# Propagation Constant Formula

• Last Updated : 28 Apr, 2022

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 (Ao) divided by the complex amplitude at a distance x (Ax). 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

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