A numerical approach to line antennas printed on dielectric materials

Computer Physics Communications

Computer Physics Communications 68 (1991) 441—450 North-Holland

A numerical approach to line antennas printed on dielectric materials Hisamatsu Nakano College of Engineering Hosei University, Koganei, Tokyo 184, Japan Received 15 August 1990

A numerical approach to line or wire antennas printed on dielectric materials is presented. The formulation is made using the method of moments, where the current on the line is expanded by piecewise sinusoidal functions. These functions are also used to form the impedance matrix elements. The numerical analyses based on this approach reveal the radiation characteristics of a bent dipole antenna, a loop antenna with perturbation elements, and an equiangular spiral antenna.

I. Introduction Radiation elements consisting of line or wire type arms, such as dipoles or loops, have appeared in many publications [1,13,14] as practical antennas. In most cases, these radiation elements have been analyzed on the assumption that they are located in free space with or without a conducting plane reflector. Needless to say, the conventional numerical techniques for line antennas in free space, of which refs. [2—5]are typical, are not able to handle line antennas in a space partially filled with dielectric material, A dielectric material may be used to provide support for practical line or wire antennas, together with a conducting plane reflector, which is a simple tool for obtaining a unidirectional radiation beam [9]. Consequently, both a dielectric and a conducting plane reflector are often incorporated into a printed-circuit antenna design. We have attacked this type of dielectric-conducting plane problems [15—17]by the real-axis numerical integration method, which is characterized by Sommerfeld-type integrals [6—8,10]. This paper reviews the generalized treatment of a curved or bent line or wire antenna printed on a dielectric substrate backed by a conducting plane reflector. The formulation to obtain the current on 0010-4655/91/$03.50 © 1991

—

a line or wire antenna is made using the method of moments. After deriving the impedance matrix, we analyze three types of antennas: a bent dipole antenna, a loop antenna with two perturbation elements, and an equiangular spiral antenna.

2. Formulation by method of moments 2.1. Assumptions and expansion functions A printed line or wire antenna of arbitrary configuration mounted on a dielectric substrate is shown in fig. 1, where the substrate thickness and relative permittivity are designated as B and Cr, respectively. We assume that the line or wire is perfectly conducting, and that its radius a is small enough for the thin-wire approximation to be applied, that is, a << A0, where A0 is the free-space wavelength. In other words, only the axial component of the current contributes to the antenna characteristics, such as the input impedance and the radiation pattern. The nonsyminetrical circumferential component may be neglected. We subdivide an antenna arm into many segments which may be regarded as linear elements [5,9], and express the current on the segments in the form 1(X) = ~‘nJ,( X), where the ~,‘s are

Elsevier Science Publishers B.V. All rights reserved

H. Nakano / Line antennasprinted on dielectric materials

442

xU

Fig. 1. Line or wire antenna configuration.

expansion functions, and the Ia’s are coefficients to be determined. Piecewise sinusoids are chosen as the expansion functions in this paper: sin [k (x x,, sin kd sin [k ( X~+ sinkd 0, —

(

J,, X)

=

—

2.2. Tangential electric field due to a single element .

A single element of length d is now picked from fig. 2a. This is shown in fig. 3. The electric field e due to the current 1( x ) I on this element is expressed as follows:

—

X,, ~

x)] ‘

n~

X

X,~,

~

‘

n+1’

otherwise.

d

e=f {(k21+ ~iI}

(1)

Yu

.I(x’)I dx’,

xu

Zn

~

Fig. 2. Coordinate system for subdivided line or wire antenna.

(2)

H. Nakano / Line antennas printed on dielectric materials

where - - -

‘,.~:f.1/

JJX

and

11Z

443

are Sommerfeld-type integrals,

at theareair—dielectric interface boundary Z = B: and obtained by applying conditions =

2u lim

f J0(pX) dA, De(A) lim f cos ~ J~(pA)~ e_~2_~

(9)

z-~B 0 =

2u(1

—

00

Cr)

2

0

z-’B

X De (~)t)ADm ~

Fig. 3. Coordinate system for a single element.

dA.

(10)

In these equations, (and eq. (16) below), originally where I = II + ,99 + 1, k = ~ and LI is the dyadic Hertz vector-potential function. It should be noted that the time dependence exp(+icot) is omitted throughout the paper. We write the current distribution on the single element as

derived by Rana [8], p is the distance between the source point (x’, y’, B) and the observation point (x, y, B), cos q = (x x’)/p, u = and —

De(A)

=

Dm(A)

I(x

sinkx’ sin kd

)=‘L

+I(/

sin k(d—x’) sinkd

,

(3)

where ‘L and I~ are temporary expressions for the expansion function coefficients. By substituting eq. (3) into eq. (2) and taking the inner product with a unit vector I tangential to the element on which an observation point (x, y, z = B) is located, the tangential component of the electric field at this observation point can be obtained:

with and

~ + P~ecoth

~ieB,

(11)

=~Er+!~te tanh !.LeB,

(12) (13)

~L=VA2_k2 !.te =

(14)

~/A2— Crk2.

From equations (5), (7) and (8), the inner product L I is expressed as

f ~~a d(

L

I

air

~

(1

1) +

a arrx ~ (

Xsin kx’ dx’ B)

=‘L

+IUikd(U•1),

(4)

where L

=

.~.

I))

a au +

Xsinkx’dx’ (15)

0

J

(TXI +

17J~)sin kx’ dx’

(5)

where LI is defined as

(6)

112(Er1)Ulim

0

art ax2

+

a2ir axaz

+

k2rF,

+

ayaz

___________

De(A)Dni(A) dA.

(16)

(7)

The treatment of the integrals of eqs. (9) and (16) is addressed in the literature [11].

(8)

Using an expression for the derivative with respect theobservation coordinate point t, which thebeginning distance between tothe and isthe

a2rrX a2rr ayax

f00Jo(pA) e~

z—.B 0

In the above equations, T~” and T’~obey the following definitions:

=

~

+(I.I)f”k2H~~sin kx’ dx’,

u=fd(r~+ryp) sin k(d—x’)dx’

TX =

art

+fdfa

sin1 kd(~1)

~

H. Nakano / Line antennasprinted on dielectric materials

444

of the observation point segment, we may simplify eq. (15) as dd

L.I=j +

I

artx

aH\

L~ I and U~•I, adding additional subscripts n 1 and n, —

dd

,)sinkx’dx’

_____

ax~_,1sinkx~_1dx~_1

Ln_i•Ifj~jk

~—~---+.~--

(1

I)fdk2ir

sin kx’ dx’.

(17)

21-rX t.n— I

+~

(21)

Xsin kx~_1dx,ç_1, Similarly, the inner product U• I in eq. (4) defined by eqs. (6), (7) and (8) is simplified to

U

•

I = jd d

( EIG1

0 ~ aHX

+

an’~

sin k(d—x~) dx~

+(I~.I)fdk2rI~~ sin k(d—x~)dx~,

sin k(d—x’) dx’

0

+(I.I)fdk2ir sin k(d—x’) dx’. (18) 0

(22) where G,~.1 and G,, obey the following definition: G11= —H~+H,1 (j=n—1, n).

(23)

2.3. Tangential electric field due to all elements It should be noted that ir and iTT from eqs. (9) The tangential component of the total electric field due to the total current flowing on all the segments can be derived using the results obtained for the single element in the previous section. We consider an electric field due to a pair of elements labelled n 1 and n, as shown in fig. 2a, where the observation point is located on the element labelled t. It is convenient to adopt the local coordinate systems (x~_1, y~_1,z~_1) and (x~, y,,, zn), in which the corresponding unit vectors are defined as (I~_~, ~ i~_~) and (I ~ z~),as shown in fig. 2b. The x-coordinate is taken in such a way that it is parallel to the element axis. The tangential component of the electric field from the two elements n 1 and n is derived from eq. (4), with ‘L = ‘U = —

—

e~,= I~(e~ .1),

and (16) are emphasized in eq. (23) by subscripts t and J’ which indicate that the observation point and source point are located on the elements labelled t and j( = n 1, n), respectively. These subscripts show that the distance p in eqs. (9) and 2 2 }i/2 (16) is given as p p,~= {(xoB x~)+y0~ in which the position of the observation point (xOB, Yon) is measured using the local coordinate system of the jth element (j = n 1, n). Also, it should be noted that in eqs. (21) and (22) the relation aH,X1/8x1 = an~1/ax; has been used. Summation of eq. (19) over all currents gives us the tangential component of the total electric field, which we write as —

‘

—

—

—

Etan = sin~kd~

(19)

where

+ fdN

sin kx~_1dx~_1

sin k(d—

x~) dx~}~

(24)

where

laG \

___

e~•1= sin kd(L~

•1+ ~.I).

(20)

Here we have made use of eqs. (17) and (18) for

N~=

(I = n

2(I +

—

1, n).

k

1.I)H~1

(25)

H. Nakano / Line antennasprinted on dielectric materials

Equation (24) is derived using piecewise sinusoidal expansion functions for the current on the printed antenna. Equation (25) is formally similar to the kernel of Pocklington’s equation for a dipole antenna in free space [2], and may be regarded as a generalized kernel for an arbitrary configuration supported by a dielectric substrate. The case of (B = ~r = 1) corresponds to an

445

j=n

where each g~ (i = m 1, m; composed of three parts. —

—

1, n) is

d gm-i.j

f

=

(cos kxm_i)(h~i,j) dxm_i

0 + (Im_i

x fd(sin

~,

antenna system in free space.

kXm_i)(h~i

j)

dx,,,_1

0

+f”(cos kx~,i)(h~i~) dxm.1, 2.4. Impedance matrix formation

(29)

0 i’d

I

g~~=

—

The boundary condition on the perfectly conducting line or wire is given by ~ + Etan = 0, or = Etan, where Eta0 is given by eq. (24) and E~0 is the tangential component of the electric field incident or impressed on the surface of the line or wire. (Note: ~ is a known function.) We obtain a set of linear equations, applying weighting functions to the abovementioned equation of boundary condition,

d

.I~)f(sin k(d— Xm)}(h~j) dXm f”{cos k(d— xm)}(h~ ) dxm (30)

+(Im

0

in which (1)

çd =

[v]

=

[z][I],

(26)

where [V], [Z] and [I] are the voltage, impedance, and current matrices, respectively, for the method of moments. The unknown currents [I] of eq. (26) are then determined by standard techniques. In Galerkin’s with testingmatrix functions defined as in eq. method, (1), the impedance element Z,,,, is calculated by

{cosk(d—xm)}(sh~j)dxm

Jo

j

—

,~

/

H;~_1cos kx0_1 dx~..1,

~

(32) d

h~_1= 2~ =

f

~

sin kx~1dx~_1, sin k(d— x’) dx’

0 (dflX

h~ l.fl

‘.11

=

j TI 1

,

/

1~,,_1cos kx~_1dx~_1, Zmn=J

0

enxm_i e,,

dxm.1 Xm

dXm,

i’d

(3)

(27)

where the local distances xm_i and xm are defined equal to X(e,, XmI,,, i ~) and respectively. asSubstituting andX (e,,Xm, Im) from eq. (20), with I = Im —l and I = I,,,, respectively, into eq. (27), we obtain —

—

I

—

~,

k ~2 sin lcd) ~gm-i.n—i +

g,,,,,),

- -

—

+

(35)

0 —

tdsiflk(d—xm) + Jo sin kd

(33) (34)

d (3)

tdsin kxm_i sin kd

(31)

d

g,,,_1•,,

(28)

/

Jo~ H,~cos k~d—x~)dx,,.

