Analysis and design of large leaky-mode array employing the coupled-mode approach

Analysis and Design of Large Leaky-Mode Array

Employing the Coupled-Mode Approach

Cheng-Nan Hu, Member, IEEE, and Ching-Kuang C. Tzuang, Fellow, IEEE

Abstract—This paper presents the coupled-mode approach to the analysis and design of a large leaky-mode array, in which an -element microstrip array above a common ground plane supports coupled leaky modes (leakyEH₁ modes) at the first higher order. Following a brief description of a conventional full-wave nonstandard eigenvalue problem for solving the complex propagation constants, we present the detailed formulation of the coupled-mode approach, clearly showing the transformation of the nonstandard eigenvalue problem into a standard one. Thus, all the eigenvalues (complex propagation constants) and eigen-vectors (modal current distributions) are solved simultaneously, regardless of the size of the array ( ). Two key issues pertinent to the successful implementation of the proposed coupled-mode approach are addressed: the determination of the coupling coef-ficients and the uniqueness of the isolated uncoupled leaky mode, which represents the leaky modal solution of a single microstrip, but is derived from a system of coupled leaky-mode solutions provided the coupled microstrips have equal width and spacing. Closed-form expressions for obtaining the coupling coefficients of orders two, three, and four are presented. Theoretical studies of closely coupled microstrip arrays of two, three, and four elements show that the magnitude of the coupling coefficient ₊ be-tween elements and + decreases at the order of 10 . These theoretical case studies also lead to the same isolated uncoupled leaky-mode solution as predicted. Furthermore, the dispersion characteristics of the microstrip array at the first higher order ob-tained by the coupled-mode approach and the full-wave approach agree excellently for all the case studies. Error analyses indicated that at least two coupling coefficients ( ₊₁ and ₊₂) are required for obtaining accurate complex propagation constants with rms errors less than 1% for most of the leaky region of the particular array under investigation. An example of applying the proposed coupled-mode approach for analyzing a corporate-fed leaky-mode array of eight elements is reported, revealing that only four out of the eight leaky modes are excited. The coupled-mode theory predicts the far-field radiation pattern in the main beam region in excellent agreement with the measured results.

Index Terms—Coupled microstrip lines, coupled-mode ap-proach, coupling coefficients, leaky-mode antenna, leaky modes.

I. INTRODUCTION

M

UTUAL coupling of guided modes has been an impor-tant phenomenon widely applied in various microwave circuits such as directional couplers and filters [1]–[6]. A system of parallel coupled microstrips placed on a substrate with an in-finite ground plane, for example, supports bound modes if

Manuscript received October 8, 1998; revised October 14, 1999. This work was supported by the National Science Council, Taiwan, R.O.C. under Grant NSC 87-2213-E-009-105 and under Grant NSC 87-2213-E-009-106.

The authors are with the Institute of Electrical Communication Engi-neering, National Chiao Tung University, Hsinchu, Taiwan, R.O.C. (e-mail: cktzuang@cc.nctu.edu.tw).

Publisher Item Identifier S 0018-9480(01)02445-0.

parallel lines are present above the common ground plane, where is a positive integer. These bound modes are usually termed dominant or quasi-TEM modes although they are hybrid modes in nature, i.e., the superposition of TE and TM waves [7], [8]. The well-known bound dominant microstrip mode can be designated as order zero, followed by first (odd), second (even),

third (odd), , higher order modes, possessing either odd or even field symmetries. Adopting this notation, the dominant modes of the parallel lines are essentially the results of the mutually coupled microstrip modes of order zero. This paper, however, departs from the conventional results and investigates the less known, but useful guiding properties of the coupled mi-crostrip modes of order one, from which similar results can be obtained for microstrip modes of order two, three, , etc.

The open microstrip modes of order greater than zero belong to the class leaky modes. A single open microstrip has been known for supporting leaky modes in various ways [9]–[13], in which the first higher order microstrip mode leaks in the form of space wave or surface wave. Coupled leaky modes of order one, however, had been reported for a two- and a 20-element array, respectively, of which the leaky modes are tightly distributed in the dispersion characteristics [14]. When , the number of elements, is increasingly larger, the solutions for leaky modes are more difficult to obtain by full-wave methods [15], [16], which result in the nonstandard eigenvalue problem [7]. Imme-diately after the complex propagation constant is obtained, the modal corresponding currents on each of the coupled lines are readily known. Thus, each corresponds to a modal current vector, describing the modal current distributions of the mi-crostrips. Thus, for coupled leaky modes of order one, there are independent current vectors.

Section II of this paper transforms the nonstandard eigen-value problem into a standard eigeneigen-value problem by adopting the coupled mode approach [17], [18] mingled with the rigorous full-wave formulation of the parallel lines supporting max-imally coupled leaky modes, where is also a positive in-teger. With this notation, becomes the complex eigenvalue and the modal current vector is the complex eigenvector of the eigenvalue problem. This paper extensively expands the work of [14] by showing how to obtain the adjacent coupling coefficient and the coupling coefficients of nonadjacent lines in Section III. Substituting these coupling coefficients into the coupled-mode formulation described in Section II, the solutions for eigenvalue and eigenvectors , for , and , are obtained simultaneously, greatly reducing the time-consuming numerical efforts encountered by the well-known full-wave analyses [7], [19]. It is interesting to note that the coupled-mode theory had very often provided vivid physical insights for explaining the 0018–9480/01$10.00 ©2001 IEEE

Fig. 1. GenericN-element microstrip leaky-mode antenna array, including a CPW corporate-fed network, CPW-to-slotline transitions, and slotlines to excite the higher orderEH modes of the microstrip lines [21]. : CPW corporate-fed network followed by N CPW-to-slotline transitions on the backside of the substrate. : Microstrip leaky-mode antennas with length of_b L, width of W , and gap of S on the top side of the substrate.

guiding properties of complex waves in a variety of waveguides [18], [20], [21].

One important application of this paper is to analyze the mi-crostrip array from the leaky-mode perspective, as described in Section IV. As shown in Fig. 1, this array may consist of cou-pled microstrip lines printed on the top surface and a coplanar waveguide (CPW) feeding network on the backside of the sub-strate. After a continuous wave (CW) signal source is applied to the CPW input end, the CPW T-junction splits the signal into two paths of equal amplitude and phase. Such power di-viding sequence continues until, finally, each radiating element receives equal amount of power and phase. Followed by the CPW-to-slotline transitions, the uniformly divided signals are coupled to the slotlines to excite the modes of the coupled microstrips [22]. Since microstrip lines can be placed closely to form a compact linear array, strong mutual couplings between coupled lines will alter the dispersion characteristics of the leaky modes. Eventually, a large linear -element array that si-multaneously supports coupled leaky modes is established. The quest for obtaining all the complex dispersion character-istics considering these mutual coupling effects is essential for the first-pass design of the array. Section V describes how to apply the novel technique to accurately predict the far-field ra-diation pattern of an experimental corporate-fed eight-element microstrip array [22], demonstrating excellent agreement in the main-lobe region between the simulated and measured results. Conclusions are finally made in Section VI.

II. MODALANALYSIS ANDEIGENVALUEPROBLEMS

A. Full-Wave Approach: Nonstandard Eigenvalue Problem

When considering the microstrip array with fewer ele-ments, e.g., two- or three-element, the leaky properties of the coupled microstrips can be characterized by invoking the rigorous full-wave integral-equation method using the Green’s impedance function approach [16]. By applying the appropriate boundary conditions at the interfaces, the tangential electric-field components at the interface of each adjacent

Fig. 2. Normalized dispersion characteristics of the symmetrical coupled microstrips at the higher orderEH mode. The data obtained by the full-wave nonstandard eigenvalue approach (in solid lines) and those by the standard eigenvalue approach (in circle symbols) agree excellently. The parameters for the coupled microstrips are also shown in the inset.

region can be expressed in terms of current distributions. One can then write

(1) where and designates regions (1)-(3), respectively (see Fig. 1). Integrating (1) along all the metal strips, the gen-eral dyadic Green’s impedance function [ ] can be obtained as follows:

(2) where

(3a)

(3b)

Let the unknown current distributions and be ex-pressed in terms of a complete set of basis functions. Following the Galerkin procedure to solve (2), a system of homogenous linear equation can be obtained. The nontrivial solution for the unknown current coefficients can be found by equating the determinant to zero. The roots of the determinant are the propagation constants for microstrip modes of order zero, one, two, three, , etc.

