Attenuators

Attenuators are used to damp a signal. Using pure ohmic resistors the circuit can be realized for a very high bandwidth, i.e. from DC to many GHz. The power attenuation $ 0 < L\le 1$ is defined as:

$\displaystyle L = \dfrac{P_{in}}{P_{out}} = \dfrac{V_{in}^2}{Z_{in}}\cdot\dfrac...
...{out}^2} = \left( \dfrac{V_{in}}{V_{out}}\right)^2 \cdot\dfrac{Z_{out}}{Z_{in}}$ (14.1)

where $ P_{in}$ and $ P_{out}$ are the input and output power and $ V_{in}$ and $ V_{out}$ are the input and output voltages.

Figure 14.1: $ \pi $-topology of an attenuator
\includegraphics[width=5cm]{picircuit}

Fig. 14.1 shows an attenuator using the $ \pi $-topology. The conductances can be calculated as follows.

$\displaystyle Y_2$ $\displaystyle = \dfrac{L - 1}{2\cdot \sqrt{L\cdot Z_{in}\cdot Z_{out}}}$ (14.2)
$\displaystyle Y_1$ $\displaystyle = Y_2\cdot\left( \sqrt{\dfrac{Z_{out}}{Z_{in}}\cdot L} - 1 \right)$ (14.3)
$\displaystyle Y_3$ $\displaystyle = Y_2\cdot\left( \sqrt{\dfrac{Z_{in}}{Z_{out}}\cdot L} - 1 \right)$ (14.4)

where $ Z_{in}$ and $ Z_{out}$ are the input and output reference impedances, respectively. The $ \pi $-attenuator can be used for an impedance ratio of:

$\displaystyle \dfrac{1}{L} \le \dfrac{Z_{out}}{Z_{in}} \le L$ (14.5)

Figure 14.2: T-topology of an attenuator
\includegraphics[width=5cm]{tcircuit}

Fig. 14.2 shows an attenuator using the T-topology. The resistances can be calculated as follows.

$\displaystyle Z_2$ $\displaystyle = \dfrac{2\cdot \sqrt{L\cdot Z_{in}\cdot Z_{out}}}{L - 1}$ (14.6)
$\displaystyle Z_1$ $\displaystyle = Z_{in}\cdot A - Z_2$ (14.7)
$\displaystyle Z_3$ $\displaystyle = Z_{out}\cdot A - Z_2$ (14.8)
$\displaystyle \textrm{with} \qquad A$ $\displaystyle = \dfrac{L + 1}{L - 1}$ (14.9)

where $ L$ is the attenuation ( $ 0 < L\le 1$) according to equation 14.1 and $ Z_{in}$ and $ Z_{out}$ are the input and output reference impedance, respectively. The T-attenuator can be used for an impedance ratio of:

$\displaystyle \dfrac{Z_{out}}{Z_{in}} \le \dfrac{(L+1)^2}{4\cdot L}$ (14.10)


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