Radiation Theory and Fundamental Theorems

Introduction

An antenna is an electromagnetic transition structure: it converts guided or circuit energy into radiated electromagnetic waves during transmission, and converts incident waves into terminal voltage during reception. Radiation theory explains how time-varying currents and charges produce fields, how power is transported away from a source, and how antenna performance is quantified. Fundamental theorems, especially the reciprocity theorem, reaction theorem, equivalence principle, and uniqueness theorem, provide rigorous tools for analysing and designing antennas.

At postgraduate level, the subject is important for deriving radiation integrals, interpreting near- and far-field behaviour, determining radiation resistance and directivity, and understanding why transmitting and receiving antennas exhibit reciprocal properties.

Learning Objectives

After completing this chapter, the reader should be able to:

1. Explain the physical origin of electromagnetic radiation.
2. Derive the vector and scalar potentials for time-harmonic sources.
3. Distinguish reactive near fields, radiating near fields, and far fields.
4. Derive the far-zone fields of an arbitrary current distribution.
5. Calculate radiated power, radiation intensity, radiation resistance, directivity, and gain.
6. State and apply the reciprocity, reaction, equivalence, and uniqueness theorems.
7. Evaluate assumptions, limitations, and practical consequences of radiation models.

1. Physical Basis of Electromagnetic Radiation

A static charge produces an electrostatic field. A steady current produces a magnetostatic field. Radiation requires time variation. When charges accelerate, their electromagnetic disturbance cannot adjust instantaneously throughout space; the resulting field disturbance propagates at the speed of light.

For a linear, homogeneous, isotropic, lossless medium, Maxwell’s equations are

Here E and H are phasor electric and magnetic fields, J is electric current density in A/m², ρ is charge density in C/m³, ε is permittivity in F/m, μ is permeability in H/m, and ω is angular frequency in rad/s. The assumed time dependence is e^jωt. With the opposite convention, signs involving j change consistently.

The wave number and intrinsic impedance are

where λ is wavelength in metres and η is impedance in ohms. In free space, η₀ ≈ 377 Ω and the wave velocity is c ≈ 3 × 10⁸ m/s.

Radiation is associated with field terms that transport net power to arbitrarily large distance. For a localized source, the electric and magnetic fields contain terms proportional to 1/r³, 1/r², and 1/r. The 1/r³ terms are predominantly reactive, the 1/r² terms represent induction effects, and the 1/r terms constitute the far or radiation field.

2. Potentials and the Retarded-Field Concept

The magnetic flux-density condition ∇ · B = 0 permits introduction of a vector potential A such that

The electric field can then be written as

where V is the scalar potential. In the Lorenz gauge,

and the potentials satisfy wave equations. For time-harmonic sources in an unbounded homogeneous medium, their solutions are expressed through the Green function

where R is the distance from a source element to the observation point. Thus, for a current distribution J(r′),

with R = |r − r′|. The phase factor shows that the field depends on the source at an earlier, or retarded, time. This is the mathematical basis of radiation from distributed sources.

For a localized source observed in the far zone, R can be approximated by

and

Consequently,

The integral is the spatial Fourier transform of the current distribution evaluated at the radiation wave vector k r̂. This observation is central to aperture theory, array theory, and electromagnetic scattering.

3. Near Field and Far Field

The boundary between near and far field is not a single universal distance; it depends on wavelength and the largest physical dimension D of the antenna. Common engineering regions are as follows.

These expressions are engineering criteria rather than exact physical discontinuities. D is the maximum antenna dimension. In the far field, E and H are orthogonal to each other and to the propagation direction, and

The time-average Poynting vector is

in W/m², where * denotes complex conjugation. For a locally plane wave,

The total radiated power is obtained by integrating outward flux over a closed surface:

For a sphere in the far field, dS = r² sinθ dθ dφ r̂.

4. Far-Zone Radiation from an Arbitrary Current

The far-zone electric field produced by a localized electric current distribution is

where

The corresponding magnetic field is

The double cross product removes the radial component of the source transform. Therefore, only the transverse component of the current contributes to far-zone radiation in a given direction.

For an electrically small current element of length dl carrying uniform current I, the radiation fields are

The element is assumed to be much shorter than λ, and its current is treated as uniform. Its radiation pattern is proportional to sinθ, with zero radiation along the element axis and maximum radiation perpendicular to it.