Fig. 2 shows the normalized complex propagation constants of a two-element coupled microstrip of order one, namely, the

even mode ( ) and the odd mode ( ), where is the free-space wavenumber equal to and is the free-space wavelength. The inset of Fig. 2 also shows that the even mode (odd mode) can be obtained by inserting an equivalent electric (magnetic) wall at the symmetry plane. An-other view for distinguishing the two leaky modes is based on

the eigenvectors of the modal current distributions [ of (3a)] on the strips, either in-phase for the even mode or out-of-phase for the odd mode. These in-phase and out-of-phase modal current distributions are two orthogonal eigenvectors of the two-element microstrip array from the perspective of the coupled-mode approach, which will become apparent in sub-sequent discussions within this paper.

B. Coupled-Mode Approach: Standard Eigenvalue Problem

By invoking the coupled-mode theory [23], [24], the transversal and longitudinal electric and magnetic fields can be expressed as follows:

(4)

(5)

where denotes the number of microstrips, represents the complex propagation constant of the th microstrip in a single isolated condition, is the unit vector in (longitudinal) direc-tion, and are the normalized eigenfunction in the transversal (denoted by ) and longitudinal (denoted by ) directions, respectively, and and represent the corre-sponding modal current and wave impedance, respectively. Sub-stituting (4) and (5) into Maxwell’s equation and taking the transverse part will lead to

(6)

Applying the orthogonality relationship to (6) allows us to ob-tain a system of linear first-order differential equations for the modal currents [23], [24] as follows:

(7)

where , and , but . Essentially , the mutual coupling coefficient between the th element and the th element, can be expressed as

(8) Notice that the conversion of Maxwell’s equations into the cou-pled-mode equations (7) is referred as the classical method. When dealing with the leaky modes, the transverse fields grow

exponentially. Therefore, (8) is never used for computing cou-pling coefficients. Equation (7) is viewed phenomenologically [25]. Section III describes in detail how to obtain the coupling coefficients for the coupled-mode equation (7). Using a matrix notation, (7) is abbreviated as

(9)

where and . When

all the coupled microstrips are in equal width, must be equal to , where is the complex propagation constant of the leaky mode of a single microstrip. On the other hand, cou-pled leaky-mode solutions should exist in the -element array, denoted as , and . For a specific mode with com-plex propagation constant , the modal solution mandates

(10) Substituting (10) into (9) for , and , we obtain

(11) The source-free modal solutions require the nontrivial solutions for the modal current vector . Therefore, (11) leads to a standard eigenvalue problem by solving

(12)

where . The procedure for

obtaining the complex propagation constants is transformed to a classical eigenvalue problem of which the eigenvalues ( ) are the complex propagation constants and the eigenvectors ( ) represent the modal currents flowing at the microstrips. The te-dious root-finding process in the full-wave nonstandard eigen-value problem is eliminated.

III. COUPLING COEFFICIENTS OF THE PARALLEL COUPLEDMICROSTRIPS

A. Formulas for the Coupling Coefficients

Combining (11) and (12) for , one may deduce the coupling coefficient of the two symmetrical microstrips at the higher order leaky mode as

(13) (14) (15) where denotes the even-mode (odd-mode) solution. The corresponding normalized eigenvector and are, re-spectively,

(16) (17) implies that the leaky modal current distributions flowing at the symmetrical coupled microstrips are in-phase

Fig. 3. Magnitude (in decibels) of the coupling coefficientC obtained by the cases ofN = 2; 3; and 4, respectively.

(out-of-phase) with equal amplitude. Therefore, a perfect elec-tric conductor (PEC) and a perfect magnetic conductor (PMC) can be inserted along the symmetry plane, as shown in the inset of Fig. 2, for the even and odd modes, respectively. Applying the full-wave data already shown in Fig. 2, we obtain the coupling coefficient using (13). Applying the full-wave analyses again for a single identical isolated microstrip, the complex leaky-mode ( ) propagation constant is obtained. Substituting this value ( ) and the coupling coefficient ( ) into (14) and (15), i.e., the complex propagation constants obtained by the standard eigenvalue problem [see (14) and (15)], we find excellent agreement between both approaches, as shown in Fig. 2.

