 Coaxial Cable Characteristic Impedance vs Frequency

April 18, 2014 Coaxial cable is characterized by a "nominal" real characteristic impedance Zo such as 50 or 75 ohm. Zo generally is defined as the ratio of a sinusoidal AC voltage phasor to the current phasor, which may be phase shifted, for a single outbound travelling wave (with no reflection from the cable end). The nominal Zo value refers to the high frequency RF value, appropriate to typical usage of this type of cable. While this description is usually sufficient for frequencies above a few MHz, Zo is in general complex (reactive) and at lower frequency, the reactive nature of Zo becomes more apparent. This is due to the frequency-dependent relative values of series resistance and inductance, combined with the skin-depth effect. Although complete solutions for Zo are available for any frequency (e.g. Schelkunoff, 1934) , the general solutions involve complex Bessel functions which are difficult to analyze for frequency trends. Instead, it is instructive to consider approximate solutions which are valid at either high or low frequency. High frequency approximations are readily available (e.g. Ramo et. al. 1984). In addition, it is not difficult to calculate the frequency dependence of Zo at lower frequency, given the material properties and dimensions of the cable. This note presents results of calculation for the Zo dependence on frequency of common RG58U "50 ohm" coaxial using high and low frequency approximate expressions. Detailed field plots based on more comprehensive modelling for a typical coaxial cable quantify the field penetration outside of coax at low frequencies. A 200 kHz reference example provides accurate reference data for all field components. A script calculator accurately computes coaxial cable parameters discussed below and transmission line parameters for any frequency.

Terminations, Reflections and Cable Input Impedance

The characteristic impedance Zo, which in general is complex, is a fundamental property of a waveguide. Zo depends on waveguide geometry and material properties. After Zo has been determined, it is easy to calculate the complex input impedance of any length of waveguide terminated with an arbitrary complex impedance. It is assumed here that only the lowest order TM0 mode is propagating which has field components Hφ, Eρ and Ez. (Between the conductors, the Ez field component is almost negligible so the this TM0 mode is often described as a TEM wave). The field form for the propogating mode, for example for the aximuthal Hφ field component is: Summarizing well known results (see references below for details), the properties describing coaxial cable are Z and Y, the distributed series impedance and shunt admittance of the cable per unit length: 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 ZL is: and the complex impedance at the input of an arbitrary length l of coaxial cable, terminated with ZL is: Material Parameters Cc, Gc, Rc, Zc

Electromagnetic analysis using Maxwell's equations shows that the shunt capacitance and conductance, which apply at any frequency, are given in MKS units by: 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 high frequency approximation for both the series resistance Rc and series inductance Lc which assumes that the skin depth δ is much smaller than the radii of both the inner and outer conductors. In this limit, the current is confined to the inner surface of the outer conductor, and the outer surface of the inner conductor, to within a few skin-depths. Rs is the surface resistivity of the conductors, in ohms. Here ri is the radius of the inner conductor, ro is the INNER radius of the outer (sheath) conductor, and roo is the OUTER radius of the outer (sheath) conductor. The inductance consists of Le the external inductance due to magnetic field between the conductors, and Li, the frequency-dependent "internal" inductance due to magnetic field inside the conductors. These results apply well for f > 1 MHz approximately:  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 minimum possible value of Rc as f is lowered: Similarly, a low frequency "limit" is reached for Lc, but in this case, a maximum inductance is reached corresponding again to uniform current in both conductors: (see Grover "Inductance Calculations", 1973 p. 42): 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.

Characteristic Impedance Calculation For Typical RG58U Coaxial Cable

The plots below show the characteristic impedance Zo for a coaxial cable with typical RG58U characteristics (cables from different manufacturers differ somewhat in the exact physical dimensions, dielectric constants etc..) using the high and low frequency approximations above for Rc and Lc. The magnitude and phase of Zo are shown. The "nominal" high frequency value of characteristic impedance is Zo = Sqrt(Le/Cc) = 51.3 Ohm for the dimensions shown. The parameters are shown in the inset of the graph along with the low frequency limits for Rc and Lc. Two sets of curves are shown. The HF (blue) curves show the predictions based on the high-frequency expressions above which are therefore accurate only for f >~ 1MHz. The LF curves (yellow) use expressions with the fixed limiting values of Rc and Lc and are therefore accurate for f < ~ 20 kHz. The transition region between 20 kHz and 1 MHz will correspond to points intermediate between the two graphs. A comparison with exact data shows that for this example Zo follows the LF approximate curve almost exactly up to ~ 100kHz and follows the HF approximate curve almost exactly for f > 100 kHz. Accurate values require use of accurate Bessel function expressions (e.g. Schelkunoff, 1934).
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.  Special Cases:

Given the characteristic impedance Zo at any frequency, the input impedance of any length of cable terminated with arbitrary complex impedance can be easily calculated using the transmission-line transformation expression for Zin above. Two cases of interest are:
• short lengths of cable (l << wavelength) terminated with infinite load (open end cable)
• short lengths of cable (l << wavelength) terminated with a short circuit
where wavelength = 2π/β and β is the phase constant.

Using the general expression above for Zin, these two cases have corresponding input impedances of:  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")
```

Physical Components of Z:

Schelkunoff (1934) provides an excellent and insightful discussion of the components (internal and external) of the distributed coaxial cable serial impedance Z. The internal components (or "surface impedances") consist of resistive (real) and reactive (imaginary) parts arising from current flow and magnetic flux created inside the inner and outer conductors of the coaxial cable. The resistive parts contribute to power absorption and attenuation with distance along the cable. In addition the external component of impedance originating from the magnetic field between the inner and outer conductors, is purely inductive and is described by the high-frequency inductance. It is interesting to display the real and imaginary parts of each of these components. The total distributed series impedance Z is the sum of the real and imaginary parts from each contributor, in units of Ω/m. The plot below shows the results of an accurate calculation using the complete Bessel function field solutions. The plots are therefore accurate over the complete frequency range shown. The usual engineering transmission-line equations can be derived directly from the fundamental field equations. The graphs below show that for f > 20kHz, the total series impedance Z becomes dominated by the reactive part Z_im due to the dominance of the external inductance Le. For f < 20 kHz, Z becomes resistive and (for the dimensions of conductors shown) dominated by the DC resistance of the inner conductor. The change in shape between 50kHz - 500 kHz is due to the skin-depth effect. At low frequencies, current is uniformly distributed in the cross sections of both conductors. At high frequencies, current is confined to a very thin surface layer in both conductors. This causes both the resistive part Z_re, which detemines attenuation, and the reactive part Z_im to increase with frequency. However the reactive part strongly dominates for f > 1MHz. This leads to the characteristic impedance Zo = √(Z/Y) being almost resistive with a slight reactive component: 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 : Reference:
• 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