T pad

An attenuator is a form of a two-port network with a generator connected to one port and a load connected to the other. In all of the circuits given below it is assumed that the generator and load impedances are purely resistive (though not necessarily equal) and that the attenuator circuit is required to perfectly match to these. The symbols used for these impedances are;

${\displaystyle Z_{1}\,\!}$ the impedance of the generator

${\displaystyle Z_{2}\,\!}$ the impedance of the load

Popular values of impedance are 600Ω in telecommucations and audio, 75Ω for video and dipole antennae, 50Ω for RF

The voltage transfer function, A, is,

${\displaystyle A={\frac {V_{\mathrm {out} }}{V_{\mathrm {in} }}}}$

While the inverse of this is the loss, L, of the attenuator,

${\displaystyle L={\frac {V_{\mathrm {in} }}{V_{\mathrm {out} }}}}$

The value of attenuation is normally marked on the attenuator as its loss, L_dB, in decibels (dB). The relationship with L is;

${\displaystyle L_{\mathrm {dB} }=20\log L\,\!}$

Popular values of attenuator are 3dB, 6dB, 10dB, 20dB and 40dB.

However, it is often more convenient to express the loss in nepers,

${\displaystyle L=e^{\gamma }\,}$

where ${\displaystyle \gamma \,}$ is the attenuation in nepers (one neper is approximately 8.7 dB).

Figure 2. A general L-section circuit with series impedance Z and shunt admittance Y.

The values of resistance of the attenuator's elements can be calculated using image parameter theory. The starting point here is the image impedances of the L section in figure 2. The image impedance of the input is,

${\displaystyle Z_{\mathrm {iT} }={\sqrt {Z^{2}+{\frac {Z}{Y}}}}}$

and the image admittance of the output is,

${\displaystyle Y_{\mathrm {i\Pi } }={\sqrt {Y^{2}+{\frac {Y}{Z}}}}}$

The loss of the L section when terminated in its image impedances is,

${\displaystyle L_{\mathrm {L1} }={\sqrt {Z_{\mathrm {iT} }Y_{\mathrm {i\Pi } }}}\ e^{\gamma _{\mathrm {L} }}}$

where the image parameter transmission function, γ_L is given by,

${\displaystyle \gamma _{\mathrm {L} }=\sinh ^{-1}{\sqrt {ZY}}}$

The loss of this L section in the reverse direction is given by,

${\displaystyle L_{\mathrm {L2} }={\sqrt {Z_{\mathrm {i\Pi } }Y_{\mathrm {iT} }}}\ e^{\gamma _{\mathrm {L} }}}$

Figure 3. A T-pad attenuator formed from two symmetrical L sections. Because of the symmetry, R₁ = R₃ in this case.

For an attenuator, Z and Y are simple resistors and γ becomes the image parameter attenuation (that is, the attenuation when terminated with the image impedances) in nepers. A T pad can be viewed as being two L sections back-to-back as shown in figure 3. Most commonly, the generator and load impedances are equal so that Z₁ = Z₂ = Z₀ and a symmetrical T pad is used. In this case, the impedance matching terms inside the square roots all cancel and,

${\displaystyle L_{\mathrm {T} }=L_{\mathrm {L1} }L_{\mathrm {L2} }=e^{2\gamma _{\mathrm {L} }}=e^{\gamma _{\mathrm {T} }}\,}$

Substituting Z and Y for the corresponding resistors,

${\displaystyle \gamma _{\mathrm {T} }=2\gamma _{\mathrm {L} }=2\sinh ^{-1}{\sqrt {\frac {R_{1}}{2R_{2}}}}\,}$

${\displaystyle Z_{0}={\sqrt {{R_{1}}^{2}+2R_{1}R_{2}}}}$

These equations can easily be extended to non-symmetrical cases.

The equations above find the impedance and loss for an attenuator with given resistor values. The usual requirement in a design is the other way around – the resistor values for a given impedance and loss are needed. These can be found by transposing and substituting the last two equations above;

${\displaystyle R_{1}=Z_{0}\tanh \left({\frac {\gamma _{\mathrm {T} }}{2}}\right)}$

${\displaystyle R_{2}={\frac {{Z_{0}}^{2}-{R_{1}}^{2}}{2R_{1}}}}$

• Π pad
• L pad

• Matthaei, Young, Jones, Microwave Filters, Impedance-Matching Networks, and Coupling Structures, pp. 41–45, 4McGraw-Hill 1964.
• Redifon Radio Diary, 1970, pp. 49–60, William Collins Sons & Co, 1969.