Transmission lines

When working with an RF circuit we can identify different operating regimes depending on the ratio between the wavelength of the signal and the size of the circuit :

1. : quasi static regime (we can use lumped parameters)
2. : Resonant regime (we can use transmission line theory)
3. : Optical regime

It is important to notice that if we use a low frequency square wave, this doesn’t mean that we can use lumped parameters because the sharp transition of the square wave require very high frequencies to be correctly represented.

Since the length of the transmission line is much bigger than its width we will consider our analysis as 1D.

Transmission line model

The transmission line is modelled as shown in the sketch below and its circuit representation is as follows:

Where

Single section

To study the behaviour of the transmission line we study a single section of length as .

We use Kirchhoff laws to find the relation between tension and current:

Warning

In the following and

Dividing by both sides and taking the limit for we get

The equations above are called Telegrapher’s equations.

We now want to get the second derivative of tension and current with respect to .

Lossless lines

We will only focus on the case where there is no dissipation in the lines (i.e. ). In this particular case we get the following, simplified, equations (obtained from the ones above):

We can then represent generic waves travelling in the transmission line, one travelling to the right and the other one to the left:

both these relations satisfy .

We now want to evaluate at a specific instant and after :

If we factor out the term in the second expression we get

It is easy to see that the blue term is the spatial displacement in the transmission line.

We now want to find the speed at which travels along the transmission line:

We can do the same for

The velocity in the transmission line is defined as

Reflections on terminated lossless lines

To properly study the behaviour of the reflections on the terminated line we can imagine a circuit like the one in the image below

where we close the switch at .

At the beginning we will have a propagating voltage and a propagating current .

At , when the wave reaches the end of the transmission line, we will need another wave to satisfy Ohm’s law. We can thus find the voltage as the sum of the incident voltage (travelling to the right) and reflected voltage (travelling to the left), needed to satisfy Ohm’s law:

where and have the same properties as the generic waves defined above:

After defining the characteristic impedance as

we can use to calculate the current following the same idea as before:

We can finally calculate as

and from this calculate the voltage reflection coefficient as