How to Calculate $V^+_0$ With the Algebraic and Smith Chart Methods


In [4]:
from IPython.display import Image
Image(filename='01e_transmissionline.png')


Out[4]:
Given:
$f$ = frequency [Hz]
$u$ = phase velocity of voltage on transmission line [m/s]
$Z_0$ = characteristic impedance of transmission line [$\Omega$]
$Z_L$ = load impedance [$\Omega$]
$l$ = length of transmission line [m]
$\tilde{V_g}$ = generator voltage phasor (in general complex) [V]
$Z_g$ = generator impedance [$\Omega$]
Step Algebraic Method Smith Chart
1 $\beta = \frac{2\pi}{\lambda}$, $\lambda = \frac{u}{f}$ same
2 $\Gamma_L = \frac{Z_L - Z_0}{Z_L + Z_0}$ calculate $z_L = \frac{Z_L}{Z_0}$ & plot Smith chart
3 $\Gamma_{in} = \Gamma_L e^{-j2\beta l}$ rotate cw by an angle of $2 \beta l$ around a circle of radius $ z_L $
4 $Z_{in} = Z_0 \frac{1 + \Gamma_{in}}{1 - \Gamma_{in}}$ read $z_{in}$ from Smith chart and calculate $Z_{in} = z_{in} Z_0$
5 $\tilde{V_{in}} = \tilde{V_g} \frac{Z_{in}}{Z_{in}+Z_g}$ same
6 $V^+_0 = \tilde{V_{in}} \frac{e^{-j\beta l}}{1 + \Gamma_{in}} $ same

In [ ]: