Lecture Notes on Voltage and Wave Propagation

Key Equation

• Starting with the equation:
  [ \frac{d^2 V}{dx^2} = \gamma^2 V ]
• Where (\gamma) is the propagation constant.

Voltage in Space and Time

• Solution leads to:
  [ V(X, T) = V_+ e^{-\gamma X} + V_- e^{\gamma X} ]
• (V_+) and (V_-) are constants.
• Voltage depends on both space (X) and time (T).

Instantaneous Value of Voltage

• For sinusoidal voltage, multiply by:
  [ e^{j \Omega T} ]
• This gives the instantaneous value of voltage.

Understanding (\gamma)

• (\gamma = \alpha + j \beta)
  • (\alpha): Attenuation constant
  • (\beta): Phase constant
• In a lossless medium, (\alpha = 0).

Key Equation After Substituting (\gamma)

• With (\alpha = 0), the equation simplifies to:
  [ V(X, T) = V_+ e^{-j \beta X} e^{j \Omega T} + V_- e^{j \beta X} e^{j \Omega T} ]

Further Simplification

• Can be expressed as:
  [ V(X, T) = V_+ \cos(\Omega T - \beta X) + V_- \cos(\Omega T + \beta X) ]

Wave Nature of Voltage

• The voltage behaves like a wave traveling in both directions:
  • Forward direction: (+X)
  • Backward direction: (-X)
• Voltage in transmission lines has wave characteristics.

Comparison with Electric Field Equation

• Recall the electric field equation:
  [ E = E_0 e^{-\alpha Z} \cos(\Omega t - \beta Z) ]
  • Shows attenuation and propagation direction.
• For lossless medium:
  [ E = E_0 \cos(\Omega t - \beta Z) ]
  • This indicates wave nature similar to voltage.

Conclusion on Current

• Current also exhibits wave nature:
  • Forward and backward traveling waves.
• Both voltage and current in transmission lines travel as waves.

Implications

• Understanding this behavior is crucial for analyzing transmission lines and wave propagation.