Cable impedance

Thanks for your answer friend. Very much appreciated. !!!

-HLB

What this is all about (reprint / rewrite of an earlier post).

Cable impedance must match the source and load impedances in order to get power to flow with maximum efficiency from source to load, when the cable electrical length exceeds about a tenth wavelength, or thereabouts. If there is mismatch anywhere, some power is reflected from the mismatch, which reduces the power delivered to the load, and (if we are looking at high power transmission), can actually damage the source.

Electrical length means the ratio of the cable length to a wavelength at the frequency of interest. The wavelength, in air, in meters, is given by:

λ = 300/f_MHz,

where f_MHz is the frequency in MHz.

However, most coax has a dielectric material between the center conductor and shield that is different than air, and has a relative dielectric constant that is different than unity. The wavelength of the signal traveling down the coax is reduced by the square root of the relative dielectric constant ε_r:

λ_m = λ_air / √ε_r,

where λ_m is the wavelength in the medium.

To give you a feel for why you use a transmission line for data consider the following. A databus runs at 100 MB/s. That translates into a wavelength in air of 3 m for the fundamental frequency, and 0.3 meters for the tenth harmonic - typically you need 10 harmonics to get a decent pulse waveform. And that's in air. A wavelength in the transmission line will be shorter by the square root of the relative dielectric constant. And you are looking at a tenth of that for your decision of when to use a transmission line , so that if you need to efficiently transmit data on a wire at 100 MB/s, if the cable length needs to be greater than a few inches, you need to use a transmission line.

The characteristic impedance of the cable is equal to the square root of the per unit length cable inductance L in units of Henries per meter divided by the per unit length capacitance in units of Farads per meter.

Z_{characteristic} = √(L/C)

Thus characteristic impedance is independent of cable length - the per unit length dependence divides out.

To give you a feel for the numbers, per unit length inductance for most transmission lines is in the tens of nanohenries per meter, while per unit length capacitance is on the order of tens of picofarads per meter.

If your inductance per meter happens to be 50 nH, and the capacitance per meter happens to be 50 pF, then the characteristic impedance is

Z_C = √(50e-9 / 50e-12) = 31.62 Ω.

Which is pretty close to 50 Ohms ( a typical coaxial transmission line impedance, with other coax and wire pairs at around 75-90 Ohms, and twin-ax at 125 or 300 Ohms), so you can see the actual values of L & C are in that ballpark.

Finally, the equation for the characteristic impedance of coax transmission line in terms of the coax physical parameters is:

Z_C = (138 / √ε_r) log₁₀ (D/d),

where D is the diameter of the coax shield, and d is the diameter of the coax center conductor. That equation is just a result of dividing the expressions of per unit length inductance and capacitance, and taking the square root.

There are similar expressions for other kinds of transmission lines, such as wire pairs.

To round out this discussion, there are also equations for how much power is reflected if you have a mismatch, based on the mismatched impedances. And transmission lines have losses which are separate from the whole matching issue. The losses are frequency dependent (they always increase exponentially with increasing frequency) and are mainly based on losses in the dielectric medium.

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Thanks a million for your answer too sir. Now I am heading in the right direction.

-HLB