There are a number of characteristics that describe an antenna as a device. Characteristics such as impedance and gain are common to any electrical device. On the other hand, a property such as radiation pattern is unique to the antenna. In this section we discuss patterns and impedance. Gain is discussed in the following section. We begin with a discussion of reciprocity.

H |

(30b) |

(31) |

(34a) (34b) |

Figure 3. Two-port device representation for coupling between antennas. |

Antenna a |

Antenna b |

This relationship incorporates several equivalencies, but most importantly the electric and magnetic quantities are scaled appropriately by ц to preserve the proper units in the dual relationship. For a 72 П strip dipole, we find the complementary slot dipole has an input impedance of Zslot = 493.5 П. Self-complementary planar structures such as spirals provide an input impedance of 188.5 П. A self-complementary structure is its own complement.

Images

Many antennas are constructed above a large metallic structure referred to as a ground plane. As long as the structure is greater than a half-wavelength in radius, the finite plane may be moldeled as an infinite structure for all but radiation behind the plane. The advantage of the infinite structure which is a perfect electric conductor (PEC) is that the planar sheet may be replaced by the images of the antenna elements in the plane. For the PEC, the images are constructed to provide a zero, tangential electric field at the plane. Figure 2 shows the equivalent current structure for the original and the image problems.

It is common to feed antennas at the ground plane through a coaxial cable. Then the equivalent voltage for the imaged problem is twice that of the source above the ground plane.

The alternate use of duality is to equate similar dual problems numerically. A classic problem is the relationship between the input impedance of a slot dipole and strip dipole. The two structures are planar complements, each filling the void of the other, and have input impedances which satisfy

-[в2F + VV F] + pV x A |

7 7 — — slot strip ^ |

1 |

Reciprocity

Circuit Form. Reciprocity plays an important role in antenna theory and can be used to great advantage in calculations and measurements. Fortunately, antennas usually behave as reciprocal devices. This permits characterization of the antenna as either a transmitting or receiving antenna. For example, radiation patterns are often measured with the test antenna operating in the receive mode. If the antenna is reciprocal, the measured pattern is identical when the antenna is in either a transmit or a receive mode. In fact, the following general statement applies: If nonreciprocal materials are not present in an antenna, its transmitting and receiving properties are identical. A case where reciprocity may not hold is when ferrite or active devices are included as a part of the antenna.

Reciprocity is also helpful when examining the terminal behavior of antennas. Consider two antennas, a and b shown in Fig. 3. Although connected through the intervening medium and not by a direct connection path, we can view this as a two-port network. Two port circuit analysis permits us to write the following:

Va — ZaaIa + ZabIb Vb — ZbaIa + ZbbIb |

f [Ja JV |

■Ea = |

Hb] dv |

(41) |

where Va and Vb are the terminal voltages and Ia and Ib are the currents of antennas a and b, respectively. Zaa and Zbb are self impedances and Zab and Zba are mutual impedances. To illustrate the use of these equations, suppose a generator of current Ia is placed on antenna a. The open circuit (Ib = 0) voltage at antenna b is then,

far field of the antenna a, then we have the radiated electric field of antenna a along the point dipole as

■Eb — Ma |

(35) |

(36) |

Vb — ZbaIa

Therefore, mutual impedance Zba provides the coupling between a transmitting antenna and a receiving antenna. Reversing the situation by using antenna b as the transmitter and antenna a as the receiver, leads to

Va — ZabIb

If we go to the extreme of taking the test antenna to the surface of the problem antenna, then Eq. (41) becomes an equation that may be used for the solution of the currents on the antenna.

