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by Kirt Blattenberger
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Chapter 7
7 FilterMixer Worksheet
A major new feature in RF Cascade Workbook 2005 is the ability to calculate frequency dependent parameters due to mixers and filters. System input power and inband gains for each stage are derived from the “Input” worksheet input parameters. Min/Max values are not used here. Figure 12 shows the parameter input area for the worksheet. By default, 75 equally spaced frequency calculation points are provided between specified minimum and maximum frequencies, but can be changed.
Bandlimiting parameters can be assigned to every stage in the cascade if you know how to model them. In doing so, gain slope across the useful band can be modeled to predict the overall system gain slope.
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 “FilterMixer” 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. Singletone mixer intermodulation products are not calculated or included in the output spectrum display. 



Figure 12 FilterMixer 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 oftenused 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 outofband rejection for an allpole transfer function. Higher attenuation requires a nonmonotonic outofband response (e.g., elliptical). Higher inband ripple results in greater outofband attenuation for a fixed order number. The cutoff frequency is defined to be where the outofband 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 outofband rejection is defined by a hyperbolic cosine function, which looks somewhat like a smashed parabola (see http://mathworld.wolfram.com/HyperbolicCosine.html), and is monotonic (no inflection points). 





Table 6 Filter Conversion Factors 

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Chapter 7 