RF Cascade Workbook© 2005 User's Guide - Chapter 7

A low order Butterworth filer might be used with a bandwidth that results in an inband slope that matched measured or datasheet values. Use a lowpass filter to model a negative gain slope or a highpass filter to model a positive gain slope. Keep in mind that gain values calculated on the “Filter-Mixer” worksheet are not used back on the “Inband” worksheet. Most users will probably only enter specifications for filter stages. As with all simulations, the output data is only as good as the input parameters.

Single-tone mixer intermodulation products are not calculated or included in the output spectrum display.

Figure 12    Filter-Mixer Input Parameter Input Area

7.1    Input Frequency

In order for Excel to calculate the frequency response, a range of frequency values must be generated to use for the gain calculations. Furthermore, since frequency translation is accommodate by using mixer stages, each position in the cascade must have its own frequency cell. This creates a lot of cells and consequently a lot of calculations. A default of 75 steps is used to strike a balance between step size, recalculation time, and file size. Parameter definitions are given in Table 4.

Table 4    Input Frequency Parameters

If a different number of steps is required, it will be necessary to add or delete columns in both the “Freq” and the “Psig” areas. Use the same methodology as described in section 2 for accomplishing the changes, keeping in mind that the equations must retain their continuity.

7.2    Filters

Models are provided for the two most often-used filter transfer functions: Butterworth and Chebyshev. If you need other response functions, you can create them simply by writing a little VBA code or even implementing the equation directly in a range of cells.

To facilitate using filters that closely match real world filters, the orders of the filters are not restricted to integer values. Seldom does a real 5th order Butterworth exactly match the theoretical profile, so you can enter 4.9 or 5.1 or any other value as needed to match your actual filter.

Table 5    Filter Parameters

Figure 13 illustrates how three different filter topologies (Butterworth, Chebyshev and Bessel) compare for equal order (5th). The Chebyshev filter inband ripple is arbitrarily set to 0.5 dB. Both gain and group delay are shown.

Figure 13    Bandpass Filter Response Comparison

The following equations are the prototype functions for Butterworth and Chebyshev filters. Use the conversions in Table 6 in place of fr for lowpass, highpass, bandpass and bandstop filter equations. These are based on infinite “Q” implementation (purely reactive components).

    Butterworth Filter

Butterworth filters are sometimes referred to as “maximally flat,” because they exhibit the flattest passband response with no ripple. The cutoff frequency is defined at the half power point (-3.01 dB). Only the amplitude response is used in RF Cascade Workbook.

    Chebyshev Filter

Chebyshev filters are sometimes referred to as “equiripple” filters because they exhibit a ripple in the inband response. Chebyshevs have the highest out-of-band rejection for an all-pole transfer function. Higher attenuation requires a non-monotonic out-of-band response (e.g., elliptical). Higher inband ripple results in greater out-of-band attenuation for a fixed order number. The cutoff frequency is defined to be where the out-of-band attenuation begins to exceed the inband ripple value.

Note that the inband ripple response is defined by a cosine function. One way to determine the order of a lowpass or highpass Chebyshev filter is to count the number of peaks and valleys in the inband response, which is equal to the order. For bandpass or bandstop, the order is half the number or peaks and valleys. The out-of-band rejection is defined by a hyperbolic cosine function, which looks somewhat like a smashed parabola (see https://mathworld.wolfram.com/HyperbolicCosine.html), and is monotonic (no inflection points).

Table 6    Filter Conversion Factors

Chapter 1, 2, 3, 4, 5, 6, 7, 8,

        9, 10, 11, 12, 13

Version 1.11

by Kirt Blattenberger

RF Cafe Website

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Chapter 7