The actual form of reaction is to suggest that the field b reaction with current a is equal to the field a reaction with current b, or

[ (Ja — Eb — Ma’Hb) dv — [ (Jb* Ea — Mb Ha) dv (42)

Jv Jv

written symbolically as |

(b, a) — {a, b) |

(43) |

It can be seen from Eqs. (35) and (36) that if the applied currents are the same (Ia = Ib = I), then reciprocity is satisfied (i. e., = Vb) if |

Zab — Zba |

(37) |

for reciprocal antennas

(44) |

(45) |

(38) |

(39) |

f [(Ja Jv |

Eb — Ma |

Hb) |

If one antenna is rotated, the output voltage as a function of rotation angle becomes the radiation pattern. Since the coupling mechanism is via mutual impedances Zab and Zba, they must correspond to the radiation patterns. For example, if antenna b is rotated in the plane of Fig. 3, the pattern in that plane is proportional to the output of a receiver connected to antenna b due to a source of constant power attached to antenna a. For reciprocal antennas Eq. (37) implies the transmitting and receiving patterns for the rotated antenna are the same.

Another interesting result follows from Eq. (34). The input impedance of antenna a is

That is, the input impedance equals the self impedance and antenna a acts as if it is in free space.

Reaction Theorem. Reciprocity may also be stated in integral form by cross multiplying Maxwell’s equations by the opposite field for two separate problems, integrating and combining to obtain

For antenna problems, the surface integral on the left of Eq. (40) is taken to an infinite radius and the integral becomes zero for finite antennas. This form is the typical field form of reciprocity. This form also suggests constructing a second problem that can be used as an auxiliary form to solve the original problem. For instance, if Mb = 0 and Jb is a point dipole (or test antenna) of vector unit length lb located in the

Va

Za = for reciprocal antennas

■La

[Ea x H. — E, x Ha] ■ ds — |

— (Jb — Ea — Mb ‘ Ha )] dv |

Za — (ZaaIa + 0)/Ia — Za |

(40) |

S |

If antennas a and b are far enough apart, such as in the far field, Zab < Za and the input impedance of antenna a becomes |

Thus, if the current distribution on the antenna is known, or may be estimated, then Eq. (45) provides a means for computing the antenna impedance Z by integrating the near-field radiated by the antenna current in free space times the current distribution itself. A common approach to this computation results in the induced-EMF method (2).

Radiation Patterns

The radiation pattern is a description of the angular variation of radiation level around an antenna. This is perhaps the most important characteristic of an antenna. In this section we present definitions associated with patterns and develop the general procedures for calculating radiation patterns.

Radiation Pattern Basics. A radiation pattern (antenna pattern) is a graphical representation of the radiation (far-field) properties of an antenna. The radiation fields from a transmitting antenna vary inversely with distance, for example, 1/r. The variation with observation angles ( в, ф), however, depends on the antenna and, in fact, forms the bulk of antenna investigations.

Radiation patterns can be understood by examining the ideal dipole. The fields radiated from an ideal dipole are shown in Fig. 4(a) over the surface of a sphere of radius r which is in the far field. The length and orientation of the

Antenna Impedance

Reciprocity may be used to obtain the basic formula for the input impedance of an antenna. If we define the two problems for Eq. (42) as (a) the antenna current distribution in the presence of the antenna structure and (b) the same antenna current in free space, then we can apply Eq. (42) to obtain

f (J> Eb) dv — f (J> Ea ) dv — IVa Jv Jv |

Z= ^ J (J-Eb)dv |

Since Va = IZ, we may write |

z |

z |

x |

z |

Figure 4. Radiation from an ideal dipole. (a) Field components. (b) E-plane radiation pattern polar plot. (c) H-plane radiation pattern polar plot. (d) Three-dimensional pattern plot.

f e—je R A* = I1J IiZ>)^R dz’ |

field vectors follow from Eq. (30a); they are shown for an instant of time for which the fields are peak. The angular variation of Ee and Hф over the sphere is sin в. An electric-field probe antenna moved over the sphere surface and oriented parallel to Ee will have an output proportional to sin в; see Fig. 4(b). Any plane containing the z-axis has the same radiation pattern since there is no ф variation in the fields. A pattern taken in one of these planes is called an E-plane pattern because it contains the electric vector. A pattern taken in a plane perpendicular to an E-plane and cutting through the test antenna (the xy-plane in this dipole case) is called an H — plane pattern because it contains the magnetic field Hф. The E — and H-plane patterns, in general, are referred to as principal plane patterns. The E — and H-plane patterns for the ideal dipole are shown in Fig. 4(b) and (c). These are polar plots in which the distance from the origin to the curve is proportional to the field intensity; they are often called polar patterns or polar diagrams.

