Antenna: Reflection Coefficient

[last updated: 2024-03-03] ...
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(link to:) antenna-theory.com

formula: https://resources.pcb.cadence.com/blog/2023-s-parameters-and-the-reflect...
http://www.mogami.com/e/cad/vswr.html
https://en.wikipedia.org/wiki/Reflection_coefficient
https://www.everythingrf.com/rf-calculators/vswr-calculator
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• This is a complicated subject that confuses me,
  so here I'll collect snippets of information,
  then try to make sense of all of it...
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• Context:
  • If you have a transmitter, and you connect a feedline to it (coax or ladder line, typ.)
    and you connect the other end of the feedline to your antenna ...
  • and if your feedline and your antenna do NOT have the same impedance,
    then you create a "constriction" or "restriction" in the signal's path,
    and some portion of your transmitter's signal will reflect off of that restriction and will be directed back towards your transmitter.
  • The measurement of the amount of signal that is reflected back is the reflection coefficient.

    OR... in other words:

  • Two devices (feedline and antenna, for example)
    that have different impedances
    will have a "constriction" in the signal path at the point where they connect.
    The source signal will be reflected back towards the source from that constriction point.

  • So this is the physical phenomenon we're trying to measure:
    The reflection of part of a signal from a point of impedance mismatch.

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• Why is it confusing?
  • It's confusing because sometimes reflection coefficient,
    or that is, the measurement of how much signal is reflected from a point of impedance mismatch,
    is reported in terms of V_SWR, or sometimes as Return Loss, or sometimes as S11, or sometimes as Insertion Loss.
  • All these things are measurements of the same phenomenon, that being the reflection of part of a signal from an impedance mismatch,
    and they track each other, that is, they are dependent variables (if one of them changes, the rest will all change),
    but they are calculated with different equations, and they are not the same number.

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• Strictly speaking...:
  • Reflection Coefficient is "gamma"
    It is a number (decimal) from 0 to +1
    (though it is often reported as between -1 and +1...???)

  • S11 ...
  • Return Loss ...
  • Insertion Loss ...

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• On a Smith chart,
  Reflection coefficient is plotted on the horizontal x-axis, and varies between -1 (far left) to +1 (far right)
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• (from various sources:)
  • Reflection Coefficient [denoted with Greek capital Gamma] = "complex ratio" between reflected voltage and incident voltage
    (or current, or power ?)
  • Since the amount of reflected signal is determined completely by the impedance mis-match between source and load,
    [Gamma] = (Z_L - Z₀) / (Z_L + Z₀)

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• from: ref-1:
  • "[reflection coefficient is] a relative fraction of the incident RF power that is reflected back due to the impedance mismatch." (between antenna and feedline)
  • Another way of describing it is with Return Loss.
    "The return loss ... is ... the [ratio] in dB between the power sent toward the antenna and the power reflected back from it. ..."
    RL=−20log10 (∣S11∣)
    where:
    RL is the return loss
    S11 is the reflection coefficient, a decimal from -1 to +1.
  • Another way of describing it is with V_SWR:
    V_SWR= (1+∣S11∣) / (1−∣S11∣)
    where:
    S11 is the reflection coefficient.

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• from: ref-4:
  • "A positive R means that the polarity of the reflected wave will be the same as that of the incident wave.
    A negative R means that the polarity of the reflected wave will be the opposite of the incident wave.

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• from: ref-2:
  • "Since S11 is sometimes used interchangeably with return loss and reflection coefficient, are they ever really the same? The answer is they are sometimes negatives of each other, and sometimes the magnitudes of all three quantities are equal in limited cases and in certain frequency ranges. The formulas shown below define return loss in terms of the reflection coefficient:
    RL = -10log(|Pref/Pinc|) = -20log(|Vref/Vinc|) = -20log(|[Gamma]|)
  • Because the reflection coefficient Γ < 1, then the return loss will have a positive dB value.
    When you look at a graph of a return loss formula, the negative sign is often omitted and return loss is sometimes used interchangeably with the S11 parameter.
    Formally, S11 is the negative of return loss and has a negative dB value:
    S11 = -RL
  • For transmission lines, and likely due to the way the data are displayed on graphs, S11 is often set equal to the reflection coefficient defined between the source/load and the transmission line characteristic impedance.
    This is only correct for a specific situation of a long transmission line. In general, we need the line's input impedance, which might be equal to the load impedance in specific circuit networks (short transmission lines).
    However, as we’ll see below, circuits with propagating waves will have S11 that eventually converges to the reflection coefficient."

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• from: ref-3:
  • The amount that's reflected is quantified with "gamma" the reflection coefficient, which is the ratio of reflected to transmitted voltage (or current): "gamma" = V- / V+
  • The refection coefficient value is: 0 > "gamma" < 1
  • The reflection coefficient is the same for both voltage and current in a given system.
  • The fraction of power that is actually transmitted = 1 - ("gamma")^2
  • The reflected power will interact with the transmitted power to create standing waves in the transmistion line. See: Standing Wave Ratio

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• Links/Refs:
  1. (link to:) mathworks
  2. (link to:) altium - when is gamma = RL = S11
  3. (link to:) MIT presentation on antennas
  4. (link to:) geometrics - seismic science

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