Bessel Filter

Typical filtering applications have a frequency specification to which the filter is designed; however, for a pulsed signal application, the data exists not in the frequency, but in the phase of the signal. To properly maintain the phase of the signal, a phase-invariant Bessel filter was designed. A Bessel filter has higher ripple in the passband than other filter types, but has the best square-wave transmission and group delay characteristics. The filter shown here is an active 5th order low-pass that drastically improves the roll off over a first order filter. Switching from passive to active filtering enabled the introduction of gain so the total measured signal would not only be free from high order noise, but of a constant magnitude. The filters were designed using a multiple-feedback topology suitable for high gain and selectivity.

Generic multiple feedback active filter topology

Generic multiple feedback active filter topology

The multiple feedback topology was chosen over the Sallen-Key because it has improved stopband rejection at the expense of one extra component. Multiple feedback cannot be used with current feedback opamps because there is always a capacitor in the feedback path. This condition confines the design to the use of voltage feedback amplifiers that are typically of a lower frequency range, but have improved stability compared to current feedback. The transfer function of the filter in the s domain (j\omega, \omega = 2\pi f_c) is:

    \[H(s) = \frac{\frac{R_2}{R_1}}{1+\omega C_1\left(R_2 + R_3 + \frac{R_2 R_3}{R_1}\right)s + \omega^2 C_1 C_2 R_2 R_3 s^2}\]

An efficient way of solving advanced filters is to use a filter table. The table has coefficient values for each filter type; Bessel, Tschebyscheff, and Butterworth. To use a filter table, the transfer function has to be separated into coefficients of standard form.

    \[H_0 = -\frac{R_2}{R_1}\]

    \[a_1 = \omega C_1 \left(R_2 + R_3 + \frac{R_2 R_3}{R_1} \right) \]

    \[b_1 = \omega^2 C_1 C_2 R_2 R_3 \]

The filter coefficients a1 and b1 associated with each filter section can be found in the table. In the table of Bessel coefficients, the values for the desired filter order are located. In addition to the filter coefficients, the table lists the Q of each filter stage and the ratio of the corner frequency of a partial filter to the whole filter, ki. The partial filter corner frequency is used to calculate the required unity gain bandwidth for the opamp. For a 5th order Bessel filter, three stages are required and the coefficients from the table are given by Kugelstadt.

i ai bi ki Qi
1 0.6656 0 1.502
2 1.1402 0.4128 1.184 0.56
3 0.6216 0.3245 2.138 0.92

The coefficients for each stage are used to determine the component values. A 5th order filter requires two 2nd order stages and one 1st order. The values for the 2nd order stages are calculated by:

    \[R_2 = \frac{a_1 C_2 - \sqrt{a_1^2 C_2^2 - 4b_1 C_1 C_2(1-H_0)}}{4\pi f_c C_1 C_2}\]

    \[R_1 = \frac{R_2}{-H_0} \]

    \[R_3 = \frac{b_1}{4\pi^2 f_c^2 C_1 C_2 R_2} \]

The values for the two capacitors, C1 and C2 on each stage have to be chosen manually before calculating the rest of the equations. To ensure the values for R2 are real (positive roots), the value for C2 must satisfy:

    \[C_2 \ge \frac{C_1 \left( 4 b_1(1-H_0) \right)}{a_1^2}\]

From the table, the 1st order stage is listed having no defined b coefficient or Q indicating that it is simply a real pole RC filter with a buffer amplifier. The values R1=1.13k and C1=47pF maintain a reasonable input impedance to keep power consumption low and reduce noise.

Simple real pole RC filter

Simple real pole RC filter

The second and third stages are full multiple-feedback filters calculated using the above equations. For the second stage: R1=464, R2=464, R3=1.13k, C1=33pF, and C2=150pF. Similarly, the third stage is calculated: R1=698, R2=698, R3=1.3k, C1=15pF, and C2=150pF. The full Bessel filter is shown below.

5th order low pass Bessel filter

5th order low pass Bessel filter

 

Active filters require fast operational amplifiers and the THS4271 from Texas Instruments meets that requirement. A good rule of thumb for choosing an op amp is that the unity gain bandwidth should be 100 times the highest Q in the filter at a given corner frequency.

    \[f_T = 100 * Gain * f_c * k_i\]

The worst case in this Bessel filter occurs in the third stage where the ratio of the corner frequency of the partial filter is highest (ki=2.138). This requires a minimum unity gain bandwidth of 472MHz. The THS4271 has a 1.4GHz bandwidth at unity gain and exceptionally low distortion of -92dB at 30MHz. One last thing to check is that the output can swing the full voltage range quickly enough (slew rate). For full-power response, the minimum slew rate is:

    \[SR = \pi V_{PP} f_C\]

Since the op amp is operating on dual +/- 5V supplies, the peak to peak output is 10V for a minimum slew rate of 62V/us. The THS4271 has a slew rate well exceeding this at 1000V/us. Filtering out the frequencies higher than 2MHz keeps the op amp well within its design limitations. Nevertheless, the bandwidth of these components is extremely high and careful consideration for the component choices and board layout is necessary to prevent these high frequency op amps from oscillating.