The average radiated power of a current element is

for RMS or peak-current conventions? This expression assumes I is the peak phasor current when the time-average Poynting factor 1/2 is used. If I is RMS current, the result is twice this value. Explicitly, with peak current I₀,

The radiation resistance is defined by P_rad = I_rms² R_r, so for a short dipole with uniform current,

when dl is the physical current-element length. A short centre-fed dipole with triangular current distribution has a different effective length and commonly yields approximately

where l is its total physical length and l ≪ λ. These formulas must not be interchanged: current distribution and the definition of length determine the coefficient.

5. Radiation Intensity, Beam Solid Angle, Directivity and Gain

Radiation intensity is the power radiated per unit solid angle:

Its unit is W/sr. Total radiated power is

The maximum radiation intensity is U_max. Directivity is

Directivity is dimensionless and measures concentration of radiation relative to an isotropic radiator having the same total radiated power.

The beam solid angle is

and therefore

This relation is exact when Ω_A is defined from the complete radiation pattern. Approximate beamwidth formulas are useful for engineering estimates but should not be treated as exact identities.

Antenna efficiency is

where P_in is accepted input power. If mismatch efficiency is included separately, total efficiency may be written

where η_rad accounts for conductor and dielectric losses, η_m for impedance mismatch, and η_c for other losses such as polarization mismatch when the system definition requires it. Gain is

Gain and directivity are often quoted in dBi using 10 log₁₀ of the dimensionless value relative to an isotropic radiator.

For an infinitesimal dipole, U(θ) is proportional to sin²θ. Since the angular average of sin²θ is 2/3, the directivity is 3/2, or approximately 1.76 dBi.

6. Fundamental Theorems

6.1 Uniqueness Theorem

For a specified frequency in a source-free region, a solution to Maxwell’s equations is unique if appropriate boundary conditions are specified on the boundary. For a perfect electric conductor, tangential E is zero; for a perfect magnetic conductor, tangential H is zero. Radiation problems also require an outward-radiation condition, commonly represented by the Sommerfeld condition.

The theorem is practically important because it validates replacement methods. If two field solutions satisfy the same differential equations, sources, boundary conditions, and radiation condition, they are the same solution. Numerical methods rely on this principle when enforcing boundary conditions on conductors and material interfaces.

6.2 Lorentz Reciprocity Theorem

Consider two electromagnetic states, labelled 1 and 2, in the same linear reciprocal medium. Their impressed electric and magnetic current densities are J₁, M₁ and J₂, M₂. Lorentz reciprocity states

For electric sources only and a volume bounded by S,

The theorem requires linear media and reciprocal constitutive relations, such as scalar ε and μ or tensors satisfying appropriate symmetry. It may fail in magnetically biased ferrites, moving media, nonlinear media, or active nonreciprocal components.

6.3 Antenna Reciprocity

A major consequence is that the transmitting and receiving patterns of an antenna are identical when polarization, coordinate direction, and normalization are properly specified. Reciprocity does not mean that transmitted and received powers are numerically identical under arbitrary conditions; it states equality of the relevant transfer characteristics of the reciprocal system.

For two antennas in free space, the Friis transmission equation is

where P_t and P_r are accepted transmitted and received powers, G_t and G_r are gains in the line-of-sight direction, R is separation, and polarization and impedance matching are assumed ideal. A polarization mismatch factor |ρ|² and mismatch factors may be added in practical systems.

6.4 Reaction Theorem

The reaction of source 1 on field 2 is defined, under a chosen convention, by

The reaction theorem states that in a reciprocal environment

Reaction is a bilinear quantity, not generally the same as complex power, which uses a complex conjugate. Confusing reaction with power is a common analytical error.

6.5 Equivalence Principle

The equivalence principle replaces sources inside a closed surface by equivalent surface currents that reproduce the external fields. For a surface S with outward normal n̂,

where J_s is equivalent electric surface current in A/m and M_s is equivalent magnetic surface current in V/m. These currents reproduce the fields outside S when the fields on the surface are known. The choice of Love’s equivalence principle sets the fields in the complementary region to zero and is widely used in aperture and reflector analysis.

The theorem is conceptually powerful but requires correct field continuity, surface orientation, and sign convention. Equivalent magnetic currents are mathematical constructs; they do not imply that magnetic monopoles exist.

7. Worked Example: Radiation Resistance of a Short Dipole

An electrically short dipole has total length l = 0.05λ and approximately triangular current distribution. Estimate its radiation resistance.