We can generalize the above discussions for a two-element case study to a large coupled microstrip array. Given a system of coupled microstrips of equal width ( ) and spacing ( ), we may rewrite (12) as

(18)

where is the complex propagation constant of the th leaky mode, for to . Expanding the determinant , and comparing order by order at both sides of (18) for , we obtain the following equations for solving the unknown cou-pling coefficients of and , representing the coupling coefficients of the adjacent elements and other-than-adjacent elements, respectively (see the inset of Fig. 3). Notably,

and since each coupled microstrip has identical width ( ) and is positioned at equal spacing ( ). Thus, we obtain the following system of equations for solving

and :

(19) where . Invoking the full-wave nonstan-dard eigenvalue approach for solving the solutions for , and , respectively, we obtain the value of , which is the arith-metic mean of the three complex propagation constants of

cou-pled modes. Such observation can be generalized by the following expression:

(20) Similarly, for , one may derive a system of equations as follows for solving the coupling coefficients , , and

, noticing that , and :

(21a)

(21b)

(21c)

B. Assessment of the Magnitudes of the Coupling Coefficients

Applying (13), (19), and (21), we obtain distinct coupling co-efficients for , , and , respectively. Assume that a system of tightly coupled microstrips of width 6.8 mm ( ) and spacing 3.4 mm ( ) with number of elements equal to two, three, and four are under investiga-tion. The magnitude of the coupling coefficients in decibel scale against frequencies of interests is plotted in Fig. 3 for , , and , respectively. values of the coupling coefficients for the case study of , coincide with those for . , one of data of the coupling coefficients for the case study of , also coincide with those for the case study of . Notice that the magnitude of is one order of magni-tude less than that of . This implies that the magni-tude of decreases very rapidly in the order of (10 ) as increases, for and ( ). When computing the eigenvalues (complex propagation constants) of a large array, the coupling coefficients obtained by assuming , i.e.,

for , are adequate for practical accuracy.

C. Numerical Results and Validity Checks

The data shown in Fig. 2 have validated the eigenvalue ap-proach for obtaining the complex propagation constants of the two-element symmetrical coupled microstrips. In this section, the confirmation of the proposed coupled-mode formulation is extended to the cases of and . Since the microstrips are all identical and placed an equal distance apart, the solu-tion for , i.e., the complex propagation constant of a single isolated microstrip of first higher order, must be the same for . is an arbitrary positive integer. Fig. 4 plots the results obtained by (20), the arithmetic mean of the coupled complex propagation constants, showing excellent agreement for all frequencies of interests in the cases of and . Notice that the full-wave solution for a single microstrip line is almost indistinguishable from the plots obtained by the cou-pled-mode approach. Such finding is important for conjecturing that the particular coupled microstrips system can be viewed as

Fig. 4. Plots of the complex propagation constant of a single isolated microstrip at first higher order obtained by (20) using the data of the coupled mode solutions of the two-, three-, and four-element coupled microstrips of identical width and spacing.

Fig. 5. Validity of the proposed coupled-mode approach is confirmed by studying a three-element array; data obtained by the full-wave analysis agree excellently with those by the coupled-mode approach considering all the coupling coefficients.

the mode coupling of a single leaky mode ( ) at each mi-crostrip.

Figs. 5 and 6 validate the coupled-mode approach that transforms the conventional full-wave nonstandard eigenvalue problem into a standard eigenvalue problem for and , respectively. The three (four) sets of data of the complex propagation constants obtained by the coupled-mode approach are virtually indistinguishable from those by using full-wave analyses, as shown in Fig. 6 for the entire leakage regime of interests. The coupled-mode approach is not only accurate for obtaining the modal solution, but also explicit for providing physical insights on the coupled lines behavior in terms of their corresponding propagation constants (eigenvalues) and modal current distributions (eigenvectors).

D. Error Analysis for Cases with Fewer Number of Coupling Coefficients

Section III-C shows the nearly error-free solutions of the cou-pled leaky modes obtained by the coupled-mode approach provided that all coupling coefficients are considered. This sec-tion investigates the percentage of rms errors if the term of cou-pling coefficients is intentionally reduced. Given a system of a coupled four-element microstrip array, as studied in the pre-vious section, the coupling coefficients of the particular case

Fig. 6. Validity of the proposed coupled-mode approach is confirmed by studying a four-element array; data obtained by the full-wave analysis agree excellently with those by the coupled-mode approach considering all the coupling coefficients.