The complete pattern for the ideal dipole is shown in isometric view with a slice removed in Fig. 4(d). This solid polar

radiation pattern resembles a doughnut with no hole. It is referred to as an omni directional pattern since it is uniform in the xy-plane. Omni directional antennas are very popular in ground-based applications with the omni directional plane horizontal. When encountering new antennas the reader should attempt to visualize the complete pattern in three dimensions.

Radiation patterns in general can be calculated in a manner similar to that used for the ideal dipole if the current distribution on the antenna is known. This calculation is done by first finding the vector potential given in Eq. (26). As a simple example consider a filament of current along the z-axis and located near the origin. Many antennas can be modeled by this line source; straight wire antennas are good examples. In this case the vector potential has only a z-component and the vector potential integral is one-dimensional

We can do this because in the far field r is very large compared to the antenna size, so r > z’ > z’ cos в. In the phase term — 3R, we must be more accurate when computing the distance from points along the line source to the observation point. The integral Eq. (46) sums the contributions from all the points along the line source. Although the amplitude of waves due to each source point is essentially the same, the phase can be different if the path length differences are a sizable fraction of a wavelength. We, therefore, include the first two terms of the series in Eq. (52) for the R in the numerator of Eq. (46) giving

z Figure 5. Geometry used for field calculations of a line source along the z-axis. |

(54) |

R & r — z’ cos 9

Using the far-field approximations Eqs. (53) and (54) in Eq. (46) yields

— je(r-z; cos 9) |

,-jP r 4n r |

j I(z’ )eJe |

f‘ |

e |

JJez’ cos в dz (55) |

Az — p I(z’) |

-dz’ = jJL |

where 3 has been used for typical radiation media. Due to the symmetry of the source, we expect that the radiation fields will not vary with ф. This lack of variation is because as the observer moves around the source such that p and z are constant, the appearance of the source remains the same; thus, its radiation fields are also unchanged. Therefore, for simplicity we will confine the observation point to a fixed ф in the yz-plane (ф = 90°) as shown in Fig. 5. Then from Fig. 5 we see that |

4л r |

where the integral is over the extent of the line source. The electric field is found from Eq. (27), which is |

E = —j(opA k(k — A) ox |

(56) |

This far-field result for a z-directed current, as in Eq. (46), reduces to |

y2 — y2 + z2 |

(47) (48) (49) |

z — r cos 9 y — r sin 9 |

E &-jo>Ae0 — jo sin 9AJ9 |

(57) |

Note that this result is the portion of the first term of Eq. (56) which is transverse to r because —ja>A = —jw( —Az sin в Q + Az cos в r). This form is an important general result for z — directed sources that is not restricted to line sources. The radiation fields from a z-directed line source (any z — directed current source in general) are Hф and Ee, and are found from Eqs. (27) and (28). The only remaining problem is to calculate Az, which is given by Eq. (26) in general and by Eq. (55) for z-directed line sources. Calculation of Az is the focal point of linear antenna analysis. We shall return to this topic after pausing to further examine the characteristics of the far-field region. The radiation field components given by Eqs. (27) and (28) yield |

Applying the general geometry of Fig. 1 to this case, r = yy + zz and r’ = zz lead to R = r — r’ = yy + (z — z’)z and then |

R = Vy2 + (z — z’)2 = y2 + z2 — 2 zz’ + (z’)2 |

(50) |

Substituting Eqs. (47) and (48) into Eq. (49), to put all field point coordinates into the spherical coordinate system, gives |

R — {r2 + [-2rz’ cos 9 + (z’ )2]}1/2 |

(51) |

This result could also be obtained by using Jz(r’) = I(z’)S(x’)S(y’) in Eq. (23) where dv’ = dx’ dy’ dz’. In order to develop approximate expressions for R, we expand Eq. (51) using the binomial theorem: R = r + — [—2rz’ cos в + (z’)2]—————— "t[—2rz’ cos в + (z’)2]2 H— 2r 8r3 . (z’)2sin2 9 (z’)3 sin2 9 cos 9 = r — z cos в H —————— 1———- —;——— h • • • |

