Propogation constant of a transmission line is :

Options

A. `(R+jomegaL)/(G+jomegaC)`
B. `sqrt ((R+jomegaL)(G+jomegaC))`
C. `sqrt ((1)/(LC))`
D. `sqrt ((R-jomegaL)(G-jomegaC))`

Answer

To find the propagation constant of a transmission line, we can follow these steps:

### Step 1: Understand the Definition
The propagation constant (γ) of a transmission line is defined as the product of the series impedance (Z) and the shunt admittance (Y) per unit length of the line. The formula is given by:
\[ \gamma = Z \cdot Y \]

### Step 2: Identify the Components
- **Z (Series Impedance)**: It is defined as:
\[ Z = R + j\omega L \]
where:
- \( R \) = resistance per unit length
- \( j \) = imaginary unit
- \( \omega \) = angular frequency
- \( L \) = inductance per unit length

- **Y (Shunt Admittance)**: It is defined as:
\[ Y = G + j\omega C \]
where:
- \( G \) = conductance per unit length
- \( C \) = capacitance per unit length

### Step 3: Substitute into the Formula
Now, substituting the expressions for Z and Y into the propagation constant formula:
\[ \gamma = (R + j\omega L)(G + j\omega C) \]

### Step 4: Expand the Expression
Using the distributive property (FOIL method), we expand the expression:
\[ \gamma = RG + jR\omega C + j\omega L G - \omega^2 LC \]
This can be rearranged as:
\[ \gamma = (RG - \omega^2 LC) + j(R\omega C + \omega L G) \]

### Step 5: Identify Real and Imaginary Parts
The propagation constant γ can be expressed in terms of its real and imaginary parts:
- Real part: \( \alpha = RG - \omega^2 LC \) (attenuation constant)
- Imaginary part: \( \beta = R\omega C + \omega L G \) (phase constant)

### Step 6: Conclusion
Thus, the propagation constant γ can be represented as:
\[ \gamma = \alpha + j\beta \]