Filters

One of the most common tasks in microwave technologies is to extract a frequency band from others. Optimized filters exist in order to easily create a filter with an appropriate characteristic. The most popular ones are:

Name	Property
Bessel filter (Thomson filter)	as constant group delay as possible
Butterworth filter (power-term filter)	as constant amplitude transfer function as possible
Chebychev filter type I	constant ripple in pass band
Chebychev filter type II	constant ripple in stop band
Cauer filter (elliptical filter)	constant ripple in pass and stop band

From top to bottom the following properties increase:

• ringing of step response
• phase distortion
• steepness of amplitude transfer function at the beginning of the pass band

The order $n$ of a filter denotes the number of poles of its (voltage) transfer function. It is:

$$= \pm\, n\cdot 20$$ (14.11)

Note that this equation holds for all filter characteristics, but there are big differences concerning the attenuation near the pass band.

LC ladder filters

The best possibility to realize a filters in VHF and UHF bands are LC ladder filters. The usual way to synthesize them is to first calculate a low-pass (LP) filter and afterwards transform it into a high-pass (HP), band-pass (BP) or band-stop (BS) filter. To do so, each component must be transformed into another.

In a low-pass filter, there are parallel capacitors $C_{LP}$ and series inductors $L_{LP}$ in alternating order. The other filter classes can be derived from it:

In a high-pass filter:

$$C_{LP} \quad \rightarrow \quad L_{HP} = \dfrac{1}{\omega_B^2\cdot C_{LP}}$$ (14.12)

$$L_{LP} \quad \rightarrow \quad C_{HP} = \dfrac{1}{\omega_B^2\cdot L_{LP}}$$ (14.13)

In a band-pass filter:

$$C_{LP} \quad \rightarrow \quad$$ parallel resonance circuit with (14.14)

$$C_{BP} = \dfrac{C_{LP}}{\Delta\Omega}$$ (14.15)

$$L_{BP} = \dfrac{\Delta\Omega}{\omega_1\cdot \omega_2\cdot C_{LP}}$$ (14.16)

$$L_{LP} \quad \rightarrow \quad$$ series resonance circuit with (14.17)

$$C_{BP} = \dfrac{\Delta\Omega}{\omega_1\cdot \omega_2\cdot L_{LP}}$$ (14.18)

$$L_{BP} = \dfrac{L_{LP}}{\Delta\Omega}$$ (14.19)

In a band-stop filter:

$$C_{LP} \quad \rightarrow \quad$$ series resonance circuit with (14.20)

$$C_{BP} = \dfrac{C_{LP}}{2\cdot\left\vert \dfrac{\omega_2}{\omega_1} - \dfrac{\omega_1}{\omega_2} \right\vert }$$ (14.21)

$$L_{BP} = \dfrac{1}{\omega^2\cdot \Delta\Omega\cdot C_{LP}}$$ (14.22)

$$L_{LP} \quad \rightarrow \quad$$ parallel resonance circuit with (14.23)

$$C_{BP} = \dfrac{1}{\omega^2\cdot \Delta\Omega\cdot L_{LP}}$$ (14.24)

$$L_{BP} = \dfrac{L_{LP}}{2\cdot\left\vert \dfrac{\omega_2}{\omega_1} - \dfrac{\omega_1}{\omega_2} \right\vert }$$ (14.25)

Where

$$\omega_1 \quad\rightarrow\quad$$ lower corner frequency of frequency band (14.26)

$$\omega_2 \quad\rightarrow\quad$$ upper corner frequency of frequency band (14.27)

$$\omega \quad\rightarrow\quad \quad \omega = 0.5\cdot (\omega_1 + \omega_2)$$ (14.28)

$$\Delta\Omega \quad\rightarrow\quad \Delta\Omega = \dfrac{\vert\omega_2 - \omega_1\vert}{\omega}$$ (14.29)

Butterworth

The $k$-th element of an $n$ order Butterworth low-pass ladder filter is:

$$\qquad C_k = \dfrac{X_k}{Z_0}$$ (14.30)

$$\qquad L_k = X_k \cdot Z_0$$ (14.31)

$$\qquad X_k = \dfrac{2}{\omega_B} \cdot \sin \dfrac{(2\cdot k + 1)\cdot\pi}{2\cdot n}$$ (14.32)

The order of the Butterworth filter is dependent on the specifications provided by the user. These specifications include the edge frequencies and gains.

$$n = \dfrac{\log{\left(\dfrac{10^{-0.1\cdot \alpha_{stop}} - 1}{10... ...}} - 1}\right)}}{2\cdot\log{\left(\dfrac{\omega_{stop}}{\omega_{pass}}\right)}}$$ (14.33)

Chebyshev I

The equations for a Chebyshev type I filter are defined recursivly. With $R_{dB}$ being the passband ripple in decibel, the $k$-th element of an $n$ order low-pass ladder filter is:

$$\qquad C_k = \dfrac{X_k}{Z_0}$$ (14.34)

$$\qquad L_k = X_k \cdot Z_0$$ (14.35)

$$\qquad X_k = \dfrac{2}{\omega_B}\cdot g_k$$ (14.36)

$$r = \sinh\left( \frac{1}{n}\cdot\text{arsinh}\dfrac{1}{\sqrt{10^{R_{dB}/10} - 1}} \right)$$ (14.37)

$$a_k = \sin \dfrac{(2\cdot k + 1)\cdot\pi}{2\cdot n}$$ (14.38)

$$g_k = \begin{cases}\begin{array}{ll} \dfrac{a_k}{r} & \textrm{ for } ... ...ac{k\cdot\pi}{n} \right)} & \textrm{ for } \quad k\ge 1 \end{array} \end{cases}$$ (14.39)

$$X_k = \dfrac{2}{\omega_B}\cdot g_k$$ (14.40)

The order of the Chebychev filter is dependent on the specifications provided by the user. The general form of the calculation for the order is the same as for the Butterworth, except that the inverse hyperbolic cosine function is used in place of the common logarithm function.

$$n = \dfrac{\textrm{sech}\left(\dfrac{10^{-0.1\cdot \alpha_{stop}}... ...}\right)}{2\cdot\textrm{sech}\left(\dfrac{\omega_{stop}}{\omega_{pass}}\right)}$$ (14.41)

Chebyshev II

Because of the nature of the derivation of the inverse Chebychev approxiation function from the standard Chebychev approximation the calculation of the order (14.41) is the same.