电扫和相控阵（科普文章）

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Electronic Scanning and the Phased Array

Introduction

The potential for increased target handling capacity available in Track While Scan radars is limited by the requirement to position the radar antenna mechanically. Existing mechanical scanning methods are inherently slow and require large amounts of power in order to respond rapidly enough to deal with large numbers of high speed maneuvering targets. With mechanically scanned systems, antenna inertia and inflexibility prevent employment of optimum radar beam positioning patterns that can reduce reaction times and increase target capacity. With electronic scanning, the radar beams are positioned almost instantaneously and completely without the inertia, time lags, and vibration of mechanical systems. In an era in which the numerical superiority of adversaries is expected to remain large, electronic scanning can offset that advantage. The specific benefits of electronic scanning include:

(1) increased data rates (reduction of system reaction time),

(2) virtually instantaneous positioning of the radar beam anywhere within a set sector (beam position can be changed in a matter of micro-seconds),

(3) elimination of mechanical errors and failures associated with mechanically scanned antennas,

(4) vastly increased flexibility of the radar facilitating multi-mode operation, automatic multi-target tracking, highly directional transmission of missile guidance and control orders, interceptor and general air traffic control from one radar at virtually the same time.

Principles of Operation

The fundamental principles underlying the concept of electronic beam steering are derived from electromagnetic radiation theory employing constructive and destructive interference.

These principles can be stated as follows:

The electromagnetic energy received at a point in space from two or more closely spaced radiating elements is a maximum when the energy from each radiating element arrives at the point in phase.

To illustrate this principle consider figure 7-1. All elements are radiating in phase, and the resultant wave front is perpendicular to the axis of the element array. Figure 7-1 and subsequent diagrams show only a limited number of radiating elements. In actual radar antenna design several thousand elements could be used to obtain a high-gain antenna with a beam width of less than two degrees.

The wave fronts remain perpendicular to the boresight axis and are considered to arrive at a point target in space at the same time. As illustrated in figure 7-2, the path lengths from the elements to point P equalize as P approaches infinity. Thus, in situations where the target range is very large compared to the distance between elements, the paths from the elements to point P are almost parallel. Under these conditions, energy will arrive at point P with the same phase relationship that existed at the array.

To achieve beam positioning off the boresight axis, it is necessary to radiated the antenna elements out of phase with one another. Figure 7-3a depicts the phase shift necessary to create constructive interference along a line joining an arbitrary point P with the center of the array. In order to achieve this

constructive interference at Point P, the energy arriving from all radiating sources must be in phase and

arrive at the same time. The energy from element e1 must travel a path length r1, while the energy from element e2 must travel the longer path length r2.

The electric field magnitudes from the two elements at point P are given by equation (1-5) as follows:

E1 = E0sin 2 (r1 - ct) + 1

E2 = E0sin 2 (r2 - ct) + 2

Since these two electric fields must be in phase at point P for constructive interference to occur, the arguments of the sine functions must be equal.

2 (r1 - ct) + 1 = 2 (r2 - ct) + 2

1 - 2 = 2 (r2 - ct) - 2 (r1 - ct)

thus

1 - 2 = = 2 (r2 - r1)

The path length difference, r2 - r1, approaches the value d sin as the distance R in figure 7-3a increases. In figure 7-3b, where point P has been moved an infinite distance from the source, the paths from the two sources have become parallel and the path length difference is exactly d sin . For points at a distance R less than infinity, d sin is still a good approximation for the path length difference, r2 - r1, as long as R is large compared to the element spacing, d. Thus, applying this distance approximation yields

r2 - r1 = dsin (R>d)

For practical radar applications, d(the distance between elements) is on the order of a few centimeters while R is on the order of kilometers, a difference of several orders of magnitude; therefore, the distance approximation is valid for all radar applications in the far radar field (ranges greater than 1 km).

Applying the distance approximation to the expression already obtained for the required phase difference between two radiating elements yield equation (7-1).

