# Strange approach to the line-fed slot antenna electromagnetic problem

• Unconscious
In summary, the conversation discusses a formula that can be used to find the electromagnetic field at any point in a given volume, with boundaries defined by surfaces. The formula includes various parameters such as charge and current densities, a Green function, and an angular frequency. It is noted that the formula can be simplified for different types of antennas, such as dipole and horn antennas. The conversation then moves on to discuss an example of a slot antenna and its corresponding formula, questioning whether the formula is accurate or not. The conversation concludes by stating that the radiation from a slot is caused by the acceleration of charges in the metal, and the ground plane plays a crucial role in this process.
Unconscious
There is a beautiful demonstration, available in the text Robert S. Elliot, Antenna theory and Design, Wiley-IEEE Press, page 17 (Stratton-Chu solution), which shows how the electromagnetic field at each point ## \mathbf { r} ## of a volume ## V ##, with boundary ## S_1, ..., S_N ##:

can be found from the following integral:

$$\mathbf{E}(\mathbf{r})=\frac{1}{4\pi}\int_V \left( \frac{\rho}{\epsilon_0}\nabla_S\psi-j\omega\psi\frac{\mathbf{J}}{\mu_0^{-1}}\right)\mathrm{d}V+\frac{1}{4\pi}\int_{S_1,...,S_N}\left[ \left(\mathbf{1}_n\cdot\mathbf{E}\right)\nabla_S\psi+\left(\mathbf{1}_n\times\mathbf{E}\right)\times\nabla_S\psi -j\omega\psi\left(\mathbf{1}_n\times\mathbf{B}\right)\right]\mathrm{d}S$$

where:

1. the point ## P ## shown in the figure corresponds to what I have called ## \mathbf {r} ## in the formula, while the surface ## \Sigma ## is a comfortable surface that only serves to demonstrate the validity of the above formula, in fact at the end it no longer appears in it;
2. in the formula ##\rho=\rho(\mathbf{r}')## is the charge density in ##V## (all, both impressed and induced);
3. ##\mathbf{J}=\mathbf{J}(\mathbf{r}')## is the current density in ##V## (both impressed and induced);
4. ##\psi=\psi(\mathbf{r},\mathbf{r}')=\frac{e^{-j k |\mathbf{r}-\mathbf{r}'|}}{|\mathbf{r}-\mathbf{r}'|}## is the Green function, the only one which is both a function of ##\mathbf{r}'## and ##\mathbf{r}##.
5. the integration is done with respect to ##\mathbf{r}'## and also ##\nabla_S## operates on ##\mathbf{r'}##;
6. ##\mathbf{1}_n=\mathbf{1}_n(\mathbf{r}')## is the unit normal vector, in each point, to the surfaces ##S_1,...,S_N##;
7. ##\omega## it is the angular frequency of the field, which is supposed to be fixed.

This formula is very useful because, for antenna problems, it allows to distinguish two types of approaches to the solution very well:

1. there are antennas for which the distribution of the currents on the metal that composes them can be assumed to be known with a good degree of approximation (eg: dipole), therefore for such problems the volume ##V## will be chosen as the entire space ## \mathbb {R} ^ 3 ##, without any exclusion surface, so in those cases the formula is reduced to:
##\mathbf{E}(\mathbf{r})=\frac{1}{4\pi}\int_V \left( \frac{\rho}{\epsilon_0}\nabla_S\psi-j\omega\psi\frac{\mathbf{J}}{\mu_0^{-1}}\right)\mathrm{d}V##

2. for other types of antennas, for which instead the distribution of the field is known with a good degree of approximation (eg: horn), one can choose a volume ## V ## delimited by a surface ## S_1 ## (on which the field is known with good approximation) such that inside ## V ## there will be neither currents nor charges (they are all inside ## S_1 ##). The formula in this case becomes:
##\mathbf{E}(\mathbf{r})=\frac{1}{4\pi}\int_{S_1}\left[ \left(\mathbf{1}_n\cdot\mathbf{E}\right)\nabla_S\psi+\left(\mathbf{1}_n\times\mathbf{E}\right)\times\nabla_S\psi -j\omega\psi\left(\mathbf{1}_n\times\mathbf{B}\right)\right]\mathrm{d}S##

For a dipole antenna, using the continuity equation for the current ## \nabla \cdot \mathbf {J} = - j \omega \rho ##, you will have:
##\mathbf{E}(\mathbf{r})=\frac{1}{4\pi}\int_V \left( -\frac{\nabla_S\cdot\mathbf{J}}{j\omega\epsilon_0}\nabla_S\psi-j\omega\psi\frac{\mathbf{J}}{\mu_0^{-1}}\right)\mathrm{d}V##
where ## \mathbf{J} ## is all the current present in the whole space (## V = \mathbb {R} ^ 3 ##), that is the surface currents distributed on the two plates of the dipole.Let us now take another example, which falls into the second category of antennas: a slot in an infinite ground plane.
The book cited above, pag. 86, analyze this situation:

taking the volume ## V ## as the one bounded by the surface ## S_1 ##: infinite hemisphere described by the plane ## x = 0 ## and all the space in front of the slot (## x> 0 ##). In this case, I would say that the field cannot be determined only with the surface integral ## S_1 ##, since that of volume does not give a null contribution (there are all the currents that are established on the ground plane). The book instead states exactly the opposite, assumes that the field on the slot is done in a certain way and proceeds to calculate only the surface integral and only on the surface of the slot.

Isn't that a mistake? Or, in the case of an implicit approximation, is it not a too strong approximation? It is like saying that we are leaving out fundamental contributions that would lead to having a field different from the one expected.

Sorry I can't do the maths, but I do know that the radiation from a slot is caused by the acceleration of charges in the metal. Wide slots and narrow slots radiate just the same. So the ground plane is the antenna.

Yes, and it is precisely for this reason that what the book says seems too strange to me. Not including the integral of the currents on ground plane means forgetting their own contribution, which is fundamental, as you also said.

## 1. What is a line-fed slot antenna?

A line-fed slot antenna is a type of antenna that is used to transmit and receive electromagnetic waves. It consists of a narrow slot cut into a conducting surface, which is fed by a transmission line. This type of antenna is commonly used in wireless communication systems.

## 2. What is the "strange approach" to the line-fed slot antenna electromagnetic problem?

The "strange approach" refers to a mathematical method used to solve the electromagnetic problem of a line-fed slot antenna. This approach involves using a complex variable transformation to convert the problem into a simpler form, which can then be solved using standard mathematical techniques.

## 3. Why is the "strange approach" used for this problem?

The "strange approach" is used because it allows for a more efficient and accurate solution to the electromagnetic problem of a line-fed slot antenna. It takes into account the effects of the slot dimensions, the feeding line, and the surrounding environment, which can all have an impact on the antenna's performance.

## 4. What are the advantages of using a line-fed slot antenna?

One of the main advantages of using a line-fed slot antenna is its compact size. It can be easily integrated into electronic devices and can provide high gain and directivity. It also has a wide bandwidth, making it suitable for a variety of applications such as wireless communication, radar, and satellite systems.

## 5. What are the limitations of a line-fed slot antenna?

Despite its advantages, a line-fed slot antenna also has some limitations. It is highly sensitive to changes in its surrounding environment, which can affect its performance. It also has a relatively narrow beamwidth, which means it may not be able to cover a wide area. Additionally, the design and fabrication of a line-fed slot antenna can be complex and require precise calculations.

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