Fig. 7. Percentage of the rms error defined by (22) for the four-element array using one term (C ), two terms (C andC ), and all coupling coefficients, respectively.

study contain the following terms , , and . Mean-while, the percentages of rms errors for the normalized phase constant and normalized attenuation constant are defined as fol-lows:

(22a)

(22b) The results of the coupled-mode approach assuming one term ( ), two terms ( and ), and three terms ( , , and ) are plotted in Fig. 7, which shows the nearly error-free solutions for and , when all coupling coefficients ( , , and ) are considered. The worst rms errors occur at the lowest frequency near 11.0 GHz on the left-hand side of Fig. 7, where 1.7% (1.86%) and 1.4% (1.55%) errors are observed for ( ) when coupling coefficients of one and two terms are considered, respectively. For a typical

leaky-wave antenna design of this study, the normalized atten-uation constant is usually chosen to be about 0.1 [21]. Corre-sponding the frequency of operation of the presented coupled microstrips is near 11.7 GHz (see Figs. 5 or 6). Notice that the two-term approximation results in a significant reduction in against one-term approximation, i.e., from 0.96% to 0.48% near 11.7 GHz. For practical use of the coupled-mode approach, at least two terms of coupling coefficients are recom-mended for obtaining solutions for all the coupled leaky modes. What follows is an array design example described in the fol-lowing section, in which we apply three coupled coefficients to ensure accuracy.

IV. MICROSTRIPLEAKY-MODEARRAYDESIGNSUBJECT TO ARBITRARYEXCITATIONS—THEORY

Given a microstrip leaky-mode array, the radiation fields are characterized by the modal current distributions (eigenvectors), , for , and . Considering an arbitrary input current excitations at the incident plane, where , (11) can be modified as

(23) Notice that the square matrix is diagonalizable. If , and are the eigenvalues of and , and are the associated linearly independent eigenvectors, the matrix

, formed by applying as the th column, can be used to diagonalize . That is,

(24) where is the diagonal matrix in the form of and is the inverse matrix of . Upon substituting (24) into (23), one derives the excited current vector

(25)

and the coefficient vector

(26) Given a known arbitrary signal , the coefficient vector can be obtained by (26) and, therefore, the resultant current distributions over the entire array are fully determined by (25). Applying the Fourier transform to the equivalent current sources on aperture, we obtain the far-zone electric field con-tributed by the th microstrip of length [26], [27] as follows:

(27)

where

is the free-space wavelength

where and are the propagation constant and attenuation constant of the th coupled leaky mode, respectively. The total far-zone electric field is the superposition of the -element microstrip leaky-mode array written as

(28)

where is the center location of th microstrip in the -direc-tion. Equations (27) and (28) explicitly relate the excited modal currents against the known current source to the far-field radiation pattern.

V. EXAMPLE OF ACORPORATE-FEDMICROSTRIPLEAKY-MODE ARRAYDESIGN

Fig. 1 shows a typical corporate-fed microstrip leaky-mode array. A proof-of-concept design of an eight-element array

with mm, mm, and mm

was fabricated on a 25-mil-thick Duroid 6010 substrate with relative dielectric constant 10.2. Careful designs were exercised for obtaining the input voltage standing wave ratio (VSWR) less than 1.2 at microstrip input feed and isolation better than 17 dB at divider outputs (made of a CPW T-junction) at the operation center frequency. Therefore, the mutual coupling due to the feeding network, as shown in Fig. 1, is minimized and can be neglected when a signal of a CW source is applied to the input port of the array during the measurement. The coupled-mode approach will explicitly show what coupled leaky modes contribute the radiation subject to a particular form of excitations. An eight-element leaky-mode array should have eight sets of eigenvectors , for , and . Each represents the modal currents of a specific leaky mode with respect to an eigenvalue (the complex propagation constant), as shown in Table I. Since the corporate-fed network results in signals of equal amplitude and phase reaching each microstrip, the incoming current source vector can be

expressed as . By invoking (26),

we obtain the coefficient vector , as tabulated in Table II, which explicitly shows that leaky modes numbered 2, 4, 6, and 8 are not excited. Only four leaky modes numbered 1, 3, 5, and 7 are presented in the corporate-fed leaky-mode array. The leaky mode denoted as determinates the corporate-fed array since the magnitude of is much greater than that of , , and . Substituting the computed coefficient vector and the eigenvalues of the excited leaky modes into (27) and (28), we obtain the far-field radiation as shown in Fig. 8, which also plots the measured radiation pattern and that obtained by the unit-cell approach described in [9] and [22]. Notice that the -plane ( – -plane) far-field patterns are normalized and plotted at the observation angle off broadside from 25 to 65 for close comparison. Although the unit-cell approach shows