= 2 d sin (7-1)

where:

= the phase shift between adjacent elements expressed in radians.

= the free space wavelength in meters.

d= the linear distance between radiating elements in meters.

= the desired angular offset in degrees.

Methods of Beam Steering

The previous discussion addressed the theory required to compute the relative phase shift between adjacent radiating elements in order to position the beam of an array-type antenna to a specific angle off of the antenna boresight axis. In practice there are three methods of accomplishing this phase difference.

Time Delay Scanning

The employment of time delay as a means of achieving the desired phase relationships between elements allows greater flexibility in frequency utilization than other methods. However, in practice the use of coaxial delay lines or other means of timing at high power levels is impractical due to increased cost, complexity, and weight.

To accomplish time delay scanning, variable delay networks are inserted in front of each radiating element. By proper choice of these time delays, the required effective phase shift can be applied to each element. The time delay between adjacent elements required to scan the beam to an angle, , is given by:

t = d sin (7-2)

c

Frequency Scanning

One of the simpler methods of phased-array radar implementation is frequency scanning. This method is also relatively inexpensive. Figure 7-6 shows the schematic arrangement of elements when frequency scanning is used to position a beam in either azimuth or elevation. The length of the serpentine wavelength line (l) is chosen such that for some center frequency, f0, the length of signal travel between elements is an integral number of wavelengths, or

= n (n = any integer greater than zero)

where

0 = wavelength in the serpentine line at frequency f0.

Thus, when the excitation frequency is f0, the serpentine line will cause all elements to radiate in phase, and the beam will be formed along the boresight axis. If the excitation frequency is increased or decreased from the center frequency, f0, the line length, l, will no longer represent an integer number of wavelengths. As the excitation energy travels along the serpentine line, it will reach each successive radiating element with a uniformly increasing positive or negative phase shift. This results in the beam being deflected by an angle from the boresight axis. Thus, by varying the radar transmitter frequency about some base frequency, the beam can be positioned in one axis. In figure 7-7 the frequency scanned array has been simplified to a two-

element system with the boresight axis normal to the plane of the elements. The feed is folded into a serpentine form to allow close element spacing while maintaining the required line length (l) between elements.

RF energy at 5,000 MHz is fed at the top of the array, and the elements are separated by distance d equal to (.03 meters). At time t = 0, the energy enters the serpentine feed line, and antenna A1 radiates immediately starting at zero phase. Since the period of the wave form is T = 1/f, then T = 1/5,000 MHz or 200 sec. Therefore, it takes 200 sec for one wavelength to propagate from A1. If the distance l traveled in the serpentine feed between A1 and A2 equals one wavelength or any integer number of wavelength (L = n where n = 1, 2, 3, . . .), then

t = l = n = n c = n = nT

c c f f

c

where

t = elapsed time, and

T = the period of the wave form

Therefore, the energy from A2 will always be in phase with A1. The beam formed by the array will be on the boresight axis. Note that this represents a broadside array (the elements transmit in phase). If the frequency is changed to 5,500 MHz, the period T becomes 181.81 sec; however, the wave form still takes t = l/c or 200 sec longer to reach A2 than to reach A1 when fed into the serpentine line as depicted in figure 7-8. Note that the wave form from A2 is no longer in phase with A1. Energy from A2 lags A1 in phase by 200 sec - 181.81 sec or 18.19 sec. The amount of phase shift can be determined by:

t =

T 2 (radians)

18.19 sec =

181.81 sec 2 radians

= (.1)(2 )

= 0.6286 radians or 36o

Since there is a phase difference, , then the beam axis can be located as follows:

= 2 d sin

0.6286 = 2 (.03m) sin

(.0545m)

sin = 0.1817

= 10.47o

Since energy from A2 lags 36o in phase, the beam will be 10.47o below the boresight (figure 7-9).

In this illustration the distance from A1 to a point on space (R1) is greater than from A2 to that point (R1). The wave forms will arrive in phase at the point because A2 lags A1 when the energy is transmitted.

