The summation of the term on the right-hand side, which is the mean power Ps,,, generated by the impressed current, reduces to

6 3 = - - Re[E,(lO, 10,15$1J,*(10, 10,15+)1. (25)

The current in the central gap can be calculated from the complex FDTD algorithm:

2

2 j w6 J,(lO, 10,15+) = -8- sin -E,(10,10,15~)

8, 2

+ dx[H,(lO, 10,153)] - dj[Hx(lO,lO, 153)l. (26)

The electric field term in (26) will not contribute to (251, as one may easily check. The following familiar result is ob- tained for (25):

Psou,,, = - Re -E,(10,10,15~) 6 2 ? .I I1 H;(10+, 10,153) 6 - H,*(9+, 10,15+) 6

-HZ(lO,lO$, 15;) 6 + H:(10,9+, 1-53) 6

1 2 (27) = - Re[Vi,IL],

where the voltage V,, and the current I,, are defined as line integrals of the electric and magnetic field.

Table 1 shows the evolution of the different power terms as calculated by formulas (23)-(27) after 10, 20, and 30 periods. For the upper part of the table the standard FDTD equations are used for the magnetic fields surrounding the antenna wire. The power balance is verified and proves to be correct up to 5 and 4 digits, respectively, with and without the surrounding conductive box, as was to be expected. The lower part of the table shows the results for a free-space antenna (without conductive box) in the case where the nonstandard FDTD equations of [4] are used for the magnetic field com- ponents that surround the wire. For this last case the radius of the wire is taken to be r = 1 mm. It is seen that the power does not balance here. In the steady state, the power calcu- lated on circuit-theoretical grounds is only 89.6% of the actual radiated power. One should be warned to use the

TABLE 1

Number of Periods 10 20 30

Standard FDTD (with conductive box)

Standard FDTD (free space antenna)

Nonstandard FDTD (free space antenna)

‘1, d Pabs

‘SO”,,,

‘Jad + Pabs

‘rad

Pm,,,,

‘rad

PS0”rCe

0.08174368 0.0495 1259 0.1290008 0.1312563

0.1 161269

0.1 1745 16

0.1212888

0.1095681

0.081 19544 0.08 115305 0.04855058 0.04857323 0.1298365 0.129721 9 0.1297460 0.1297263

0.1166692 0.1166147

0.1166591 0,1166257

0.1214689 0.1214119

0.1087900 0.1087681

circuit-theoretical approach to compute the power in such cases.

IV. CONCLUSION A discretized version of Poynting’s theorem was derived for the standard FDTD algorithm. The discretized form of Poynt- ing’s theorem was numerically verified by verifying the power balance for an antenna problem. It was also shown that the theorem is violated when used in connection with the modi- fied FDTD equations of [41.

REFERENCES 1. K. S. Yee, “Numerical Solution of Initial Boundary Value Prob-

lems Involving Maxwell’s Equations in Isotropic Media,” ZEEE Trans. Antennas Propagat., Vol. AP-14, 1966, pp. 302-307.

2. J. Toftghd, S. N. Hornsleth, and J. B. Andersen, “Effects on Portable Antennas of the Presence of a Person,’’ ZEEE Trans. Antennas Propagat., Vol. AP-41, 1993, pp. 739-746.

3. R. J. Luebbers and J. Begs, “FDTD Calculation of Wide-Band Antenna Gain and Efficiency,” IEEE Trans. Antennas Propagat.,

4. A. Taflove, K. R. Umashankar, B. Beker, F. Harfoush, and K. S. Yee, “Detailed FDTD Analysis of Electromagnetic Fields Pene- trating Narrow Slots and Lapped Joints in Thick Conducting Screens,” IEEE Tmns. Antennas Propagat., Vol. AP-40, 1992, pp.

Vol. 40, 1992, pp. 1403-1407.

357-366.

Received 10-25-94

Microwave and Optical Technology Letters, 8/5, 257-260 0 1995 John Wiley & Sons, Inc. CCC 0895-2477/95

MODAL CONTROL OF A CORNER REFLECTOR TO MAXIMIZE FAR-FIELD POWER Gregory Washington and Larry Slhrerberg Mars Mission Research Center North Carolina State University Raleigh, North Carolina 27695

KEY TERMS Comer refictor, autofocus control, adaptwe array

ABSTRACT The far-field power transmitted from a comer rejlector was maximized by varying its direction angle and comer angle. The optimal direction angle and comer angle are found by the modal autofocus method. The method is shown to conueee over a wide range of initial angles. The results are demonstrated in a physical experiment abng with complimentary numer- ical predictions. 0 1995 John Wiley & Sons, Im.