Eg = °fH, = пНф |

(58) |

where ц = V^/e is the intrinsic impedance of the medium. An interesting conclusion can be made at this point. The radiation fields are perpendicular to each other and to the direction of propagation r and their magnitudes are related by Eq. (58). These are the familiar properties of a plane wave. They also hold for the general form of a transverse electromagnetic (TEM) wave which has both the electric and magnetic fields transverse to the direction of propagation. Radiation from a finite antenna is a special case of a TEM wave, called a spherical wave which propagates radially outward from the antenna and the radiation fields have no radial components. Spherical wave behavior is also characterized by the e—j374nr |

2r2 |

2r |

(52) |

The terms in this series decrease as the power of z’ increases if z’ is small compared to r. This expression for R is used in the radiation integral Eq. (46) to different degrees of approximation. In the denominator of Eq. (46) (which affects only the amplitude) we let |

(59) |

(60) |

(D/2)2 2r /Г |

(61) |

(62) |

,ff. |

factor in the field expressions; see Eq. (55). The e j phase factor indicates a traveling-wave propagating radially outward from the origin and the 1/r magnitude dependence leads to constant power flow just as with the infinitesimal dipole. In fact, the radiation fields of all antennas of finite extent display this dependence with distance from the antenna.

Another way to view radiation field behavior is to note that spherical waves appear to an observer in the far field to be a plane wave. This local plane wave behavior occurs because the radius of curvature of the spherical wave is so large that the phase front is nearly planar over a local region.

If parallel lines (or rays) are drawn from each point on a line current as shown in Fig. 6, the distance R to the far field is geometrically related to r by Eq. (54), which was derived by neglecting high order terms in the expression for R in Eq. (52). The parallel ray assumption is exact only when the observation point is at infinity, but it is a good approximation in the far field. Radiation calculations often start by assuming parallel rays and then determining R for the phase by geometrical techniques. From the general source shown in Fig. 6, we see that

R = r — r’ cos a

Using the definition of dot product, we have

R = r — r • r’

This form is a general approximation to R for the phase factor in the radiation integral. Notice that if r’ = z’z, as for line sources along the z-axis, Eq. (60) reduces to Eq. (54).

The definition of the distance from the source where the far field begins is where errors due to the parallel ray approximation become insignificant. The distance where the far field begins, rff, is taken to be that value of r for which the path length deviation due to neglecting the third term of Eq. (52) is a sixteenth of a wavelength. This corresponds to a phase error (by neglecting the third term) of 2w/A X A/16 = пУ8 rad = 22.5°.

If D is the length of the line source, rff is found by equating the maximum value of the third term of Eq. (52) to a sixteenth of a wavelength; that is, for z’ = D/2 and в = 90°, the

Figure 6. Parallel ray approximation for far-field calculations of radiation from a general source. |

third term of Eq. (52) is

X

16

Solving for rf gives

2D2

~X~

The far-field region is r > rff and rff is called the far-field distance, or Rayleigh distance. The far-field conditions are summarized as follows:

2D2 ■ >—— X |
(63a) |

r » D |
(63b) |

r » X |
(63c) |

The condition r > D was mentioned in association with the approximation R « r of Eq. (53) for use in the magnitude dependence. The condition r > A follows from /3r = (2nr/A) > 1 which was used to reduce Eq. (46) to Eq. (55). Usually the far field is taken to begin at a distance given by Eq. (62) where D is the maximum dimension of the antenna. This is usually a sufficient condition for antennas operating in the ultra high frequency (UHF) region and above. At lower frequencies, where the antenna can be small compared to the wavelength, the far-field distance may have to be greater than 2D2/A in order that all conditions in Eq. (63) are satisfied.

