The new algorithm teaches you to implement filter low pass to high pass conversion

Filters are generally described using low-pass prototypes because low-pass is standard. How to convert low-pass poles into high-pass poles? ADI provides you with a conversion algorithm. Speed ​​up get up~

Affected by the zero value of the origin, its DC response is not flat, but an ascending response of n × (20 dB/decade), where n is the number of poles. At the corner frequency, the above-mentioned rising response increases the response of n × (–20 dB/decade) due to the pole. The result is a flat response outside the corner frequency.

A low-pass prototype can be converted to a high-pass filter by scaling the transfer function in 1/s. In practice, this is usually equivalent to an inductor that has a capacitance of 1/C, and for a passive design, it is equivalent to a capacitor with a value of 1/L. For active designs, the resistor becomes a 1/R capacitor and the capacitor becomes a 1/C resistor. This applies only to the frequency setting resistor and not to the gain setting resistor (ie not for each resistor or capacitor in the circuit).

Another way to look at the transformation is to study the transformation of the s-plane. The complex pole pair of the low-pass prototype consists of a real part α and an imaginary part β. The normalized high-pass pole is calculated by the following formula

as well as

In this way, a simple pole α0 is converted into

The low-pass zero (ωZ, LP) is converted by the following formula

In addition, the zero point equal to the number of poles will be increased at the origin.

After normalizing the low-pass prototype poles and zeros into high-pass, the reverse normalization process is then performed in the same manner as low-pass (ie, with frequency and impedance).

As an example, here is a 1 kHz, 3-pole, 0.5 dB Chebyshev filter. The reason for choosing the Chebyshev filter is that it can be displayed more clearly if the response is not correct; in this case, Butterworth may be too loose. The 3-pole filter is chosen to convert a pole pair and a single pole separately.

Pole position

The pole position of the low-pass prototype comes from the design table.

Table 1

The first level is the pole pair and the second level is the single pole. Note that it is not advisable to use alpha to represent two completely different parameters. The α and β on the left side are the pole positions on the complex plane. These are the values ​​used in the conversion algorithm. The α on the right is 1/Q, which is what the physical filter design equation would like to see. The result of the conversion will produce the results shown in Table 2.

Table 2

It should be reminded that since a customary practice of describing Chebyshev filters (ie, the approach used here) is to reference the end of the error band rather than the 3 dB frequency, F 0 must be divided by the (high pass) ripple band and The ratio of the 3 dB bandwidth.

The filter is constructed using the Sallen-Key high-pass topology. The schematic is shown in Figure 1.

Figure 1. High-pass conversion

Figure 2 shows the response of the low-pass prototype and high-pass conversion. Note that they are symmetrical around the 1 kHz cutoff frequency. It should also be noted that the 0.5 dB error band is at 1 kHz instead of the −3 dB point, which is a feature of the Chebyshev filter. The symmetry of the response verifies the accuracy of the conversion.

Figure 2. Low pass and high pass response

From the low-pass to the high-pass filter conversion, have you got the above algorithm?