Tag: Fraunhofer region

Far-field retarded potentials

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

\[\begin{aligned} \Phi(r) &= \frac{e^{-jkr}}{4 \pi \epsilon} \int_V \frac{\rho(z')e^{ j k \hat{r} \cdot z' }}{r} d^3 z' \\ A(r) &= \frac{\mu e^{-jkr}}{4 \pi} \int_V \frac{J(z')e^{j k \hat{r} \cdot z' }}{r} d^3 z' \end{aligned} \quad \quad (5)\]
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.

Rate this article: Note: There is a rating embedded within this post, please visit this post to rate it.

Books by the author


Wireless Communication Systems in Matlab
Second Edition(PDF)
Note: There is a rating embedded within this post, please visit this post to rate it.
Checkout Added to cart

Digital Modulations using Python
(PDF ebook)
Note: There is a rating embedded within this post, please visit this post to rate it.
Checkout Added to cart

Digital Modulations using Matlab
(PDF ebook)
Note: There is a rating embedded within this post, please visit this post to rate it.
Checkout Added to cart
Hand-picked Best books on Communication Engineering
Best books on Signal Processing

Near-field – far-field : Where is the boundary ?

Antennas are radiation sources of finite physical dimension. To a distant observer, the radiation waves from the antenna source appears more like a spherical wave and the antenna appears to be a point source regardless of its true shape. The terms far-field and near-field are associated with such observations/antenna measurement. The terms imply that there must exist a boundary between the near field and far field.

Essentially, the near field and far field are regions around an antenna source. Though the boundary between these two regions are not fixed in space, the antenna measurements made in these regions differ significantly. One method of establishing the boundary between the near-field and far-field regions is to look at the acceptable level of phase error in the antenna measurements.

An antenna designer is interested in studying how the phase of the radiation waves launched from the antenna source is affected by the distance between the antenna source and the receiver (observation point). As the distance between the antenna and the receiver increases, there exists a phase difference between the measurements taken along the two lines shown. This phase difference contribute to antenna measurement errors, it also affects retarded potentials and radiation fields.

Near-field and far-field approximations

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. The antenna is of standard dimension of length l. The figures show two rays – one from the origin to the observation point P (on the yz plane) and the other from the mid-point of distance z’=l/2 from the origin towards the observation point P.

In Figure(2)(b), the observation point P is at a distance that is very far from the antenna source element. The term ‘far’ implies that the distance r is much greater than the spatial extent of the current distribution of the antenna element, that is, r >> z’. Also, the two rays appear parallel to each other.

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

The essence of the following exercise is to determine the boundary between the ‘near’ and the ‘far’ field regions of the antenna. Once that boundary is established, we can determine whether far field approximation can be used on the antenna measurements or for the calculation of retarded potentials/fields produced by the antenna.

Let’s take a quick look at the retarded potentials derived for a single frequency wave emanating from the antenna source.

\[\begin{aligned} \Phi(r) &= \frac{1}{4 \pi \epsilon} \int_V \frac{\rho(z')e^{-j k R }}{R} d^3 z' \quad\quad (1) \\ A(r) &= \frac{\mu}{4 \pi} \int_V \frac{J(z')e^{-j k R }}{R} d^3 z' \quad\quad (2) \end{aligned}\]

We note that we cannot arbitrarily set R=r, because any small relative difference between R and r, will result in phase errors in the retarded potentials such that e-j k R ≠ e-j k r . Solving for the relationship between R and r is the crux of the radiation boundary problem.

From Figure (1)(a), applying law of cosines, the distance R can be written as

\[R = \sqrt{r^2 – 2 r z' cos \theta + z'^2} \quad \quad (3)\]

which can be expanded using the following Binomial series expansion,

\[ \begin{aligned}(x+y)^{n}&=\sum _{k=0}^{\infty }{n \choose k}x^{n-k}y^{k}\\&=x^{n}+nx^{n-1}y+{\frac {n(n-1)}{2!}}x^{n-2}y^{2}+{\frac {n(n-1)(n-2)}{3!}}x^{n-3}y^{3}+\cdots .\end{aligned}\]

Setting x = r2 and y= – 2 r z’ cos θ + z’ 2 , equation (3) can be expanded as

\[\begin{aligned} R & = r^{2(\frac{1}{2})}+\frac{1}{2}r^{2(\frac{1}{2}-1)} \left( -2 r z’ cos \theta + z’^2 \right )+\cdots .\\ & = r+ \frac{1}{2r} \left( -2 r z’ cos \theta + z’^2 \right ) – \frac{1}{8 r^3} \left(2 r z’ cos \theta \right )^2 + \cdots .\\ & = r – z’ cos \theta + \frac{z’^2}{2r} sin^2 \theta + \cdots . \end{aligned}\]

Neglecting the higher order terms,

\[\begin{aligned} R & \simeq r – z’ cos \theta + \frac{z’^2}{2r} sin^2 \theta \quad \quad (4) \end{aligned}\]

Truncation of equation (4) means we are dealing with the following maximum error in the antenna measurements:

\[\frac{z’^2}{2 r} sin^2 \theta = \frac{z’^2}{2 r} , \quad \quad \text{for } \theta=\frac{\pi}{2}\quad (5)\]

On the other hand, from Figure (1)(b), the distance \(R\) is given by

\[R = r – z’ cos \theta \quad \quad (6)\]

As r → ∞, equation (4) approaches exactly the parallel ray approximation given by equation (6). However, for finite values of r (due to the additional term z’ 2/2r sin2 θ and also the additional terms that were neglected) there exists an error between parallel ray approximation and the actual value of R computed using equation (4).

So the question is: What is the minimum distance over which the parallel ray approximation can be invoked ?

According to text books, for the maximum extent of the antenna (z’ = l/2), when the maximum phase difference is π/8, it produces acceptable errors in antenna measurements.

\[k \frac{z’^2}{2 r} \simeq \frac{\pi}{8} \quad \quad (7)\]

which gives

\[\boxed{r = \frac{2 l^2}{ \lambda}} \quad \quad (8)\]

In these equations, k = ω/c = 2 π/ λ is the free-space wavenumber.

Equation (8) defines the minimum distance (a.k.a the boundary between near and far field regions) over which the parallel ray approximation can be invoked. This minimum distance is called far-field distance – the boundary beyond which the far-field region starts. The quantity l is the maximum dimension of the antenna.

The far-field region, also known as Fraunhofer region, is dominated by radiating terms of the antenna fields. The far-field region is

\[\boxed{\frac{2 l^2}{ \lambda} < r < \infty }\quad \quad (9)\]
Figure 2: Far-field distance and far-field region

Rate this article: Note: There is a rating embedded within this post, please visit this post to rate it.

Books by the author


Wireless Communication Systems in Matlab
Second Edition(PDF)
Note: There is a rating embedded within this post, please visit this post to rate it.
Checkout Added to cart

Digital Modulations using Python
(PDF ebook)
Note: There is a rating embedded within this post, please visit this post to rate it.
Checkout Added to cart

Digital Modulations using Matlab
(PDF ebook)
Note: There is a rating embedded within this post, please visit this post to rate it.
Checkout Added to cart
Hand-picked Best books on Communication Engineering
Best books on Signal Processing