The concept of field regions was introduced in an earlier section and illustrated with the fields of an ideal dipole. We can now generalize that discussion to any finite antenna of maximum extent D. The distance to the far field is 2D2/A. This zone was historically called the Fraunhofer region if the antenna is focused at infinity; that is, if the rays at large distances from the antenna when transmitting are parallel. In the far-field region the radiation pattern is independent of distance. For example, the sin в pattern of an ideal dipole is valid anywhere in its far field. The zone interior to this distance from the center of the antenna, called the near field, is divided into two subregions. The reactive near-field region is closest to the antenna and is that region for which the reactive field dominates over the radiative fields. This region extends to a distance 0.62VDVA from the antenna, as long as D > A. For an ideal dipole, for which D = Дz ^ A, this distance is A/2u. Between the reactive near-field and far-field regions is the radiating near-field region in which the radiation fields dominate and where the angular field distribution depends on distance from the antenna. For an antenna focused at infinity the region is sometimes referred to as the Fresnel region. We can summarize the field region distances for cases where D > A as follows:

Region Distance from antenna (r)

Reactive near field OtoO,62y/D3/X (64a)

Radiating near field 0.62^D3/X to 2D2/X (64b)

Far field 2D2/X to <x) (64c)

Steps in the Evaluation of Radiation Fields. The derivation for the fields radiated by a line source can be generalized for application to any antenna. The analysis of the line source, and its generalizations, can be reduced to the following three step procedure:

1. Find A. Select a coordinate system most compatible with the geometry of the antenna, using the notation of Fig. 1. In general, use Eq. (23) with R « r in the magnitude factor and the parallel ray approximation of Eq. (60) for determining phase differences over the antenna. These yield

Example: The Uniform Line Source. The uniform line source is a line source for which the current is constant along its extent. If we use a z-directed uniform line source centered on the origin and along the z-axis, the current is

I(z’) = |

(73) |

l0 x’ = 0,/ = 0, z’ < ^

0 elsewhere

where L is the length of the line source; see Fig. 5. We first find Az from Eq. (67) as follows:

o—jpr fL/2 |

I |

в dz’ = і |

I |

— M’ |

— ID r |

e |

o |

Jee rr’ dv’ |

(fiL/2) cos в |

4n r |

A = |

(65) |

L/2 |

/X- |

4n r |

r sin[(fiL/2) cos в] 4n r |

(74) |

For z-directed sources |

The electric field from Eq. (69) is then

r |

e |

> Jfir. лвіп[(pL/2)cos6}~ sm0—. ;—в (75) |

Jze>e r r’ dv’ |

A = |

(66) |

E = jcoAz sin вв = j(o/iI0L |

4nr jy For z-directed line sources on the z-axis —je r |

(PL/2) cos в |

4n r |

The magnetic field is simply found from this form using Нф = Ee/V. Radiation Pattern Definitions. Since the radiation pattern is the variation over a sphere centered on the antenna, r is constant and we have only в and ф variation of the field. It is convenient to normalize the field expression such that its maximum value is unity. This is accomplished as follows for a z-directed source which has only a в-component of E |

f I(z’ )j Jz |

lz cos вdz’ |

A = z/i |

(67) |

4n r |

which is Eq. (55). 2. Find E. In general, use the component of E = —joA |

(68) |

which is transverse to the direction of propagation, r. This result is expressed formally as |

Ee |

F (в, ф) = |

(76) |

Ee (max) |

E = —joA + jo(r • A)r = —jo(Ae<) + Афф) |

(69) |

where F(e, ф) is the normalized field pattern and Ee(max) is the maximum value of Ee over a sphere of radius r. In general Ee can be complex-valued and, therefore, so can F(e, ф). In this case the phase is usually set to zero at the same point the magnitude is normalized to unity. This is appropriate since we are only interested in relative phase behavior. This variation is, of course, independent of r. An element of current on the z-axis has a normalized field pattern of |

which arises from the component of A tangent to the far-field sphere. For z-directed sources this form be- |

comes |

E = joAz sin вв (70) which is Eq. (57). 3. Find H. In general, use the plane-wave relation H = — r xE (71) n This equation expresses the fact that in the far field the directions of E and H are perpendicular to each other and to the direction of propagation, and also that their magnitudes are related by ^. For z-directed sources |

