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WKB Analysis of Impedance-Level Modulated MicrostripStructuresAmiya Kumar MallickPublished online: 15 Dec 2010.

To cite this article: Amiya Kumar Mallick (2001) WKB Analysis of Impedance-Level Modulated Microstrip Structures, Electromagnetics, 21:1, 51-72, DOI:10.1080/02726340119028

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WKB Analysis of Impedance-Level ModulatedMicrostrip Structures

AMIYA KUMAR MALLICK

The propagation characteristics of sinusoidally impedance-level modulated microstripstructures are investigated by solving a standard second-order differential equation usingthe WKB method (named after Wentzel, Kramers, and Brillouin). The WKB method, whenmodified by the successive approximation technique, is found to remove all the drawbacksof the method (zero order) and is able to predict accurately the behaviour of the disper-sion characteristics of the propagation phenomenon. In addition, the dependence of thewave equation solution-both magnitude and phase, on frequency is also explored. Fur-thermore, convergence tests and experimentation establish the validation of the proposedmethod. Investigation is carried out to explore the dependency of the propagation proper-ties on the modulation parameters of the structure.

Introduction

Many researchers (Nair & Mallick, 1984; Choudhuri & Bandyopadhyay, 1981; Glandorf &Wolff, 1987; Mallick & Bhattacharrya, 1989) have carried out the investigation of microstripstructures with periodically varying strip width in the last decade. The studies clearly disclosethat the instantaneous impedance level of the structure is a strong function of the instantaneousstrip width. More precisely, the sinusoidal variation of width of the metallic patch creates anonsinusoidal but periodic variation in impedance level. On the other hand, to achieve sinu-soidal axial variation in impedance level, the width of the planar structure on dielectric sub-strate needs to be suitably modulated periodically and certainly not in sinusoidal fashion.

In this paper, a sinusoidal impedance-level modulated structure has been considered foranalysis with the aim of exploring various aspects of propagation characteristics of the peri-odic structure.

The analysis of such an impedance-level modulated structure necessitates a solution ofa second-order differential equation (Scott, 1953) with periodic coefficients satisfied by theline voltage or current. In the present problem, we intend to apply the WKB method (namedafter Wentzel, Kramers, and Brillouin) for the solution of the differential equation becausethe coefficients of the equation, which happens to be of Hill’s form, are reasonably slowlyvarying functions along the axial coordinate. The main advantages of the method are that itis less time-consuming and presents an easily obtainable close-formed expression revealing

Electromagnetics, 21:51 –72, 2001

Copyright © 2001 Taylor & Francis

0272-6343 /01 $12.00 + .00

51

Received 30 July 1999; accepted 20 March 2000.Address correspondence to Amiya Kumar Mallick, Department of Electronics and Electrical

Communication Engineering , Indian Institute of Technology, Kharagpur - 721302, West Bengal,India. E-mail: akm@ece.iitkgp.ernet.i n

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all properties inherent to the equation. It is noted that the zero-order approximation of themethod suffers from certain limitations so far as the Mathieu–Hill-type equation is con-cerned and hence does not lead to practical results. The WKB method, therefore, requiressuitable modification for better approximation to the solution of the equation yielding a realpragmatic approach to the problem. Starting with a suitable wave equation (of Hill type) theWKB method is applied to a solution that is modified by employing successive approxima-tion (Brillouin, 1950). This is found to accommodate expected alternate pass band and stopband characteristics of the periodic waveguide. Experimental data have been incorporated tojustify the analytical process through numerical simulation.

Analysis

The impedance-level modulated line is governed by the following differential equation (Scott,1953; Ghose, 1963)

(1)

where V(x) is the voltage variation along the line,z(x) = z0 [ 1 + m cos b P x ] = series impedance / unit length (Datta Roy, 1965),y(x) = y0 / [ 1 + m cos b P x ] = shunt admittance / unit length (Datta Roy, 1965),b P = 2p / P,m = depth of modulation, andP = periodicity of modulation.

It may be noted that the propagation constant of the line andis independent of the axial distance x, while the x-dependent characteristic impedance of theline

Note that is the average impedance level about which the “instanta-neous” impedance level is a sinusoidal modulation.

Intrinsically, the propagation constant g and the average characteristic impedance level Z0

of the unmodulated uniform microstrip line are functions of the angular frequency x , relativedielectric constant e r of the substrate, thickness of the substrate h, and the width of the stripconductor W. In this article, for the sake of simplicity, zero-thickness metallization has beenassumed.

With fairly good accuracy, a useful empirical relation between the characteristic imped-ance Z0 and the physical parameters W/h of an unperturbed uniform microstrip line may beobtained from Wheeler’s (1977, 1965) works. By suitably sampling the sinusoidal variationof the characteristic impedance Z0(x) for different values of x over a period P, W/h for each xis obtained using Wheeler’s inverse relation. These values are finally consolidated to realizethe entire axially width-modulated microstrip structure. For the sake of accuracy, the width Wmay be adjusted for strip thickness and mixed dielectric (Wheeler, 1977).

It may be noted that for the sinusoidal nature of Z0(x), the corresponding strip widthW(x)/h of the metallic patch requires an axial variation of some periodic function of x that isnot, essentially, sinusoidal in nature.

