# RF filter design basics

### - basics of RF filter design indicating basic calculations and methodologies required..

The exact filter design parameters are dependent upon the type of filter used. However the basic methodology and parameters used within RF filter design have a common approach whatever form of filter used.

## RF filter design basics

In general terms, filters modify the amplitudes and phases of sinusoidal waveforms that pass through them. This change varies according to the frequency of the individual sinusoids within the overall waveform.

Most filters are what is termed linear filters. As such they have no non-linear actions in which the response is proportional to the input. Instead the signals pass through and their amplitude and phase is altered in a linear fashion according to their frequency.

From this it is possible to determine some of the key RF filter design parameters which are the factors by which the signal is changed, namely the gain, G and the phase shift θ. As both the gain G and the phase shift θ are dependent upon the frequency, i.e. they are functions of frequency, they can be expressed as follows:

These two functions represent the magnitude response (often referred to as the frequency response) and the phase response of the filter respectively.

These two functions govern the major features that need to be known about the filter. In being able to determine the functions it is possible to design the RF filter.

Normally the example of the low pass RF filter design is used in the first instance and then this is expanded to include other forms of filter. Accordingly we will look at low pass filter design first.

## Real and ideal filters

When designing an RF filter it would be ideal if the filter would allow signals within the pass-band through without any change in amplitude or phase. Filters like this could have a rectangular response, falling straight to their stop-band and giving the required level of stop-band attenuation.

Unfortunately it is not possible to design RF filters like this, and real RF filter designs can only approximate to the ideal response curves and parameters. These approximations can then be used as the different types of filter that exist. These include the Butterworth, Bessel, Chebyshev, Elliptical, Gaussian and many more.

Using a mathematical approach for RF filter design, it is possible to use a mathematical relationship. It can be proved that the square of the response for all realisable filters can be expressed as the ratio of two even, i.e. rational polynomials. This means that a generic mathematical expression for all RF filter designs is:

## Key filter RF design parameters

There are a number of key parameters that are of great importance for RF filter design. These are some of the highlight requirements for any RF filter design.

• Pass-band:   This is the region in which the signal passes through relatively un-attenuated. It is the band in a low pass filter, extends up to the cut-off frequency. For high pass filters it is designated as the band above which signal pass through, or for a band pass filter, it is the band between the two cut-off frequencies.
• Cut-off frequency:   This is normally taken to be the point at which the response of the filter has fallen by 3 dB. With certain filters, typically equi-ripple types such as the Chebyshev or inverse Chebyshev, the cut-off point has to be defined differently. It is often designated fc.
• Ripple band:   Within the pass-band, the filter response may show variations in its response - ripples. The variation is known as the ripple band
• Transition band:   Once the RF filter response has gone beyond the cut-off point, the response falls away in a region known as the transition band. It is the region between the pass-band and the stop-band. This region is also sometimes referred to as the "skirt."
• Stop-band:   This is the band where the filter has reached its required out of band rejection. The stop-band rejection may be defined as a required number of decibels.
• Number of poles :   A pole is a mathematical term. There is one pole for each capacitor or inductor in a filter.
• Roll-off :   Each filter has an ultimate roll-off rate. It is governed by the number of poles in the filter. The ultimate roll-off is 6⋅n dB where n is the number of poles. Different types of filter may reach their ultimate roll off rate at different rates, but they all reach the same ultimate roll-off.
• Phase shift:   The phase shift is another important factor for any RF filter design. It is accommodated into the overall response of the filter by considering the calculations for H(s) where s = jω. The phase response can be of importance to a waveform because the waveform shape will be distorted if the phase changes within the pass-band. A constant time delay corresponds to the phase shift increasing linearly with frequency. This gives rise to the term linear phase shift referred to in many RF filter designs.
• Impedance:   Filters have a characteristic impedance in the same way that as an antenna feeder. For them to operate correctly the input and output must be properly matched.

## RF filter design & normalisation

While relatively straightforward equations are available for Butterworth filters, other forms of filter require more complicated calculations. The approach to RF filter design that has been used for many years uses what are termed normalised filters. A normalised filter would have a cut-off frequency of 1 radian per second, i.e. 0.159Hz and an impedance of 1Ω. It was then possible to tables of pre-calculated values which could then be scaled for use at the required frequency and impedance. In this way the tedious and involved mathematics required for RF filter design was reduced to little more than determining the requirements and then finding the relevant table of values.

The requirements that first need to be chosen include parameters such as the filter type (Butterworth, Chebyshev, etc.), level of ripple, etc, filter order (i.e. the number of inductors and capacitors), etc.

Once these are chosen the relevant table can be found and the values for the elements in the filter determined.

## RF filter design & scaling

Once the filter design has been realised in its normalised form, it is then necessary to transform the values to the required frequency and impedance. In the normalised format the filter design has a cut-off of 0.159 Hz, i.e. 1 radian per second and it is designed to work into a load resistance of 1 Ω.

Where:
C = real capacitor value
L = real inductor value
Cn = normalised capacitor value
Ln= normalised inductor value
R = required load resistor value
fc = required cut-off frequency

## RF filter design process

There are a number of steps or stages in the RF filter design process. Following these in order helps the RF filter to be designed in a logical fashion. These steps are for the low pass filter design - further stages for transferring this to a high pass or band pass filter are given on the following pages.

While some computer programmes may allow direct design, often design using tables, etc. is still widely used. If a computer programme is used, the filter design process can be modified accordingly.

1. Define response needed:   the first stage in the process is to actually define the response required. Elements such as cut-off point, attenuation at a given point, etc..

2. Normalise frequencies:   In order to be able to use the various tables and diagrams of filter curves, it is necessary to convert all frequencies so that the cut-off point is at 1 radio per second and any other points are relative to this.

3. Determine maximum pass-band ripple:   One of the major steps in the RF filter design is to understand how much in-band ripple can be tolerated. The more ripple, the greater the level of selectivity that can be obtained. The greater the selectivity the faster the transition from pass-band to ultimate roll off will be .

4. Match required attenuation curves with those from filter:   With a knowledge of the characteristics, both in terms of ripple and rejection required at particular points, it is possible to determine the filter type and also the order or number of elements required within the filter design.

5. Determine element values:   Using the relevant look up tables the normalised filter component values can be determined

6. Scale normalised values:   Finally the values need to be scaled for the required cut-off frequency and resistance.

Filter values and curves can be found in a number of filter design books including "handbook of Filter Synthesis" by Zvrev, pub Wiley. Later we hope to develop some curves and tables ourselves.

By Ian Poole

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