# Conductance in a transmission line

I'm just trying to clarify the transmission line concepts in my head. Now wikipedia (that great source of always reliable information) defines the conductance in the transmission line as

The conductance G of the dielectric material separating the two conductors is represented by a shunt resistor between the signal wire and the return wire (siemens per unit length).

Now just to confirm this would be in effect the 1/R where R is the internal resistance of the capacitor? If so what is the benefit of modelling it in this way as opposed to just a second resistor in series with the capacitor? Is it purely mathematical for the telegraphers equations - or is there something deeper?

The conductance term represents leakage current between the signal wire and the return wire.

this would be in effect the 1/R where R is the internal resistance of the capacitor?

The conductance term represents leakage current through the dielectric material. In a typical discrete capacitor, the resistance term represents resistance of the wires connecting the capacitor to the circuit, although other energy loss terms might be lumped in as well, if it improves the accuracy of the model.

If so what is the benefit of modelling it in this way as opposed to just a second resistor in series with the capacitor?

Very basically, a resistance in series with the capacitor would not allow any dc leakage current. A conductance in parallel with a capacitance does allow dc leakage.

Also a series resistance term would generally lead to different behavior as frequency increased. The shunt term would be

$Y_{\mathrm{sh}}=\dfrac{j\omega{}C}{1+j\omega{}CR}$

which would be increasing in magnitude at low frequencies and then flat above the pole frequency at $\omega=1/RC$. Whereas with the more usual model we have

$Y_{\mathrm{sh}}= G + j\omega{}C$

which has a zero but no poles.

are we saying therefore if the resistance term of the parallel wires then needs to be lumped in with the resistance in series with the inductor?

No, in this context, R and G represent two different things. R represents loss caused by resistance in the wires. G represents loss caused by leakage through the dielectric. We could have used $R_1$ and $R_2$, but we chose to call them R and G instead.

as I understood it G=1/R - so in that case doesn't substituting that into your equations make them equivalent?

No. You could substitute 1/Rshunt for G, but that doesn't make a parallel connection equivalent to a series connection.

• OK that makes a lot of sense - thanks for the long and detailed answer! So are we saying therefore if the resistance term of the parallel wires then needs to be lumped in with the resistance in series with the inductor? Also as I understood it G=1/R - so in that case doesn't substituting that into your equations make them equivalent? – George Wilson Apr 20 '14 at 1:09
• R here is exclusive to series loss and G here is exclusive to load and/or leakage. Dont mix these. You are trying to equate R1 to R2 – user38637 Apr 20 '14 at 8:09

The wiki article you linked says this: -

The transmission line model represents the transmission line as an infinite series of two-port elementary components, each representing an infinitesimally short segment of the transmission line:

• The distributed resistance R of the conductors is represented by a series resistor (expressed in ohms per unit length).
• The distributed inductance L (due to the magnetic field around the wires, self-inductance, etc.) is represented by a series inductor (henries per unit length).
• The capacitance C between the two conductors is represented by a shunt capacitor C (farads per unit length).
• The conductance G of the dielectric material separating the two conductors is represented by a shunt resistor between the signal wire and the return wire (siemens per unit length).

There are four components that fully describe a transmission line - they are independant of each other and non-transferrable. The wiki article you linked also shows this diagram: -

Clearly these are four separate components and G is clearly not in series with C

The formula for the transmission line characteristic impedance is this: -

$Z_0 = \sqrt{\dfrac{R+j\omega L}{G+j\omega C}}$

Look at the bottom line where G is - note also that the term involving capacitance does not show capacitive reactance ($\frac{1}{j\omega C}$) but the inverse ($j\omega C$).

G is the parallel leakage, not the series resistance of C. There is symmetry in the format of comparing GL and RC in the equation but there is no deeper meaning.

As these are two different resistors and either or both could be represented as Rx = 1/Gx where x is just a reference designation for two elements, where for simplicity they were omitted and given unique letters instead.

These time constants, where if they were equal and the transmission line is said to be "matched" with no overshoot on pulse inputs or no reflected power. This requires and external load in parallel with G. Normally G is very small leakage, and transmission line impedance becomes ratio of root L/C must be terminated by this value for optimal transient response. As this does not happen in power lines, transient suppression must be added. Perhaps this external matching is the deeper significance you were looking for. But actual G is a leakage loss factor for power in long distance lines often measured by tan delta.

• This was probably supposed to be as comment. – Rev1.0 Apr 20 '14 at 12:59