Z0 5 z0 y0

Z0 x 5z x

y x5 Z0 1 1 m cos b px .

c 5 z x y x 5 z0y0

d2V x

dx2 21

z x dz x

dx dV x

dx2 y x z x V x 5 0,

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Further, for lossless structure, c = j b , where b is the phase constant of a uniform line.According to Wheeler (1977) and Schneider (1969), b = 2 p / k eff, where k 0 = free space wavelength and

due to a mixed dielectric around the strip conductor. Therefore, the propagation constant of anunperturbed lossless line may be expressed as where c = 2.9987 3 108

m/s.

Hill’s Equation

The wave equation (1), in general form, may be expressed as a linear second-order differen-tial equation with varying coefficients as follows:

(2)

where variable coefficients are given by

(3)

Equation (2) belongs to the category of the Sturm–Liouville class. However, for the sakeof convenience, the general form of (2) can be reduced to its normal form (Schroedinger-typeequation) by suitably eliminating the second term of equation (2) in the following way, with-out losing the generality of the equation.

It may be seen in (3) that no pole in either A (x) or B (x) can exist in the real domain of x,as the value of the depth of modulation, m, is always less than unity. Hence, A (x) and B (x)are analytic and every point in the domain is an ordinary point. If x0 is such an ordinary pointand x is a variable point, then the following transformation may be assumed as

(4)

Substitution of (4) and (3) in (2) yields the normal form (Whittaker & Watson, 1962; Ince,1926) of (1) or (2) as

(5)

where F (x) for the lossless cases (i.e., c = j b ) is given by

(6)

Further, the solution of (1) or (2) expressed in the format of (4) can be more explicitlywritten by putting the expression of A (x) of (3) in (4) as

F x 5 b 2 234

m2b 2

P sin2 b Px

1 1 m cos b Px 2 21 2m b 2

P cos b Px

1 1 m cos b Px .

d2f x

dx2 1 F x f x 5 0,

V x 5 f x exp 2 1 2 #x

x0

P n d n .

A x 5m b p sin b px

1 1 m cos b px and

B x 5 2 z0y0 5 2 c 2.

d2V

dx2 1 A x dVdx

1 B x V 5 0,

b 5 2 p f e ref f c,

e ref f 5e r 1 1

21

e r 2 12

1 110hW

2 1 2

k ef f 5 k 0 e ref f ;

WKB Analysis of Impedance-Level Structures 53

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The above explicit transformation shows that the phase factors of the variables V(x) and f(x)of (4) are identical, while their magnitudes are different. Thus, for the purpose of analysis ofthe propagation characteristics of the modulated structure, it is enough to study the phase fac-tor of the transformed variable f(x) that is basically a solution of (5).

In (5), the variable coefficient F(x) given in (6) can be expanded in terms of an infinitebut convergent series of differential periodic coefficients having even functions of x. In theparlance of mathematics, such a differential equation (5) is known as Hill’s equation (Hill,1886; Brillouin, 1953; McLaclan, 1951). However, solutions of this type of equation requiredetermination of an infinite but very slowly convergent determinant involving a difficult andtime-consuming computation process.

WKB Method

We therefore propose to solve the transformed differential equation (5) using an approximateanalytical technique known as the WKB (Morse & Feshbach, 1953; Burke & Manheimer,1990; Froman & Froman, 1965; Pauli, 1980) or phase integral method (Heading, 1962). In thespirit of WKB approximation, F (x) of (5) is assumed to be a slowly varying function of x.Further, it may be noted that the WKB approximation attains an instability in the vicinity ofthe turning points (Morse & Feshbach, 1953; Ghatak, Gallawa, & Goyal, 1991, 1992; Xiang& Yip, 1994), where F(x) becomes zero. The method suffers from the lack of analytic con-tinuation between the solutions on either side of the turning point, deviating the solutions fromtheir exact value.

However, because of its inherent simplicity, the use of the WKB method is very popularand well established in physics (Chiang, 1995) and mathematics (Moriguchi, 1959) in spiteof its limitations and inaccuracy in certain regions. Further, the method has been subsequentlyupgraded and improved by researchers like Moriguchi (1959), Wang (1991), and others(Morse & Feshbach, 1953; Ghatak, Gallawa, & Goyal, 1991; Brillouin, 1948).

Turning points and cutoff frequency. It is, therefore, important and necessary to find outthe turning points and, for that matter, to discuss the nature of the functions F(x) and H(x) ofthe problem under study.

It may be noted that the expression of F(x) of (6) may be rewritten as

F(x) = b 2 - H(x), (6a)

where,

and

As is apparent from (6a), F(x) is dependent on frequency through b , as also on the physicalparameters like m and P of the guiding structure, through H(x). In the case when frequency iszero, F (x) = - H (x), and therefore F (x) has to cross through zeros as H (x) does. The zero-crossings of F (x) are known as the turning points. It may be carefully noted here that even if

b 52 p f

ce ref f .

