# Transmission Lines in the Sinusoidal Steady State

|## Equations of transmission lines

The partial derivative equations of transmission lines with instantaneous functions of voltage and current, *v(z,t)* and *i(z,t)*, are the general equations for transmission lines.

If there is a harmonic dependency with the time, the use of phasors simplifies these ordinary differential equations. Taking the cosine as a reference, we can write:

where the phasors *V(z)* and *I(z)* have functions that only depend on the spacial coordinate, z, and they can be complex.

This way, we can obtain the following ordinary differential equations for the phasors*V(z)* and *I(z), *bearing in mind that **we are working with lossless lines**.

These are the transmission line equations with harmonic dependency in time.

If we combine the previous equations, we can obtain V(z) and I(z):

It’s important to notice that the term corresponds with the propagation constant:

In our case, for a lossless transmission line, the attenuation factor would be zero, . Therefore, R and G will also be zero, so

The solutions to the previous equations are:

## Secondary line constants

Now, we can define the **secondary line constants**:

In general, the **load impedance** is:

## Reflection coefficient at the load

The **reflection coefficient at the load** is the ratio of complex amplitudes of the reflected and incident waves in the load and it’s defined as:

which can also be written from:

resulting in:

where c1 and c0 are the reflected and incident waves’ voltages, respectively, and their ratio is also written as . *K*, in this case, is the **phase constant**.

Generally, the reflection coefficient is a complex quantity which absolute value is equal or less than 1.

In the following diagram, we can observe a line which ends in a load impedance, ZL and this is where the possible waves’ reflection will be originated.

Finally, we can define the complex impedance, given by the ratio V(z’)/I(z’), which is the impedance facing towards the load at a distance of z’ from it:

## Final touches…

In our next posts, this topic will start becoming more interesting and we will learn how the Stationing Wave Patterns are built and some **Matlab examples with solutions**! 😀

You can also have a look to our previous post, where we discussed **normal incidence over conductors**, an specific case for the parameters derived in today’s post.

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