A Transmission Line
Two-wire transmission lines can be modeled explicitly in AN-SOF. In this example, the line will have a single wire but there will be a ground plane below it, so we have the mirror image of the wire as the return of the line.
Step 1 | Setup
Go to the Setup tab and select Single in the Frequency panel >. Set a frequency of 100 MHz. Then, go to the Environment panel > and set a perfect ground plane at Z = 0, Fig. 1.
Step 2 | Draw
Go to the Workspace tab, right click on the screen, and select Line from the pop-up menu >. Draw a horizontal line with the coordinates indicated in Fig. 2. Next, connect the ends of the line to the ground plane by drawing two vertical wires. You can right click on the line and select the commands Start point to GND and End point to GND to connect the ends to ground.
Set 40 segments for the horizontal wire and 1 segment for each of the vertical wires. Note that dimensions in Fig. 1 are in millimeters. To change the unit of length, go to Tools menu > Preferences > Units tab >.
Right click on the vertical wire at (0,0,0), select Source/Load from the displayed pop-up menu and put a 1 Volt voltage source on it. Refer to Adding Sources > to add the voltage source.
Step 3 | Run
Go to the Run menu and click on Run Currents. Since we are only interested in the current distribution and the input impedance, it is not necessary to calculate the radiated field (you can do it to check that it is practically negligible). Click on the Zin (List Input Impedances) button on the toolbar to display a table where the input impedance is shown as a function of frequency, Fig. 3. Refer to Listing Input Impedances >.
The impedance obtained is practically reactive, j512 Ohm. The small real part is the radiation resistance, since the line radiates a small amount of power, which is negligible but not zero.
This is a short-circuited line. Now right click on the vertical wire at (0,500,0) mm and select Delete from the pop-up menu to remove it. You will get an open-circuited line in this way. Rerun the calculations with the Run Currents command in the Run menu. The input impedance will now be -j105 Ohm.
Summarizing, we have,
- Zin (short-circuited line) = j512 Ohm.
- Zin (open-circuited line) = -j105 Ohm.
According to transmission line theory, the characteristic impedance can be calculated as the geometric mean of the short-circuit and open-circuit line input impedances, hence
On the other hand, the expression for the characteristic impedance of a line above a ground plane is given by:
where a is the wire cross-section radius and h is the line height above the ground plane. As we can see, the agreement between the characteristic impedance obtained from AN-SOF and that from theory is quite good. The difference is since the theory neglects the radiation of the line, and the logarithmic formula is an approximation that is valid when h >> a.