Subsections

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:

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

slope of asymptote$\displaystyle = \pm\, n\cdot 20$   dB/decade (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:

$\displaystyle C_{LP} \quad \rightarrow \quad$ $\displaystyle L_{HP} = \dfrac{1}{\omega_B^2\cdot C_{LP}}$ (14.12)
$\displaystyle L_{LP} \quad \rightarrow \quad$ $\displaystyle C_{HP} = \dfrac{1}{\omega_B^2\cdot L_{LP}}$ (14.13)

In a band-pass filter:

$\displaystyle C_{LP} \quad \rightarrow \quad$ parallel resonance circuit with (14.14)
  $\displaystyle C_{BP} = \dfrac{C_{LP}}{\Delta\Omega}$ (14.15)
  $\displaystyle L_{BP} = \dfrac{\Delta\Omega}{\omega_1\cdot \omega_2\cdot C_{LP}}$ (14.16)
$\displaystyle L_{LP} \quad \rightarrow \quad$ series resonance circuit with (14.17)
  $\displaystyle C_{BP} = \dfrac{\Delta\Omega}{\omega_1\cdot \omega_2\cdot L_{LP}}$ (14.18)
  $\displaystyle L_{BP} = \dfrac{L_{LP}}{\Delta\Omega}$ (14.19)

In a band-stop filter:

$\displaystyle C_{LP} \quad \rightarrow \quad$ series resonance circuit with (14.20)
  $\displaystyle C_{BP} = \dfrac{C_{LP}}{2\cdot\left\vert \dfrac{\omega_2}{\omega_1} - \dfrac{\omega_1}{\omega_2} \right\vert }$ (14.21)
  $\displaystyle L_{BP} = \dfrac{1}{\omega^2\cdot \Delta\Omega\cdot C_{LP}}$ (14.22)
$\displaystyle L_{LP} \quad \rightarrow \quad$ parallel resonance circuit with (14.23)
  $\displaystyle C_{BP} = \dfrac{1}{\omega^2\cdot \Delta\Omega\cdot L_{LP}}$ (14.24)
  $\displaystyle L_{BP} = \dfrac{L_{LP}}{2\cdot\left\vert \dfrac{\omega_2}{\omega_1} - \dfrac{\omega_1}{\omega_2} \right\vert }$ (14.25)

Where

$\displaystyle \omega_1 \quad\rightarrow\quad$ lower corner frequency of frequency band (14.26)
$\displaystyle \omega_2 \quad\rightarrow\quad$ upper corner frequency of frequency band (14.27)
$\displaystyle \omega \quad\rightarrow\quad$ center frequency of frequency band$\displaystyle \quad \omega = 0.5\cdot (\omega_1 + \omega_2)$ (14.28)
$\displaystyle \Delta\Omega \quad\rightarrow\quad$ $\displaystyle \Delta\Omega = \dfrac{\vert\omega_2 - \omega_1\vert}{\omega}$ (14.29)

Butterworth

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

  capacitance:$\displaystyle \qquad$ $\displaystyle C_k =$ $\displaystyle \dfrac{X_k}{Z_0}$ (14.30)
  inductance:$\displaystyle \qquad$ $\displaystyle L_k =$ $\displaystyle X_k \cdot Z_0$ (14.31)
  with$\displaystyle \qquad$ $\displaystyle X_k =$ $\displaystyle \dfrac{2}{\omega_B} \cdot \sin \dfrac{(2\cdot k + 1)\cdot\pi}{2\cdot n}$ (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.

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

  capacitance:$\displaystyle \qquad$ $\displaystyle C_k$ $\displaystyle = \dfrac{X_k}{Z_0}$ (14.34)
  inductance:$\displaystyle \qquad$ $\displaystyle L_k$ $\displaystyle = X_k \cdot Z_0$ (14.35)
  with$\displaystyle \qquad$ $\displaystyle X_k$ $\displaystyle = \dfrac{2}{\omega_B}\cdot g_k$ (14.36)
    $\displaystyle r$ $\displaystyle = \sinh\left( \frac{1}{n}\cdot\text{arsinh}\dfrac{1}{\sqrt{10^{R_{dB}/10} - 1}} \right)$ (14.37)
    $\displaystyle a_k$ $\displaystyle = \sin \dfrac{(2\cdot k + 1)\cdot\pi}{2\cdot n}$ (14.38)
    $\displaystyle g_k$ $\displaystyle = \begin{cases}\begin{array}{ll} \dfrac{a_k}{r} & \textrm{ for } ...
...ac{k\cdot\pi}{n} \right)} & \textrm{ for } \quad k\ge 1 \end{array} \end{cases}$ (14.39)
    $\displaystyle X_k$ $\displaystyle = \dfrac{2}{\omega_B}\cdot g_k$ (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.

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


This document was generated by Stefan Jahn on 2007-12-30 using latex2html.