Lumped parameter vs transmission line model for a piece of cable

Question

I would like to model a piece of cable (less than 5m) over a frequency range of DC to \$1\mathrm{MHz}\$. I am however doubting if I can approximate it by a lumped parameter model (mainly at \$1\mathrm{MHz}\$) or if I need to model it as a continuous transmission line. If I am correct, lumped parameters can be used as long as the physical dimensions of the system (here: cable length) are sufficiently below the wavelength lambda. Since lambda (in free space) can be calculated using the speed of light \$c\$ and \$f\$, for \$1\mathrm{MHz}\$ I get: $$\lambda = \frac{300000000\text{ km/s}}{1000000\text{ Hz}} = 300\mathrm{ m}\text{ (approximately)}. $$ Since my physical system is much smaller, I suppose a lumped parameter representation is sufficiently accurate. However, in transmission line theory, the wavelength is derived through the propagation constant $$ \gamma = \alpha + j\cdot\beta. $$ \$\gamma\$ is calculated through the cable's impedance and admittance and the corresponding wavelength can be calculated as $$ \lambda = 2\cdot\frac{\pi}{\beta}. $$ That would thus mean that the physical properties (\$R,L,C\$) of my cable will determine the wavelength. Does that also mean that two different cables of the same length would possibly need to be modeled as a continuous or a lumped parameter model, just because we would choose thicker wires or have more spacing between the wires? I am a bit confused on the approach that should be followed and I am happy to receive your input.

Answer 1

The speed of propogation in a 'normal' coaxial cable, that is straight cylindrical conductors with an annular dielectric spacer, is entirely controlled by the dielectric constant of the spacer. For most solid plastics, this a bit more than 2, so the speed in typical RF cables is 0.67c, regardless of the other dimensions. Foamed or other air-containing spacer structures will increase the speed closer to c.

5m of cable has an electrical length of about 8m of air, about 25nS, which at 1MHz is 8/300 = 0.027 wavelengths.

This is less than the \$\lambda/20\$ or so that we normally regard as 'short' for a transmission line, so it would be quite reasonable to represent an open line as a lumped capacitance, or a shorted line as a lumped inductance, or the line itself as a ladder of Cs and Ls.

You could design the LC ladder by simply using the rated impedance per length of the line. The smaller the individual Cs and Ls, the better will be the approximation. An alternative is to design a low order lowpass filter, either CLC or LCL, choosing the bandwidth such that it gives you the right phase shift at 1MHz.

Answer 2

The telegrapher's equations can be simplified to a lumped parameter model for the case ∆z<<λ thereby we ignore the dependency of current and voltage on distance traveled by electric and magnetic field and model them as perfect sinusoidals. Considering the dimensions you are considering, it is absolutely perfect to work with lumped parameters. Seconldy, it is wrong to consider that the product of C' and L' depends on thickness of wire or the seperation of wire. Rather they are decided my material parameters which are basically derived using Electric and Magnetic fields. For a reference:

The propagation constant: results from maxwell's equation or more precisely from Helmholtz equation and is thereby a function of material parameters e.g. permitivity, permiability and has no dependency on width or seperation. therefore wavelength is unaffected by geometric alterations.