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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...