Bert Simonovich's Design Notes

In my last Design Note on coaxial transmission geometry,  I mentioned it was one of three unique cross-sectional geometries that have exact equations for inductance and capacitance. The other two are twin-rod and rod-over-plane.  All three relationships assume the dielectric material is homogeneous and completely fills the space when there are electric fields.

A common application for twin-rod geometry is twin-lead ribbon cable; once used for RF transmission between antenna and TV sets. With the popularity of cable and satellite TV over the years, twin-lead has given way to coaxial cable due to its superior noise rejection and shielding effectiveness.

If we look at Figure1, we can see the electromagnetic field relationship of a twin-rod geometry when it is driven differentially. As current propagates along one rod, an equal and opposite current flows in the opposite direction along the other.

The right half of Figure 1 shows the magnetic-field loops and direction of rotation around each rod. Only one loop is shown for clarity, but the number of loops is a function of the amount of current and the length of the rods. The counter-rotating  loops of current forms a virtual return at exactly one half of the space between the two rods. We call this a virtual return because if we were to put a conducting plane in the same position, the electromagnetic fields would look exactly the same.

Figure 1 Twin-rod geometry showing electromagnetic field relationship.

In his book, “Signal Integrity Simplified”,  Eric Bogatin defines the loop inductance as, “the total number of field line loops around a conductor per amp of current”, and the loop self-inductance as, “the total number of field line loops around a conductor per amp of current in the same loop” . Applying these definitions to the figure, the loop inductance (L) is the inductance between the two rods, and the loop self-inductance (L/2) is the loop self inductance to the virtual return plane; equal to one half the loop inductance.

Likewise, the left half of Figure 1 shows the electric field with a capacitance (C) between the two rods, and twice the capacitance (2C) from each rod to the virtual return plane.

The relationships between capacitance, inductance and impedance of a twin-rod geometry are described by the following equations:

Where:

Ctwin = Capacitance between twin-rods – F

Ltwin = Loop Inductance between twin-rods – H

Zdiff = Differential impedance of twin-rods – Ω

Dk = Dielectric constant of material

Len = Length of the rods – inches

r = Radius of the rods – inches

s = Space between the rods – inches

Because the electro-magnetic fields create a virtual return plane at exactly one half of the spacing between the rods, each rod behaves like a single rod-over-plane geometry as illustrated in Figure 2.

Figure 2 Electromagnetic fields comparison of Twin-rod (left) vs. Rod-over-plane (right) geometries.

Whenever an AC current carrying conductor is in close proximity to a conducting plane, as is the case for rod-over-plane, some of the magnetic-field lines penetrate it.  When the current changes direction, the associated magnetic-field lines also change direction; causing small voltages to be induced in the plane. These voltages create eddy currents, which in turn produce their own magnetic-fields.

Eddy current-induced magnetic-field line patterns look exactly like magnetic-field lines from an imaginary current  below the plane; located the same distance as the real current  above the plane. This imaginary current is called an image current, and has the same magnitude as the real current; except in the opposite direction [1]. The image current creates associated image magnetic-field lines in the opposite direction of the real field lines. As a result, the real magnetic-field lines are compressed between the rod and the plane. Since the rod-over-plane geometry has only one rod, the loop inductance is the same as the loop self-inductance.

For a twin-rod geometry, the odd mode capacitance is the capacitance of each rod to virtual return plane and is equal to twice the capacitance between rods.

Likewise, the odd mode inductance is the inductance of each rod to virtual return plane and equal to one half the inductance between rods.

The odd mode impedance of each rod is half of the differential impedance, and is equivalent to the rod-over-plane impedance.

[1] “Signal Integrity Simplified”, Eric Bogatin