~36

Inspection of eqs. (29) and (30) allows us to make the following observations: (1) Because of the presence thesecond inner prod1~),ofthe term ucts (Imon -i the I) antenna and (Im depends configuration. (2) The third integrals in each of the eqs. (29) and (30) above, with their h~. 11 and h~1 dependence, are the terms associated with the dielectric material, and vanish when the relative permittivity Cr is one. Therefore, an antenna of arbitrary configuration in free space, backed by a plane conductor, may be analyzed using just the first and

H. Nakano / Line antennas printed on dielectric materials

446

second terms. This gives us an alternative approach to such an antenna system. (3) The impedance matrix element has diagonal symmetry; that is, Zmn is equal to Znm. This property saves computer time in the [Z] calculation. In addition, a particular configuration may possess other symmetry properties which further reduce the computational burden, as illustrated later.

—

E 6

r ~,.

30’ ~~6a’

-6~’~

3. Numerical analyses

-90’

____________

90’

We analyze three antennas based on the previous formulation. The antenna characteristics are evaluated by the currents which are numerically

s=o’

0’ 30’

obtained. For a delta-gap source excitation, input impedance is given as Z10 = V/I,0, where I~is the current flowing between the terminals where the infinitesimal gap is excited with an input voltage V. The far-field radiation may be calculated using the stationary phase method [12] or directly calculated by integration of the obtained current distribution, as seen from eq. (2). From the farfield radiation the axial ratio is calculated.

80

~E0’

-go’

90’

~=90’plane Fig. 5. Radiation pattern of a bent dipole antenna with a bend angle ofT = 30°.

bend angle 1=30’ curren

—

plane

60

-

140

-

20

________ _________ --‘-________

3.1. A bent dipole antenna

ii.

The inset of fig. 4 shows a bent dipole antenna

1

of total wire length 0.4 A0, which is bent at permittivity = ±0.1A0. ~r, and The the wire substrate radius thickness a, the relative B are a = io—~A0, Cr = 2.0, and B = 0.1016 A0, respec-

-

0 ~

a

~

feed point -20

+

~--

-

—

~

erty tively. of Since line symmetry the bent configuration with respect tohas thethe X~, propaxis, the impedance matrix elements have the relation

— — — —

bend point)

________________________________ -0.2A 0 0 +0,2~~ _______

~.wire length Fig. 4. Current distribution of a bent dipole antenna with bend angle of r = 30°.

= ZN+l_n,N+I_m for an N-by-N matrix. Tocalculations of Zm~may gether with the Zmn =2 Znm symmetry property, this means that the N be reduced to (N+ 1)(N+ 1)/4 and N(N+ 2)/4 for odd and even values of N, respectively. (Note: This type of reduction holds true for a configuration with point symmetry or origin symmetry.)

Zmn

-

)B.P.

~

H. Nakano / Line antennasprinted on dielectric materials

447

vu =

R

1~ + jX1~

e

~end~gleT Xin

Fig. 7. A loop antenna with two perturbation elements.

-50

Fig. 6. Input impedance of a bent dipole antenna versus bend angle.

T

0

5

3.2. A loop antenna The loop whose perimeter is of the order of one wavelength is usually used as a resonant antenna (with a standing-wave-type current distribution) radiating a linearly polarized wave. However, if the current distribution is changed to a travelingwave-type along the loop, a circularly polarized wave (CPW) can be generated. In this section, we establish a traveling-wave-type current distribution using two perturbation elements. Figure 7 shows a loop antenna with two perturbation elements, each having a length of 8. The wire loop and perturbation elements are printed on a dielectric substrate of Cr = 2.5, with a wire radius of 0.001A0. Figure 8 shows the normalized perturbation length 8/P as a function of the nor-

-

0.10

-

0.05

-

-

‘.,

A typical current distribution of the bent dipole is shown in fig. 4, in which the bend angle T is 3o0 Using this current distribution, we obtain the radiation pattern shown in fig. 5. It can be said that, although cross polarization is caused by bending the arm, it is very small. Figure 6 shows the input impedance versus bend angle. The bent dipole antenna does not have an impedance of pure resistance for = 300, but it does have a nearly pure resistance at a bend angle of 900. In other words, the antenna is resonant in the vicinity t ang e.

0.15

0.75

./‘

0,73 2~

~

G-.

