# Filters

Filters are two port networks used to control frequency response.

## Insertion Loss Method

We utilise the insertion loss method to design microwave filters.

We define a filter response by it's power loss ratio, the ratio of power available from the source to that delivered to the load.

The insertion loss (in dB) is then

As is an even function of , it can be expressed as a polynomial in :

By choosing coefficients of and , we can design filters with a specific frequency response.

### Maximally Flat Response

Also known as binomial or Butterworth response. For a given filter order, it provides the flattest response in the passband. For a low pass filter of order with cutoff frequency :

• At the cutoff frequency, the power loss ratio is .
• If this is chosen as the -3 dB point then ,
• Usually the case
• The first derivatives are zero at
• For , the insertion loss increases at a rate of dB/decade

### Equal Ripple Response

A Chebyshev polynomial is used to specify the insertion loss:

• Results in a sharper cutoff
• Passband response will have ripples of amplitude , as oscillates between for
• determines the passband ripple level
• For large ,
• For , the power loss ratio is
• Increases at the same rate of dB per decade
• At any given , the power loss ratio is greater than that of the binomial filter for

### Linear Phase Response

A linear phase response in the passband is important where signal distortion is to be avoided. A sharp-cutoff response is generally incompatible with a good phase response. Linear phase response can be achieved by:

• is the phase of the voltage transfer function of the filter
• is a constant

## Normalised Design

We can normalise impedance and frequency values to simplify the design of filters.

### Maximally Flat Response

Consider an LC circuit as shown below, with a source impedance of 1, a load impedance , and a cutoff frequency normalised to 1. The desired power loss ratio will be for .

The power loss ratio of this filter can be derived from it's input impedance and reflection coefficient:

This equation solves to give , for the case .

The same process can be repeated for different values of to give the element values for the ladder-type circuits show. The values are numbred from source impedance to load impedance for a filter with reactive elements alternating between series and shunt connections.

The graph shows attenuation vs normalised frequency for filter prototypes

### Equal Ripple Response

For Chebyshev polynomials, when is odd, and when even, so there are two cases for the power loss ratio depending on . Considering the same LC circuit shown above, for even it can be shown that is not unity, so there will be an impedance mismatch if the load has a unity impedance, which can be corrected with a transformer. For odd this is not an issue: it can be shown that .

The tables for equal ripple responses depend on the passband ripple level.

### Scaling

In the prototype designs above, the source and load resistances are all unity. A source resistance of is obtained by multiplying all the impedances of the prototype design by

To change the cutoff frequency from unity to , replace by

Applying both impedance and frequency scaling, the new reactive element values are:

### High Pass Transformation

The substitution is used to convert a low pass to high pass response. This maps and vice-versa.

The impedance and frequency scaling for mapping a normalised prototype to a high pass filter are:

## Filter Implementation

Lumped elements are fine at low frequencies but usually don't work at RF. Richards' transformations can be used to convert lumped elements to transmission line sections:

The stub length of the lines is at with unity impedance.

The Kuroda identities can convert shunt to series. Each box represents a transmission line of the indicated characteristic impedance at length at . The inductors and capacitors represent short and open circuit stubs, respectively.

## Stepped-Impedance Low Pass Filters

Low pass filters can be implement in microstrip using alternating sections of high and low impedance lines. For a low-pass filter prototype, the series indcutors can be replaced by high impedance line sections , and low impedance . The ratio should be as large as can possibly be fabricated. The lengths of the lines can then be determined from:

Where is the filter impedance, and and are the normalised element values from the prototype. To obtain the best response, the lengths should be evaluted at the cutoff frequency.