Propagation Constant Formula: Formula & Examples

For a sinusoidal electromagnetic wave, the propagation constant is defined as the measure of the change in its amplitude and phase of the wave as it propagates in a given direction. 

The quantity which can be measured can be voltage, the current in a circuit, or a field vector such as electric field strength or flux density. 

• The propagation constant measures the change per unit length, but it is a dimensionless quantity.
• The value of propagation constant is expressed logarithmically to the base e, rather than the more usual base 10 that is used in telecommunications in other situations.

Key Terms: amplitude, electromagnetic wave, frequency, wavelength, transmission lines, attenuation, phase constant, inductance, capacitance.

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Propagation Constant

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Propagation constant is a measure of changes in amplitude and phase of a sinusoidal electromagnetic wave, while propagating through a medium. The medium here can be a transmission line, free space or vacuum. 

• The phase of the sinusoidal wave varies with distance which results in the propagation constant being a complex number.
• The Propagation constant is a dimensionless quantity.
• Propagation constant is also known as propagation parameter, transmission parameter, propagation coefficient, transmission function and transmission constant.
• Propagation constant strongly varies with angular frequency ⍵.
• For two port networks and their cascades, the measurement of the change in the source quantity as it propagates from one port to another is called propagation constant.

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Propagation Constant Formula

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For a given system, the propagation constant is defined by the ratio of the complex amplitude at the source of the wave (A₀) to the complex amplitude at distance x (A_x). Such that,

\(\frac{A_0}{A_x} =e ^{\gamma x}\)

Where, γ is Propagation Constant.

Since, the propagation constant is a complex quantity, therefore, we can write

γ = α + iβ

Where, 

• ⍺, the real part of the equation is called the attenuation constant.
• β, the imaginary part of the equation is called the phase constant.

The propagation constant for conducting lines can be calculated as,

γ = \(\sqrt{ZY}\)

Where, Z = R + iωL, the series impedance of the line per unit length and

Y = G + iωC, the shunt admittance of the line per unit length.

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Attenuation Constant

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Attenuation constant is the decrease in the signal amplitude of an electromagnetic wave propagating through a medium per unit distance from the source.

• It is denoted by α and this is the real part of the propagation constant.
• It is measured in Np/m (nepers/meter).
• 1 neper is approximately equal to 8.7 dB.
• It is also called attenuation parameter or attenuation coefficient.
• Attenuation constant can be given by the amplitude ratio

\(|\frac{A_0}{A_x}| = e ^{\alpha x}\)

• The attenuation constant for conductive lines is given by

α = \(\sqrt{RG}\)

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Phase Constant

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In electromagnetic theory, phase constant represents the change in phase per unit length along the path traveled by the wave at any instant.

• It is the imaginary part of the equation of propagation constant.
• It is equal to the real part of the angular wavenumber of the wave.
• It is also called phase change constant, phase change parameter or phase change coefficient.
• It is denoted by ꞵ and its unit is radians per meter.
• For transverse electromagnetic wave transmission, the wavenumber and the propagation phase constant are equal and it is given by

β = k = \(\frac{2 \pi}{\lambda}\)

• The Phase constant for conductive lines is given by

β = ω \(\sqrt{LC}\)

Where, C and L are capacitance and inductance per unit length of the transmission line.

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Solved Examples

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Ques. The propagation constant of a transmission line is 0.15 x 10^-3 + i1.5 x 10^-3. What is the wavelength of a traveling wave?

Ans. Propagation constant of a transmission line is given by

γ = α + iβ

Now, given equation of propagation constant is

γ = 0.15 x 10^-3 + i1.5 x 10^-3

Comparing both the equation, we get

β = 1.5 x 10^-3

But, β = \(\frac{2 \pi}{\lambda}\)

Where, λ is the wavelength of the traveling wave.

⇒ λ = \(\frac{2 \pi}{\beta}\) = 21.5 x 10^-3 m 

Ques. A lossless transmission line has the distributed circuit parameters of inductance and capacitance per meter as 625 nH/m and 64 pF/m respectively. What is the phase constant of the line at 100 MHz?

Ans. Phase constant for a lossless transmission line is given by

β = ω \(\sqrt{LC}\)

Where, L and C are inductance and capacitance of transmission line

But angular frequency, ω = 2πf

Where, f is frequency of the wave.

⇒ β = 2πf \(\sqrt{LC}\)

Given, f = 100 MHz= 100 x 10⁶ Hz

L = 625 nH/m = 625 x 10^-9 H/m

And, C = 64 pF/m = 64 x 10^-12 F/m

⇒ β = 2 x 100 x 10⁶ x \(\sqrt{625 \times 10^{-96} \times 4 \times 10^{-12}}\)

⇒ β = 6.28 x 10⁸ x \(\sqrt{40 \times 10^{-18}}\)

⇒ β = 6.28 x 10⁸ x \(\sqrt{6.32 \times 10^{-9}}\)

⇒ β = 3.97 rad/s

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Things to Remember

• The measure of the change in its amplitude and phase of the wave as it propagates in a given direction is called Propagation constant of electromagnetic waves. 
• Propagation constant varies with angular frequency ⍵.
• The decrease in the signal amplitude of an electromagnetic wave propagating through a medium per unit distance from the source is called Attenuation constant.
• The attenuation constant for conductive lines is given by

α = \(\sqrt{RG}\)

•  Phase constant is the change in phase per unit length along the path traveled by the wave at any instant.
• Phase constant and wavenumber are equal for transverse electromagnetic wave transmission and it is given by

β = k = \(\frac{2 \pi}{\lambda}\)

• The Phase constant for conductive lines is given by

β = ω \(\sqrt{LC}\)