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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:

Attenuator k-min equation - RF Cafe,   for Z1 > Z2 in circuits shown below

Express in decibels as: Attenuator k-min equation (dB) - RF Cafe


In the attenuator formulas below:

Attenuator formula (k) - RF Cafe, 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:

Attenuator formula (k dB) - RF Cafe  <———>  Attenuator formula (k linear) - RF Cafe

An online attenuator calculator is provided at the bottom of the page.

Unbalanced Tee (T) Attenuator

These equations apply to the two forms of Tee attenuators at the left.

"T" Attenuator R1 Equation - RF Cafe

"T" Attenuator R2 Equation - RF Cafe

"T" Attenuator R3 Equation - RF Cafe

If  Z1 = Z2, then:

"T" Attenuator (Z1 = Z2) R1 & R2 Equation - RF Cafe

"T" Attenuator (Z1 = Z2) R3 Equation - RF Cafe

Unbalanced T Attenuator - RF Cafe
Balanced Tee (T) Attenuator
Balanced T Attenuator - RF Cafe
Unbalanced Pi (π) Attenuator

These equations apply to the two forms of Pi attenuators at the left.

"Pi" Attenuator R1 Equation - RF Cafe

"Pi" Attenuator R2 Equation - RF Cafe

"Pi" Attenuator R3 Equation - RF Cafe

If  Z1 = Z2, then:

"Pi" Attenuator (Z1 = Z2) R1 & R2 Equation - RF Cafe

"Pi" Attenuator (Z1 = Z2) R3 Equation - RF Cafe

Unbalanced Pi Attenuator - RF Cafe
Balanced Pi (π) Attenuator
Balanced Pi Attenuator - RF Cafe

Note: Only enter values in the yellow cells or risk overwriting formulas!

Input Resistance: Output Resistance:

Attenuation: dB


k = (Pin/Pout)
 
  Tee Attenuator        Pi Attenuator
   Ω    

Rin        

   Rout

Rin        

 Rout

  Ω    

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). 

Attenuators with unequal terminations - RF Cafe

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