Transmission Line Model for Coax wire

Question

I have got 2 questions on transmission line. In coax line, the centre wire is Copper, and the outer line is Aluminium, as far as I can see in the coax that I have. I have got couple of questions on that.

Question 1:

Here this model assumes uniform resistance, conductance, capacitance ... per unit length of both wires, and then went on to derive the equations and intrinsic impedance. But this is not the case for coax line, as centre wire (Copper), and the returning wire (Aluminium) both have different resistance, capacitance ... per unit length. Say top wire and bottom wire in the diagram are Copper and Aluminium respectively.So how come we are allowed to use $$Z_0 = \frac{R + j\omega L}{\gamma}$$. Where R can either be the per unit resistance of Copper wire OR Aluminium wire, same goes for the value of inductance. So what I am reading wrong here?

The second question is on the application side of this question.

Question 2:

Say I am feeding the signal through centre wire (lets assume antenna, and coax are perfectly matched). Braid is at ground. Now as antenna is a transmission line, the upper antenna gets the entire signal, but the bottom antenna is connected to ground. In this particular case the upper half radiates, but the bottom half of the antenna does not radiate as it is at ground. But this a topology which I have seen in some of the places. This does not make sense to me, as the antenna is intended to be a half wavelength dipole antenna, and same V/I characteristic should produced in both the parts of the antenna. Instead this design puts the bottom half at flat Voltage. I must be interpreting something wrong here.

Your guidance will be greatly appreciated.

Answer 1

The resistance of the outer and inner conductors are important – but just not in the way I think you think they are.

The resistive element in the equivalent circuit doesn't exist because the energy has to flow in the conductors from start of the coax line to the end.

In a a waveguide such as a coax cable, energy is transported as wave within the dielectric – and losses within that isolatig material are usually at least as important as the losses of the edge currents in the surrounding conductor.

When you start looking at the depth of how much the wave on the inside of the coax cable permeates into the center conductor, you'll find it's often a couple µm only – and hence, you can find a lot high-quality cables where that conductor is actually made of steel or aluminum, and is only coated in a thin layer of copper.

The same applies to the outer conductor. Due to geometry, the field strengths are typically significantly smaller than at the center conductor, and hence, losses in the conductor don't contribute to overall loss as much; that's why it's common to find thin foils as outer conductors of coax.

Answer 2

It's highly likely that the coax in the picture is being used as an "unbal" ie unbalanced at the transistor and balanced at the antenna. Pick the right length of coax and make use of it as an impedance transformer and you get this effect: -

On the left above you see the "conventional" way of turning an unbalanced drive into a balanced antenna feed. This makes use of a dummy piece of coax set at a length of a quarter wave. By applying a further outer shield (as you appear to have in your question), an alternative "unbal" approach is realized.

As for aluminium versus copper, it's the loop resistance per metre that is important and anyway, aluminium and copper aren't that different in conductivity: -

Answer 3

The R and L are equivalent resistance and inductivity per length unit, like ohms/m. The equivalent means the sum of all in that section, then you get a single one. If you wish you could make a new model R_wire, R_braid and change the R to R = R_wire + R_braid.

The depicted antenna is wrongly connected. As the dipole is a balanced load, while the coax is unbalanced. Therefore at the end of coax you should mount a BALUN (BALanced / UNbalanced).