Far-field retarded potentials

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Key focus: Far-field region is dominated by radiating terms of antenna fields, hence, knowing the far field retarded potentials is of interest.

Introduction

The fundamental premise of understanding antenna radiation is to understand how a radiation source influences the propagation of travelling electromagnetic waves. Propagation of travelling waves is best described by electric and magnetic potentials along the propagation path.

The concept of retarded potentials was introduced in this post.

The electromagnetic field travels at certain velocity and hence the potentials at the observation point (due to the changing charge at source) are experienced after a certain time delay. Such potentials are called retarded potentials.

The retarded potentials at a radial distance r from an antenna source fed with a single frequency sinusoidal waves, is shown to be

\[\begin{aligned} \Phi(r) &= \frac{1}{4 \pi \epsilon} \int_V \frac{\rho(z’)e^{-j k R }}{R} d^3 z’ \\ A(r) &= \frac{\mu}{4 \pi} \int_V \frac{J(z’)e^{-j k R }}{R} d^3 z’ \end{aligned} \quad \quad (1)\]

where, the quantity k = ω/c = 2 π/λ is called the free-space wavenumber. Also, ρ is the charge density, J is the current density, Φ is the electric potential and A is the magnetic potential that are functions of both radial distance.

Far-field region

Figure 1 illustrates the two scenarios: (a) the receiver is ‘nearer’ to the antenna source (b) the receiver is ‘far away’ from the antenna source.

Figure 1: Radiation fields when antenna and receiver are (a) near and (b) far away
Figure 1: Radiation fields when antenna and receiver are (a) near and (b) far away

Since the far-field region is dominated by radiating terms of the antenna fields, we are interested in knowing the retarded potentials in the far-field region. The far field region is shown to be

\[\frac{2 l^2}{ \lambda} < r < \infty \quad \quad (2)\]

where l is the length of the antenna element and λ is the wavelength of the signal from the antenna.

In the process of deriving the boundary between far-field and near-field, we used the following first order approximation for the radial distance R.

\[R = r – z’ \; cos \theta= r – \hat{r} \cdot z’ \quad \quad (3)\]

Far field retarded potential

Substituting this approximation in the numerator of equation (1) and replacing R by r in the denominator

\[\begin{aligned} \Phi(r) &= \frac{1}{4 \pi \epsilon} \int_V \frac{\rho(z’)e^{-j k \left( r – \hat{r} \cdot z’ \right) }}{r} d^3 z’ \\ A(r) &= \frac{\mu}{4 \pi} \int_V \frac{J(z’)e^{-j k \left( r – \hat{r} \cdot z’\right) }}{r} d^3 z’ \end{aligned} \quad \quad (4)\]

The equation can be written as

\[\begin{aligned} \Phi(r) &= \frac{e^{-jkr}}{4 \pi \epsilon} \int_V \frac{\rho(z’)e^{ j k \hat{r} \cdot z’ }}{r} d^3 z’ \\ A(r) &= \frac{\mu e^{-jkr}}{4 \pi} \int_V \frac{J(z’)e^{j k \hat{r} \cdot z’ }}{r} d^3 z’ \end{aligned} \quad \quad (5)\]
Spherical coordinate system over Cartesian coordinate system
Figure 2: Spherical coordinate system on a cartesian coordinate system

Antenna radiation patterns are generally visualized in a spherical coordinate system (Figure (2)). In a coordinate system, each unit vector can be expressed as the cross product of other two unit vectors. Hence,

\[\begin{aligned}\hat{r} &= \hat{\theta} \times \hat{\phi} \\ \hat{\theta} &= \hat{\phi} \times \hat{r} \\ \hat{\phi} &= \hat{r} \times \hat{\theta} \end{aligned} \quad \quad (6) \]

Therefore, the far-field retarded potentials in equation (5) can be written in terms of polar angle (θ) and azimuthal angle (ɸ)

\[\boxed{\begin{aligned} \Phi(r) &= \frac{e^{-jkr}}{4 \pi \epsilon r} \int_V \rho(z’)e^{ j k \left( \hat{\theta} \times \hat{\phi} \right) \cdot z’ } d^3 z’ \\ A(r) &= \frac{\mu e^{-jkr}}{4 \pi r} \int_V J(z’)e^{j k \left( \hat{\theta} \times \hat{\phi} \right) \cdot z’ } d^3 z’ \end{aligned}} \quad \quad (7)\]

We note that the term inside the integral is dependent on polar angle (θ) and azimuthal angle (ɸ). It determines the directional properties of the radiation. The term outside the integral is dependent on radial distance r. These terms can be expressed separately

\[\begin{aligned} \Phi(r) &= \frac{e^{-jkr}}{4 \pi \epsilon r} \mathbf{Q} \left(\theta, \phi \right) \\ A(r) &= \frac{\mu e^{-jkr}}{4 \pi r} \mathbf{F} \left(\theta, \phi \right) \end{aligned} \quad \quad (8)\]

The terms that determine the directional properties: Q(θ,ɸ) & F(θ,ɸ) are called charge form-factor and radiation vector respectively. The charge form-factor Q(θ,ɸ) and the radiation vector F(θ,ɸ) are three dimensional spatial Fourier transforms of charge density ρ(z’) and current density J(z) respectively.

\[\boxed{\begin{aligned} \mathbf{Q} \left(\theta, \phi \right) & = \int_V \rho(z’)e^{ j k \left( \hat{\theta} \times \hat{\phi} \right) \cdot z’ }d^3 z’ \quad \quad \text{(charge form-factor)}\\ \mathbf{F} \left(\theta, \phi \right) &=\int_V J(z’)e^{j k \left( \hat{\theta} \times \hat{\phi} \right) \cdot z’ } d^3 z’ \quad \quad \text{(radiation vector)} \end{aligned}}\quad \quad (9) \]

The charge-form factor and radiation vector can also be written in terms of direction of the unit vector of radial distance.

\[\boxed{\begin{aligned} \mathbf{Q} \left(\mathbf{k} \right) & = \int_V \rho(z’)e^{ j \mathbf{k}\cdot z’ }d^3 z’ \quad \quad \text{(charge form-factor)} \\ \mathbf{F} \left(\mathbf{k}\right) &=\int_V J(z’)e^{j \mathbf{k}\cdot z’ } d^3 z’ \quad \quad \text{(radiation vector)} \end{aligned}} \quad \quad \boxed{\mathbf{k} = k\hat{r}} \quad \quad (10) \]

Recap

We are in the process of building antenna models. In that journey, we started with the fundamental Maxwell’s equations in electromagnetism, then looked at retarded potentials that are solutions for Maxwell’s equations. Propagation of travelling waves is best described by retarded potentials along the propagation path. Then, the boundary between near-field and far-field regions was defined. Since most of the antenna radiation analysis are focused in the far-field regions, we looked at retarded potentials in the far-field region.

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