Propagation Constant Formula

Electromagnetic waves travel in a sinusoidal pattern. The propagation constant is defined as the change in amplitude and phase per unit distance. It can be measured as a field vector, such as electric flux density or electric field strength, or it can be measured as the current or voltage in the circuit. It is a dimensionless quantity and changes significantly with angular frequency ω. It is denoted by the Greek alphabet γ. It helps in measuring the change per unit length.

Propagation Constant Formula

Propagation constant is defined as the complex amplitude at the wave source (A_o) divided by the complex amplitude at a distance x (A_x). It is equal to the change in amplitude and phase of a sinusoidal electromagnetic wave as it propagates through a medium. The phase of a sinusoidal wave varies as it propagates, and the propagation parameter becomes a complex number. In this situation, the complex/imaginary component is caused by the phase change. Its formula has two components:

1. Attenuation constant: It decreases the signal amplitude when it is propagating through a transmission line. It is denoted by the symbol α.

2. Phase constant: It is the imaginary component of the propagation constant. It provides the signal phase along a transmission line at a consistent time and is denoted by the symbol β. It is equal to the ratio of 2π to the wavelength of the sinusoidal wave (λ).

Propagation constant (γ) = α + iβ

or

γ = α + i (2π/λ)

Sample Problems

Problem 1. Find the propagation constant if the attenuation constant is 2 × 10^-2 and the phase constant is 3.5 × 10^-2.

Solution:

We have,

α = 2 × 10^-2

β = 3.5 × 10^-2

Using the formula we have,

Propagation constant (γ) = α + iβ

= (2 × 10^-2) + i (3.5 × 10^-2)

Problem 2. Find the propagation constant if the attenuation constant is 0.5 × 10^-4 and the phase constant is 1.5 × 10^-4.

Solution:

We have,

α = 0.5 × 10^-4

β = 1.5 × 10^-4

Using the formula we have,

Propagation constant (γ) = α + iβ

= (0.5 × 10^-4) + i (1.5 × 10^-4)

Problem 3. The wavelength of a travelling wave is 3.51 × 10³ m. Find the propagation constant if the attenuation constant is 1.5 × 10^-3.

Solution:

We have,

λ = 3.51 × 10³

α = 1.5 × 10^-3

Calculate the phase constant using the formula β = 2π/λ.

β = 2π/(3.51 × 10³)

= 1.8 × 10^-3

Using the formula we have,

Propagation constant (γ) = α + iβ

= (1.5 × 10^-3) + i (1.8 × 10^-3)

Problem 4. The propagation constant of a wave is (1.8 × 10^-2) + i (3.2 × 10^-2). Find the wavelength of the travelling wave.

Solution:

We have,

γ = (1.8 × 10^-2) + i (3.2 × 10^-2)

Using the formula γ = α + iβ, we get

=> β = 3.2 × 10^-2

Calculate the wavelength using the formula β = 2π/λ.

λ = 2π/β

= 2π/(3.2 × 10^-2)

= 1.96 × 10² m

Problem 5. The propagation constant of a wave is (2.8 × 10^-3) + i (4.5 × 10^-3). Find the wavelength of the travelling wave.

Solution:

We have,

γ = (2.8 × 10^-3) + i (4.5 × 10^-3)

Using the formula γ = α + iβ, we get

=> β = 4.5 × 10^-3

Calculate the wavelength using the formula β = 2π/λ.

λ = 2π/β

= 2π/(4.5 × 10^-3)

= 1.39 × 10³ m

Problem 6. The propagation constant of a wave is (5.5 × 10^-3) + i (7.2 × 10^-3). Find the angular frequency of the travelling wave if its velocity is 2 × 10^-3 m/s.

Solution:

We have,

γ = (5.5 × 10^-3) + i (7.2 × 10^-3)

v = 2 × 10^-3 m/s

Using the formula γ = α + iβ, we get

=> β = 7.2 × 10^-3

Calculate the frequency using the formula β = ω/v.

ω = βv

= (7.2 × 10^-3) (2 × 10^-3)

= 14.4 × 10^-6 s^-1

Problem 7. The propagation constant of a wave is (3.2 × 10^-3) + i (5.6 × 10^-3). Find the velocity of the travelling wave if its angular frequency is 4 × 10^-3 s^-1.

Solution:

We have,

γ = (3.2 × 10^-3) + i (5.6 × 10^-3)

ω = 4 × 10^-3

Using the formula γ = α + iβ, we get

=> β = 5.6 × 10^-3

Calculate the frequency using the formula β = ω/v.

v = β/ω

= (5.6 × 10^-3)/(4 × 10^-3)

= 1.4 m/s