Electromagnetic Field Theory
Review of: Vector Analysis- different coordinate systems, gradient, divergence and curl, Divergence and
Stokes theorem, Electrostatic field and magnetostatic fields.
Time Varying Fields: Continuity equation, Displacement current, Faraday's Law of Electromagnetic
Induction, Maxwell's Equations (point form and integral form). Potential Functions, Electromagnetic Boundary
Conditions, Time Harmonic Fields
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Plane Electromagnetic Waves propagation: Wave Equations & Their Solutions, Plane Waves in Lossless
&Lossy Media, Group Velocity, Poynting Vector &Poynting Theorem, Refractions and Reflections at Normal
and Oblique Incidence at Plane Conducting and Plane Dielectric Boundary.
Transmission Lines: Transmission line equations, parameters- primary and secondary constants, Analogy of
transmission lines, Determination of α, β,γ, and vp, characteristics impedance, Input impedance of a lossless
line, open and short circuited lines, distortion-less lines, reflection coefficient and standing wave ratio,
matched transmission line, Impedance matching, Smith-chart and its applications
Waveguides- Rectangular waveguide, Circular Waveguides. Solution of the wave equation in rectangular and
cylindrical co-ordinates, Derivation of field equations for TE & TM modes, degenerate and dominant mode,
Power Transmission and Power loss, Excitation of waveguides.
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1-Waveguides- Rectangular waveguide
Rectangular Waveguide — A rectangular waveguide is a hollow metallic tube with a rectangular cross-section used
to guide electromagnetic waves, typically at microwave frequencies.
🔹 Structure
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●​ Made of a good conductor (usually copper or aluminum)
●​ Dimensions:
○​ a → wider side (width)
○​ b → shorter side (height), where a > b
🔹 Principle of Operation🔹 Modes of Propagation-
EM waves propagate by reflecting from the conducting walls. Unlike ordinary wires,
waveguides do not carry signals via current flow but via field propagation.
Waveguides support specific field patterns called modes:
1.​ TE (Transverse Electric)
○​ No electric field in the direction of propagation
○​ Most common in practice
2.​ TM (Transverse Magnetic)- No magnetic field in the propagation direction
3.​ TEM mode- Not supported in hollow waveguides
🔹 Dominant Mode
●​ TE₁₀ mode is the dominant (lowest cutoff frequency)
●​ Safest and most efficient mode for transmission
🔹 Cutoff Frequency-
A wave propagates only if:
🔹 Key Characteristics
●​ Very low loss at high frequencies
●​ High power handling capability
●​ Frequency-dependent behavior
●​ Bulky at lower frequencies
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Applications
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●​ Radar systems
●​ Satellite communication
●​ Microwave links
●​ RF & microwave labs
●​ Antenna feeds
Advantages- Low attenuation, High power capacity, Excellent for microwaves
Limitations- Large physical size, only high-frequency operation
2-Circular Waveguides
Circular Waveguides — A circular waveguide is a hollow metallic tube with a circular cross-section used to
transmit microwave and RF electromagnetic waves.
🔹 Structure
●​ Cylindrical conducting tube (copper/aluminum)
●​ Dimension:→ radius of the waveguide
Working Principle: Electromagnetic waves propagate through multiple reflections from the inner metallic
surface. Energy travels as field patterns (modes) rather than current.
Modes of Propagation Circular waveguides support:
1.​ TE (Transverse Electric) Modes-No electric field along the propagation direction
2.​ TM (Transverse Magnetic) Modes- No magnetic field along propagation direction
3.​ TEM Mode- Not supported in hollow waveguides
🔹
🔹
🔹 Dominant Mode
●​ The TE₁₁ mode is the dominant mode
●​ Has the lowest cutoff frequency
●​ Most commonly used for transmission
🔹 Cutoff Frequency: Wave
propagation occurs only if:
Cutoff frequency depends on radius and mode:
where:
●​ X ₙ → Mode constant (from Bessel functions), c → Speed of light, a → Waveguide radius
For dominant TE₁₁ mode:
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Field Characteristics
●​ Fields vary according to Bessel functions., Symmetrical field distribution., Degeneracy may occur (two identical
modes)
Key Features
●​ Suitable for high-frequency signals, Can handle high power, Lower losses at microwave frequencies, No sharp
corners → Less breakdown risk
Applications
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●​ Satellite communication, Radar systems, Rotary joints, Microwave antennas, High-power RF systems
Advantages- High power handling, Low attenuation, Good symmetry
Limitations- Mode degeneracy, Complex mode behavior, Larger size for the same cutoff vs rectangular
3-Solution of the wave equation in rectangular and cylindrical co-ordinates
1. Wave Equation in Rectangular Coordinates
The wave equation in three-dimensional rectangular (Cartesian) coordinates (x,y,z) is given by:
where u(x,y,z,t) is the wave amplitude, and v is the wave velocity.
