Week Three | Notion

Impedance Transformations

Phasor Notation of Voltage and Current on TL

Used to represent voltage and current along the transmission line, accounting for phase shifts due to distance $z$:

$$ \tilde{V}(z) = V_0^+ e^{-j\beta z} + V_0^- e^{j\beta z} \newline \tilde{I}(z) = \frac{V_0^+}{Z_0} e^{-j\beta z} - \frac{V_0^-}{Z_0} e^{j\beta z} $$
These equations describe the traveling waves along a lossless transmission line, where $\beta$ is the phase constant

Reflection Coefficient as a Function of Distance

Reflection coefficient depends on the position along the line

$$ \Gamma(z) = \Gamma_L e^{-j2\beta z} $$
This equation shows how the reflection coefficient varies along the line, where $\Gamma _L$ is the reflection coefficient at the load

Maximum and Minimum Voltages along the Line

$$ |\tilde{V}(z)|_{max} = |V_0^+| (1 + |\Gamma|) \newline

|\tilde{V}(z)|_{min} = |V_0^+| (1 - |\Gamma|)

Maxima: correspond to even multiples of $\frac{\lambda}{2}$

$$ z = n \frac{\lambda }{2} \text{ for } n = 0,1,2 \cdots $$
- Corresponds to positions along the line where the phase difference between the waves is an even multiple of $\pi$, resulting in constructive interference
Minima: correspond to odd multiples of $\frac{\lambda}{4}$

$$ z = (n + \frac{1}{2}) \frac{\lambda}{2} $$
- Corresponds to positions along the line where phase difference between the waves is an odd multiple of $\pi$, resulting in destructive interference

Input Impedance of the TL

$$ Z_{in}(z) = Z_0 \frac{1 + \Gamma_L e^{-j2\beta z}}{1 - \Gamma_L e^{-j2\beta z}} \newline Z_{in} = Z_0 \frac{Z_L + jZ_0 \tan (\beta l)}{Z_0 + j Z_L \tan (\beta l)} $$

This equation transforms the impedance at any point along the line back to the source, aiding in matching and analysis of the line behaviour

Special Cases for Input Impedance