Thus, as frequency is varied, the beam axis will change, and scanning can be accomplished in one axis (either elevation or azimuth). The principles are employed in the AN/SPS-48 and AN/SPS-52 series radars as well as in the older AN/SPS-39. Variation in frequency tends to make these radars more resistant to jamming than they would be if operated at a fixed frequency, and it also provides a solution to the blind speed problem in MTI systems. Frequency scanning does impose some limitations in that a large portion of the available frequency band is used for scanning rather than to optimize resolution of targets. Additionally, this imposes the requirement that the receiver bandwidth be extremely wide or that the receiver be capable of shifting the center of a narrower bandwidth with the transmitted frequency. Equation (7-3) gives the relationship between the percentage variation in frequency (bandwidth) and the scan angle, which is referred to as the Wrap-up Ratio.

Wrap-up Ratio f - f0 l = dsin (7-3)

f

or

l = fsin

d f - f0

The wrap-up ratio is the ratio of the sine of the maximum scan angle to the percentage change in frequency required to scan.

Phase Scanning

In a phase-scanned radar system, the radiating elements are fed from a radar transmitter through phase-shifting networks or "phasers." This system is shown in figures 7-11, 7-12, and 7-13. The aim of the system is again to position the beam at any arbitrary angle, , at any time. In this case the means of accomplishing the phase shift at each element is simply to shift the phase of the incoming energy to each element. These phasers are adjustable over the range 0 to + 2 radians. The task of the system is to compute the phase shift required for each element, and set each phaser to the proper value to accomplish the desired beam offset. While phase scanning is more expensive than frequency scanning, it is much less expensive (in dollars, weight, and power losses) than time delay steering. To a first approximation, the

bandwidth of a phase-scanning antenna in % of f0 is equal to the normal beamwidth in degrees. Thus a 1o beamwidth antenna working at 10 GHz can radiate over a 100 MHz bandwidth (i.e., +50 MHz) without distortion.

Energy reception. To receive energy transmitted in a steered beam by any of the three scanning methods, the applied frequency, time, or phase relationships are maintained at each element, which has the effect of making the radar sensitive to energy from the direction of transmission. Thus, each of the scanning methods is completely reversible and works equally well in reception of energy as in transmission. Unfortunately, some beam positions do not have the same phase shift in the reverse direction. When non-reciprocal phase shifters are used, it is necessary to change the phase-shifter setting between transmit and receive to maintain the same phase shift on receive as was used on transmission.

Computation of Required Phase Relationships

No matter which of the three possible methods of phase scanning is used in a phased array system, the objective is a relative phase shift of the energy being radiated by each element in the array. The incremental phase shift required between two adjacent elements is given by equation (7-1). When using this equation is will be assumed for consistency that represents the phase lead given to each element with respect to its adjacent element in the direction of the chosen reference element. Thus, when positioning the beam to the same side of the boresight axis as the reference element, each array element must lead the next element closer to the reference by the same amount, >0. When positioning the beam on the opposite side of the boresight axis from the reference by the same amount, <0 (negative or effective phase lag). This convention can be extended to include a sign convention for the angle . Choose the reference element as the top most element and the farthest right element when looking from the antenna along the boresight axis. Also choose elevation angles as positive above the boresight axis and negative below, and choose azimuth angles as positive in a clockwise direction (scanning to the right) and negative in a counterclockwise direction. (Do not confuse the above definitions of lead/lag in the spatial domain with the electrical definitions of lead/lag in the radian domain.)

Figure 7-14 illustrates this convention. In order to position the beam above the boresight axis, the angle will be positive and thus sin will be positive also. This yields a positive between elements. To determine the phase applied to each element , simply use the relationship

e= e , e = 0,1,2, . . . (7-4)

Similarly, to position the beam below the boresight axis requires the use of a negative angle . This yields a negative and equation (7-4) again yields the applied phase of each element.