We seek a solution using the method of separation of variables, assuming the solution can be written as the product of
four functions, each depending on only one variable:
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4-Derivation of field equations for TE & TM modes
Starting with Maxwell’s Equations (Phasor Form)
Wave Equation- From Maxwell equations:
TE Mode (Transverse Electric Mode)
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TM Mode (Transverse Magnetic Mode)
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Cutoff Frequency
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Key Differences Between TE & TM
5-degenerate and dominant mode
1. Dominant Mode
Definition-The dominant mode in a waveguide is the mode that has the lowest cut-off frequency. Because it
requires the least frequency to propagate, it is the first mode that can propagate through the waveguide when the
operating frequency increases from zero.
Dominant mode: The electromagnetic mode having the minimum cut-off frequency in a waveguide.
Cut-off Frequency of Rectangular Waveguide-For a rectangular waveguide, the cut-off frequency is-
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Where- c = velocity of light, a = broad dimension of the waveguide,
b = narrow dimension of the waveguide, m,n = mode numbers (integers)
Why TE₁₀ is the Dominant Mode
1.​ The value of mmm and nnn produces the minimum cut-off frequency.
2.​ Electric field distribution is simple.
3.​ It provides maximum power transmission efficiency.
4.​ Higher modes such as TE₂₀, TE₁₁, TM₁₁ require higher frequencies to propagate.
Thus, waveguides are usually designed to operate only in the TE₁₀ dominant mode.
Field Distribution in Dominant Mode (TE₁₀)
In TE₁₀ mode:
●​ Electric field exists mainly in the y-direction.
●​ Magnetic field exists in x and z directions.
●​ Electric field varies sinusoidally across the width a.
Electric field expression:
Advantages of Dominant Mode Operation
1.​ Minimum signal distortion.
2.​ Efficient power transmission.
3.​ Reduced attenuation.
4.​ Stable propagation in microwave communication systems.
2. Degenerate Mode
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Definition-Degenerate modes occur when two or more modes have the same cut-off frequency but different field
configurations.
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Degenerate modes: Two or more modes that possess identical cut-off frequencies are called degenerate modes.
Physical Meaning
Degenerate modes:
●​ Have equal cut-off frequency
●​ Can propagate at the same operating frequency
●​ Have different field orientations
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●​ Usually occur in symmetric structures such as square or circular waveguides.
Degeneracy in Circular Waveguide- In circular waveguides, degeneracy occurs because of symmetry in the
structure. For example, the TE₁₁ mode can exist in different orientations but with the same cut-off frequency.
Difference Between Dominant Mode and Degenerate Mode
Parameter
Mode with lowest cut-off frequency
Degenerate Mode
Modes with equal cut-off frequency
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Definition
Dominant Mode
Number of Modes
Single
Two or more
Occurrence
All waveguides
Mainly symmetric waveguides
Example
TE₁₀ in rectangular waveguide
TE₁₀ and TE₀₁ in square waveguide
Propagation
First mode to propagate
Modes propagate at same frequency
Summary
●​ Dominant mode is the mode with the lowest cut-off frequency, and it is the first mode to propagate in a
waveguide.
●​ In rectangular waveguides, TE₁₀ is the dominant mode.
●​ Degenerate modes occur when two or more modes have identical cut-off frequencies, usually due to
symmetry in the waveguide structure.
●​ Example: TE₁₀ and TE₀₁ in a square waveguide.
6-Power Transmission and Power Loss
Power Transmission and Power Loss in Waveguides
In waveguides, electromagnetic waves carry power from one point to another. During propagation, some power is
successfully transmitted, while some power is lost due to conductor resistance and dielectric losses. These concepts
are explained as Power Transmission and Power Loss.
1. Power Transmission
Definition-Power transmission is the amount of electromagnetic power carried by the wave through the waveguide.
In electromagnetic theory, the power flow is represented by the Poynting vector.
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Factors Affecting Power Transmission
1.​ Frequency of operation
2.​ Waveguide dimensions
3.​ Mode of propagation
4.​ Material properties
5.​ Field intensity
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2. Power Loss
Definition- Power loss is the amount of electromagnetic power that is dissipated as heat or radiation during wave
propagation inside the waveguide.
Loss mainly occurs due to finite conductivity of the waveguide walls and dielectric losses.
Types of Power Loss
1. Conductor Loss- Occurs because the waveguide walls are not perfect conductors.
●​ Current flows on the inner surface of the waveguide.