H x 5 3 4m b P sin b Px

1 1 m cos b Px

2

1 1 2 m b 2

P cos b Px

1 1 m cos b Px

V x 5 f x1 2 m cos b Px0

1 2 m cos b Px

1 2

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F (x) is pushed up more and more towards the positive side by the increase of signal frequency,turning points persist until a certain frequency level is reached. It is, therefore, justified to definethe cutoff frequency, fc, as the frequency at which the negative peak point of F (x) just attains alevel equal to zero. So, for frequencies above fc, F (x) is always positive for all values of x. Log-ically, fc further links to the maximum level of H (x), i.e., H (x)max, where the first derivative ofH (x) is zero. Because of F (x) being zero at fc, = H(x)max or, in other words,

(6b)

Considering the fact that the WKB approximation technique does not behave properlyand provides inaccurate solutions in the neighborhood of the turning points (Brillouin, 1950)and exponentially decayed solutions for the negative values of F(x) entering into the contin-uous stop band, it will be proper to select the frequency range above fc for the present analy-sis for which F(x) is always positive and the WKB method is quite valid and safely applicable.

In the present problem, the design restriction imposed on the physical dimensions of themicrostrip line, which always keeps the (mW0/P) ratio well below unity (i.e., W0 m/P <<1),allows F(x) to remain as a “slowly varying” function of x. Moreover, F(x) does not possessany pole or singularity, as m is always less than unity. However, in order to ensure that the F(x) value is always greater than zero, the range of frequency of operation for a given set of lineparameters present in the H(x) function or vice versa must be suitably adjusted.

Keeping this in mind, the following parameters of the modulated structure have been cho-sen for the purpose of theoretical and experimental investigations:

� cu-cladded sheet: e r = 10.2 and h = 0.0635 cm;� unperturbed strip width: W0 = 0.25 cm (fixed for matching purposes).

Two samples of guiding structure are considered:

Sample 1: P = 1 cm, m = 0.2, 0.4, and 0.6.Sample 2 : P = 2 cm, m = 0.2, 0.4, and 0.6.

Results of the computer computation of (6a) is depicted in Figure 1 showing the varia-tion of H(x) as a function of x within the period P for Sample 2. It is clear from this figure thatH(x) is a periodic function of x with period P and perfectly symmetric around the point P/2,having positive values over the regions on either side around x = 0 and x = P, while negativevalues around its center (i.e., at P/2). Interestingly, with increasing values of m, the gradualappearance of a small positive peak on either side of the almost flat positive response of thecurve is observed. Here, the peak is prominent in m = 0.6, while it is not so in m = 0.2. Fur-ther, it is also observed that with the increasing value of m, the positive peak slowly driftstowards the center from either side. The picture is the same for the other sample as well.

The negative peak, on the other hand, is normally broader in width and smaller in mag-nitude at lower values of m and gradually becomes narrower and larger as m increases in steps.This study, further, reveals that all these features associated with m, although essentially iden-tical for different values of P, the peaks of H(x), both positive and negative, assume inflatedmagnitudes for lower values of P and vice versa.

It may be worthwhile to mention here that the positive peaks of H(x) play an importantrole in deciding the cutoff frequency of the guiding structure because of the relation given in(6b). Investigation on F(x) for various values of m and P at and around cutoff frequency man-ifests that F(x) is always positive throughout the period when the signal frequency is abovecutoff. Consolidated results of this study, in this regard, are given in Tables 1 and 2. Turningpoints for frequencies below cutoff are also indicated.

f c 5 2.9987 3 108 p e ref f2 1 H x max Hz.

b c2

WKB Analysis of Impedance-Level Structures 55

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56 A. K. Mallick

Figure 1. Plot of H(x) as a function of x (cm) for Sample 2, i.e., P = 2 cm, m = 0.2, 0.4, 0.6 .

Table 1. Locations and peak values of H(x) of the guiding structures. All x- values are given in cm; W0 = 0.25 cm, h = 0.0635 cm, e r = 10.2.

P m (+) side of H(x) (–) side of H(x)

(cm) at x = 0&P Hmax Peak locations Hmax at P/2

1 0.2 3.290 3.290 0.00, 1.00 – 4.3950.4 5.640 5.796 0.14, 0.86 –13.1600.6 7.402 10.690 0.26, 0.74 –29.610

2 0.2 0.823 0.823 0.00, 2.00 –1.2340.4 1.410 1.449 0.28, 1.72 –3.2900.6 1.851 2.673 0.52, 1.48 –7.402

Table 2. Cutoff frequencies and turning points of the guiding structures. Locations x aregiven in cm and frequencies in GHz; W0 = 0.25 cm, h = 0.0635 cm, e r = 10.2.

P m Cutoff frequency Turning points at

(cm) fc (GHz) Locations (cm) Frequency (GHz)

1 0.2 3.05537 0.239, 0.761 2.000.4 4.05272 0.322, 0.678 2.000.6 5.50443 0.375, 0.625 2.00

2 0.2 1.52715 No turning point 2.000.4 2.02636 0.377, 1.623 2.000.6 2.75221 0.697, 1.303 2.00

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Calculation shows that the cutoff frequency, fc = 4.05 GHz for P = 1 cm and m = 0.4, while,fc = 2.02 GHz for P = 2 cm and m = 0.4 ; fc further reduces for higher values of P and instantvalues of m. So, X-band (8.2 to 12.4 GHz) may safely be used.

Assuming a case with an incident wave of unit amplitude and without any reflected wave,(5) will have a solution obtained through the use of the WKB method (Morse & Feshbach,1953; Burke Manheimer, 1990), as

(7)

In a periodic structure with periodicity P, the variation of phase of the wave, as seen in(7), depends solely on S. It has been computed for different values of m and P. However, Fig-ure 2 exhibits one of such results in which the phase shift per period, b 0 P, is found to varywith frequency for a range from 2 to 12 GHz. The periodicity P is kept constant at 2 cm andm is varied from 0.2 to 0.6 at a stepwidth of 0.2. The dispersion curves (zero order) of the sam-ples, in general, show clearly two distinct regions: (i) a region of no propagation, which isimmediately followed by (ii) a region of continuous propagation similar to high-pass filters.

F(x) is negative up to a certain range of frequency making S imaginary, hence the spec-tral region results in a stop band. Thereafter, S becomes real, causing an attenuation-less prop-agation. However, the theory, surprisingly, does not predict and support the alternate pass bandand stop band behavior of the structures that is inherently present in all types of periodicwaveguides. Hence, the solution, (7), is highly approximate and needs modification and isreferred to as the zero-order solution.

Modification of zero-order WKB solution. It may noted that the appearance of interspersedstop bands in the spectrum is due to the phenomenon of cumulative reflections; i.e., each cell

f x 5 F x 2 1 4exp 2 jS , S 5 #x

0F x 1 2dx.

WKB Analysis of Impedance-Level Structures 57

Figure 2. Dispersion characteristics (zero-order WKB) for sample 2, i.e., P = 2 cm, m = 0.2, 0.4,0.6. (Note: No stop band.)

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of the periodic structure reflects back a certain amount of the incident wave, and all these ele-mentary wavelets propagating backwards produce a large reflected wave. Such a result maybe obtained through the application of the successive approximations.

Successive approximation (Brillouin, 1950, 1948). For the sake of convenience, let thesolution of (7) be written with new notation:

(8)

here G(x) is a given real periodic function with period P. In comparison with (5), one findsthat

f(x) = G2 – (3/4) (G9 /G)2 + (1/2) G 0 /G, (9)

where prime indicates differentiation. Equation (8) gives a reasonable solution to (5). Thezero-order WKB approximation is evidently obtained by taking G = G0 = F1/2, omitting G 9and G0 , considered earlier.

A systematic set of successive approximations can be worked out by assuming an expan-sion (Brillouin, 1950):

(10)

where successive terms would be of order d 2, d 4, ..., and

(10a)

The terms in the right-hand side of (10) are given as

(11)

Therefore, the expansion G of (10) when put into (8) would yield the first independent solu-tion of (5). Another independent solution of (5) is the complex conjugate of U, given by

(12)

In transmission line problems, U corresponds to a propagating wave to the right and V tothe left directions. Two cases may arise depending on whether F(x) is greater or less than zero:

(i) S = real, which corresponds to propagating waves;(ii) G and S are imaginary, which corresponds to attenuated wave without propagation.It may be noted that a condition (Brillouin, 1948): UV 9 – VU9 = 2 j must hold betweenthe two independent solutions.

It may be noted here that the solution thus obtained is based on the assumptions indicatedin (10a) and that the method fails in the following three cases:

V 5 U* 5 G2 1 2 exp 1 jS .

G0 5 F 1 2,

G1 5 3 8 G¢ 2

0

G30

2 1 4 G²

0

G20,

G2 51

2G03 2

G¢0

G0

2 G¢1

G¢0

2G1

G02 1 2

G²0

G0

G²1

G²0

2G1

G²0

2 G21

G90

G20

< d , G²

0

G30

< d 2, d 2 ½ 1.

G 5 G0 1 G1 1 G2 1 …

U 5 G x 2 1 2exp 2 jS , S 5 #x

0

G x dx;

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(i) G0 Þ 0 corresponds to total reflection at the end of the line and is often referred to asthe turning point.(ii) G09 Þ ¥ corresponds to sudden changes in the properties of the line, with propaga-tion on both sides of the discontinuity in F(x). This is often called “partial reflection.”(iii) G00 Þ ¥ results in partial reflection as in (ii), but the discontinuity occurs in the deriv-ative of F(x), i.e., in F9 (x), and there is a sharp change in angle in the curve F(x).Now, to consider the case of cumulative reflection by taking into account all reflections

on possible irregularities of the periodic line, U and V (= U *) are taken to be the solutions ofthe differential equation

(13)

where,

and differs very little from the known function F(x).The G(x) function is periodic and exhibitsthe same period P as the original Hill’s equation f 0 (x) + F(x) f(x) = 0 (see (5)). It is assumedthat the function G(x) has been obtained through (10) and (11).

Considering U as a first approximation, f = U + g, which is substituted in (5). Then (13)is used to obtain

(14)

The function U of (8) gives a solution of (13) exhibiting pure propagation and reflections.Using (13), a set of equations with right-hand terms representing a continuous distribution ofsources along the periodic line can be built. This results in an emission of secondary waveletspropagating in both the directions and starting from all points along the line, particularly frompoints where (H – F) is not negligibly small. With this in view, g can be expanded as

where

(15)

Schelkunoff (1946) presented practical methods for the integration of (15), which may bewritten in the following form:

(16)

The solution of (16) may be obtained assuming g and g9 to be zero at x0. Hence, the solutionof the equation becomes

(17)

where U and V are the two independent solutions of the homogeneous equation (13), with thecondition that UV 9 – VU 9 = 2 j. The hn functions represent the fictitious sources distributedalong the line, resulting in additional waves propagating to the right (i.e., U function) or to theleft (i.e., V function).

gn x , x0 5U x

2j #y 5 x

y 5 x0

hn y V ¢ y dy 2V x

2j #y 5 x

y 5 x0

hn y U ¢ y dy,

d2gn

dx2 1 Hgn 5dhn x

dx5 h ¢

n 5 H 2 F gn 2 1.

g ²1 1 Hg1 5 H 2 F U,

g ²2 1 Hg2 5 H 2 F g1,…

g ²n 1 Hgn 5 H 2 F gn 2 1

g 5 g1 1 g2 1 g3 1 g4 1 . . .

f ² x 1 F x f x 5 g ² 1 F x g 1 F x 2 H U 5 0

H x 5 G2 2 3 4 G¢

G

2

1 1 2 G²

G

U ² 1 H x U 5 0,

WKB Analysis of Impedance-Level Structures 59

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Integrating by parts, (17) becomes

(18)

Solutions and propagation constant. Using equation (18) and keeping in mind that h19 (y)= (H – F)U, two independent solutions of (5) can be obtained as

(19a)

and

(19b)

For a periodic structure with period P, identical set of solutions can be obtained at x + P as itis at x. Hence, another set of independent solutions based on linear relations of other solutionsmay be formed as given below:

(20)

It may be noted that the condition UV 9 – VU 9 = 2 j specifies the following relation.However, the application of Floquet’s theorem leads us to

(21)

where c 0 is the propagation constant of the modulated structure. Hence, in order to explore thephase variation, only coefficient a1 is to be determined.

Again, considering (19a) at x = x0+ P and applying Floquet’s theorem, it may be shownthat

(22)

Further, recalling that at x = x0, the wave of unit amplitude is incident on the structurewithout causing any reflection, it leads to the fact that C = 1 and D = 0. Hence, U(x0) equalsQ(x0); i.e., U(x0) = Q(x0).

Now, substituting the values of the constants in (19a) and working on a few steps, onemay get

(23)

As a consequence,

where J is real and given by

(24)J 5 #x0 1 P

x0

H 2 F UU*dy.

cosh c 0P 5 cos GavP 1 1 2 J ·sin GavP ,

a1 5 exp 2 jGavP 1 1j

2 #x0 1 P

x0

H 2 F UU*dy .

c 0P 5 2 j #P

0

Gdx i.e., c 0 5 2 jGav, where Gav 5 1 P #P

0

Gdx .

cosh c 0P 5 1 2 a1 1 a*1 5 Re a1

*Q x 1 P 5 a1Q x 1 b1Q* x ,

Q* x 1 P 5 b*1Q x 1 a*

1Q* x .

Q* x 5 U* x 1 g*1 x 5 D* x 1 C* x U*

Q x 5 U x 1 g1 x 5 U 1 212j #

x

x0

H 2 F UU*dy 1U*

2j #x

x0

H 2 F U2dy

Q x 5 U x 1 g1 x 5 C x U 1 D x U*,

gn x, x0 5 2U x

2j #x

x0

H 2 F gn 2 1U*dy 1

U* x

2j #x

x0

H 2 F gn 2 1Udy.

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The procedure thus developed provides us with a practical WKB method to yield a use-ful solution of Hill’s equation (5).

It may be recalled that only the first step in the series of successive approximation has sofar been considered. It may be found necessary to go a few steps further and to use the expres-sion (15). In such a case, (19) can still be maintained. Then, in order to accommodate a num-ber of successive approximations, the expression for J in (24) is modified as

(25)

when the approximation considering only the first n terms of (15) is used. J is no longer a realquantity. Hence, the expression for cosh ( c 0 P ) finally boils down to

(26)

Dispersion characteristic and frequency bands. From the knowledge of the values of Gav

from (21), Jr and JI from (25), the phase shift per period, b 0 P of the line now, can be deter-mined using equation (26) for different frequencies of operation, and thus the dispersion dia-gram of the structure may be obtained. Interestingly, the study shows the occurrences ofalternate pass bands and stop bands interspersed in the frequency spectrum as expected inany periodically loaded structure. It may therefore be concluded that (26) is capable of ensur-ing a more accurate and detailed description of the frequency spectrum of the modulatedstructure with the most desirable and expected filter-like characteristics that are otherwiselacking in the zero-order WKB method.

It may be noted that the following conditions are usual and apparent for the determina-tion of the bandwidths of the pass bands and stop bands of the periodic structure:

(i) –1 < cosh ( c 0 P) < +1 corresponds to pass bands;(ii) ú cosh ( c 0 P) ÷> 1 corresponds to stop bands;(iii) cosh ( c 0 P) = ± 1 corresponds to cutoff frequencies.

Further, for a lossless case, c 0 = j b 0, and therefore cosh ( c 0P) = cos ( b 0 P).

Numerical Analysis and Computation

Convergence Analysis

For the sake of practical and useful analysis of the problem it is highly desirable that the solu-tion expressed in terms of a sum of a series of a large number of elements, viz., U, g1, g2, g3 ...etc. in (25), must converge to a constant value. Therefore, in order to ensure an effective andmeaningful computation of (25) and (26) numerically, the convergence behavior of the solutionis investigated. Based on that, an optimum value of the correction order, n of gn is estimated,keeping in mind to achieve the maximum possible accuracy of the results in minimum possi-ble computer time.

Figure 3 displays the convergence behavior of the phase shift per period, b 0 P, as againstn (correction order) and m (modulation depth) of the structure. In this case, m values of thestructures have also been varied from 0.2 to 0.6 with a stepwidth of 0.2 at a fixed operatingfrequency 6 GHz that is within the pass band yet somewhat close to the cutoff. The purpose

cos h c 0P 5 1 2J i

2cosh GavP 1 1 2 J r sin GavP .

J 5 #x0 1 P

x0

H 2 F U 1 g1 1 g2 1 g3 1 … 1 gn U*dy

J 5 J r 1 jJ 1

WKB Analysis of Impedance-Level Structures 61

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of the study is to observe, if any, the effects of m on the convergence behavior. It is found that,in general, the parameter under investigation, b 0 P, rapidly and gracefully converges to a fixedvalue with n. It is, further, noted that there is no need for any correction order, n (i.e., n = 0) ifthe operating frequency is far off from the cutoff values, say, at the middle of the pass band,where the modification of the WKB method is found to be redundant. On the other hand, then-value has to be nonzero if the operating frequency in the pass band is close to the cutoff,requiring correction. In short, the closer the cutoff, the larger the n-value is in respect of theoperating frequency. However, reviewing all aspects of the complications involved in theproblem, especially at cutoff, the n-value equal to 11 is chosen to be optimal for all subsequentcomputations.

The procedure that has been proposed can be summed up as follows: G0 may be calcu-lated from the given function F(x) of equation (6). Then G and H are determined from (10)and (13), respectively. They are used to find the two solutions U and V (= U ), and gn. The first11 terms in g (i.e., n = 11) are considered for calculation. Finally, the phase shift per period,b 0 P, has been computed from (26).

Dispersion Characteristics

The computed phase shift per period, b 0P (degree), under the modified WKB scheme (26) isplotted over the frequency range of 2 to 12 GHz for Samples 1 and 2, respectively, and shownin Figures 4(a) and 4(b). In these figures, TP represents the region in the spectrum where F(x)is always negative. fc, the cut-off frequency, is obviously the upper limit of the region abovewhich F(x) is always positive. The SB, on the other hand, is the width of the stop band of the

62 A. K. Mallick

Figure 3. Convergence behavior of phase shift per period, b 0 P, for Sample 2, i.e., P = 2 cm and m= 0.2, 0.4, 0.6 at frequency f = 6 GHz.

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WKB Analysis of Impedance-Level Structures 63

Figure 4. (a) Dispersion characteristics (modified WKB) for Sample 1, i.e., P = 1 cm, m = 0.2, 0.4,0.6. (Note: No pass band for m = 0.6 from 2 to 11.5 GHz.) (b) Dispersion characteristics (modifiedWKB) for Sample 2, i.e., P = 2 cm, m = 0.2, 0.4, 0.6.

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propagation of the wave. It may be noted that for Sample 1 (m = 0.6 only), there is no passband present in the spectral region from 2.0 to 11.5 GHz. It starts appearing from 12 GHzonly. A frequency difference of 0.5 GHz is apparently observed in all the band transitionsbecause of identical frequency step size being taken in present computations. An accurateband transition in all the cases may be achieved by reducing the step size suitably. In addition,the detailed analyses of the filter characteristics of the modulated structures are given in Tables3 and 4. What has been discovered from the analyses is the following:

1. fc , the cutoff frequency, the widths of the stop bands and pass bands, and their locationson the spectrum are strongly dependent on the periodicity P and the modulation indexm of the structure.

2. fc , representing the upper limit of the region for F(x) < 0, decreases steadily with theperiodicity P, irrespective of m-values.

3. On the contrary, fc increases with m for a fixed value of P. But this increment with mgets retarded a bit with the increase of P.

4. A wide stop band always follows fc. The location of the stop band center graduallyshifts with m, towards the higher side of the spectrum. Interestingly, the width of thestop band is found to steadily increasing with m, but strikingly declines with P.

5. In subsequent pass bands, the phase shift per period, b 0P, does not show any apprecia-ble deviation in magnitude when m is varied from 0.2 to 0.6 irrespective of P-values(Figures 4 (a) and 4(b)).

6. Interestingly, when the effects of m and P are integrated into a new dimensionless para-meter such as W0m/P, the widths of the stop bands immediately after fc are seen tobuild up progressively as the new parameter advances upwardly (Table 4).

7. Unlike the zero-order solution, the modified solution by successive approximationundoubtedly exhibits the presence of the pass bands and stop bands interspersed inthe spectrum of the propagation characteristics of the periodic structure.

64 A. K. Mallick

Table 3. Sample specifications: W0 = 0.23 cm, h = 0.0635 cm, e r = 10.2; frequency range: 2 to 20 GHz.

Period First stop Pass band Stop band Pass band P (cm) m fc (GHz) band (GHz) (GHz) (GHz) (GHz)

0.25 0.2 12.0 12.5 – >20.0 — — —0.4 >20.0 — — — —0.6 >20.0 — — — —

0.50 0.2 6.0 6.5 – 10.5 11.0 – >20.0 — —0.4 8.0 8.5 – 15.0 15.5 – >20.0 — —0.6 11.0 11.5 – >20.0 — — —

1.00 0.2 3.0 3.5 – 5.0 5.5 – >20.0 — —0.4 4.0 4.5 – 7.5 8.0 – 10.0 10.5 11.0 – >20.00.6 5.5 6.0 – 11.5 12.0 – 15.0 16.0 16.5 – >20.0

2.00 0.2 — <2.0 – 2.5 3.0 – >20.0 — —0.4 2.0 2.5 – 3.5 4.0 – >20.0 — —0.6 2.5 3.0 – 5.5 6.0 – 7.5 8.0 8.5 – >20.5

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Wave Equation Solutions

Investigation continues, in this section, to study the nature, behavior and dependence of theaxial distribution of the wave equation solutions on frequency. The analytical steps establishedin the sections on the “WKB Method” and “Numerical Analysis and Computation” have beentranslated into a computer program, and the complex solutions, both u(x), the zero order solu-tion (7), and U(x), the modified version (19), are evaluated for their magnitudes and phases asfunctions of the axial length, x, at typical representative signal frequencies. As a test piece,Sample 2 (P = 2 cm, m = 0.2) has been chosen for this study. The typical frequencies selectedfor the study are based on the filter characteristics of the sample and are the following: 2.0GHz (mid–stop band), 4.0 GHz (pass band closed to cutoff), 10.0 GHz (pass band far fromcutoff). Figures 5–7 depict these instant magnitude and phase distributions. Here, cutoff rep-resents the transition between a stop band and its next pass band and is found to be around 3.0GHz. Interestingly, it is observed that at mid–stop band (2 GHz), the correction factors gns rep-resenting the sources of secondary wavelets propagating on either side of the guiding struc-ture are predominantly present and as such, the zero-order solution u(x) is massively modifiedby gns all through the period, causing U(x) to completely contrast with u(x). Figure 5(a)depicts a very nice symmetrical magnitude distribution of u(x) with two peaks crossing above1.0, while the same for U(x), in Figure 5(b), reveals a more realistic distribution with the mag-nitude strengthening in steps to a very high value of the order of 1013 at P. It is now absolutelyclear that the secondary backscattered wavelets predominantly react with the propagatingwave to effectively create a stop band. Wherever it is a reality, the process of activating andregulating the presence of correcting factors improves the results, which is otherwise lackingin the zero-order WKB method. Studies also reveal that whenever there is suppression of ashort stop band due to the inherent insensitivity of the zero-order WKB method, the proposedmodified WKB technique automatically discloses the exact details of the spectrum andupgrades the performance with required corrections.

However, at higher frequencies above cutoff (3.0 GHz), the correction process continuesto prevail for some more depth into the pass band region of the spectrum, yet keeping magni-tude peaks of the variations well within the limit of 1 which is evidently clear in Figure 6(a).With further increase of frequency within the pass band, the correction process graduallydiminishes and finally disappears when the frequency is far away from the cutoff, say, at 10GHz as shown in Figure 7(a), where the magnitude distributions of u(x) and U(x) are foundto gradually merge and remain confluent throughout the period. The phase distributions alsocorroborate the same phenomenon all through the spectrum, as illustrated in Figures 5(c),6(b), and 7(b).

WKB Analysis of Impedance-Level Structures 65

Table 4. Stop Band Widths

W0 m / P Stop band width (GHz)

0.025 1.000.050 2.500.075 3.500.100 4.500.150 6.500.200 7.500.300 9.50

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66 A. K. Mallick

Figure 5. (a) Plot of the magnitude of u (zero order) as a function of x at freq. = 2.0 GHz. P = 2cm, m = 0.2 (mid-stop band). (b) Plot of the magnitude of U (modified ) as a function of x at freq. =2.0 GHz. P = 2 cm, m = 0.2 (mid-stop band). (c) Plot of the phase angles of u (zero order) and of U(modified ) as functions of x at freq. = 2 GHz.

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Further increase of frequency in the pass band shows a gradual reduction in the magni-tude variations for both the solutions. The magnitudes of both the solutions tend to becomeconstant with respect to x as b 2 takes over control of F(x) by setting aside the effects of H(x)at frequencies as high as, say, 40 to 50 GHz.

In comparison with the zero-order solution, the present modified WKB method thereforesuggests a better approximation and more pragmatic solution of the system. The successiveapproximations of WKB procedure provides the correct system behavior with adequately bet-ter accuracy in terms of the close form expressions. It is also less time consuming.

