At this point the gain of the filter is given as: Q × A = 14, or about +23dB, a big difference from the calculated value of 2.8, (+8.9dB).īut many books, like the one on the right, tell us that the gain of the filter at the normalised cut-off frequency point, etc, etc, should be at the -3dB point. We can see that the peakiness of the frequency response curve is quite sharp at the cut-off frequency due to the high quality factor value, Q = 5. If the filters characteristics are given as: Q = 5, and ƒc = 159Hz, design a suitable low pass filter and draw its frequency response.Ĭharacteristics given: R1 = R2, C1 = C2, Q = 5 and ƒc = 159Hzįrom the circuit above we know that for equal resistances and capacitances, the cut-off frequency point, ƒc is given as: Second Order Filter Example No1Ī Second Order Low Pass Filter is to be design around a non-inverting op-amp with equal resistor and capacitor values in its cut-off frequency determining circuit. So for a Butterworth second order low pass filter design the amount of gain would be: 1.586, for a Bessel second order filter design: 1.268, and for a Chebyshev low pass design: 1.234. Then to bring the second order filters -3dB point back to the same position as the 1st order filter’s, we need to add a small amount of gain to the filter. In other words, the corner frequency, ƒr changes position as the order of the filter increases. However, this -3dB cut-off point will be at a different frequency position for second order filters because of the steeper -40dB/decade roll-off rate (blue line). This point is generally referred to as the filters -3dB point and for a first order low pass filter the damping factor will be equal to one, ( ζ = 1 ). We can see from the normalised frequency response curves above for a 1st order filter (red line) that the pass band gain stays flat and level (called maximally flat) until the frequency response reaches the cut-off frequency point when: ƒ = ƒr and the gain of the filter reduces past its corner frequency at 1/√ 2, or 0.7071 of its maximum value. Then for a low pass second order filter, both Q and ζ play a critical role. One final note, when the amount of feedback reaches 4 or more, the filter begins to oscillate by itself at the cut-off frequency point due to the resonance effects, changing the filter into an oscillator. This is when the filter is “critically damped” and occurs when ζ = 0.7071. Then somewhere in between, ζ = 0 and ζ = 2.0, there must be a point where the frequency response is of the correct value, and there is. When ζ = 0, the filters output peaks sharply at the cut-off point resembling a sharp point at which the filter is said to be “underdamped”. When ζ = 1.0 or more (2 is the maximum) the filter becomes what is called “overdamped” with the frequency response showing a long flat curve. The amplitude response of the second order low pass filter varies for different values of damping factor, ζ. The basic configuration for a Sallen-Key second order (two-pole) low pass filter is given as: Second order low pass filters are easy to design and are used extensively in many applications. Second order (two-pole) active filters whether low pass or high pass, are important in Electronics because we can use them to design much higher order filters with very steep roll-off’s and by cascading together first and second order filters, analogue filters with an n th order value, either odd or even can be constructed up to any value, within reason. Most active filters consist of only op-amps, resistors, and capacitors with the cut-off point being achieved by the use of feedback eliminating the need for inductors as used in passive 1st-order filter circuits. The Sallen-Key filter design is one of the most widely known and popular 2nd order filter designs, requiring only a single operational amplifier for the gain control and four passive RC components to accomplish the tuning. All these types of filter designs are available as either: low pass filter, high pass filter, band pass filter and band stop (notch) filter configurations, and being second order filters, all have a 40-dB-per-decade roll-off. Most designs of second order filters are generally named after their inventor with the most common filter types being: Butterworth, Chebyshev, Bessel and Sallen-Key.
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