Lossy transmission line example

Lossy TLs Dispersionless TLs Special Cases for General TLs.
Real transmission lines – such as coaxial cables – have losses that will, among other effects, attenuate the signal as it propagates along the TL. As we’ve learned, there are two types of current (conduction and displacement) and both of these are supported on a transmission line and are necessary for its operation. There are loss mechanisms associated with each type of current:
. R is due to the conductor losses in the metal parts of the TL.
. G is due to the losses in the dielectric material surrounding the conductors in the TL.
The equivalent circuit for an infinitesimally short section (Ñz) of such a lossy TL is:

Following a procedure very similar to the lossless TLs, we can derive the phasor domain form of the telegrapher’s equations as

The phasor form of the wave equations for V (z) and I (z) can be derived from these telegrapher’s equations quite easily. For example, taking the derivative of (2)

Substituting (2) into this equation gives the phasor-domain form of the wave equation for V (z) to be

The general solution for V(z) in (3) is
where g is the propagation constant defined as

We see from this definition of g that it is a complex quantity, so we’ll define its real and imaginary parts as
g ºa+ jb[m-1]
where a[ is called the attenuation constant [Np/m] and b[ is the phase constant [rad/m].
Consequently, with
V(z) = V0+ e -az e-jbz + V0- e +az e +jbz
We can surmise that the voltage signal will attenuate as it propagates along the TL:

As shown in the text:

Z0 is the same characteristic impedance concept for the TL we’ve been using for lossless TLs. However, for a lossy TL we see from (8) that Z0 is a complex number. Lastly, the solution for the current waves on a lossy TL is

The current waves attenuate as they propagate, just as the voltage waves.
Signal Dispersion
Another non-ideal characteristic of general (lossy) TLs is signal dispersion. This occurs when the signal velocity is a function of frequency.
With m = w /b and from (6) and (5), we surmise that b is not
simply w lc . Consequently, u is a function of frequency.
This can be a very undesirable effect since the different frequency components of a signal will propagate at different speeds. This can lead to distortion of the signal, which gets worse the further the signal travels along the TL.

Dispersionless Transmission Lines
Oliver Heaviside (in the late 1800s) discovered the amazing fact that it is possible to design a lossy TL so that it presents no signal dispersion! For this to happen, he found the PUL parameters of the TL must satisfy

When this condition is met, then a and m are not functions of frequency! To verify this, note from (5) with r l = g c that:

Now, using this result in (11) compute the velocity, a, and Z0:

This is not a function of frequency. It’s actually the same result as for lossless TLs!

We see that a is not a function of frequency. That’s a great result. While there is definitely some attenuation of the signal as it propagates (a = 0), this attenuation is no frequency dependent. Consequently, we could position linear amplifiers along the channel if needed.

This is the same result as for a lossless TL. It’s also not a function of frequency.
Special Cases for General TLs
There are two special – and extreme – cases of general, lossy TLs that are worth discussing. These are the large reactance and large resistance approximations.
1. Large Reactance Approximation. In this case wl >> r and g » 0.


Consequently, in the large reactance limit

Both are independent of frequency! Recall Heaviside’s dispersionless line conditions from

When g is small, it is difficult to make r/l small to satisfy this condition. However, with the large reactance line we have achieved nearly the same condition. Neat! For a practical example, telephone companies achieve nearly this large reactance state by adding lumped inductor coils every mile or so:

Say d = 1 mile = 1,609 m = 0.1 l. Then this implies that 0.1 l = u/f = 0.67 c0. Therefore, fmax » .3*10 10 20 MHz. Consequently, 1 mile is small with respect to wavelength for frequencies up to approximately 20 MHz.
2. Large Resistance Approximation. On the other extreme is when r >>wl and g» 0. In this case

Using that fact that

From this expression, we can identify

Both a and m are functions of frequency (actually f ) and the TL is highly dispersive. This is not a good communications “channel”. The first transatlantic cable was, unfortunately, an example of such a “high resistance” line and was found to be useless for communications. There is an interesting description of this on pp. 82-83 in your EE 322 text The Electronics of Radio by D. Rutledge