# The private and public scope of coaxial cable

Let’s start by reviewing the concept of inductance.

## Inductance

Inductance of a conductor is the property that a change in current in a conductor causes a electro motive force (emf or voltage) to be induced in a conductor.

We can speak of self inductance where the voltage is induced in the same conductor as the changing current, or mutual inductance where the changing current in one conductor induces a voltage in another conductor.

### Self inductance of a solid round conductor

At low frequencies, the current is distributed uniformly inside the conductor, and its self inductance can be calculated readily (formula is in most good text books). For example, the self inductance of a 2mm diameter round copper conductor is about 1370nH/m. Note that this includes the effect of flux within the copper conductor, internal inductance, it contributes 50nH/m.

### Self inductance of a thin round tubular conductor

At low frequencies, the current is distributed uniformly inside the conductor, and its self inductance can be calculated readily (formula is in most good text books). For example, the self inductance of a 10mm diameter thin round copper conductor is about 998nH/m. There is no magnetic flux inside the tube as there is no current flowing there to create flux.

### Coaxial conductors

If we locate a round conductor concentrically or coaxially within a hollow tubular conductor (shield), there is not only the self inductance L1 and L2 of each conductor respectively at play, but the mutual inductances M12 and M21. M12 and M21 are equal to each other, and by virtue of the fact that all of the flux of L2 is shared with L1, M12 and M21 are equal to L2.

If we consider the series path of current flowing in the inner conductor and returning via the outer conductor, we have L=L1-M21+L2-M12 and given M12=M21=L2, we can write L=L1-L2 which tells us that the inductance is due to the flux inside the shield, there is no flux outside the the shield.

If the example 2mm and 8mm conductors were arranged coaxially, the return circuit inductance would be L1-L2=1370-998=372nH.

## Skin effect

The effect of self inductance is to cause current to concentrate in the area where self inductance is least. This effect is more pronounced as frequency increases and because tends to flow mainly near the surface it is commonly known as skin effect. We can think of the current density as decaying exponentially with increase depth, and although it is more complicated than that, this is an adequate model for this discussion. We speak of the skin depth δ as the depth that carries 63% of the total current and often make the assumption that the total current is carried in a depth of 3δ, it that there is insignificant current at greater depth.

Skin effect implies that at sufficiently high frequency, the current on the inner conductor flows mainly near the outer surface. A consequence is that the internal inductance approaches zero.

Skin effect implies that at sufficiently high frequency, the current on the outer conductor flows mainly near the inner surface.

With well developed skin effect, the outer conductor behaves almost like two independent / isolated tubes being the inner and outer surfaces with negligible current flowing in the region between those ‘layers’.

## TEM mode

Coaxial cable is usually used in TEM (Transferse Electro Magnetic) mode, magnetic flux is circular within the coax, and radial electric field exists between the outside surface of the inner conductor and inside surface of the outer conductor (there is no magnetic flux due to inside currents outside the coax). Other modes are possible at some frequencies, but they cause higher loss and are usually discouraged.

In TEM mode in the presence of well developed skin effect, a current I flowing at a point along the coax on the outer surface of the inner conductor is accompanied by an equal current flowing in the opposite direction on the inner surface of the outer conductor. This is a really important attribute to consider when analysing a system.

## The private scope

In the presence  of well developed skin effect, the combination of self inductance and mutual inductance of each of the outer surface of the inner conductor and inner surface of the outer conductor result in no external magnetic or electric fields, and the constraint that a current I flowing at a point along the coax on the outer surface of the inner conductor is accompanied by an equal current flowing in the opposite direction on the inner surface of the outer conductor.

These if you like define what is going on inside the coax as a private scope that is affected only by the transmission line characteristics, load impedance and source. Calculators such as RF Transmission Line Loss Calculator provide solutions to this problem.

## The public scope

In the presence  of well developed skin effect, the outside surface of the outer conductor can carry currents independent of what his happening inside the coax. The fields due to these currents are entirely external to the outside surface of the outer conductor, and the outer conductor is free to interact with other sources of magnetic and electric fields.

## Treatment of shield end

In the presence  of well developed skin effect, the inner and outer surfaces of the outer conductor are effectively isolated, but they connect to each other at the ends of the coaxial structure.

One can analyse the configuration by considering that there is a node formed at the shield end, and connected to that node are the inner and outer surfaces of the shield and any other external conductors. At the node, Kirchoff’s Current Law applies.

## Antenna system implications

Two scenarios will be analysed to demonstrate an approach to the problems.

### Small / ideal resistor load

An ideal resistor for this discussion is one that is electrically so small that we can ignore distributed capacitance and inductance and phase change of current through the resistor.

If a ideal resistor is attached to the cut of of a coax cable and excited from the other end, the current flowing from the inner conductor to the resistor flows entirely to the inner surface of the outer conductor, there is no residual to flow to the outer surface of the outer conductor. This is independent of whether there are standing waves on the inside of the coax (ie whether the load resistor equals Zo).

Nothing above prevents current flowing on the outer surface of the outer conductor due to other excitation, but none will flow into the inner of the coax.

### Practical half wave dipole

The configuration is a coaxial cable directly attached to the centre of a practical half wave dipole. For the purpose of discussion, the dipole is not perfectly symmetric and the current flowing into one leg is not exactly equal to that from from the other leg.

Lets call the current from the centre conductor to one dipole leg I1, and the current from the other leg I2. I2 flows into the node formed by the connection of the inner surface of the other conductor, other surface of the outer conductor and the dipole leg. By Kirchoff’s Current law, the current flowing into the outer surface of the outer conductor at the node is I2-I1,  and that current (often termed common mode current) gives rise to external fields (including radiation).

Likewise, voltage induced into the outer surface of the outer conductor by external source will flow into the same node and divide between the attached dipole leg and inner surface of the outer conductor. The current flowing to the dipole leg causes a current in the other dipole leg providing the complementary differential current in the interior of the coax, eventually delivering energy to the remote load.