Cutoff frequency of multi-section filters. Encyclopedia of radio electronics and electrical engineering
Encyclopedia of radio electronics and electrical engineering / Radio amateur designer
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The author of the article talks about some aspects of building multi-link frequency RC filters and choosing elements for them.
As you know, the low-pass filter (LPF), the circuit of which is shown in Fig. 1a, passes signals from direct current to the cutoff frequency (passband) into the load and attenuates signals of higher frequency (attenuation band). The cutoff frequency f0 (Fig. 1b) is taken to be the one at which the curve of the amplitude-frequency response (AFC) of the filter decreases by 3 dB (more precisely, in √2 once). This level is chosen because with such a change in voltage, the signal power P=U2/r decreases by 2 times.
The cutoff frequency of a single-link (1st order) RC filter is f0=1/2πτ, where τ = RC. Its frequency response is very flat, in the attenuation band, the characteristic decay is only 6 dB per octave (when the frequency is doubled) or 20 dB per decade (ten times).
To increase the steepness of the recession, you can use several RC links connected in series. In order for the links not to influence one another, they are separated by buffer voltage followers, as shown in Fig. 2, a.
For a two-link filter, the frequency response decay will be approximately 12 dB / oct, for a three-link filter - 18 dB / oct, for a four-link filter - 24 dB / oct, and so on (Fig. 2,b; the values of the normalized transfer coefficient are plotted along the vertical axis). In the general case, the frequency response of a multi-section filter will be the product of the frequency response of its sections, and the cutoff frequency will decrease in α = 1/√n√2-1 times, where n is the number of links.
On fig. 2b shows a decrease in the cutoff frequency f0 with an increase in the number of filter sections. The table shows the values of a for filters with the number of links from 2 to 10.
Such a filter can be easily implemented on any amplifiers with any gain (greater and less than unity), even with a different gain sign (i.e., with or without inversion). Just the overall voltage transfer coefficient Ku in the passband will be equal to the product of the coefficients of all links.
Another advantage of such a filter is a very "soft", without fluctuations and emissions, response to input signal level jumps.
If a high-pass filter is built on the RC-links (Fig. 3, a), then the cutoff frequency will not decrease, but increase when adding each link (Fig. 3, b), but the numerical value of the coefficient a remains the same.
Such a change in the cutoff frequency must be taken into account when designing a device with several steps and an appropriate number of coupling capacitors. If, for example, the number of coupling capacitors is 10, and the lower cutoff frequency is to be 20 Hz, then each low-pass filter formed by the capacitor, together with the input resistance of the stage following it, should have a cutoff frequency of approximately 5,4 Hz.
The same applies to the choice of corrective capacitors OS, forming the characteristic of the low-pass filter. If there are several such capacitors or there are RC circuits between the nodes of the device that limit the frequency band from above, the cutoff frequency of each of them must be chosen higher than the upper operating frequency of the entire device as a whole.
Author: A. Brazhniov, Penza
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