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:

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

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

$$Y_2 = \dfrac{L - 1}{2\cdot \sqrt{L\cdot Z_{in}\cdot Z_{out}}}$$ (14.2)

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

$$Y_3 = Y_2\cdot\left( \sqrt{\dfrac{Z_{in}}{Z_{out}}\cdot L} - 1 \right)$$ (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:

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

Figure 14.2: T-topology of an attenuator

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

$$Z_2 = \dfrac{2\cdot \sqrt{L\cdot Z_{in}\cdot Z_{out}}}{L - 1}$$ (14.6)

$$Z_1 = Z_{in}\cdot A - Z_2$$ (14.7)

$$Z_3 = Z_{out}\cdot A - Z_2$$ (14.8)

$$\textrm{with} \qquad A = \dfrac{L + 1}{L - 1}$$ (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:

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