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:

v(z,t)=Re[V(z){\rm e}^{j\omega t}  ]                              i(z,t)=Re[I(z){\rm e}^{j\omega t}  ] 

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 phasorsV(z) and I(z), bearing in mind that we are working with lossless lines.

\frac{-dV(z)}{dz} = j \omega LI(z)

\frac{-dI(z)}{dz} = j \omega CV(z)

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):

\frac{-d^2V(z)}{dz^2} = j \omega^2 LI(z)

\frac{-d^2I(z)}{dz^2} = j \omega^2 CV(z)

It’s important to notice that the term \omega^2 LC corresponds with the propagation constant:

In our case, for a lossless transmission line, the attenuation factor would be zero, \alpha=0 . Therefore, R and G will also be zero, so

\gamma=j\beta 

 

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 \rho(0) . 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:

complex impedance

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.

Was this useful? 🙂 If there is something that is not clear or you need more material, just let us know!

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