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

By BYJU'S Exam Prep

Updated on: September 25th, 2023

EM Wave Propagation study notes cover the important topics such as introduction, Wave propagation in Lossy dielectrics, plane waves in free space, Plane waves in lossless Dielectrics, Plane waves in good conductors, the incidence of EM wave and polarization. Preparation of EM Wave Propagation study notes is helpful for GATE, BARC, ISRO, ESE and other Electronics and Communication Engineering 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 Es, Hs 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 θis 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: θc is the minimum angle of incidence at which total internal reflection starts.
• The condition for total internal reflections:

•  Brewster’s Angle (θB): 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 = + 90o, the rotation is in a clockwise direction, then the wave is said to be left hand circularly polarized.
• If Φ = – 90o, the rotation is in anti-clockwise direction, then the wave is said to be right hand circularly.
• Elliptical Polarization

• Note:
• If Φ = + 90o, the rotation is clockwise, then the wave is said to be left hand elliptically polarized.
• If Φ = – 90o, 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.

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: