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# CHAPTER 7

## 7.1 INTRODUCTION

Containment of unintentional electromagnetic radiation is based on principles that have already been discussed, but they will now be differently applied. For example, a conducting plane in which a nearby current establishes its negative image can simply be viewed as a reflector of the radiation of that current. Based on that viewpoint then, a radiation that is fully reflected back toward its source from all directions will be fully contained. Previously it was seen that radiations can be effectively controlled by properly placing a reflector so that incident and reflected radiations cancel each other. In implementing containment, however, radiated fields are simply reflected back in the direction of their source. There is generally no attempt to cause any cancellation with the reflections.

Thus, the principles on which control and containment are based are the same and, with respect to the surrounding environment, the desired end results are the same. However, the methods of implementation are quite different from one another, and the implications of those differences are well worth examining.

## 7.2 FULLY CONTAINED RADIATING CURRENTS

The idea of containing circuit-current radiations is based primarily on the principle that electromagnetic fields do not exist in the interior of a perfect conductor. As discussed in Chapter 1, that makes the tangential electric field and the normal magnetic field both zero at the surface of a perfect conductor. Therefore, a radiating current placed inside a fully closed, perfectly conducting container can cause no tangential electric field, and no normal magnetic field, at the outer surface(s) of its container, or beyond.

Consider, for example, an element of current i(t) = |I| sin(ωt) centered along the z axis of the coordinate system. The radiated electric field in the θ direction, which is everywhere tangential to the surface of an imaginary sphere of radius d in which the current element is centered, would be

where

However, as shown in Fig. 7-1, if the current element were also centered in a smaller, perfectly conducting spherical container, then on the sphere of radius d the tangential electric field is Eθ(t) = 0.

On the other hand, the normal electric field and the tangential magnetic field at the outer surface of a perfectly conducting container of a radiating current will not necessarily be zero. Those field components are independent of the nonexistence of fields in the interior of a perfect conductor. As previously discussed in Chapter 1, those field components are determined by the charge and the current on the conductor's surfaces. Any charge or current induced on the inner surface of a perfectly conducting container will change the charge and possibly cause a current on the outer surface. In other words, if the total charge on a container is zero, the instantaneous charges on its inner and outer surfaces will always be equal and opposite, so that their sum is zero. However, the charge on either surface will generally change when radiation is incident on the other, and a nonzero surface charge will cause a nonzero electric field normal to the surface. The implications of this for a sinusoidal current element that is fully contained by a perfectly conducting sphere, in comparison to when it is not contained, are illustrated in Fig. 7-2.

Thus, it can be seen from Figs. 7-1 and 7-2 that a fully closed, perfectly conducting, spherical container would change the direction of the radiated electric field of a current element. Also, because there are no fields between the surfaces of a perfect conductor, in otherwise field-free space the time-varying charge on the outer surface would always be evenly distributed. Therefore, the magnitude of Er(t), the radially directed electric field in Fig. 7-2b, would be equal in all directions. In some directions Er(t) would be reduced compared to the original field Eθ(t), and in other directions it would be increased.

To reduce the radiation caused by surface charges induced on the outer surface of a perfectly conducting container of a radiating current, the surface can be connected to ground. However, while this will return the outer surface charge to zero whenever it varies, that cannot happen instantaneously. The surface charge will be held to zero much of the time, but it will still cause radiation when it must be returned to zero. What is more, to return the outer surface charge to zero, there will have to be a current from the surface to the point at which it is grounded. Thus, there will be time-varying charges and currents on the outer surface of the container, and currents in the conductor connecting it to ground, all of which will radiate. This is illustrated in Fig. 7-3 for a current element centered in a perfectly conducting sphere the surface of which is connected to ground.

Figure 7-1. The electric fields Eθ(t) from a sinusoidal current element (a) in free space, and (b) in a perfectly conducting, completely closed, spherical container, in otherwise free space.