I

0.71

I

0.10

0.15 0.20 0.25 thickness B/Ag Fig. 8. Normalized perimeter P/A0 and normalized perturbalion element length 8/P versus normalized substrate thickness B/A5, when a loop antenna radiates a circularly polarized wave.

malized substrate thickness B/A , where P is the perimeter at which a CPW (with an axial ratio 0.4 dB) is radiated from the loop, and Ag is a guided wavelength defined as Ag = A0/ ~ A typical current distribution for B/A = 0.1 in fig. 8 is depicted in fig. 9. It is clearly shown that a traveling-wave-type current flows along the loop.

30 S ~

~

15

/ ~

0

~

feed Poin~\ ‘s,. ~

,,1

/

-15

\,,~r i~

1

+

\

,‘

J~. 1

03714 perirneter~ Fig. 9. Current distribution for B

0.3714 =

O.lA5.

H. Nakano / Line antennasprinted on dielectric materials

448

~Rtn+JXt~~

F::

~

0

!~

-

~>1rrit~

0.95

I 0.10

I

I

I

0.15

0.20

0.25

—4———

0

..—‘ø.—

0,95

arm length,)

0

thickness B/Xe

(a)

Fig. 10. Input impedance versus normalized thickness B/A5.

=

0°

=

:2O~~•.EL

_

/

90°

(b)

J

-20

Fig. 13. Current distribution (a) and radiation pattern (b) of an equiangular spiral antenna.

~.

.

Fig. 11. Radiation pattern for B = 0.1A5. ER, main polarization (right-hand circularly polarized wave); EL, cross polarization (left-hand circularly polarized wave).

The two points where the discontinuity of the current is observed are the locations at which the perturbation elements are connected. Figure 10 shows the input impedance Z~,,as a function of the normalized thickness B/A under the condition that a CPW (with an axial ratio 0.4 dB) is radiated. For B/A = 0.1, the input impedance is calculated to be = 40 The radiation pattern for this case is shown in fig. —

u

0.)

________________

Finally, we analyze a spiral antenna printed on a substrate of ~r = 2.5 and B = O.25Ag = 0.25 x A0/ ~ The antenna arm, wound by an equiangular spiral function r = 0.03 A0 exp(0.25c1), is approximated by linear segments which are consecutively connected, as shown in fig. 12 where the wire radius is 0.0011 A0. The calculated current distribution and its corresponding radiation patterns are shown in fig. 13.

I[~ i

radiation patterns reveal that the spiral radiates a broad, +48°.circularly polarized beam, whose HPBW is

/

u

-

________________

3.3. A spiral antenna

The current shows a gradual decay from the feed point along the arm, as in the case of an Archimedean spiral antenna in free space [5,9]. The

/ \

_~/

11. It is found that the loop radiates a broad, circularly polarized beam whose half-power beamwidth (HPBW) is about ±450

S

/

~

Fig. 12. An equiangular spiral antenna,

S

x~

Since the current distribution is of a decaying traveling wave type the spiral can have fairly wideband characteristics for the input impedance and axial ratio. This is shown in figs. 14 and 15.

H. Nakano

~ 0

F

100 0

210

81n

+

ix~~ ~

•

~

-

-

— -

...

——

~ ~1O0 0.9

1.0

1.1

1.2

449

is also found that a loop antenna can radiate a circularly polarized wave (CPW) with the help of two perturbation elements. Finally, the numerical results clearly explain that the CPW radiation from an equiangular spiral antenna is related to a smoothly decaying current distribution, as in the case of an Archimedean spiral antenna.

3~ 200

1-’

/ Line antennasprinted on dielectric materials

1.3

normalized frequency fn

Fig. 14. Input impedance versus normalized frequency ~ = f/300 MHz.

6 ~ lii-

~2j______________________ E

3~ 0

‘

0.9

1.0

1.1

1.2

normalized frequency

Acknowledgements

The author would like to acknowledge the various forms of generous assistance given by his colleagues at The University of California, Los Angeles and Hosei University, Tokyo. In particular, appreciation is extended to Professor N.G. Alexopoulos for his constructive suggestions. Appreciation is also expressed to Dr. S. Kerner, Dr. J. Yamauchi and K. Hirose for their numerical

1.3

and experimental work.

f~

Fig. 15. Axial ratio versus normalized frequency f,,. MHz.

f,,

=

f/300

The bandwidth over which the axial ratio is less than 3 dB is calculated to be 21%, with an impedance of nearly pure resistance 250 fi.