Experimental Validation

Besides the checks on the convergence characteristics of the numerically obtained solutions,the successive approximation technique is further validated by comparing the analytical-numerical data with the corresponding experimental observations. For that purpose, twomicrostrip samples of the following specifications have been constructed on a double-sidedcu-cladded dielectric substrate with e r = 10.2 and h = 0.0635 cm:

� P: 1 cm,� W0: 0.25 cm,� the number of sinusoidal periods used: 8,� m = 0.2,� m = 0.4.

A Network Analyzer (HP 8410A) and a Reflection/Transmission Test Set (HP8743A) inconjunction with a sweep oscillator are used. However, the modulated microstrip line under

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test is made to resonate by short circuiting the line at a plane of symmetry. Apparently, whenthe output plane of the line is shorted to ground, the input plane of the line, under resonance,will show the low impedance level. Under the condition of this simulated resonance, all theresonant frequencies are noted. When the number of periods of the line is q (here, q = 8), thephase shift per period becomes kp /q, where k is an integer. The exact value of k, in this case,is determined by probing the number of nulls along the short-circuited line; i.e., the resonatorso formed.

The experimental observations are shown in Figure 8 by the -o-o-o-o- line for m = 0.2 andthe – – D – D – D – D – line for m = 0.4, superimposed on the corresponding analytical plots of the

68 A. K. Mallick

Figure 6. (a) Plot of the magnitudes of u (zero order) and U (modified ) as functions of x at Freq.=4 GHz, P = 2 cm, m = 0.2. (Note: Pass band closed to cutoff: freq. 2.6373 GHz.) (b) Plot of thephases of u (zero order) and U (modified ) as functions of x at freq. = 4 GHz. P = 2 cm, m = 0.2.(Note: Pass band, closed to cutoff freq. 2.6373 GHz.)

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phase shift per period (degree) over the frequency range of 4 to 10 GHz. The trend of varia-tion of the parameter is in good agreement with theoretical and experimental data.

Measured data on phase shift per period of periodic structures with several other valuesof m and P show distinctly a spectral gap when approach is made from two consecutive passband regions. The trend of the extrapolated curves of these pass bands creates the gap thatdoes not essentially possess any real value of phase shift, b 0 P. This gap in the cutoff regionbecomes more and more prominent with larger values of m/P ratio for a fixed value of W0 andprovides the values of cutoff frequencies of the periodic structure under the experimentalstudy.

WKB Analysis of Impedance-Level Structures 69

Figure 7. (a) Plot of the magnitudes of u (zero order) and U (modified) as functions of x at freq. =10 GHz. P = 2 cm, m = 0.2. (Note: Pass band, very far away from cutoff.) (b)Plot of the phases of u(zero order ) and U (modified ) as functions of x at freq. = 10 GHz. P = 2 cm, m = 0.2. (Note: Passband, very far away from cutoff.)

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Conclusions

The wave propagation through a microstrip periodic structure with sinusoidal impedance-level modulation has been successfully investigated in this paper. For this purpose, an appro-priate second-order differential equation has been solved using the WKB method. Becausezero-order solution of the equation was found to be inadequate to reveal the details of thepropagation characteristics of the structure at and around the cutoff frequencies and also in thestop bands, a rigorous modification to the method employing successive approximation hasbeen suggested. An exhaustive study has been carried out in this paper. The analysis is foundto be extremely efficient and accurate in exhibiting all the features of the propagation charac-teristics, e.g., axial variation of the complex solution of the wave equation, propagation con-stant as a function of frequency for various physical parameters of the structure, etc. The x - bdiagram of the structure shows the typical pass band stop band characteristics of the line,which are otherwise lacking in the zero-order WKB method. The analytical technique sug-gested and established here has also been corroborated with rigorous experimental verifica-tion data.

The turning points of the structure— inherent characteristics of the system—have alsobeen thoroughly discussed. It was found that in the spectrum below a specific turning point,the system does not respond, as F (x) goes negative. Further, the spectrum just above the turn-ing point where F (x) becomes fully positive and the system is made capable always inheritsa wide stop band, which is normally followed by a series of alternate pass bands and stopbands interspersed. All possible mathematical conditions for system failures at the turning

70 A. K. Mallick

Figure 8. Experimental verification : Phase shift per period (in degree ) as a function of frequency(GHz) over the range 4 to10 GHz for the sample with P = 1 cm and m = 0.2, 0.4.

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points have been clearly brought out and thoroughly discussed. The absolute cutoff frequencyof the structure may be determined based on the position of the turning point.

The convergence test of the modified WKB method has been meticulously carried out andconvincingly and comfortably achieved even for higher values of m and lower values of P overthe X-band. This test certainly establishes the feasibility, practicality, and validity of the methodemployed without demanding excessive computer time and memory. The WKB method, a pop-ular and widely used one, after being refined through successive approximations as in the presentcase, may find wide application in analyzing arduous and intricate problems of identical nature.

The investigation of the planar structure inflicted with sinusoidal impedance-level mod-ulation discloses that the propagation characteristics of such structures are strong functions ofm and P. These parameters are found to have ample commands on the bandwidths and thepositions of the bands in the spectrum. This property of the structure is readily and skillfullycapitalized on in the filter design. It is also observed that the periodic impedance-level mod-ulation enables ones to design microstrip components with greater ease and flexibility becauseof inclusion of two extra design parameters, m and P.

References

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