmonics in the power supply voltage, and so require power conditioning for proper operation. If a distorted sine-wave voltage behaves like a series of harmonic waveforms added to the fundamental frequency, then it should be possible to construct a filter circuit that only allows the fundamental waveform frequency to pass through, blocking all (higher-frequency) harmonics

TYPES OF FILTERS :

Notch / Band Reject filters : Crystal or Discrete component filter that passes all frequencies except those in a stop band centered on a center frequency.Band Pass Filters : Filter that passes a certain band of frequencies. It has two cutoff frequencies, a high cutoff frequency and a low cutoff frequency.High Pass Filters : Discrete component filter that passes high frequency but alternates frequencies lower then the cut off frequency.Low Pass Filters : Discrete component filter that passes low frequency signals but alternates signals with frequencies higher then the cut off frequency.

FILTER DESIGNS :Chebyshev : The transfer function of the filter is derived from a chebychev equal ripple function in the passband only. These filters offer performance between that of Elliptic function filters and Butterworth filters. For the majority of applications, this is the preferred filter type since they offer improved selectivity, and the networks obtained by this approximation are the most easily realized.Butterworth : The transfer function of the filter offers maximally flat amplitude. Selectivity is better then Gaussian or Bessel filters, but at the expense of delay and phase linearity. For most bandpass designs, the VSWR at center frequency is extremely good. Butterworth filters are usually the least sensitive to changes in element values.Bessel/Linear Phase : The transfer function of the filter is derivedfrom a Bessel polynomial. It produces filter with a flat delay around center frequency. The more poles used, the wider the flat region extends. The roll-off rate is poor. This type of filter is close to a Gaussian filter. It has poor VSWR and loses its maximally flat delay properties at wider bandwidths.Elliptic : The passband ripple is similar to the Chebyshev but with greatly improved stopband selectivity due to the addition of finite attenuation peaks. The network complexity is increased over the Butterworth or Chebyshev, but still yields practical realizations over nearly the entire operating region.Gaussian : The transfer function of the filter is derived from a Gaussian function. The step and impulse response of a Gaussian filter has zero overshoot. Rise times and delay are lowest of the traditional transfer functions. These characteristics are obtained at the expensive of poor selectivity, high element sensitivity, and a very wide spread of element values. Gaussian filter is very similar to the Bessel except that the delay has a slight hump at center frequency and the rate of roll-off is slower. Because of the delay response, the ringing characteristics are better then the Bessel. Realization restrictions also apply to these filters.

THEORY AND DESCRIPTION :The bandpass filter passes one set of frequencies while rejecting all others. The band-stop filter does just the opposite. It rejects a band of frequencies, while passing all others. This is also called a band-reject or band-elimination filter. Like bandpass filters, band-stop filters may also be classified as (i) wide-band and (ii) narrow band reject filters.The narrow band reject filter is also called a notch filter. Because of its higher Q, which exceeds 10, the bandwidth of the narrow band reject filter is much smaller than that of a wide band reject filter.(i) Wide Band Reject Filters

A wide band-stop filter using a low-pass filter, a high-pass filter and a summing amplifier isshown in figure. For a proper band reject response, the low cut-off frequency fL of high-pass filter must be larger than the high cut-off frequency fH of the low-pass filter. In addition, the passband gain of both the high-pass and low-pass sections must be equal.

(ii) Narrow Band Reject Filters

This is also called a notch filter. It is commonly used for attenuation of a single frequency such as 60 Hz power line frequency hum. The most widely used notch filter is the twin-T network illustrated in fig. (a). This is a passive filter composed of two T-shaped networks. One T-network is made up of two resistors and a capacitor, while the other is made of two capacitors and a resistor.One drawback of above notch filter (passive twin-T network) is that it has relatively low figure of merit Q. However, Q of the network can be increased significantly if it is used with the voltage follower, as illustrated in fig. (a). Here the output of the voltage follower is supplied back to the junction of R/2 and 2 C. The frequency response of the active notch filter is shown in fig (b).

Notch filters are most commonly used in communications and biomedical instruments for eliminating the undesired frequencies.A mathematical analysis of this circuit shows that it acts as a lead-lag circuit with a phase angle, shown in fig. (b). Again, there is a frequency fc at which the phase shift is equal to 0. In fig. (c), the voltage gain is equal to 1 at low and high frequencies. In between, there is a frequency fc at which voltage gain drops to zero. Thus such a filter notches out, or blocks frequencies near fc. The frequency at which maximum attenuation occurs is called the notch-out frequency given byfn = Fc = 2RCNotice that two upper capacitors are C while the capacitor in the centre of the network is 2 C. Similarly, the two lower resistors are R but the resistor in the centre of the network is 1/2 R. This relationship must always be maintained.

INTRODUCTION TO SPICE :SPICE is a general purpose analog simulator which contains models for most circuit elements and can handle complex nonlinear circuits. The simulator can calculate dc operating points, perform transient analyses, locate poles and zeros for different kinds of transfer functions, find the small signal frequency response, small signal transfer functions, small signal sensitivities, and perform Fourier, noise, and distortion analyses.PSpice is a PC version of SPICE. PSpice has analog and digital libraries of standard components (such as NAND, NOR, flip-flops, and other digital gates, op amps, etc) which makes it a useful tool for a wide range of analog and digital applications.AIM-Spice is a new version of SPICE running under the Microsoft Windows and Linux operating systems. AIM-Spice for Windows is capable of displaying graphically the results of a simulation in progress, a feature that allows the operator to terminate a run based on an instant information on intermediate simulation results. The development of AIM-Spice was motivated by the need of a more user friendly interface, and as a vehicle for the new set of advanced device models for circuit simulation.

RESULTS AND DISCUSSION :