A note on the modeling of transmission-line losses

We consider uniform lossy transmission lines characterized by their primary parameters. Exact and approximate formulas for the characteristic impedance and propagation coefficient are reviewed and discussed for low-loss lines. Approximating the characteristic impedance by its real part can lead to erroneous results for the input impedance of short- and open-circuited stubs. This problem is analytically demonstrated on electrically short stubs. Results obtained using the exact and approximate expressions are compared with numerical solutions that are generated by various circuit simulation software.