# 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.

$P_{LR}=P_{load}P_{inc} =1−∣Γ(ω)∣_{2}1 $

The insertion loss (in dB) is then

$IL=10g_{10}P_{LR}$

As $∣Γ(ω)∣_{2}$ is an even function of $ω$, it can be expressed as a polynomial in $ω_{2}$:

$P_{LR}=1+N(ω_{2})M(ω_{2}) $

By choosing coefficients of $M$ and $N$, 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 $N$ with cutoff frequency $ω_{c}$:

$P_{LR}=1+k_{2}(ω_{c}ω )_{2N}$

- At the cutoff frequency, the power loss ratio is $1+k_{2}$.
- If this is chosen as the -3 dB point then $k=1$,
- Usually the case

- If this is chosen as the -3 dB point then $k=1$,
- The first $2N−1$ derivatives are zero at $ω=0$
- For $ω≫ω_{c}$, the insertion loss increases at a rate of $20N$ dB/decade

### Equal Ripple Response

A Chebyshev polynomial $T_{N}(x)$ is used to specify the insertion loss:

$P_{LR}=1+k_{2}T_{N}(ω/ω_{c})$

- Results in a sharper cutoff
- Passband response will have ripples of amplitude $1+k_{2}$, as $T_{N}(x)$ oscillates between $±1$ for $∣x∣≤1$
- $k_{2}$ determines the passband ripple level
- For large $x$, $T_{N}(x)≈(2x)_{2N}/2$
- For $ω≫ω_{c}$, the power loss ratio is $(k_{2}/4)(2ω/ω_{c})_{2N}$
- Increases at the same rate of $20N$ dB per decade

- For $ω≫ω_{c}$, the power loss ratio is $(k_{2}/4)(2ω/ω_{c})_{2N}$
- At any given $ω$, the power loss ratio is $(2_{2N})/4$ greater than that of the binomial filter for $ω≫ω_{c}$

### 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:

$ϕ(ω)=Aω[1+p(ω_{c}ω )_{2N}]$

- $ϕ(ω)$ is the phase of the voltage transfer function of the filter
- $p$ 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 $R$, and a cutoff frequency normalised to 1. The desired power loss ratio will be $1+ω_{4}$ for $N=2$.

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

$P_{LR}=2(Z_{in}+Z_{in})∣Z_{in}+1∣_{2} =1+ω_{4}$

This equation solves to give $L=C=2 $, for the case $N=2$.

The same process can be repeated for different values of $N$ to give the element values for the ladder-type circuits show. The values are numbred from $g_{0}$ source impedance to $g_{N+1}$ load impedance for a filter with $N$ reactive elements alternating between series and shunt connections.

The graph shows attenuation vs normalised frequency for filter prototypes

### Equal Ripple Response

For Chebyshev polynomials, $T_{N}(0)=0$ when $N$ is odd, and $T_{N}(0)=0$ when even, so there are two cases for the power loss ratio depending on $N$. Considering the same LC circuit shown above, for even $N$ it can be shown that $R$ is not unity, so there will be an impedance mismatch if the load has a unity impedance, which can be corrected with a $λ/4$ transformer. For odd $N$ this is not an issue: it can be shown that $R=1$.

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 $R_{0}$ is obtained by multiplying all the impedances of the prototype design by $R_{0}$

$L_{′}=R_{0}LC_{′}=C/R_{0}R_{s}=R_{0}R_{L}=R_{0}R_{L}$

To change the cutoff frequency from unity to $ω_{c}$, replace $ω$ by $ω/ω_{c}$

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

$L_{k}=ω_{c}R_{0}L_{k} C_{k}=R_{0}ω_{c}C_{k} $

### High Pass Transformation

The substitution $ω←(−ω_{c}/ω)$ is used to convert a low pass to high pass response. This maps $ω=0→±∞$ and vice-versa.

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

$C_{k}=R_{o}ω_{c}L_{k}1 L_{k}=ω_{c}C_{k}R_{0} $

## 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:

$jX_{L}=jLtan(βl)jB_{c}=jCtan(βl)$

The stub length of the lines is $λ/8$ at $ω_{c}$ with unity impedance.

The Kuroda identities can convert shunt to series. Each box represents a transmission line of the indicated characteristic impedance at length $λ/8$ at $ω_{c}$. The inductors and capacitors represent short and open circuit stubs, respectively.

$n_{2}=1+Z_{2}/Z_{1}$

## 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 $Z_{0}=Z_{h}$, and low impedance $Z_{0}=Z_{l}$. The ratio $Z_{h}/Z_{l}$ should be as large as can possibly be fabricated. The lengths of the lines can then be determined from:

$βl=Z_{h}LR_{0} βl=R_{0}CZ_{l} $

Where $R_{0}$ is the filter impedance, and $L$ and $C$ are the normalised element values from the prototype. To obtain the best response, the lengths should be evaluted at the cutoff frequency.