# 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