INTRODUCTION Since its inception by P. W. Howells in the late 1950s adap- tive antenna technology has been confined mainly to phased array and adaptive array methods. This growth stemmed from a military need for antennas that automatically respond to various unknown interference environments by beam nulling and pattern manipulation [l]. Because the need was of na- tional interest and some adaptive arrays had the ability to make nulls appear in the directions of the side-lobe jammers the methodology was researched heavily and still is to this

260 MICROWAVE AND OPTICAL TECHNOLOGY LElTERS / Vol. 8, No. 5, April 5 1995

day. A primary problem with adaptive arrays is the cost associated with signal amplification, digital processing, and transmitting/receiving equipment. A relatively inexpensive alternative capable of achieving beam shaping, such as beam scanning and ground swath varying, lies in adapting the physical shape of the reflector surface.

This study employs shape adaptation in order to maximize the power transmitted by a corner reflector to one or more points in the far field. The associated power maximization algorithm presented in this article was previously applied to cylindrical reflectors [2]. The algorithm, called the modal autofocus algorithm, varies the shape of the reflector through a series of autofocus modes. In the case of the corner reflector the autofocus modes degenerate to two rigid body modes of the reflector, specifically a direction mode and a corner mode. The algorithm proceeds by letting the two amplitudes of the autofocus modes, also called the modal amplitudes, vary in sequence over their full range. After the first mode is varied over its full range the maximum power and the corresponding modal amplitude are determined. Next, while holding the first mode at its optimal modal amplitude, the second mode is varied over its full range and an updated maximum power and corresponding optimal amplitude of the second mode are determined. The updated maximum power obtained at the completion of this step is a global maximum.

The next section of the article reviews the equations governing the motion of the reflector and the associated transmitted power. The modal autofocus method is presented in the third section, and the fourth section describes the experimental setup followed by a discussion of the results.

GOVERNING EQUATIONS

Consider a corner reflector modeled as two rectangular plates hinged in the middle. This corner reflector is governed by the two linear ordinary differential equations of motion,

where I , and I , represent the moments of inertia of the rectangular plates, O , ( t ) and O , ( t ) denote the angular dis- placements of the rectangular plates, and MI and M 2 denote the external moments acting on the two rectangular plates. The angular displacements can be written in vector form and can then be expressed as a linear combination of the rigid body modes of the reflector

where = (1, l)T, and +, = (1, -1IT denote the two cor- ner reflector modes and ql ( t ) and q,(t) denote the corre- sponding modal amplitudes [3]. Inverting Eq. (2) yields

Referring to Figure 1, the first modal amplitude q J t ) repre- sents the reflectors' direction angle and the second modal amplitude q2(t) represents the reflectors' corner angle. The variation of ql( t ) over its full range of amplitudes is now physically interpreted as a scanning process and the variation of q,(t) over its full range of amplitudes is regarded as a focusing or ground swath varying process.

Figure 1 Comer reflector (top view)

The corner reflector was fed by a dipole antenna. The electric field emanating from the dipole was given by

(4)

where Z , denotes intrinsic impedance, k,, denotes the free- space wave number, I denotes the current on the dipole, y represents the angle between the far-field vector r and the z axis, and r represents the distance to the far-field point [4]. The electric field emanating from the dipole, once calculated, is multiplied using the method of images in a manner that yields the total electric field. The method of images con- structs a finite number of image dipoles when the corner angle is an integer multiple of 180". The other angles can be calculated through interpolation, by a method of images that uses an infinite series of image dipoles as well as by standard numerical methods in physical optics.

