# Quadrant Filters / Second Order Filters

Since op-amp is used as a voltage-controlled voltage source amplifier, the second-order Filters, also called VCVS filters, are another important type of active filter design. Because the active first level RC filters that we have looked at before can be designed using higher level filter circuits.

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In this section of analog filters, we looked at both passive and active filter designs. We found that filters of the first order can be easily converted from second-order filters using an additional RC network within the input or feedback path. The filters from the second order can be defined simply as follows: " two 1st order filters cascading with**amplification". **

Most designs of second-order filters are often named after their inventors, and the most common types of filters are: Butterworth, Chebyshev, Bessel and Sallen-Key. All these filter designs are available as follows. They all have a rounding of 40 db/decade, including low-pass filter, high-pass filter, tape-passing filter and tape stop (notch) filter configurations, and quad-degree filters.

The Sallen-Key filter design is one of the most widely known and popular 2nd degree filter designs, requiring only a single operational amplifier for gain control and four passive RC components to perform the adjustment.

Most active filters consist of only op-amps, resistors and capacitors with the breakpoint obtained using feedback that eliminates the need for inductors used in passive 1st-level filter circuits.

Active filters from the second level (bipolar) are important in electronics, whether low-pass or high-pass. Because we can use them together to design filters of a much higher order. By combining filters of the first and second order, analog filters with a n. level value can be built to any value within reasonable limits, single or even.

## Quad-Low Pass Filter

Filters that pass lower than the second level are easy to design. It is widely used in many applications. The basic configuration for a Sallen-Key second-order (bipolar) low-pass filter is given as follows:

## Quad-Low Pass Filter

This second-order low-pass filter circuit has two RC network, R1 – C1 and R2 – C2, which give the filter frequency response characteristics. The filter design is based on an inverted op-amp configuration, thereby increasing the gain of filters. A will always be greater than 1. In addition, the op-amp has a high input impedance, which means that it can be easily cascaded with other active filter circuits to achieve more complex filter designs.

The normalized Frequency response of the filter, which is lower than the second order, is fixed by the RC network and is usually the same as that of the first-order type. The main difference between the transition filter lower than 1st and 2nd degree is that as the operating frequency rises above the εc cutting frequency, the rolling of the stopband will be twice as high as the filters from 40db/decade (12dB/octave) 1st degree.

## Normalized Low Pass Frequency Response

The Bode chart is basically the same as a 1st degree filter. The difference this time is the steepness of the rolling with -40dB /decade on the stop band. However, filters of the second order may show various reactions depending on the voltage magnification factor of the Q circuits at the cutting frequency point.

In active second-order filters, the damping factor ζ (Zeta), which is the opposite of Q, is normally used. Both Q and ζ are independently determined by the gain of the amplifier. Thus, as Q decreases, the damping factor increases. In simple words, a low-pass filter will always be low-pass in nature. But it may exhibit a resonance Peak near the cutting frequency, meaning that earnings may increase rapidly due to the resonant effects of the gain of amplifiers.

Then Q, the quality factor, represents the "peak" of this resonance peak, that is, its height and spacing around the cutting frequency point. But a filter gain also determines the amount of feedback and therefore has a significant effect on the frequency response of the filter.

## Quadnion Filter Amplitude Response

The amplitude response of the filter, which is low from the second order, varies for different values of the damping factor. ζ. When = 1.0 or more (2 is the maximum), the filter becomes what is called "over-damped" with frequency response showing a long straight curve. When = 0, the output of the filters reaches a sharp peak at the cutting point, which resembles a sharp point, where the filter is said to be "insufficiently damped".

Then, somewhere between ζ = 0 and ζ = 2.0, there should be a point where the frequency response has the correct value and is there. This occurs when the filter is "critically dampened" and ζ = 0.7071.

## Low-Pass Filter Circuit from The Second Level

We can see that the peak of the frequency response curve is quite sharp in the cutting frequency due to its high quality factor value of Q = 5. At this point, the gain of the filter is given as follows: Q × A = 14 or about +23dB, a difference greater than the calculated value of 2.8 (+8.9 dB).

But many books, such as the one on the right, say that the gain of the filter at the normalized cutting frequency point should be at point-3dB, etc. By significantly reducing Q to 0.7071, a = 1,586 gains and a maximum flat Frequency response in the transition band with a weakening of -3db at the breakpoint. It's like a second-degree butterworth filter response.

So far, we have found that the cutting frequency points of filters of the second order can be adjusted to any desired value, but with the damping factor ζ, it can move away from this desired value. Active filter designs combine filter sections to allow the order of the filter to change to any value within reasonable limits.

## Circumsted from Quadnk Pass Filter

There is little difference between the filter configuration that goes lower than the second level and the filter configuration that is higher than the second level, the only thing that changes is the position of the resistors and capacitors as shown.

Since second-degree high-pass and low-pass filters are the same circuits other than changing the positions of resistors and capacitors, the design and frequency scaling procedures for the high-pass filter are exactly the same as those for the previous low-pass filter. Therefore, for a filter with a transition higher than 2nd degree, the bode chart is given as follows:

## Normalized High Transition Frequency Response

With the previous low-pass filter, the rolling steepness in the stop band is -40dB /decade.

In the two circuits above, the value of the op-amp voltage gain is adjusted by the feedback network of (Av) amplifiers. This only adjusts the gain for frequencies within the filter's transition band. We may choose to upgrade the output and set this earnings value to any amount that is appropriate for our purpose and define this gain as fixed,

Sallen-Key filters from level 2 are also called positive feedback filters because they feed back into the positive terminal of the output op-amp. This type of active filter design is popular because it requires only one op-amp. So it's relatively inexpensive.