4. Conclusions The numerical approach to a line or wire antenna printed on a dielectric substrate backed by a conducting plane reflector is formulated using the method of moments. The unknown current on the antenna arm is expanded using piecewise sinusoidal functions. Adopting the local coordinate systems facilitates the derivation of the tangential component of the electric field at the line or wire surface. Some observations are made for the derived impedance matrix elements, The obtained formulation is used to analyze three antennas. The numerical analyses reveal the antenna characteristics, such as the current distribution, radiation pattern, and input impedance. A bent dipole antenna shows that changing the bend angle results in an impedance of pure resistance. It

References [I] R.C. Johnson and H. Jasik, eds., Antenna Engineering Handbook (McGraw-Hill, New York, 1984) chs. 4 and 5. [21 W.L. Stutzman and G.A. Thiele, Antenna Theory and Design (Wiley, New York, 1981) p. 310. [3] H. Nakano, M. Tanabe, J. Yamauchi and L. Shafai, Spiral slot antenna, in: Proc. Fifth lEE Int. Conf. Antennas and Propagation ICAP87 (1987) pp. 86—89. 14] K.K. Mci, On the integral equations of thin wire antennas, IEEE Trans. Antenna Propag. 13 (1965) 374. [5] H. Nakano and J. Yamauchi, Characteristics of modified spiral and helical antennas, Proc. lEE Pt. H 129 (1982) 232. [6] JR. Mosig and FE. Gardiol, The near field of an open microstrip structure, IEEE Int. Antennas Propag. Soc. Symp. Digest, vol. 1(1979) p. 379. [7] A.J.M. Soares and A.J. Giarola, The effect of a dielectric cover on the current distribution and input impedance of printed dipoles, IEEE Trans. Antennas Propag. 32 (1984) 1149. [8] I.E. Rana and N.G. Alexopoulos, Current distribution and input impedance of printed dipoles, IEEE Trans. Antennas Propag. 29 (1981) 99. [9] H. Nakano, K. Nogami, S. Arai, H. Mimaki and J. Yamauchi, A spiral backed Propag. by a conducting plane reflector, IEEE antenna Trans. Antennas 34 (1986) 791. [10] A. Sommerfeld, Partial Differential Equations in Physics, vol. VI (Academic, New York, 1949).

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H. Nakano

/ Line antennas printed on dielectric materials

[11] D.R. Jackson and N.G. Alexopoulos, An asymptotic cxtraction technique for evaluating Sommerfeld-type integrals, IEEE Trans. Antennas Propag. 34 (1986) 1467. [12] RH. Clarke and J. Brown, Diffraction Theory and Antennas (Wiley, New York, 1980). [13] H. Nakano, J. Yamauchi, K. Kawashima and K. Hirose, Effects of arm bend and asymmetric feeding on dipole antennas, Int. J. Electron. 55 (1983) 353. [14] H. Nakano, H. Tagami, A. Yoshizawa and J. Yamauchi, Shortening ratios of modified dipole antennas, IEEE Trans. Antennas Propag. 32 (1984) 385.

[15] H. Nakano, S. Kerner and N.G. Alexopoulos, The moment solution for printed antennas of arbitrary configuration, IEEE Trans. Antennas Propag. 36 (1988) 1667. [16] H. Nakano, K. Hirose, T. Suzuki, S. Kerner and N.G. Alexopoulos, Numerical analyses of printed line antennas, Proc. lEE Pt. H 136 (1989) 98. [17] H. Nakano, N. Tsuchiya, T. Suzuki and J. Yamauchi, Loop and spiral antennas at microstrip substrate surface, in: Proc. Sixth lEE Int. Conf. Antennas and Propagation ICAP89 (1989) pp. 196—200.