EM Wave Propagation Study Notes for GATE and Electronics & Communication Exams

1. Introduction of EM Waves Propagation

Waves are means of transporting energy or information. Examples of EM waves include radio waves, TV signals, radar beams, & light rays. All forms of EM energy share three fundamental characteristics as they all travel at high velocity in travelling, they assume the properties of waves & they radiate outward
from a source without the benefit of any discernible physical vehicles.

• Characteristics of EM medium:
  • Free space: 
  • Perfect dielectric: and μ can have any value.
  • Good dielectrics: 
  • Perfect conducting: and μ have any value.
  • Good conductor: 
• Where σ is Conductivity and ω is the Angular frequency of the wave.

2. Wave propagation in Lossy Dielectrics

A lossy dielectric is a medium in which an EM wave, as it propagates, loses power owing to the imperfect dielectric. A lossy dielectric is a partially conducting medium.

• Vector wave equation or vector Helmholtz’s equation

    (i) For field:       

    (ii) For field:    

• Where γ is the Propagation constant of the medium and E_s, H_s is Source electric and magnetic fields, respectively.

         

• Where α is the Attenuation constant (Neper/m) and β is the Phase constant (rad/m).

         

• Field equation of EM wave in s-domain

         

• Field equation of EM wave in the time domain

        

3. Intrinsic Impedance

• It is the ratio of Electric Field intensity to the corresponding Magnetic Field Intensity for an electromagnetic wave.

where,   

• It is found to be independent of the location of measurement.
• It is a complex quantity.

          

• For general lossy medium

        

• Note:

 

        , , 

• Phase Velocity

        

• Where V is Wave velocity and c is Free space velocity.
• also,                
• Where λ is Wavelength
• Phase velocity decreases as loss tangent increases from in free space to a few m/s in the good conductor.

4. Loss tangent/ Dissipation factor

• It is the ratio of the conduction current density to the displacement current density and is given by

      

• Where δ is the loss angle.
• Note: If θ_n is the intrinsic angle, then the loss tangent is given by tan2θn.
• Complex Permittivity

         

• Where ε’ = ε is a Real component of complex permittivity and ε” = σ/ε is an Imaginary component of permittivity.
• Loss Tangent

        

5. Plane Waves in Free Space

• In free space

        

               

• where the Direction of wave propagation

6. Plane Waves in Lossless Dielectrics/Perfect Dielectric

• In lossless dielectric

       

•  E and H are in the time phase with each other

7. Plane waves in Good Conductors

• In good conductor

      

• Note: In good conductors, electric and magnetic fields are out of phase by 45° in time. 

8. Skin Depth

The skin depth is a measure of the depth to which the amplitude of an EM wave will reduce to 36.8% of initial value.

     

• Surface or Skin Resistance

      

 

• Note:
  • The skin depth is useful in calculating the ac resistance due to the skin effect.

         

• The perfect dielectric medium behaves like a perfect transmitter of EM waves.
• The perfect conducting medium behaves like a perfect reflector of an EM wave.
• Poynting’s Theorem: The theorem states that “the net power flowing out of a given volume V is equal to the sum of the Ohmic losses and the time rate of decrease in the energy stored with in volume V.”

       

• Poynting Vector

       

The poynting vector represents the following parameter associated with the EM wave.

(i) Power density associated with EM wave in watt/m²

(ii) The power flow for the given EM wave, and this is the direction of the unit vector normal to the plane containing E × H, according to right hand system. So this is also the direction of propagation of the EM wave.

• Instantaneous and Average Power Density

(i) Instantaneous power density

    

(ii) Average power density

    

(iii) Average power density in a lossless medium

    

9. Incidence of EM Wave

• Normal Incidence

  

• Reflection Coefficient
  • It is the ratio of the electric field of the reflected wave to the electric field of the incident wave

     

• where the η₁, η₂ is the Intrinsic impedance of medium-1 & medium-2 respectively.
• Note: Reflection coefficient is a complex quantity and can be written as

      

10. Transmission Coefficient (T)

• It is defined as the ratio of a transmitted electric field to the incident electric field.

        

• Standing Wave Ratio

        

• Note:

        

11. Oblique Incidence

     

• Snell’s Law

       

• where  are the refractive indices of the media?
• Snell’s law for EM waves is

        

• Total internal Reflection of EM wave

        

• Note: θ_c is the minimum angle of incidence at which total internal reflection starts.
• The condition for total internal reflections:

          

•  Brewster’s Angle (θ_B): This is the angle of incidence for complete transmission of EM wave.

         

• Note: When a circular or elliptically polarized wave is incident at Brewster’s angle, the reflected and transmitted wave is linearly polarized. Therefore this angle is also known as the polarization angle.
• Reflection on the Surface of Conductors:

          Because η₂=0 for conductors.

• At the surface of the conductor, an electric field is a minimum while the magnetic field is a maximum.

12. Polarization

This is the orientation of the electric vector at a fixed position in space with respect to time.

• Linear Polarization

       

       a) Parallel Polarization

         

       b) Perpendicular Polarization

         

• Circular Polarization

         

• Note:
  • Wave is assumed to be propagating in the +z direction.
  • If Φ = + 90^o, the rotation is in a clockwise direction, then the wave is said to be left hand circularly polarized.
  • If Φ = – 90^o, the rotation is in anti-clockwise direction, then the wave is said to be right hand circularly.
• Elliptical Polarization

         

• Note:
  • If Φ = + 90^o, the rotation is clockwise, then the wave is said to be left hand elliptically polarized.
  • If Φ = – 90^o, the rotation is in anti-clockwise direction, then the wave is said to be right hand elliptically polarized.
•  S and P Polarized Waves

           

•  In S-polarization, the electric field lies in the plane perpendicular to the at of interface. The reflection coefficient is given as

           

• where 
• In P-polarization, the E field lies in the plane of the interface. The reflection coefficient is given as

         

• Note: Brewster’s angle is valid for P-polarized waves only.

 

For more information about

EM wave propagation, you can refer to the following video on Byju Exam Prep’s official youtube channel.

 

Candidates can practice 150+ Mock Tests with BYJU’S Exam Prep Test Series for exams like GATE, ESE, and NIELIT from the following link:

Click Here to Avail Electronics Engineering Test Series (150+ Mock Tests)

Get unlimited access to 24+ structured Live Courses and all 150+ mock tests to boost your GATE 2021 Preparation with Online Classroom Program:

Click here to avail Online Classroom Program for Electronics Engineering

Thanks