F(в) = sin в |

(77) |

and there is no ф variation. The normalized field pattern for the uniform line source is from Eq. (75) in Eq. (76) |

sin[(/!L/2)cos0] (J3L/2)cos9 |

F(в) = sin в |

(78) |

(79) |

(72)

which is Eq. (58). The most difficult step is the first, calculating the radiation integral. To develop an appreciation for the process, we present an example. This uniform line source example will also serve to provide a specific setting for introducing general radiation pattern concepts and definitions.

and again there is no ф variation. The second factor of this expression is the function sin(w)/M. It has a maximum value of unity at u = 0; this corresponds to в = 90° where u = (3L/2) cos в. Substituting в = 90° in Eq. (78) gives unity and we see that F(d) is properly normalized.

In general, a normalized field pattern can be written as the product

F (в, ф) = g(в, ф) f (в,,

z (c) Endfire |

z |

z |

Figure 7. Polar plots of uniform line source patterns. (a) Broadside. (b) Intermediate. (c) Endfire.

(80) |

where g( e, ф) is the element factor and f( e, ф) is the pattern factor. The pattern factor comes from the integral over the current and is strictly due to the distribution of current in space. The element factor is the pattern of an infinitesimal current element in the current distribution. For example, for a z-directed current element the total pattern is given by the element factor:

F (9) — g(9) — sin 9

Frequently the directional properties of the radiation from an antenna are described by another form of radiation pattern, the power pattern. The power pattern gives angular dependence of the power density and is found from the в, ф variation of the r-component of the Poynting vector. For z-directed sources Hф = Ee/^ so the r-component of the Poynting vector is ШАф = |Ee|2/(2^) and the normalized power pattern is simply the square of its field pattern magnitude Р(в) = |F( в)|2. The general normalized power pattern is

(83) |

(84) |

P(9) — |

(85) |

sin9 |

(86) |

for a z-directed current element. Actually this factor originates from Eq. (57) and can be interpreted as the projection of the current element in the в-direction. In other words, at в = 90° we see the maximum length of the current, whereas at в = 0° or 180° we see the end view of an infinitesimal current which yields no radiation. The sin в factor expresses the fraction of the size of the current as seen from the observation angle в. On the other hand, the pattern factor f(fi, ф) represents the integrated effect of radiation contributions from the current distribution, which can be treated as being made up of many current elements. The pattern value in a specific direction is then found by summing the parallel rays from each current element to the far field with the magnitude and phase of each included. The radiation integral of Eq. (65) sums the far-field contributions from the current elements and when normalized yields the pattern factor.

Antenna analysis is usually easier to understand by considering the antenna to be transmitting as we have here. However, most antennas are reciprocal and thus their radiation properties are identical when used for reception; as discussed in the section on reciprocity.

For the z-directed uniform line source pattern Eq. (78) we can identify the factors as

Frequently patterns are plotted in decibels. It is important to recognize that the field (magnitude) pattern and power pattern are the same in decibels. This follows directly from the definitions. For field intensity in decibels |

and for power in decibels P(9, ф)^в — 10logP(9, ф) — 10log F(9, ф)2 — 20log F(9, ф) (87) |

The normalized power pattern for a z-directed current element is |

F(9, ф)ав — 20log F(9,ф) |

sin[(/!L/2)cos0] (J3L/2)cos9 |

Р(9,ф) — F (9,ф)2 |

P(9, ф) — sin2 9 |

and for a z-directed uniform line source is |

(81) |

g(9) — sin 9 |

and we see that

and |

(88) |

P(9^)dB — IF (9,ф)Лв

f (9) — |

(82) |

sin[(/!L/2) cos0] (J3L/2)cos9

For long line sources (L > A) the pattern factor of Eq. (82) is much sharper than the element factor sin в, and the total pattern is approximately that of Eq. (82), that is, F(0) » f(0). Hence, in many cases we need only work with f(0), which is obtained from Eq. (67). If we allow the beam to be scanned as in Fig. 7, the element factor becomes important as the pattern maximum approaches the z-axis.