Figure 7-2. A current element's radiated electric Held normal to the imaginary sphere of radius d is (a) zero when the element is in free space, and (b) nonzero when the element is in a closed, perfectly conducting, spherical container.

It should be clear, then, that full containment will generally not completely solve a radiation problem; it will alter the problem. Containment may bring about improvements in some cases, but in others it may worsen the problem. Thus, containment may sometimes be useful to maximize the reduction of unintentional radiations, but it is by no means a primary solution to the problem. Furthermore, in practice, the results expected from the nonspherical containment of the radiations of numerous current elements will be very difficult to predict.

Next considered is what to expect when containment can only be partially implemented.

Figure 7-3. The radiated electric field in the direction normal to a grounded perfectly conducting sphere containing a current element would not be zero, nor would the field of the ground wire.

## 7.3 PARTIALLY CONTAINED RADIATING CURRENTS

Complete containment of radiating currents—placement in a highly conductive container with no openings—is generally difficult, if not impossible, to implement in practice. And openings in conductive containers will often cause stronger radiations than there would be with no container at all. To see that openings in highly conductive containers are capable of radiating as effectively as current-carrying conductors, or antennas, consider the following.

### 7.3.1 The Slot Element

The concept of an isolated short length of current, the current element, was introduced in Chapters 1 and 2, and from that concept descriptions of the radiations of numerous real antennas were obtained. Similarly, a slot element—a short, narrow opening containing a transverse electric field, in an infinite, perfectly-conducting plane—yields descriptions of the antenna-like behavior of longer slots in an infinite conducting plane.

The slot element and current element and their associated geometries are illustrated in Fig. 7-4 for comparison. The radiated electric field of the current i(t) = |I| sin(2πft) from a current element of length Le, is given in Eq. 7-1. The radiated electric field of a slot element with the electric field E(t) = |E|sin(2πft) across the slot's width w is

where

In these expressions, Ls is the slot length, , the relative sizes of Ls and w are such that , and d >> Ls is the distance from the center of the slot to the observation point. The source of the current in a current element is not specified, nor is the source of the electric field in a slot element.

In comparing Eqs. 7-1 and 7-2, it is seen that radiation pattern factors of the slot element and the current element are identical except for the factors w|E|L and Z0|I|L. In Fig. 7-4 it is seen that the radiated electric field of a vertical slot element has the φ direction, whereas the radiated electric field of a vertical current element has the θ direction. Thus, at any common observation point the directions of their fields differ by 90°, but the relative magnitudes of their fields are the same. In other words, except in the conducting plane that contains it, the slot element has the same doughnut-shaped radiation pattern as the current element, as shown in Fig. 7-5. Thus, if w|E|L = Z0|I|L, radiations of these two elementary sources will be the same except for the 90° difference in the directions of their fields.

Figure 7-4. The concept of (a) a slot element in an infinite conducting plane compared to that of (b) a current element in free space.

Figure 7-5. Radiation characteristics in the right half-space of a slot element along the z axis when the yz plane is a good conductor.

The radiations of longer slots can be characterized with slot elements the same as the radiations of longer conductors were characterized with current elements. For example, the radiations of slots longer than a slot element can be described in the same way the radiations of current segments longer than current elements were described in Chapter 2. The results for a slot of length L on the z axis, obtained by replacing Z0|I|L with w|E|L in Eqs. 2-14 and 2-18, are

when the electric field propagates upward in the slot, and

when the electric field propagates downward in the slot. Here, of course, vp is the propagation velocity of the transverse electric field along the length of the slot.

The characteristics of radiating slots or unintentional slot antennas can now be examined.

### 7.3.2 Unintentional Slot Antennas

A slot in a conductor need not be intentionally excited for it to radiate. If an opening or a slot exists in any conductive enclosure that has surface currents on it, containment of the radiations causing those currents will seldom be achieved. Any container with a time-varying surface current on it, the direction of which is not parallel to a slot, will cause the slot to radiate much like an antenna. The fields induced in a slot by a current parallel to it are equal and opposite, as shown in Fig. 7-6, and cancel each other. However, a surface current in any other direction will have a component perpendicular to the slot, as shown in Fig. 7-7. And perpendicular currents will cause differing charge densities on opposite sides of a slot and electric fields that do not cancel.