If the reference element has a phase of zero, compute the phase applied to element five when the beam is scanned 40o above the boresight axis.

Find : = c = .06m

Find : = 2 d sin (7-1)

= 2 (.03) sin 40o

.06

= 2.02 radians

Find : = e (7-4)

= 5(2.02 radians)

= 10.10 radians

Note that this result is greater than 2 radians. In practice, the phasers can only shift the phase of the energy going to an element by an amount between -2 and 2 radians. Expressed mathematically

-2 < < 2

Thus, the phase shift applied to element five must be

= 10.10 radians - 2 radians

= 3.82 radians

In other words, element five must lead the reference element (element zero) by an amount of 3.82 radians.

Example 2. A Hypothetical Three-Dimensional Search Radar

In this example, azimuth information is obtained by rotating the antenna mechanically in a

continuous full-circle search pattern. Range information is obtained in the standard way by timing the pulse travel to and from the target. The phase shifters (0), (1), (2), (3) control the elevation position of the beam. It is desired to control the elevation of the beam in 0.0872 rad (5o) steps from +1.047 rad (60o) to -0.2617 rad (15o) with respect to the antenna boresight axis, which is +0.2617 rad (15o) displaced from the horizontal. The system operational parameters are as follows:

Antenna rotational speed---10 rpm

Pulse repetition rate (PRR)---400 pps

No. of elevation beam positions---16

No. of pulse per beam position---2

For each beam position the amount of phase shift is calculated for each radiating element. The resultant phase shifts are applied, and then two pulses are transmitted and received. The elevation scan logic is illustrated by the flow diagram, figure 7-17. The resultant scan pattern is illustrated in figure 7-18.

This example is, of course, hypothetical. Some operational 3-D radars, such as the Marine AN/TPS-59 function similarly. It is important to note that the concept of controlling beam position by varying the

relative phase of radiating elements is common to frequency-scanned, phase-scanned, and time-delay scanned arrays. The difference is in the methods employed to achieve the proper phase relationships between the radiating elements.鉴于某人的媚外，贴英文版，大家顺便练练外语


Electronic Scanning and the Phased Array

Introduction

The potential for increased target handling capacity available in Track While Scan radars is limited by the requirement to position the radar antenna mechanically. Existing mechanical scanning methods are inherently slow and require large amounts of power in order to respond rapidly enough to deal with large numbers of high speed maneuvering targets. With mechanically scanned systems, antenna inertia and inflexibility prevent employment of optimum radar beam positioning patterns that can reduce reaction times and increase target capacity. With electronic scanning, the radar beams are positioned almost instantaneously and completely without the inertia, time lags, and vibration of mechanical systems. In an era in which the numerical superiority of adversaries is expected to remain large, electronic scanning can offset that advantage. The specific benefits of electronic scanning include:

(1) increased data rates (reduction of system reaction time),

(2) virtually instantaneous positioning of the radar beam anywhere within a set sector (beam position can be changed in a matter of micro-seconds),

(3) elimination of mechanical errors and failures associated with mechanically scanned antennas,

(4) vastly increased flexibility of the radar facilitating multi-mode operation, automatic multi-target tracking, highly directional transmission of missile guidance and control orders, interceptor and general air traffic control from one radar at virtually the same time.

Principles of Operation

The fundamental principles underlying the concept of electronic beam steering are derived from electromagnetic radiation theory employing constructive and destructive interference.

These principles can be stated as follows:

The electromagnetic energy received at a point in space from two or more closely spaced radiating elements is a maximum when the energy from each radiating element arrives at the point in phase.

To illustrate this principle consider figure 7-1. All elements are radiating in phase, and the resultant wave front is perpendicular to the axis of the element array. Figure 7-1 and subsequent diagrams show only a limited number of radiating elements. In actual radar antenna design several thousand elements could be used to obtain a high-gain antenna with a beam width of less than two degrees.

