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Phase delay formula derivation

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New for February 2022.  Someone aske us this question: How is the equation for time delay, which is frequency divided by unwrapped tranmission phase, derived?

Propagation constant of a wave along a transmission line is described exactly as:

γ=√(Z^′ Y^′ )=√((R′+jωL′)(G′+jωC′) )=α+jβ

Where α is the attenuation constant and β is the phase constant

For near-lossless transmission line, (R’<<ωL’ and  G’ << ωC’ ):

α≈R^′/2∙√(C^′/L^′ )+G^′/2∙√(L^′/C^′ )=R′/(2Z_0 )+(G′Z_0)/2 (Nepers/meter)

β≈ω√(L^′ C^′ )  (radians/second)

Wavelength is distance where phase changes by 2π radians:

λ=2π/β (meters)

Phase velocity is defined as wavelength multiplied by frequency:

v_p=λf=2π/β f=ω/β=ω/(ω√(L^′ C^′ ))=1/√(L^′ C^′ ) (meters/second)

Time delay is defined as length of line divided by phase velocity:

•  Time delay=l/v_p  (seconds)

Noting that phase velocity and time delay are:

  v_p=2π/β f (m/s)  Time delay=l/v_p  (seconds)

By substitution:

    Time delay=l/v_p =βl/2πf   (seconds)

 “Electrical Length” at fixed frequency is  related to βl: [2]

  EL=βl×360/2π (degrees)

Replacing  βl with EL in time delay equation:

Time delay=(EL 2π/360)/2πf=EL/(f×360)

Noting that the unwrapped phase of a transmission line is:

  AngU(S(2,1))=-EL 

Time delay, a.k.a. phase delay, is expressed as:

  Time delay=(-AngU(S(2,1)))/(f×360)

One problem is that nwrapped phase angle is often ambiguous, especially if low frequency data are missing, like in a bandpass filter. YOu have to put on your Sherlock Holmes detective hat to extract the correct value! We'll explian that later...

 

Author : Unknown Editor

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