Fixed Pi and Tee Attenuators - Equations

Fixed attenuators can be designed to have either equal or unequal impedances and to provide any amount of attenuation (theoretically) equal to or greater than the configuration's minimum attenuation - depending on the ratio of Z₁/Z₂. Attenuators with equal terminations have a minimum attenuation of 0 dB. Unequal terminations place a lower limit on the attenuation as follows:

,   for Z₁ > Z₂ in circuits shown below

Express in decibels as:

In the attenuator formulas below:

, which is the linear attenuation ratio of the two powers or voltages (note that "k" has a minimum value if Z₁ and Z₂.are not equal).

If, as is typical, the attenuation is given in decibels (K dB vs. k), then convert to a ratio as follows:

  <———> 

Unbalanced Tee (T) Attenuator	These equations apply to the two forms of Tee attenuators at the left. If  Z1 = Z2, then:
Balanced Tee (T) Attenuator
Unbalanced Pi (π) Attenuator	These equations apply to the two forms of Pi attenuators at the left. If  Z1 = Z2, then:
Balanced Pi (π) Attenuator

An RF Cafe visitor wrote to say that he thought the above equations might be in error when unequal source and load termination resistances are used. The image below shows the mathematical steps that prove the equations are correct. It uses a source resistance of 50 ohms and a load resistance of 100 ohms, with an attenuation of 10 dB. Resistor values for both the "T" and ""Pi" attenuators were determined using the attenuator calculator on RF Cafe (which uses these equations).