Active Filters, Physics tutorial

Active filter design criteria:

In designing Active Filter establish specifications to meet desired performance criteria. The filter circuit configurations given will acts as template to guide design.

Sallen and Key, and VCVS filters provide low dependency on accuracy of components that mean low tolerance components can be utilized to make filter. State variable and biquadratic filters, Twin T Active filter, Dual Amplifier Bandpass, Wien notch, Multiple Feedback Filter, Fliege which gives the lowest component count for 2 operational amplifiers but with good controllability over frequency and kind and the Akerberg Mossberg filter which allows you completely and independently control gain, frequency, and filter envelope.

For design proper, determine passband or desired range of frequencies that filter is expected to filter through. Then decide on shape of frequency response that specifies variety of filter and centre or corner frequencies.

Also specify input and output impedances as these will limit circuit topologies available for design. To support this fact note that not all active filter topologies let you to buffer output to provide low output impedance that is desirable for driving most loads. The frequency dependence and frequency limitations of active devices intend to utilize mainly for high Pass Filter design.

Remember the given points when deciding level of attenuation in stopband:

  • If design is for narrow-band bandpass filters, Q value determines -3dB bandwidth and degree of rejection of frequencies far eradicated from centre frequency; if the two requirements are in conflict then the staggered-tuning bandpass filter may be required.
  • If design is for notch filters, degree to which unwanted signals at notch frequency should be rejected finds out accuracy of components, but not Q value that is administered by desired steepness of notch. This signifies that bandwidth around notch before attenuation will be small.
  • If design is for high-pass, low-pass and band-pass filters far from centre frequency, the required rejection may find out slope of attenuation needed, and therefore order of filter.
  • Permissible ripple within passband of high-pass and low-pass filters, and shape of frequency response curve near corner frequency, determines damping factor and phase/time response to the square-wave input.

Damping factors to which you might refer have well established responses. Tschebyscheff Filter is characterized by the slight peaking with ripple in passband before corner frequency while Butterworth filters give flattest amplitude response. Linkwitz-Riley filter are best suitable for audio crossover applications when critically damped and Paynter filter that is alternatively called as transitional Thompson- Butterworth and compromise filter has the faster fall-off than Bessel filters. Bessel filter give best time-delay and best overshoot response while Elliptic filter or Cauer filters add the notch just outside passband, to give the much greater slope in this region than combination of order and damping factor without the notch.

Ideal filter approximations:

Ideal filter response curve is the rectangular shape, indicating the abrupt boundary between passband and stopband with the infinitely steep roll off slope. Ideal response curve, were it attainable would let us to totally divide signals at different frequencies from one another. Such amplitude response curves are not physically realizable and best approximation which will still meet requirements for the given application is frequently applied out of numerous options and deciding on best approximation includes making the compromise between different properties of filter's transfer function.

Properties of filter's transfer function are:

Order: Order of a filter is directly related steepness of its roll off slope, number of components in filter, its cost, physical size and its complexity.

Ultimate Rolloff Rate: Amount of attenuation for the given ratio of frequencies is 20 dB/decade for every low and pass filter pole and 20 dB/decade for every bandpass filter pole pair is called as ultimate roll off rate.

Attenuation Rate near Cutoff Frequency: Sharp cutoff characteristic is needed when frequency to be rejected lies adjacent to the frequency to be passed by filter requiring high attenuation rate.

Transient Response: Response of the filter to step function and filters will sharper cutoff characteristics or higher Q will have more pronounced ringing as contrasted with smooth reaction to the step input signal of filters with lower roll off.

Monotonicity: The monotonic filter's amplitude response never changes sign and gain always increases with increasing frequency or always decreases with increasing frequency in high and low pass filters, but bandpass or notch filters can be monotonic on either side of their centre frequency.

Passband Ripple: For non monotonic filters, transfer function inside passband will show one or more undulations called as ripple.

Amax: Amax is maximum allowable change in gain within passband.

Amin: Amin is minimum allowable attenuation within stopband.

Butterworth filter:

The best known ideal filter approximation is Butterworth low-pass filter; or else known as maximally flat filter and that is unique in its ability to give maximum passband flatness and they provide superior performance as anti-aliasing filter in data converter applications where precise signal levels are needed across the whole passband. It shows the nearly flat passband with no ripple. Roll off is smooth and monotonic, with low-pass or high-pass roll off rate.

1154_Butterworth low Pass Filter Optimisation.jpg

The higher the filter order, the longer the passband flatness, and the closer the filter characteristics to the idealized low pass filter.

Tschebyscheff filter:

Alternatively called as equal ripple filter, Tschebyscheff low-pass filter gives impressively high gain roll-off above cutoff frequency, major drawback being that passband gain is not monotonic but includes ripples of constant magnitude and for given filter order, higher passband ripples result in higher filter roll off.

Observe that influence of magnitude of ripples on filter roll off diminishes with increasing filter order and that every ripple accounts for one second-order filter stage. Tschebyscheff filter of order n will have n-1 peaks or dips in passband response. Tschebyscheff lowpass Filters having even order numbers produce ripples above 0-dB line, while filters with odd order numbers produce ripples below 0 dB. The nominal gain of Tschebyscheff filter is equivalent to filter's maximum passband. Tschebyscheff filter Optimization find helpful application in filter banks, where frequency content of the signal is of more significance than constant amplification.

1198_Tschebyscheff Low-Pass Filter Optimisation.jpg

Bessel filter:

Frequency dependent phase shift is the expected characteristic of filters and can offer problems in definite situations. Linear phase increase with frequency has effect of delaying output signal by constant time period but if phase shift is not directly proportional to frequency, components of input signal at one frequency will observe at output shifted in phase with respect to other frequencies and distorts non-sinusoidal waveforms.

Where linear phase response is of utmost significance, Bessel lowpass filter that is alternatively known as Thompson filter has the clear benefit over Butterworth and Tschebyscheff filters. This linear phase response shown over the wide frequency range, results in constant group delay in two plots below in comparison with those of filters.

558_Comparative Phase Responses of Discussed Low-Pass Filters.jpg

1603_Relative Group Delay.jpg

Due to its wideband linear phase response, Bessel filters show optimum square-wave transmission behaviour. In comparison, passband gain of Bessel low-pass filter is not as flat as that of Butterworth low-pass filter and transition from passband to stopband is not as sharp as that of Tschebyscheff low-pass filter.

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