Sallen Key Circuit for Low Pass Bessel Filter.

Cuthbert Nyack
The applet on this page uses 2nd and 3rd order Sallen Key circuits to construct LP Bessel Filters with orders 2 to 8.
Even order filters 2, 4, 6, 8 can be constructed from 1, 2, 3, 4 second order sections. Odd order filters 5, 7 can be constructed from 1 third order and 1, 2 second order sections.

The applet calculates normalized and denormalized components and illustrates the magnitude and phase error introduced when any of the components are out by ± 20%.

The Circuits used are shown above.
Second and third order Circuit and Filter transfer functions can be written as shown above. Circuit components are calculated by equating the coefficents of powers of s.
In the applet below, the filter shown is determined by Fn which is set by scrollbar (0). Fn = 1 to 7 shows filters with orders 2 to 8.
The normalized component values shown are calculated from the equations above and are shown by the green text on the applet.
The denormalized component values are calculated from the normalized values, the 3dB frequency set by scrollbars(38, 39 and 40) and the impedance scaling factor set by scrollbars(41 and 42). Denormalized component values are shown by the yellow text on the applet.
Scrollbars 11 to 30 are used to show the effect that inaccuracies in the component values can have on the circuit transfer function. The green text at the top show which component are affected by which scrollbar. eg R1 is changed by scrollbar (11) and C1 by scrollbar(21). The % change from the correct value is shown for each component.
Scrollbars (45 to 48) are used to change the gain of the error and T/F phase and magnitude.
All resistor values are in ohms and capacitor values are in farads.

The Image below shows a 6th order filter (Fn = 5) implemented with 3 sections and 3 opamps(6 resistors and 6 capacitors). Normalized components are shown in green at the top and denormalized components are shown in yellow at the bottom. In this case, the 3dB frequency is 1kHz and the Impedance scaling factor is 1000.0.
Pink and Cyan plots show the magnitude and phase error introduced when the capacitor C2 in section 2 is 20% larger than its correct value. 20% change is shown in the normalized components at the top but does not affect the denormalized components at the bottom.
Numerical values of the errors can be seen at the bottom by using scrollbar 43 to move the white line.

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COPYRIGHT 2010 Cuthbert Nyack.