The wave fronts remain perpendicular to the boresight axis and are considered to arrive at a point target in space at the same time. As illustrated in figure 7-2, the path lengths from the elements to point P equalize as P approaches infinity. Thus, in situations where the target range is very large compared to the distance between elements, the paths from the elements to point P are almost parallel. Under these conditions, energy will arrive at point P with the same phase relationship that existed at the array.

To achieve beam positioning off the boresight axis, it is necessary to radiated the antenna elements out of phase with one another. Figure 7-3a depicts the phase shift necessary to create constructive interference along a line joining an arbitrary point P with the center of the array. In order to achieve this

constructive interference at Point P, the energy arriving from all radiating sources must be in phase and

arrive at the same time. The energy from element e1 must travel a path length r1, while the energy from element e2 must travel the longer path length r2.

The electric field magnitudes from the two elements at point P are given by equation (1-5) as follows:

E1 = E0sin 2 (r1 - ct) + 1

E2 = E0sin 2 (r2 - ct) + 2

Since these two electric fields must be in phase at point P for constructive interference to occur, the arguments of the sine functions must be equal.

2 (r1 - ct) + 1 = 2 (r2 - ct) + 2

1 - 2 = 2 (r2 - ct) - 2 (r1 - ct)

thus

1 - 2 = = 2 (r2 - r1)

The path length difference, r2 - r1, approaches the value d sin as the distance R in figure 7-3a increases. In figure 7-3b, where point P has been moved an infinite distance from the source, the paths from the two sources have become parallel and the path length difference is exactly d sin . For points at a distance R less than infinity, d sin is still a good approximation for the path length difference, r2 - r1, as long as R is large compared to the element spacing, d. Thus, applying this distance approximation yields

r2 - r1 = dsin (R>d)

For practical radar applications, d(the distance between elements) is on the order of a few centimeters while R is on the order of kilometers, a difference of several orders of magnitude; therefore, the distance approximation is valid for all radar applications in the far radar field (ranges greater than 1 km).

Applying the distance approximation to the expression already obtained for the required phase difference between two radiating elements yield equation (7-1).

= 2 d sin (7-1)

where:

= the phase shift between adjacent elements expressed in radians.

= the free space wavelength in meters.

d= the linear distance between radiating elements in meters.

= the desired angular offset in degrees.

Methods of Beam Steering

The previous discussion addressed the theory required to compute the relative phase shift between adjacent radiating elements in order to position the beam of an array-type antenna to a specific angle off of the antenna boresight axis. In practice there are three methods of accomplishing this phase difference.

Time Delay Scanning

The employment of time delay as a means of achieving the desired phase relationships between elements allows greater flexibility in frequency utilization than other methods. However, in practice the use of coaxial delay lines or other means of timing at high power levels is impractical due to increased cost, complexity, and weight.

To accomplish time delay scanning, variable delay networks are inserted in front of each radiating element. By proper choice of these time delays, the required effective phase shift can be applied to each element. The time delay between adjacent elements required to scan the beam to an angle, , is given by:

t = d sin (7-2)

c

Frequency Scanning

One of the simpler methods of phased-array radar implementation is frequency scanning. This method is also relatively inexpensive. Figure 7-6 shows the schematic arrangement of elements when frequency scanning is used to position a beam in either azimuth or elevation. The length of the serpentine wavelength line (l) is chosen such that for some center frequency, f0, the length of signal travel between elements is an integral number of wavelengths, or

= n (n = any integer greater than zero)

where

0 = wavelength in the serpentine line at frequency f0.

Thus, when the excitation frequency is f0, the serpentine line will cause all elements to radiate in phase, and the beam will be formed along the boresight axis. If the excitation frequency is increased or decreased from the center frequency, f0, the line length, l, will no longer represent an integer number of wavelengths. As the excitation energy travels along the serpentine line, it will reach each successive radiating element with a uniformly increasing positive or negative phase shift. This results in the beam being deflected by an angle from the boresight axis. Thus, by varying the radar transmitter frequency about some base frequency, the beam can be positioned in one axis. In figure 7-7 the frequency scanned array has been simplified to a two-

element system with the boresight axis normal to the plane of the elements. The feed is folded into a serpentine form to allow close element spacing while maintaining the required line length (l) between elements.

