Quick And Practical Design Of A High-Pass Third-Order Bessel Filter

In the design of a proximity switch using PIN diodes as the receiving element, the signal is first amplified and then must be filtered from any noise before proceeding to the several signal-conditioning stages of the proximity switch. Typically, to reject any noise resulting from 50-Hz supplies, neon lamp frequencies, 100 Hz harmonics, and any other anticipated sources of noise, a unity-gain third-order Bessel filter is a good choice for giving a flat response without damping at the corner frequency (this doesn’t happen with Butterworth or Chebyshev filters).

The filter shown provides a zero response at dc and a flat response from corner frequency f_cp, up to f_u (Fig. 1). The f_u frequency is where Avc gain crosses unity (Fig. 2).

The filter is a high-pass type only for the region between f_cp and f_u. In practice, f_u varies with temperature and the filter will be subjected to more error drift near f_u.

The following procedure, which uses equally sized capacitors, enables the selection and calculation of the wider range of resistors that are available as opposed to capacitors:

Estimate which frequencies are to be rejected and determine the corner frequency f_cp.

Select an op amp with a range of frequencies in which the open-loop gain A_vlo ≥ 100. This value must be maintained from dc up to the unitygain frequency f_u, in order to keep the actual frequency response with 0.1 dB of the theoretical response.

Select a value for the three equally sized capacitors C1 = C2 = C3.

From the chosen corner frequency, calculate the capacitors initial value C’.

For equally sized capacitors, calculate the scaling factor K.

Calculate the three final resistor values according to: R1 = KR1’; R2 = KR2’; R3 = KR3’, where R1’ = 0.7027; R2’ = 1.012; and R3’ = 3.940 are the initial resistor values for a high-pass Bessel filter.

The following is an example set up for a PIN diode signal as the input to the filter:

The value of the required corner frequency is: f_cp = 800 Hz.

An HA-4625 op amp, with A_vlo = 200, from dc up to 70 MHz is selected.

For the three equally sized capacitors, 2.7-nF, 1% film is selected.

From the above f_cp, the initial value of capacitors, C’ is calculated:

The scaling factor K is then calculated:

The final resistor values are calculated, according to the high-pass Bessel filter initial values: R1’, R2’, R3:

R1 = KR1’ = 52k

R2 = KR2’ = 75k

R3 = KR3’ = 287k