Suppose the surface current density J(t) = i(t)/m2 (amperes/meter2) at some point along the edge of a slot is J(t) = |J| cos(2πft), then the surface charge density at that point, in coulombs/meter2, will be

Figure 7-6. The pattern of (a) a surface current parallel to a slot and (b) the equal and opposite electric fields caused by equal charge densities on opposite sides of the slot.

That surface charge density causes an electric field normal to the slot edge that is

where έ is the permittivity in the slot in farads/meter.

Referring to Fig. 7-8, it is seen that the electric field propagating to the right in the left half of the slot of length 2L causes the radiated electric field

a distance d + L/2cos φ d >> 2L from the center of the slot. The electric field propagating to the right in the right half of the slot causes the radiated electric field

Figure 7-7. The pattern of (a) a surface current that is perpendicular to the slot on which it is incident and (b) the electric fields caused by the resulting charge densities on the sides of the slot.

Figure 7-8. The radiation geometry and an expanded view of a slot and its electric fields when the slot is on the x axis and the conducting plane is the xz plane with a surface current in the z direction.

at the distance dL/2cos φ d. Thus, recalling that cos(ab) + cos(a + b) = 2 cos a cos b, the sum of those two fields at that observation point is seen to be

Similarly, the electric field propagating to the left in the left half of the slot causes the radiated electric field

at the same observation point. The electric field propagating to the left in the right half of the slot causes the radiated electric field

at that observation point. Thus, the sum of the fields that propagate to the left in the slot is

The total electric field radiated by the slot, therefore, is

where, using Eq. 7-3, with φ replacing θ because the slot is on the x axis,

The latter expression of Eq. 7-7 follows by applying the trigonometric identities sin(a ± b) = sin a cos b ± sin b cos a and cos(a ± b) = cos a cos b sin a sin b.

The expression arrived at for the radiation pattern Es(φ) in Eq. 7-7 is based on the assumption that the slot has length 2L << d. The equivalent expression for the radiation pattern of any slot of length L << d is

Assuming that , the radiation pattern for any slot of length L << d is

Radiation patterns obtained with Eq. 7-9 for slots of several different lengths Lλ in infinite conducting planes are shown in Fig. 7-9. The xy-plane radiation patterns are shown, but the pattern cross section is the same in any plane containing the x axis, except the xz plane, which is the conducting plane.

### 7.3.3 Partial Containment

A partial container of radiations is a highly conductive container with holes or openings in it that can in many cases be described as slots. Given the above descriptions of the radiations of slots in infinite planes it should be clear that any slot in a conducting surface is liable to be excited and radiate like an antenna, even though that surface is not infinite. Therefore, it should also be clear that partial containment of radiation may worsen the problem it is meant to solve.

Figure 7-9. The radiation patterns of slots centered along the x axis with the lengths shown assuming the xz plane is infinite and perfectly conducting and has the same z-directed surface current in each case.

## 7.4 SUMMARY AND CONCLUSIONS

Based on all of the above, it should be clear that containment is not the best way to reduce and control unintentional electromagnetic radiations. However, there are at least two more reasons that containment should be carefully considered before attempting to implement it.

The first of those reasons follows from the principle of reciprocity. That principle says that if electromagnetic radiations are reflected back toward their source they are very likely to interfere with the proper operation of that source. In other words, containment is highly likely to induce currents in the device contained that will corrupt its performance.

The second reason results from the difficulty in predicting the overall effects to expect from a particular container. Because of that difficulty, trial and error will generally be the only way to proceed. Each container will have to be implemented and its effectiveness measured. As a result, any attempt to contain radiations will generally be both time-consuming and costly.

The basic conclusion to be drawn from this chapter should be clear. Time and money will be better spent on efforts to control unwanted radiations rather than efforts to contain them.

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