Attenuators

[last updated: 2022-05-02]
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• Attenuators are used to reduce signal strength. This can be needed eg:
  • when you want to connect a transmitter's high-wattage output to some sensitive device, eg. a spectrum analyzer, whose input can't handle high power signals,
  • or when fox-hunting, when you'd connect your directional antenna's signal through an attenuator to your receiver, to reduce the signal strength and increase sensitivity to low-level signals from the fox.

• from dB gain equation:
  (Voltage) Attenuation_dB = 20 * log₁₀(V_in / V_out)
  Note: Gain ratio (ie. amplified) is V_out / V_in, but Attenuation ratio is V_in / V_out
• In the calculations below, the attenuation ratio V_IN / V_OUT is set to = K.
  K = V_IN / V_OUT
  -------------------------------------------------------

• pi network attenuators:
  • If shunt resistors (R₂) are equal, then pi networks are balanced, with Z_in = Z_out
    
    
    

    pi-network attenuator

    -------------------------------------------------------

  • To calculate the input impedance of the whole network, it helps to rearrange the schematic
    to emphasize that it's just a 2-leg parallel circuit, with different resistors in each leg.
  
  
  equivalent circuit
  • It's easy to see that: R_LEG-1 = R₂.
  • But R_LEG-2 must include the load impedance Z_out.
    
    
    

    LEG-2

    R_LEG-2 = R₁ + (R2 || Z_OUT)
    Parallel circuit formula (product over the sum) gives:
    (R2 || Z_OUT) = (R₂ * Z_OUT) / (R₂ + Z_OUT)

    It will be helpful in another calculation to treat this as a separate quantity:
    R_OL = R_OutputLeg = (R2 || Z_OUT)

    So now:
    R_LEG-2 = R₁ + R_OL

  • The total impedance of the network,
    that is, the input impedance that your signal source will see,
    is again product over sum for the two parallel legs:
    Z_IN = (R_LEG-1 * R_LEG-2) / (R_LEG-1 + R_LEG-2)

  • The attenuation that the network gives,
    • will be defined by the ratio of the input and output voltages,
      specifically the K voltage ratio given above:
      K = V_IN / V_OUT
    • And from dB calculation formulas:
      A_dB = 20 * log(K)

    • To calculate attenuation-dB, when you are given the network resistances,
      It's helpful to realize that LEG-2 is a voltage divider,
      with V_IN feeding it,
      and V_OUT taken at R_OL above ground.
    
    
    pi-network attenuator
    • The voltage attenuation ratio will then be equal to the resistance ratio:
      V_OUT / V_IN = R_OL / R_LEG-2
    • And the inverse of this gives:
      K = V_IN / V_OUT = R_LEG-2 / R_OL

    • A_dB = 20 * log(R_LEG-2 / R_OL)
      -------------------------------------

    • If, OTOH, you want a particular amount of attenuation and/or a particular in/out impedance,

      To Find resistances desired, given A_dB and Z:

      • A_dB = 20 * log₁₀( K )
      • K = 10^(Adb/20)

      • R₁ = Z * (K² - 1) / 2 * K
      • R₂ = Z * (K + 1) / (K - 1)
      • Note: These equations came from somewhere on the web.
        I have not been able to derive them, but they work.
  • Switchable cascade attenuator network:
    
    
    

    switchable attenuator element
    
    

    cascade switchable attenuator network

    -------------------------------------

  • Resistance table for desired attenuation and impedance:
  

  -------------------------------------

  • Calculation spreadsheet:
    ...

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  • Links:
    https://wcar.ca/pg/bafh/
    http://lea.hamradio.si/~s53mv/dds/attenuator.html
    https://pa1b-qrp.blogspot.com/p/fast-in-qsos-simple-way-to-reduce-power....
    https://edadocs.software.keysight.com/display/genesys2010/Attenuators
    https://www.allaboutcircuits.com/textbook/semiconductors/chpt-1/attenuat...
    https://www.electronics-tutorials.ws/attenuators/pi-pad-attenuator.html
    DPDT switches Amazon $0.80 https://www.amazon.com/mts-202-125VAC-Position-Toggle-Switch/dp/B0799KM8...
    https://kc9on.com/product/fox-hunt-offset-attenuator/
    Adjustable passive coax attenuator: https://www.dxzone.com/dx19846/simple-adjustable-passive-attenuator.html
    http://theleggios.net/wb2hol/projects/rdf/p_atten.htm

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