Bounce Diagrams

Voltage Reflection Coefficient

$$ \Gamma = \frac{Z_L - Z_0}{Z_L + Z_0} $$

Voltage Transmission Coefficient

$$ T = 1 + \Gamma $$

Voltage at a point on the line

$$ V(x, t) = V_0 \cdot \Gamma^n $$

• $n$ is the number of reflections, and $V_0$ is the initial voltage
• The pulse behaviour shown in the bounce diagram involves the pulse width, delays, and reflections at intervals based on the TL length and signal propagation speed

Special Cases of TLs

Input impedance for short circuited line

$$ Z_{\text{in}} = j Z_0 \tan\left(\frac{\beta l}{\lambda}\right) $$

• $\beta$ is the phase constant, $l$ is the line length

Input impedance for open-circuited line

$$ Z_{\text{in}} = -j Z_0 \cot\left(\frac{\beta l}{\lambda}\right) $$

Quarter-wave transformer

$$ Z_{\text{in}} = \frac{Z_0^2}{Z_L} $$

• $Z_L$ is the load impedance
• $Z_0$ is characteristic impedance

Standing wave ratio

$$ SWR = \frac{1 + |\Gamma|}{1 - |\Gamma|} $$

• Used to characterize the efficiency of the TL

Power Flow in TLs

Power delivered to the load

$$ P_L = \frac{V_L^2}{R_L} $$

Input power with reflections

$$ P_{\text{in}} = \frac{|V_{\text{in}}|^2}{Z_0} (1 - |\Gamma|^2) $$

Reflection coefficient

$$ \Gamma = \frac{Z_L - Z_0}{Z_L + Z_0} $$

Average power absorbed by the load

$$ P_{\text{avg}} = \frac{V_{\text{in}}^2}{2 Z_0} \cdot (1 - |\Gamma|^2) $$