How can I deal with the traveling waves associated with lumped impedance? [closed]

Question

The answer I want to get is how to find the reflection coefficient.

I wanted to know how to find the reflection coefficient and so this is a different question as the question was written to get that.

I am having a hard time dealing with lumped impedance in problems related to traveling waves.

How can I handle problems where both characteristic impedance and lumped impedance are present, as in the example below?

Z0, Z2 are characteristic impedances and Z1 is lumped impedance. When a 100[v] DC incident wave enters, what is the reflection coefficient at the boundary between Z0 and Z1?

Answer 1

what is the reflection coefficient at the boundary between Z0 and Z1?

Here is an example of what I mean ...
Note that there is a "coefficient emission" of 1/2=50/(50+50) at the beginning.

You can see that "vout" is "only" 100 ns displaced in time with "vout1".
See the curves "black" and "green" color.
At Vm and Vm1, see the "red" and "dotted black" curves.

This is for the first bouncing (here, there is only one, z1 and z2 are matched).

You must calculate Kr with Laplace and then take the inverse Laplace to get equations.
Note that there is a "transmitted" wave together with the reflected wave.
Boundary condition (at Vm) ... the voltage at the left (Wi+Wr) equal the voltage at the right (W12).

Made with microcap v12.

Answer 2

How can I handle problems where both characteristic impedance and lumped impedance are present, as in the example below?

In your previous question you wrote this (that I believe applies in this question): -

The line is terminated to the right so that there is no reflection off the end of the line.

If the line marked \$Z_2\$ is terminated in its characteristic impedance then, \$Z_2\$ will appear to be infinitely long and will present a constant impedance equal to its characteristic impedance.

_{I'm assuming that the transmission lines marked \$Z_0\$ and \$Z_2\$ are ideal and lossless of course}

In short, \$Z_2\$ will behave like a \$\color{red}{\text{resistor}}\$.

You can then use \$Z_2\$'s characteristic impedance (a \$\color{red}{\text{resistor}}\$) in series with lumped impedance \$Z_1\$. Call it \$Z_3\$ for convenience: -

This forms the load to transmission line \$Z_0\$ and, you use \$Z_0\$'s characteristic impedance and \$Z_3\$ to calculate to reflection coefficient at the boundary of \$Z_0\$ and \$Z_3\$

Answer 3

There's nothing mentioned of how long (=delay) the rightmost line has and of is there some load, too. I guess you want only a hint how to handle the case for the time period when there's no reflection arrived from the right end to the lumped impedance.

The incident wave which comes from the left branch to the lumped impedance is a 100 V step function. The reflection could be handled quite easily (see NOTE) if the lumped impedance was given in Laplace domain; for an inductor that is Z(s) = sL. You can consider the right line branch and the lumped impedance as a load which has Laplace domain impedance = Z(s) + Z2. Then you can apply the same reflection factor formula as you did in your previous question.

The Laplace domain reflection factor can be seen as a transfer function. The input is the incident wave and the reflected wave is the output. Multiply the Laplace transform of the input with the transfer function to get the Laplace transform of the output i,e, the reflected wave.

The inverse L-transform of the output is valid only as long as the reflection from the right end has not arrived to the lumped impedance. The reflected wave can be a tricky waveform, no matter the incident wave is a simple step. The reflection factor cannot be a single number except when the lumped impedance is a resistor.

Another approach is to go back to the basics and handle the case with differential equations. The lumped impedance also must be given as a differential equation between its voltage and current. Very likely you can finally formulate the reflection factor as a differential operator. That (Heaviside approach) is equivalent with the Laplace domain method. I skip it.

Hopefully you have about one year of (properly done) university level engineering math studies behind you to handle the things mentioned above.

If you have Zo, Z2 and the lumped impedance as numerical values you can check the result by performing a transient analysis in a circuit analysis program. There's still available Micro Cap (now freeware) which has transmission line as a component.

NOTE: It's easy only as a principle, the calculations can be massive.

Answer 4

A few transmission line fundamentals allow a straight forward algebraic approach for calculating the voltage reflection coefficient caused by the Z0 to Z1 junction.

1. Recall that an infinitely long transmission line will exhibit an input impedance equal to its Z_O regardless of how it is, or is not, terminated.
2. Recall that the formula for a voltage reflection coefficient, Γ_L, caused by a load, Z_L, terminating a transmission line of a characteristic impedance, Z_O is given as Γ_L = (Z_L-Z_O)/(Z_L+Z_O). We should presume all values to be complex numbers.

By combining these basic fundamentals, we can find the solution. We treat Z2 as an infinitely long transmission line. This then simply manifests as itself as its characteristic impedance which is in series with Z1. We calculate Z_L as Z1+Z2 and plug the this into the above formula. This will give you the voltage reflection coefficient caused by the Z0 to Z1 junction.

Keep in mind, however, that there may be an additional non-zero voltage reflection coefficient as a result of the termination, or lack thereof, of Z2. So your overall solution may require more consideration than simply the voltage reflection coefficient caused by Z1.

If this is a real world problem, the phase delays and losses of the respective transmission lines should also be factored into the solution.