 # Propagation Constant of a Wave and transmission line - Definition, Formula, Explanation

## Propagation Constant Definition

Electromagnetic waves propagate in a sinusoidal fashion. The measure of the change in its amplitude and phase per unit distance is called the propagation constant. Denoted by the Greek letter 𝜸. The terminologies like Transmission function, Transmission constant, Transmission parameter, the Propagation coefficient, Propagation parameter are synonymous with this quantity. Sometimes 𝜶 and 𝜷 are referred collectively as Propagation parameters or Transmission parameters.

### Propagation Constant Measurement

It can be measured as a field vector, such as electric flux density or electric field strength or the current, the voltage in the circuit. It also measures the change per unit length, but otherwise, it is a dimensionless quantity. It strongly varies with angular frequency 𝝎.

Generally, Its value is expressed logarithmically, with the base e. Sometimes the telecommunication community uses base 10. It is also expressed as a sinusoidal Phasor when measured as a voltage in the circuit. Sinusoidal wave changes its phase when it propagates, as a result, the propagation parameter turns out to be a complex number. Here the complex/imaginary part is caused by the phase change.

## Propagation Constant Formula

For any given system, the Propagation constant can be mathematically expressed as-

$$Propagation\; Constant\left ( \gamma \right )=\frac{Complex\;amplitude\;at\;the\;source\;of\;the\;wave\left ( A_{0} \right )}{Complex\;amplitude\;at\;distance\;x\left ( A_{x} \right )}$$ $$\frac{A_{0}}{A_{x}}=e^{\gamma x}$$

The complex entity can be written as $$\gamma =\alpha +i\beta$$

Where,

𝜶, is the real part called the attenuation constant.

𝜷, is the imaginary part called phase constant.

The phase can be calculated using Euler’s formula as –

$$e^{i\theta }=cos\theta +isin\theta$$

The equation is sinusoidal in nature. Here phase varies according to 𝜽 whereas, the amplitude remains invariant as $$\left | e^{i\theta } \right |=\sqrt{cos^{2}\theta +sin^{2}\theta} =1$$

The angles are measured in radian.

### Propagation Constant of a Transmission line

The propagation constant for any conducting lines (like copper lines) can be calculated by relating the primary line parameters.

$$\gamma =\sqrt{ZY}$$

Where,

$$Z=R+i\omega L$$ Series impedance of line per unit length.

$$Y=G+i\omega C$$ The shunt admittance of line per unit length.

We hope you got a clear idea about propagation parameters. For more such articles follow