Solving Smith Chart Homework: Find Input Impedance & Admittances

[likephysics]

Homework Statement

The characteristic impedance of a given lossless transmission line is 75 ohms. use a smith chart to find the input impedance at 200Mhz of such a line that is (a)1m long and open circuited and (b) 0.8m long and short circuited. Then (c) determine the corresponding input admittances for the lines in part (a) and (b).

Homework Equations

The Attempt at a Solution

I can find the answer for the short circuit case. The R=0 circle is the short ckt circle. I just started from the left most part of the smith chart and then moved a distance of (l/lambda)=0.53 , to find the input impedance.

But for open circuit, the circle degenerates to a point. I am not sure how to proceed.

Also, the analytical results don't agree with the graphical results. So I am going wrong somewhere. Any help?

 

[KataKoniK]

Hello!

For the open circuit case, the input impedance will be purely imaginary, with no real component. This means that it will lie on the imaginary axis of the Smith chart. To find the exact location, you can use the formula Zin = jZ0*tan(beta*l), where Z0 is the characteristic impedance and beta is the propagation constant (2*pi/lambda). In this case, beta*l = pi/2, so the input impedance will be j*Z0. On the Smith chart, this will be represented as a point at the intersection of the imaginary axis and the circle with radius Z0.

For the short circuit case, you can use the same formula but with a beta*l value of pi, which will give an input impedance of -j*Z0. This will also be represented as a point on the imaginary axis, but in the opposite direction from the open circuit case.

To find the input admittances, you can simply take the reciprocal of the input impedances found above. So for the open circuit case, the input admittance will be -j/Y0, and for the short circuit case, it will be j/Y0. On the Smith chart, these will be represented as points on the imaginary axis, but with the opposite sign from the input impedances.

I hope this helps! Let me know if you have any further questions.

 

[Mene]

To solve this problem, we can use the Smith Chart to plot the normalized impedance of the transmission line at 200Mhz. For the open circuit case, the impedance will be purely imaginary and will lie on the R=0 circle. We can then use the Smith Chart to find the normalized input impedance at a distance of (l/lambda) = 0.53. This will give us the normalized input impedance, which can then be converted back to the actual input impedance by multiplying by the characteristic impedance of the line.

For the short circuit case, we can use the same approach, but the impedance will lie on the X=0 circle. Again, we can find the normalized input impedance at a distance of (l/lambda) = 0.53 and then convert it back to the actual input impedance.

To find the input admittances, we can use the reciprocal property of the Smith Chart. The admittance at any point on the chart is the reciprocal of the impedance at the same point. So, for the open circuit case, the admittance will be purely imaginary and will lie on the G=0 circle. For the short circuit case, the admittance will lie on the B=0 circle.

It is important to note that the analytical results may not always agree with the graphical results due to the approximations made in using the Smith Chart. However, the graphical results can provide a good estimation and can also help in visualizing the problem.