Dual-Dielectric Coaxial Cable

Question:

Once upon a time, I had an equation for computing effective dielectric constant for a coax line with mixed dielectrics. Can anyone supply the equation and/or reference material?

The Professor Answers.....

When a coaxial cable is constructed using two concentric dielectrics as shown above, the wave propagation is no longer purely TEM, as in the single-dielectric coax, but instead is quasi-TEM. Therefore, in general the characteristic impedance and effective dielectric constant are functions of frequency. However, the static Zo and Keff terms can be derived easily, and they are useful over a fairly wide range of frequencies. The derivation of these terms is as follows:

In the static (DC) case, all of the electric flux lines are perpendicular to the center conductor - dielectric 1, dielectric 1 - dielectric 2, and dielectric 2 - outer conductor interfaces, as shown:

To calculate the static capacitance between the inner and outer conductors, imagine a zero-thickness conducting cylinder placed inside the cable, at the interface between the two dielectrics. Since all of the electric field components are already perpendicular to this interface, this does not violate the boundary conditions of the initial problem. With this conducting cylinder in place, two capacitance values can now be calculated: 1) the capacitance (per meter length of cable) between the inner conductor and the cylinder; and 2) the capacitance between the cylinder and the outer conductor. The capacitance per meter of cable between the inner and outer conductors is then the series combination of the above two capacitance values.

The capacitance per length of a single-dielectric coaxial cable (dimensions shown above) is as follows:

Therefore, the capacitance per meter length between the inner conductor and the cylinder is:

The capacitance per meter length between the cylinder and the outer conductor is:

The capacitance per meter length of the cable itself is the series combination of the above two capacitances:

Leaving out the algebraic steps, this comes out to:

The effective dielectric constant of the structure is the ratio of the capacitance of the structure to the capacitance if the dielectrics were replaced with a vacuum (or air), i.e.:

where:

This gives the following expression for the effective dielectric constant:

With a bit of additional effort, the characteristic impedance can also be found. For this we need the inductance per meter length, which is unaffected by the dielectric(s) in the structure. The expression for the inductance is:

The characteristic impedance of any transmission line is equal to the square root of the ratio of the inductance per length to the capacitance per length. Again leaving out the algebraic steps,

I validated these expressions using HFSS for the following geometry: r0 = 0.040",
r1 = 0.100", r2 = 0.140", Er1 = 2.0, and Er2 = 5.0. For those dimensions, the above expressions yield Z0 = 48.65, and Keff = 2.384. HFSS results for the same configuration are as follows:

As seen from the numbers, the expressions for the static Z0 and Keff are valid at low frequencies, and for the dimensions used in this example, the dispersion is mild below 15 GHz. Smaller dimensions and closer values of Er1 and Er2 would reduce the dispersion further.

I hope you find this helpful. The derivation of the expressions for Z0 and Keff in the case in which the division between Er1 and Er2 is angular, rather than radial, is probably more complicated.

Regards,

The Professor

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