MODAL AUTOFOCUS METHOD

In previous work. the modal autofocus method was developed in theory and applied in simulation to a cylindrical reflector antenna [2]. This study applies the same algorithm to a corner reflector. Like the case of the cylindrical reflector, the modal autofocus method begins by exerting two discrete external forces on the corner reflector, causing the reflector to undergo its full range of rigid body deformations associ- ated with the first autofocus mode. While undergoing its first mode deformations, the far-field power P1(q l ) at a reference point in the far field is calculated. The maximum power is denoted by Pi = P1(qf), where 77:: represents the amplitude at which the maximum occurs [2]. Next the algorithm repeats this process for the second mode, holding the first mode at its optimal modal amplitude. The equations describing the modal autofocus algorithm are as follows:

The measurements of P,! ( Y = 1,2) can be regarded as one cycle in the modal autofocus method. Unlike the case of the cylindrical reflector, the algorithm converges here in one cycle if the first rigid body mode is taken as the direction mode. The algorithm converges in two cycles when the first rigid body mode + , ( t ) is taken to be the corner mode. Equation (5a) represents the initial displacement of the cor- ner reflector, Eq. (Sb) represents the first mode sweep, and Eq. (Sc) represents the second mode sweep. After completing the two sweeps, that is, the first cycle, the globally optimal shape of the corner reflector is obtained.

MICROWAVE AND OPTICAL TECHNOLOGY LE-ERS / Vol 8, No. 5, April 5 1995 261

EXPERIMENTAL SETUP Modal control of the comer reflector is carried out in two experiments. The first experiment consisted of a corner re- flector having a fixed comer angle and fed by a transmitting quarter-wave monopole, extended over a conducting ground plane as shown in Figure 2.

The one-degree-of-freedom (1DOF) system was attached to a rotating base, and the sides were fixed to 90" to one another. A stationary receiving comer reflector was located in the far field. The transmitting antenna was rotated in discrete steps and the receiving antenna recorded the power at each individual step. The transmitting signal was set to 245 GHz. Unwanted reflections were guarded against by encasing the structure in a chamber constructed from a lossy foam that attenuates electromagnetic signals.

The second experiment considered an identical corner reflector that employed a quarter-wave monopole, but now both sides were free to vary (see Figure 3). The motion of each reflector side is now controlled by a motor, thereby allowing this 2DOF system to undergo both beam scanning and varying ground swath.

Figure 2 lDOF antenna system (top view)

Figure 3 2DOF antenna system (top view)

Direction Angle Mode

4.0

3.0

20

1.0

ao

-1.0

-20

-3.0 - 3 - 2 - 1 0 I 2 3 4

4.0

10

20

1.0

ao

-1.0

-20

-3.0

RESULTS AND DISCUSSION Each side of each corner reflector was of length 6.819 in. and height 4.0 in. The transmitting feed and the receiving feed were monopoles of height of 1.205 in. These values were calculated based on an aperture opening of two wavelengths at a reflector angle of 90" [5]. The distance between the transmitting feed and the receiving feed was seven wave- lengths. The two reflector modes of vibration and their effect on power are shown in Figure 4. The first mode was the

Direction Angle Power

Corner Angle Mode Corner Angle Power

E 3

4 4 -2 D 2 4 6

X (in)

5.0

0.0

-5.0

-10

- I S

-20

Figure 4 Rigid body modes and the associated far-field power

262 MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 8, No. 5, April 5 1995

direction mode and the second mode was the corner mode. In Figure 5, the power versus direction angle for both the computational and the experimental cases are shown. Both plots show a normalized maximum at an angle of 0". Figure 6 shows the power versus corner angle for both the computa- tional and the experimental cases. The maximum power in the computational model occurs at 86", and the maximum in the experimental model occurs at 76".

Figure 7 shows the computational and experimental re- sults for the modal autofocus algorithm associated with the lDOF system. The monopole of the 90" corner reflector was initially pointed -45" (135" from the horizontal). At the completion of the control, the reflector pointed broadside in

Power vs Dlredlon Angle (Comp) 1.2 . . . , . . . . , . . . . . .

1

0.8

0.6

0.4

0.2

0 4 0 60 80 100 120 140

Dirslia A q L (D.rrr )

the direction of maximum transmission. Figures 8 and 9 show the numerical and experimental results for the modal autofo- cus algorithm associated with the 2DOF system, in which we distinguished between two cases. In the first case, which is shown in Figure 8, the initial angles of the two sides of the corner reflector are 75" and 120", respectively. At the comple- tion of the control, the new angles are approximately 52.5" and 127.5", respectively. This yields a corner angle of 75" just 1" shy of the maximum of 76". In Case 2, the initial angles of the two sides of the comer reflector are 55" and 165", respectively. At the completion of the control, the new angles reduce to the same values as in Case 1. In Figure 9 the values of the two angles in both cases converged to 47.5" and 133.5")

Power vs. Direction Angle (Exp) 1.2 . . .