For a short centre-fed dipole,

Substitution gives

Thus, R_r ≈ 0.49 Ω. This is only radiation resistance; conductor loss, dielectric loss, and mismatch loss are not included. The low value explains why short antennas often have poor efficiency unless their loss resistance is made very small.

8. Applied Case: Reciprocal Link at 2.4 GHz

Background

Consider a point-to-point wireless sensor link operating at 2.4 GHz. Each antenna has gain 5 dBi, the separation is 100 m, and the antennas are aligned with matching polarization. Assume free-space propagation and matched terminals.

Analysis

The wavelength is

The linear gain corresponding to 5 dBi is

Using Friis’ equation,

The factor in parentheses is approximately 9.95 × 10⁻⁵; its square is approximately 9.90 × 10⁻⁹. Hence

The corresponding path loss is approximately 70.0 dB. Reciprocity means that interchanging the transmitting and receiving antennas does not change this link ratio, provided the same frequency, orientation, polarization, matching, and propagation conditions are maintained.

Practical Interpretation

The result is an ideal free-space estimate. Cable loss, radome loss, polarization misalignment, multipath, terrain obstruction, and receiver noise can significantly reduce link performance. At 2.4 GHz, objects and human bodies can produce substantial shadowing because their dimensions are comparable with several wavelengths.

Learning Points

1. Gain must be converted from dBi to a linear ratio before using Friis’ equation.
2. Reciprocity concerns transfer behaviour, not equality of transmitter and receiver hardware power ratings.
3. A free-space model is a baseline, not a complete propagation prediction.
4. Antenna polarization and impedance matching are system-level design variables.

9. Limitations, Design Practice and Contemporary Relevance

Radiation formulas often assume an isolated antenna in an infinite homogeneous medium. Real antennas are mounted above ground planes, on vehicles, in arrays, near dielectric covers, or beside other radiators. These environments alter current distribution, impedance, efficiency, and pattern.

A robust design workflow should therefore:

• define the phasor convention and coordinate system;
• identify whether peak or RMS quantities are being used;
• separate accepted power, radiated power, and delivered source power;
• check electrically small or electrically large assumptions;
• verify far-field distance before using pattern data;
• include conductor, dielectric, mismatch, and polarization losses;
• validate analytical results using full-wave simulation or measurement;
• report uncertainty, calibration, chamber reflections, and cable effects.

Computational electromagnetics uses the same theoretical foundations. The method of moments applies integral equations derived from equivalence and boundary conditions; finite-element and finite-difference methods solve field equations in discretized domains; physical-optics methods approximate induced currents on electrically large surfaces. Machine-learning models can assist surrogate modelling and optimization, but training data, physical constraints, extrapolation error, and reproducibility must be critically assessed. AI does not remove the need to satisfy Maxwell’s equations or conservation of energy.

Regulatory practice also matters. Antenna systems must meet spectrum masks, effective isotropic radiated power limits, human-exposure requirements, electromagnetic compatibility rules, and site-specific licensing conditions. A theoretically efficient antenna is not automatically legally deployable.

Chapter Summary

Radiation is produced by time-varying charge and current distributions and is represented mathematically by retarded potentials and Green-function solutions. The far field varies as 1/r, is transverse, and carries power through the Poynting vector. Radiation intensity, directivity, gain, efficiency, and radiation resistance provide complementary measures of antenna performance.

The uniqueness theorem establishes when a field solution is determined. Lorentz reciprocity explains the equivalence of transmitting and receiving transfer characteristics in reciprocal media. The reaction theorem provides a bilinear source-field relation, while the equivalence principle replaces enclosed sources by electric and magnetic surface currents. Correct use of these results requires attention to constitutive assumptions, boundary conditions, phasor conventions, polarization, losses, and field-region validity.

Key Terms

Radiation: Propagation of electromagnetic energy away from a time-varying source.

Retarded potential: Potential determined by the source at an earlier time.

Phasor: Complex representation of a sinusoidally varying field.

Green function: Fundamental response used to construct fields from distributed sources.

Reactive near field: Region dominated by stored, non-propagating electric and magnetic energy.

Radiating near field: Region where radiation exists but pattern varies with distance.

Far field: Region with transverse fields, 1/r amplitude decay, and stable angular pattern.

Poynting vector: Vector describing electromagnetic power-flow density.

Radiation intensity: Radiated power per unit solid angle.

Radiation resistance: Equivalent resistance representing power radiated by an antenna.

Directivity: Ratio of maximum radiation intensity to average radiation intensity.