TABLE I

EIGENVECTORSI ANDEIGENVALUES(COMPLEXPROPAGATIONCONSTANTS)OF ANEIGHT-ELEMENTCORPORATE-FEDARRAY

(W = 3:304 mm, S = 13:38 mm, h = 25 mil, " = 10:2,ANDf = 12:16 GHz

TABLE II

COEFFICIENTVECTOR[]OF THEPARTICULAREIGHT-ELEMENTARRAYUNDERSTUDY

Fig. 8. An almost replica of the measured radiation pattern in the main beam from 33to 65is obtained by the coupled mode analyses of the eight-element leaky-mode array. The well-known unit-cell approach [9], [21], however, predicts a narrower bandwidth with a slight shift in the beam-pointing angle.

close agreement with the measured data, the coupled-mode approach, which accounts for all the contributing coupled leaky modes, offer further improvement. An excellent agreement is observed for the radiation pattern in the main beam region from 33 to 65 at the -plane cut, between the measured data and those obtained by the coupled-mode approach. Importantly, the mutual coupling affects the array on the radiation pattern by slightly shifting the beam pointing direction from 43.2 to 44.7 and increasing 3-dB beamwidth from 13.5 to 14.7 . The latter effect can be attributed to the contribution of the excited leaky modes, possessing slightly different phase propagation constants. This effect cannot be predicted by the unit-cell approach (a single leaky-mode analysis) but, however, can be accurately predicted by using the coupled-mode approach (four excited leaky-mode analyses).

VI. CONCLUSION

This paper has presented and validated the coupled-mode approach for the analysis and design of the large leaky-mode array from the perspective of mode coupling of the leaky modes of higher order. Detailed formulation is derived to transform the well-known nonstandard eigenvalue problem using the full-wave approach to the standard eigenvalue problem by the coupled-mode approach, which simultaneously obtains all the eigenvalues (complex propagation constants) and eigenvectors (modal current distributions) regardless of the size of the array ( ). Consequently the proposed coupled-mode approach greatly reduces the tedious root-searching process of the nonstandard eigenvalue problem, which may become unman-ageable for a large array problem. Closed-form equations for obtaining the coupling coefficients inhering in the microstrip array are derived and the theoretical results indicate that the magnitude of the coupling coefficient between elements and decreases at the order of 10 as increases. Exten-sive study for validating the new approach is conducted both numerically and experimentally, clearly indicating that at least two coupling coefficients are required for obtaining accurate complex propagation constants with rms errors less than 1% for most leaky region of the particular array under investigation. Finally, the investigation of a corporate-fed eight-element array using the coupled-mode approach explicitly shows which leaky modes are excited at what levels and demonstrates that the effect of mutual coupling on the array performance can be accurately determined.

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Cheng-Nan Hu (S’97–M’99) was born in Taiwan,

R.O.C., on April 17, 1960. He received the B.S. de-gree in electronic engineering from the Chung Cheng Institute of Technology, Taiwan, R.O.C., in 1982, and the M.S. and Ph.D. degrees from the National Chiao Tung University, Taiwan, R.O.C., in 1989 and 2000, respectively.

Since 1982, he has been with the Chung Shan In-stitute of Science and Technology (CSIST), Taiwan, R.O.C. His research activities involve the design and development of phased-array antennas, reflector tennas, adaptive antennas, and active and passive microstrip leaky-mode an-tennas.

Ching-Kuang C. Tzuang (S’84–M’86–SM’92–

F’99) received the B.S. degree in electronic engi-neering from the National Chiao Tung University, Hsinchu, Taiwan, R.O.C., in 1977, the M.S. degree from the University of California at Los Angeles, in 1980, and the Ph.D. degree in electrical engineering from the University of Texas at Austin, in 1986.

From 1981 to 1984, he was with TRW, Redondo Beach, CA, where he was involved with analog and digital monolithic microwave integrated circuits. Since 1986, he has been with the Institute of Communication Engineering, National Chiao Tung University, Hsinchu, Taiwan, R.O.C. His research activities involve the design and development of millimeter-wave and microwave active and passive circuits and the field theory analysis and design of various complex waveguiding structures and large-array antennas. He has supervised 53 M.Sc. and 14 Ph.D. students.

Dr. Tzuang assisted in the formation of the IEEE Microwave Theory and Techniques Society (IEEE MTT-S), Taipei Chapter. He has served as secre-tary, vice chairman, and chairman in 1988, 1989, and 1990, respectively. Since 1994, he has also been on the Asia–Pacific Microwave Conference International Steering Committee, where he was the international liaison officer representing the Taipei Chapter.