●​ Due to resistance, energy converts into heat.
Surface power loss per unit area:
Where
●​ Rs​= surface resistance
●​ Ht​= tangential magnetic field​
2. Dielectric Loss
Occurs when the medium inside the waveguide is not perfectly lossless.Energy is lost due to polarization and
dielectric heating.
Dielectric loss depends on:
●​ Loss tangent tanδ
●​ Permittivity of medium
3. Radiation Loss
Occurs due to:
●​ Bends in waveguide
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●​ Imperfect joints
●​ Discontinuities
Part of the electromagnetic energy leaks out.
3. Attenuation Due to Loss
Loss causes attenuation of the signal.
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Total attenuation constant:
Where
●​ αc​= conductor attenuation
●​ αd​= dielectric attenuation
4. Difference Between Power Transmission and Power Loss
Parameter
Power Transmission
Power Loss
Meaning
Power carried by electromagnetic wave Power dissipated during propagation
Cause
Electric and magnetic fields
Resistance, dielectric heating
Result
Signal propagation
Signal attenuation
Representation Poynting vector
Loss equations
7-Excitation of waveguides
Excitation of Waveguides
Introduction-Excitation of a waveguide refers to the method of introducing electromagnetic energy from a source
(such as a microwave generator or transmission line) into a waveguide so that the desired mode of propagation can
be established inside the waveguide.
In microwave systems, power is usually generated by devices like klystrons, magnetrons, or Gunn diodes, and this
power must be coupled efficiently into the waveguide.
1. Purpose of Waveguide Excitation
The main objectives of excitation are:
1.​ To transfer microwave power from a source to the waveguide.
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2.​ To generate the desired propagation mode (usually the dominant mode).
3.​ To minimize reflection and power loss.
4.​ To ensure efficient impedance matching between the source and waveguide.
2. Basic Principle of Excitation
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Excitation works by placing a coupling element inside the waveguide so that it interacts with the electric or magnetic
field of the propagating mode.
Depending on which field is used for coupling, excitation methods are divided into:
1.​ Electric field excitation
2.​ Magnetic field excitation
3. Methods of Waveguide Excitation
1. Probe (Coaxial Probe) Excitation
Description-In this method, a coaxial probe is inserted through the waveguide wall so that the inner conductor
extends into the waveguide cavity.
The probe acts like a small antenna and couples with the electric field inside the waveguide.
Working
●​ The probe is placed at a point where the electric field is maximum.
●​ It excites the TE₁₀ dominant mode in rectangular waveguides.
Characteristics
●​ Couples with an electric field.
●​ Commonly used for rectangular waveguides.
Advantages- Simple structure, Easy to implement
Disadvantages- Improper placement may cause reflection and a mismatch.
2. Loop Excitation
Description: In loop excitation, a small conducting loop is placed inside the waveguide.
Working
●​ The loop interacts with the magnetic field of the wave.
●​ It acts as a magnetic dipole antenna.
Characteristics
●​ Couples with a magnetic field.
●​ Loop plane is perpendicular to the magnetic field lines.
Applications
●​ Often used for magnetic field coupling in microwave circuits.
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3. Aperture (Slot) Excitation
Description-In this method, power is coupled through a small slot or aperture between a transmission line and the
waveguide.
Working
●​ The slot allows electromagnetic energy to pass through.
●​ Energy couples into the waveguide fields.
Advantages
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●​ Provides good isolation between circuits.
●​ Used in directional couplers and antenna feeds.
4. Position of Excitation
For efficient excitation:
●​ The coupling element should be placed where field intensity is maximum.
●​ For TE₁₀ mode, the electric field is maximum at the center of the broad wall.
Improper placement may lead to:
●​ Mode distortion
●​ Power loss
●​ Signal reflection
5. Conditions for Proper Excitation
1.​ Source frequency must be greater than the cutoff frequency.
2.​ Proper impedance matching should be maintained.
3.​ Coupling element should be placed at maximum field location.
4.​ Only the desired propagation mode should be excited.
6. Applications of Waveguide Excitation
Waveguide excitation is used in:
●​ Microwave communication systems
●​ Radar systems
●​ Satellite communication
●​ Microwave antennas
●​ Microwave ovens
●​ RF and microwave measurement equipment
7. Summary
●​ Excitation of a waveguide is the process of coupling microwave power into the waveguide.
●​ It is achieved by using a probe, a loop, or an aperture coupling.
●​ Proper excitation ensures efficient power transfer and correct mode propagation.
●​ In rectangular waveguides, excitation is usually designed to produce the dominant TE₁₀ mode.
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