1

0.8

0.6

0.4

0.2

0 40 60 a0 100 120 140

biPi..A.3r*lr)

Figure 5 Power versus direction angle

Power vs. Corner Angle (Comp)

. . . . . . . . . .

. . . . .

20 25 30 35 40 45 5 0 5 5 6 0 c- hu13. (W-)

Power vs. Corner Angle (Exp)

. . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I , , I , I . . . . I

20 25 3 0 35 4 0 45 50 55 60 C- (W-)

Figure 6 Power versus corner angle

1 DOF Modal Autofocus Control (Comp) 1 W F Modd Autofacur Control (Exp)

-8 -6 -4 -2 0 2 4 6 X (u

.......... 5 . l ' . ' l - - . l . . -

4 --bib1 i ................................

2 ......... ...........(........... + ........ 3 ,

....... L. ........ - 1 ......... i ........... ......... ................ ........... ......... -2 ........ i ........... i ......... -3 . . . . . . . . . . . . . . . . . . . . . . . . . . .

- - - -Final Pwition ................................. - ......... ........

: \

-8 -6 -4 -2 0 2 4 6 X (4

Figure 7 Modal autofocus control (1DOF)

MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 8, No. 5, April 5 1995 263

2 DOF Modal Autofocus (Case 1, a m p ) 2 DOF Modal Autofocus (Case 2. Comp)

9 c

-5 0 x (i.)

5 -6 -4 -2 0 2 4 6 x 6.)

Flgure 8 Modal autofocus control (2DOF computational)

2 DOF Modal Auto-Focus (Case 1,Exp) 2 DOF Modal Auto-focus Case 2 (Exp)

- 4 - 3 - 2 - 1 0 1 2 3 4 -6 -4 -2 0 2 4 X (im) x 6.)

Figure 9 Modal autofocus control (2DOF experimental)

respectivcly. This difference in maximum configuration stems from the difference in maximum values as illustrated in a comparison of the plots in Figure 6. All in all, these figures reveal that the algorithm converges to the configuration that gives the maximum power regardless of whether the initial angle is acute or obtuse (indicated by Cases 1 and 2, respec- tively).

In summary, the autofocus control algorithm converged to the configuration that gives the maximum power whether the initial corner angle was acute (Case 1) or obtuse (Case 2). More spccifically, the algorithm converged to the experimen- tal maximum far-field power and not the theoretical maxi- mum far-field power. In the field, this property can be re- garded as a self-correcting feature that accommodates for atmospheric disturbances and the presence of conducting elements near the antenna.

ACKNOWLEDGMENTS The authors would like to give thanks to Dr. W. T. Joines, Professor of Electrical Engineering at Duke University, for his valuable assistance and support.

REFERENCES 1. Special Issue on Adaptive Antennas, IEEE Trans. Antennas Propa-

gat., Vol. AP-24, Sept. 1976, p. 575. 2. L. Silverberg and G. N. Washington, “Modal Control of Reflector

Surfaces Using Far Field Power Measurements,” Microwaue Opt. Technol. Lett., Vol. 7, No. 12, Aug. 20, 1994, p. 588.

3. L. Meirovitch, Analytical Methods in Vibrations, Macmillan, New

4. R. E. Collin, Antennas and Radiowave Propagation, McGraw-Hill,

5. L. Blake, Antennas, Munro Publishing, Silver Spring, MD, 1991.

York, 1967.

New York, 1985.

Receitted 10-20-94

Microwave and Optical Technology Letters, 8/5, 260-264 0 1995 John Wiley & Sons, Inc. CCC 0895-2477/95

TIME-DOMAIN ANALYSIS OF TRANSMISSION LINES WITH ARBITRARY NONLINEAR LOADS Chan-Eui Yun and Jung-Woong Ra Department of Electrical Engineering Korea Advanced Institute of Science and Technology 373-1, Kusung-dong, Yusung gu Taejon 305-701, Korea

KEY TERMS Transmission line, nonlinear loads, transient response, linearization, time stepping

ABSTRACT A direct time-domain method is presented for calculating uoltage waves at terminals and on transmission line terminating in an arbitrary nonlin-

264 MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 8, No. 5, April 5 1995