RF energy at 5,000 MHz is fed at the top of the array, and the elements are separated by distance d equal to (.03 meters). At time t = 0, the energy enters the serpentine feed line, and antenna A1 radiates immediately starting at zero phase. Since the period of the wave form is T = 1/f, then T = 1/5,000 MHz or 200 sec. Therefore, it takes 200 sec for one wavelength to propagate from A1. If the distance l traveled in the serpentine feed between A1 and A2 equals one wavelength or any integer number of wavelength (L = n where n = 1, 2, 3, . . .), then

t = l = n = n c = n = nT

c c f f

c

where

t = elapsed time, and

T = the period of the wave form

Therefore, the energy from A2 will always be in phase with A1. The beam formed by the array will be on the boresight axis. Note that this represents a broadside array (the elements transmit in phase). If the frequency is changed to 5,500 MHz, the period T becomes 181.81 sec; however, the wave form still takes t = l/c or 200 sec longer to reach A2 than to reach A1 when fed into the serpentine line as depicted in figure 7-8. Note that the wave form from A2 is no longer in phase with A1. Energy from A2 lags A1 in phase by 200 sec - 181.81 sec or 18.19 sec. The amount of phase shift can be determined by:

t =

T 2 (radians)

18.19 sec =

181.81 sec 2 radians

= (.1)(2 )

= 0.6286 radians or 36o

Since there is a phase difference, , then the beam axis can be located as follows:

= 2 d sin

0.6286 = 2 (.03m) sin

(.0545m)

sin = 0.1817

= 10.47o

Since energy from A2 lags 36o in phase, the beam will be 10.47o below the boresight (figure 7-9).

In this illustration the distance from A1 to a point on space (R1) is greater than from A2 to that point (R1). The wave forms will arrive in phase at the point because A2 lags A1 when the energy is transmitted.

Thus, as frequency is varied, the beam axis will change, and scanning can be accomplished in one axis (either elevation or azimuth). The principles are employed in the AN/SPS-48 and AN/SPS-52 series radars as well as in the older AN/SPS-39. Variation in frequency tends to make these radars more resistant to jamming than they would be if operated at a fixed frequency, and it also provides a solution to the blind speed problem in MTI systems. Frequency scanning does impose some limitations in that a large portion of the available frequency band is used for scanning rather than to optimize resolution of targets. Additionally, this imposes the requirement that the receiver bandwidth be extremely wide or that the receiver be capable of shifting the center of a narrower bandwidth with the transmitted frequency. Equation (7-3) gives the relationship between the percentage variation in frequency (bandwidth) and the scan angle, which is referred to as the Wrap-up Ratio.

Wrap-up Ratio f - f0 l = dsin (7-3)

f

or

l = fsin

d f - f0

The wrap-up ratio is the ratio of the sine of the maximum scan angle to the percentage change in frequency required to scan.

Phase Scanning

In a phase-scanned radar system, the radiating elements are fed from a radar transmitter through phase-shifting networks or "phasers." This system is shown in figures 7-11, 7-12, and 7-13. The aim of the system is again to position the beam at any arbitrary angle, , at any time. In this case the means of accomplishing the phase shift at each element is simply to shift the phase of the incoming energy to each element. These phasers are adjustable over the range 0 to + 2 radians. The task of the system is to compute the phase shift required for each element, and set each phaser to the proper value to accomplish the desired beam offset. While phase scanning is more expensive than frequency scanning, it is much less expensive (in dollars, weight, and power losses) than time delay steering. To a first approximation, the

bandwidth of a phase-scanning antenna in % of f0 is equal to the normal beamwidth in degrees. Thus a 1o beamwidth antenna working at 10 GHz can radiate over a 100 MHz bandwidth (i.e., +50 MHz) without distortion.