Gain: Directivity multiplied by antenna efficiency.

Beam solid angle: Solid angle associated with the complete normalized radiation pattern.

Reciprocity: Equality of transfer characteristics under source and observation interchange in reciprocal media.

Reaction: Bilinear interaction between a source and an electromagnetic field.

Equivalence principle: Replacement of enclosed sources by equivalent surface currents.

Uniqueness theorem: Principle that appropriate equations and boundary conditions determine one field solution.

Intrinsic impedance: Ratio of electric to magnetic field magnitude for a plane wave.

Polarization: Orientation and time variation of the electric-field vector.

Friis equation: Relation predicting received power in an ideal free-space link.

Review Questions

Short Answer Questions

1. Why is time variation necessary for electromagnetic radiation?
2. State the relationship between far-zone electric and magnetic fields.
3. Define radiation intensity and give its unit.
4. What physical condition is represented by radiation resistance?
5. State one important assumption required for antenna reciprocity.

Long Answer Questions

1. Derive the far-zone electric field produced by an arbitrary localized electric current distribution.
2. Explain reactive near field, radiating near field, and far field, including the role of antenna dimension and wavelength.
3. State and explain the Lorentz reciprocity theorem and discuss situations in which it may fail.
4. Derive the expressions for directivity, beam solid angle, and gain, and explain their engineering significance.
5. Explain the equivalence principle and derive the equivalent electric and magnetic surface-current expressions.

Analytical / Application Questions

1. A short centre-fed dipole has l/λ = 0.02 and triangular current distribution. Calculate its approximate radiation resistance and explain why loss resistance is important.
2. Two reciprocal antennas operate at 900 MHz, each with gain 8 dBi, and are separated by 2 km in free space. Derive the received-to-transmitted power ratio using Friis’ equation, clearly stating all assumptions.
3. An antenna has maximum radiation intensity 40 W/sr and total radiated power 500 W. Calculate its directivity and beam solid angle, then discuss what the values imply about pattern concentration.

Multiple Choice Questions with Answers

Question 1

Which field dependence represents the radiation field of a localized antenna in the far zone?

A. 1/r³

B. 1/r²

C. 1/r

D. Independent of r

Correct Answer: C. 1/r

Question 2

For a plane wave in a homogeneous medium, the ratio of electric-field magnitude to magnetic-field magnitude is

A. ε/μ

B. √(μ/ε)

C. √(ε/μ)

D. με

Correct Answer: B. √(μ/ε)

Question 3

The radiation intensity U is related to radial power density P by

A. U = P/r²

B. U = rP

C. U = r²P

D. U = P/r

Correct Answer: C. U = r²P

Question 4

The directivity of an antenna is defined as

A. P_in/P_rad

B. 4πU_max/P_rad

C. P_rad/(4πU_max)

D. η_ant/U_max

Correct Answer: B. 4πU_max/P_rad

Question 5

For a short centre-fed dipole with triangular current distribution, the approximate radiation resistance is

A. 80π²(l/λ)² Ω

B. 20π²(l/λ)² Ω

C. 20π²(λ/l)² Ω

D. 120π(l/λ) Ω

Correct Answer: B. 20π²(l/λ)² Ω

Question 6

Which theorem establishes that a field solution is uniquely determined when the governing equations and appropriate boundary conditions are specified?

A. Uniqueness theorem

B. Reaction theorem

C. Poynting theorem

D. Sampling theorem

Correct Answer: A. Uniqueness theorem

Question 7

In the equivalence principle, the equivalent electric surface current is

A. J_s = n̂ · H

B. J_s = n̂ × H

C. J_s = −n̂ × E

D. J_s = E × H

Correct Answer: B. J_s = n̂ × H

Question 8

A reciprocal antenna has the same transmitting and receiving

A. battery voltage under all conditions

B. physical dimensions only

C. pattern and transfer characteristics under corresponding conditions

D. radiated power regardless of input power

Correct Answer: C. pattern and transfer characteristics under corresponding conditions

Question 9

At 2.4 GHz, using c = 3 × 10⁸ m/s, the wavelength is approximately

A. 1.25 m

B. 0.125 m

C. 0.0125 m

D. 12.5 m

Correct Answer: B. 0.125 m

Question 10

The Friis transmission equation assumes, among other conditions,

A. strongly reactive near-field coupling

B. free-space far-field propagation

C. nonlinear antenna materials

D. zero antenna gain

Correct Answer: B. free-space far-field propagation