Radiation Pattern Parameters. A typical antenna power pattern is shown in Fig. 8 as a polar plot in linear units (rather than decibels). It consists of several lobes. The main lobe (or main beam or major lobe) is the lobe containing the direction of maximum radiation. There is also usually a series of lobes smaller than the main lobe. Any lobe other than the main lobe is called a minor lobe. Minor lobes are composed of side lobes and back lobes. Back lobes are directly opposite the main lobe, or sometimes they are taken to be the lobes in

half the maximum value: |

(89) |

HP 9HPleft 9HP right |

the half-space opposite the main lobe. The term side lobe is sometimes reserved for those minor lobes near the main lobe, but is most often taken to be synonymous with minor lobe; we will use the latter convention.

The radiation from an antenna is represented mathematically through the radiation pattern function, F(fi, ф) for field and P(в, ф) for power. This angular distribution of radiation is visualized through various graphical representations of the pattern, which we discuss in this section. Graphical representations also are used to introduce definitions of pattern parameters that are commonly used to quantify radiation pattern characteristics.

A three-dimensional plot as in Fig. 4(d) gives a good overall impression of the entire radiation pattern, but cannot convey accurate quantitative information. Cuts through this pattern in various planes are the most popular pattern plots. They usually include the E — and H-plane patterns; see Figs. 4(b) and (c). Pattern cuts are often given various fixed ф values, leaving the pattern a function of в alone; we will assume that is the case here. Typically the side lobes are alternately positive and negative valued. In fact, a pattern in its most general form may be complex-valued. Then we use the magnitude of the field pattern |Р(в)| or the power pattern P^).

A measure of how well the power is concentrated into the main lobe is the (relative) side lobe level, which is the ratio of the pattern value of a side lobe peak to the pattern value of the main lobe. The largest side lobe level for the whole pattern is the maximum (relative) side lobe level, frequently abbreviated as SLL. In decibels it is given by

F (SLL)

SLL — 20 log

F (max)

(90)

where 0HPl<* and 0HPright are points to the left and right of the main beam maximum for which the normalized power pattern has a value of one-half (see Fig. 8). On the field pattern |Р(в)| these points correspond to the value 1/V2. For example, the sin в pattern of an ideal dipole has a value of 1/V2 for в values of 6ві-Рieft = 135° and ви^^ы = 45°. Then HP = |135° — 45°| = 90°. This is shown in Fig. 4(b). Note that the definition of HP is the magnitude of the difference of the half-power points and the assignment of left and right can be interchanged without changing HP. In three dimensions the radiation pattern major lobe becomes a solid object and the half-power contour is a continuous curve. If this curve is essentially elliptical, the pattern cuts that contain the major and minor axes of the ellipse determine what the Institute of Electrical and Electronics Engineers (IEEE) defines as the principal half-power beamwidths.

Antennas are often referred to by the type of pattern they produce. An isotropic antenna, which is hypothetical, radiates equally in all directions giving a constant radiation pattern. An omnidirectional antenna produces a pattern which is constant in one plane; the ideal dipole of Fig. 4 is an example. The pattern shape resembles a doughnut. We often refer to antennas as being broadside or endfire. A broadside antenna is one for which the main beam maximum is in a direction normal to the plane containing the antenna. An endfire antenna is one for which the main beam is in the plane containing the antenna. For a linear current on the z-axis, the broadside direction is в = 90° and the endfire directions are 0° and 180°. For example, an ideal dipole is a broadside antenna. For z-directed line sources several patterns are possible. Figure 7 illustrates a few |Дв)| patterns. The entire pattern (in three dimensions) is imagined by rotating the pattern about the z-axis. The full pattern can then be generated from the E-plane patterns shown. The broadside pattern of Fig. 7(a) is called fan beam. The full three dimensional endfire pattern for Fig. 7(c) has a single lobe in the endfire direction. This single lobe is referred to as a pencil beam. Note that the sin в element factor, which must multiply these patterns to obtain the total pattern, will have a significant effect on the endfire pattern. Intermediate scan angles are also possible, as shown in Fig. 7(b).