Energy reception. To receive energy transmitted in a steered beam by any of the three scanning methods, the applied frequency, time, or phase relationships are maintained at each element, which has the effect of making the radar sensitive to energy from the direction of transmission. Thus, each of the scanning methods is completely reversible and works equally well in reception of energy as in transmission. Unfortunately, some beam positions do not have the same phase shift in the reverse direction. When non-reciprocal phase shifters are used, it is necessary to change the phase-shifter setting between transmit and receive to maintain the same phase shift on receive as was used on transmission.

Computation of Required Phase Relationships

No matter which of the three possible methods of phase scanning is used in a phased array system, the objective is a relative phase shift of the energy being radiated by each element in the array. The incremental phase shift required between two adjacent elements is given by equation (7-1). When using this equation is will be assumed for consistency that represents the phase lead given to each element with respect to its adjacent element in the direction of the chosen reference element. Thus, when positioning the beam to the same side of the boresight axis as the reference element, each array element must lead the next element closer to the reference by the same amount, >0. When positioning the beam on the opposite side of the boresight axis from the reference by the same amount, <0 (negative or effective phase lag). This convention can be extended to include a sign convention for the angle . Choose the reference element as the top most element and the farthest right element when looking from the antenna along the boresight axis. Also choose elevation angles as positive above the boresight axis and negative below, and choose azimuth angles as positive in a clockwise direction (scanning to the right) and negative in a counterclockwise direction. (Do not confuse the above definitions of lead/lag in the spatial domain with the electrical definitions of lead/lag in the radian domain.)

Figure 7-14 illustrates this convention. In order to position the beam above the boresight axis, the angle will be positive and thus sin will be positive also. This yields a positive between elements. To determine the phase applied to each element , simply use the relationship

e= e , e = 0,1,2, . . . (7-4)

Similarly, to position the beam below the boresight axis requires the use of a negative angle . This yields a negative and equation (7-4) again yields the applied phase of each element.

If the reference element has a phase of zero, compute the phase applied to element five when the beam is scanned 40o above the boresight axis.

Find : = c = .06m

Find : = 2 d sin (7-1)

= 2 (.03) sin 40o

.06

= 2.02 radians

Find : = e (7-4)

= 5(2.02 radians)

= 10.10 radians

Note that this result is greater than 2 radians. In practice, the phasers can only shift the phase of the energy going to an element by an amount between -2 and 2 radians. Expressed mathematically

-2 < < 2

Thus, the phase shift applied to element five must be

= 10.10 radians - 2 radians

= 3.82 radians

In other words, element five must lead the reference element (element zero) by an amount of 3.82 radians.

Example 2. A Hypothetical Three-Dimensional Search Radar

In this example, azimuth information is obtained by rotating the antenna mechanically in a

continuous full-circle search pattern. Range information is obtained in the standard way by timing the pulse travel to and from the target. The phase shifters (0), (1), (2), (3) control the elevation position of the beam. It is desired to control the elevation of the beam in 0.0872 rad (5o) steps from +1.047 rad (60o) to -0.2617 rad (15o) with respect to the antenna boresight axis, which is +0.2617 rad (15o) displaced from the horizontal. The system operational parameters are as follows:

Antenna rotational speed---10 rpm

Pulse repetition rate (PRR)---400 pps

No. of elevation beam positions---16

No. of pulse per beam position---2

For each beam position the amount of phase shift is calculated for each radiating element. The resultant phase shifts are applied, and then two pulses are transmitted and received. The elevation scan logic is illustrated by the flow diagram, figure 7-17. The resultant scan pattern is illustrated in figure 7-18.

This example is, of course, hypothetical. Some operational 3-D radars, such as the Marine AN/TPS-59 function similarly. It is important to note that the concept of controlling beam position by varying the

relative phase of radiating elements is common to frequency-scanned, phase-scanned, and time-delay scanned arrays. The difference is in the methods employed to achieve the proper phase relationships between the radiating elements.