April 18, 2014

Coaxial cable is characterized by a "nominal" real

Summarizing well known results (see references below for details), the properties describing coaxial cable are Z and Y, the distributed

where Rc is the series resistance, Lc is the series inductance (including both internal and external inductance), Gc is the shunt conductance and Cc is the shunt capacitance, all per unit length. The material parameters Rc and Lc both depend intrinsically on frequency due to the skin-depth effect. Gc depends on frequency for typical dielectric loss effects and Cc is independent of frequency to a very good approximation for all frequency. All the parameters depend on the cable cross-sectional dimensions.

Once Rc, Lc, Gc and Cc are determined using standard electromagnetic analysis, transmission line analysis shows that the characteristic impedance is given in terms of Z and Y by:

and the complex propagation constant γ , which also provides the waveguide loss rate α and waveguide "phase constant" β is:

Finally, the voltage reflection coefficient at the end of the line, when terminated by an arbitrary complex impedance

and the complex impedance at the input of an arbitrary length

where εr and εi are the real and imaginary parts of the complex dielectric constant. εi accounts for loss in the dielectric. Most published results provide a

At high frequency, the inductance becomes just the fixed external inductance Le as field is excluded from the interior of the conductors thereby lowering Li. Note that Rc at high frequency increases as √f due to skin-effect current confinement. In fact, the internal inductive reactance jωLi and Rc are equal at high frequency. But with Lc ~ Le being constant in this regime, the inductive reactance, dominated by Le, increases linearly with f so that the inductive reactance will always dominate the distributed series impedance Z at high enough frequency. For the example below, Rc and jωLc are equal at about 20 kHz.

However at lower frequency, the skin depth increases until the current completely fills both inner and outer conductors of the coaxial cable. This leads to a

Similarly, a low frequency "limit" is reached for Lc, but in this case, a

where ξ is a small parameter that depends on the ratio ro/roo and is ~ 0.083 for dimensions of RG58U coaxial cable. Note that at very low frequency (less than ~ 20kHz), Rc is constant, and Lc is constant but the inductive reactance jωLc continues to drop as the frequency is lowered. This means the phase angle of the cable distributed series impedance Z decreases as Z starts to become more resistive than inductive as Z approaches Rc. Since the capacitive admittance jωCc is much larger than Gc and the capacitive admittance drops as the frequency is lowered, the net effect is that the characteristic impedance Zo increases strongly as the frequency is lowered and the phase angle becomes more negative or capacitive in character.

Note that the inductance Lc has both a low and high frequency limit corresponding to current completely filling both conductors and current flowing in a very thin skin layer at the outer/inner surfaces of the inner/outer conductors. For the example below, the low/high frequency values for Lc are: 0.331uH/m and 0.259uH/m, a range of only ~ 25%. Compare this to the distributed resistance Rc which has a low frequency limit, but increases continuously without limit as the frequency is increased.

Some values from the curves are:

Zo (nominal) == Sqrt(Le/Cc) = 51.3 Ohm Zo(100MHz) = 51.5 / -0.2 deg Ohm Zo(1MHz) = 53.3 / - 2.1 deg Ohm Zo(1kHz) = 240 / -43 deg

The second graph shows the skin depth for copper as a function of frequency.

- short lengths of cable (l << wavelength) terminated with infinite load (open end cable)
- short lengths of cable (l << wavelength) terminated with a short circuit

Using the general expression above for

For the open-ended line, Gc is always << ωCc (since εi << εr for insulating dielectrics). Therefore the input impedance of a short length of cable is just the impedance of a capacitor of value Ccl. This familiar result is therefore a very good approximation at ANY frequency.

For the shorted line, the situation at low frequency is different from high frequency. At high frequency, jωLcl becomes much greater than Rl and in this high-frequency limit, the input impedance of the short shorted line is simply jωLcl, purely inductive with an inductance value of Lcl, another familiar result for high frequency.

However, at low frequency, the Rc and Lc both approach constant values (as discussed above), but the inductive reactance continues to drop at low frequency. Eventually Rc will exceed ωLc and the input impedance of the shorted line will be resistive.

Using the same parameters as above for typical RG58U coaxial cable and a cable length of 3m (9' 10"), typical Zin values for these two cases are:

Zin_open(1MHz) = 536.0 / -89.97 deg ("capacitive") Zin_open(1kHz) = 537771.9 / -89.97 deg ("capacitive") Zin_short(1MHz) = 5.31 / +85.8 deg ("inductive") Zin_short(1kHz) = 0.107 / +3.33 deg ("resistive")

The plot below shows an exact calculation for the characteristic impedance Zo= √(Z/Y) using the same parameters as above. The real and imaginary parts of Zo are shown with the imaginary part being negative. As expected, at high frequency, Zo becomes almost perfectly resistive with an asymptotic value Zo(hf) ~ √(Le/C) = 51.3 Ω. At low frequency, Zo becomes strongly complex with equal resistive and reactive parts (phase angle -45 deg). The magnitude of Zo in this limit is √(Rdc/ωC), increasing with decreasing frequency. The transition region arising from the skin-depth effect is evident between 10kHz and 100 kHz :

**Fields and Waves in Communication Electronics**, S. Ramo, J. Whinnery and T. Van Duzer, 2nd Edn. 1984, John Wiley & Sons, pp. 249-252

**Electromagnetic Theory**, J. Stratton, 1941, McGraw Hill, pp. 551-554

**The Electromagnetic Theory of Coaxial Transmission Lines**, S. Schelkunoff, 1934, Bell System Technical Journal, p. 532

**Inductance Calculations**, F. Grover, 1946, 1973, Dover, Instrument Society of America, p. 42, pp.262 - 282.

**Waveguide Handbook**, N. Marcuvitz, 1986, Peter Peregrinus Ltd. pp. 17-25, p. 72