Dumbbell-shaped defected ground structure

Dumbbell-Shaped Defected Ground Structure

Sushim Mukul Roy, Nemai C. Karmakar, Isaac Balbin

Department of Electrical and Computer Systems Engineering, Monash University, Clayton Campus,VIC 3800 Australia

Received 5 August 2005; accepted 4 April 2006

ABSTRACT: A simplified equivalent circuit model of a dumbbell-shaped defected ground

structure (DGS) is developed. The equivalent circuit model is derived from the equivalent

inductance and capacitance, which develop because of the perturbed return current path on

the ground and the narrow gap, respectively. The theory is validated against the commer-

cial full-wave solver CST Microwave Studio. Finally, the calculated results are compared

with the measured results. Good agreements between the theory, the commercially available

numerical analyses and the experimental results validate the developed model. VVC 2007 Wiley

Periodicals, Inc. Int J RF and Microwave CAE 17: 210–224, 2007.

Keywords: defected ground structure; electromagnetic bandgap; photonic bandgap; lowpass

filter; frequency spectrum; bandstop filter; high impedance surface

I. INTRODUCTION

Wireless communications have been playing a vital

role in mankind’s way of living since their inception

in 1895 with the trans-Atlantic transmission by Mar-

coni. In the last few decades, the impact of wireless

communications has increased tremendously because

of the high penetration of mobile phones and wireless

computers. High frequency microwave signals are

the key element of the modern wireless technology.

Effective transmission and reception of signals as

well as rejection of interference are only possible

with efficient design procedures. The memory hungry

emerging communications technologies have de-

manded ever larger bandwidth from very compact

designs. Nowadays ‘‘Internet and Inbox in your pock-

ets’’ are not a dream, but a reality due to this efficient

design. The contrasting requirements of large band-

width, the largest ever possible functionality per unit

volume, the reconfigurability of the circuit for multi-

frequency operations and compactness in design

have imposed tremendous pressure on RF and Micro-

wave designers.

These contrasts are not only limited to the me-

chanical and electrical requirements of the micro-

wave circuits, but also in overcoming the physical

principles of electromagnetic wave propagations. For

example, the limited bandwidth, gain, axial ratio,

signal-to-noise ratio, bit-error-rates and inter-modula-

tion products are limited by the inherent losses of

dielectric materials, their dielectric constants and the

material properties of active devices such as diodes

and transistors. Throughout the last decade, tremen-

dous efforts have been invested to overcome these

limitations in RF and microwave circuits. Electro-

magnetic bandgap structures (EBGSs), also known as

photonic crystals at microwave frequencies, have

been playing a vital role in mitigating these challeng-

ing issues in microwave active and passive designs.

EBGSs are a class of periodic dielectrics, which are

the photonic analogs of semiconductors. EBGSs ex-

hibit wide band-pass and band-rejection properties at

microwave and millimeter-wave frequencies and

have offered tremendous applications in active and

passive devices. While various configurations have

Correspondence to: N. C. Karmakar; e-mail: [email protected]

DOI 10.1002/mmce.20215Published online 6 February 2007 in Wiley InterScience (www.

interscience.wiley.com).

VVC 2007 Wiley Periodicals, Inc.

210

been proposed in literature, only the planar etched

EBG configurations have attracted much interest due

to their ease of fabrication, integration with other cir-

cuits and compatibility with the hybrid microwave

integrated circuits and Microwave Monolithic Inte-

grated Circuits. The passband of an EBGS is used as

slow wave medium that is useful for compact design.

The stopband is useful for suppressing surface waves,

leakage and spurious transmission. Because of these

unique properties of EBG structures, they have

potential applications in filter, antennas, waveguides,

phased arrays and many other microwave devices

and components [1, 2].

Conventional EBGs are 2-D periodic structures

that satisfy Bragg’s condition; the inter-cell separa-

tion (period) is close to a half guided wavelength and

they are not suitable for higher order implementa-

tions in compact filters and amplifiers. To alleviate

the problem Yang et al. [3] proposed a compact uni-

planar PBG (UC-PBG) structure. Some results are

produced on the suppression of higher order harmon-

ics in UC-PBG engineered bandpass filter (BPF).

Besides these periodic structures, defected ground

structures (DGSs) [4, 5] are designed by connecting

two square PBG cells with a thin slot. This DGS

yields lowpass performance with very wide stopband.

The frequency of operation can be changed with the

DGS dimensions. Garde et al. [6] proposed non-uni-

form ring patterned dumbbell-shaped DGS to design

LPF similar to the author’s proposition of the non-

uniform distribution of EBG [7]. Liu et al. [8]

reported a LPF with multilayer fractal PBGS. Signifi-

cant ripples appear in the passband. Although the

LPF performance reported in [7] and [4] is impres-

sive yet the designs need to take care of both the bot-

tom and top layouts that may be contrast to high-

level implications.

As mentioned above, the DGSs are designed by

connecting two square Electromagnetic Band Gap

(EBG) cells within a thin slot. The evolution of EBG

to DGS is fully studied by the author [7]. Figures

1(a) and 1(b) show the isometric view of a dumbbell-

shaped DGS as proposed by Ahn et al. [4]. As can be

seen in Figure 1, ‘‘h’’ is the height of the dielectric

substrate, ‘‘w’’ is the width of the microstrip line, ‘‘a’’and ‘‘b’’ are the arm lengths and ‘‘g’’ is the width of

the gap under the microstrip line on the ground plane.

The frequency of operation can be changed with the

DGS dimensions.

Design and analysis are a challenging problem for

DGSs. The easy availability of commercially avail-

able EM solvers is the main resource for the design

and analysis of DGS. The full-wave analysis [4] is

very involving and does not give any physical insight

of the operating principle of the DGS. The following

flow chart in Figure 2 shows the conventional design

and analysis methods of DGSs. It can be seen in Fig-

ure 2, the design phase of a DGS starts with the

design specifications of stopband frequencies. The

dielectric material is selected for the design. The

full-wave solver is used to find the S-parameters vs

frequency behavior of the DGS. If the results are sat-

isfactory, only then will the S-parameters be con-

verted to ABCD and Z-parameter matrices and the

equivalent LC resonant structure is derived [9]. The

physical insight is understood based on the equiva-

lent circuit model of the DGS. The other disadvant-

Figure 1. (a) Isometric view of unit cell DGS under a

microstrip line (b) Unit cell DGS in individual layers.

Dumbbell-Shaped DGS 211

International Journal of RF and Microwave Computer-Aided Engineering DOI 10.1002/mmce

age of this method is that there is no direct correlation

between the physical dimensions of the DGS and the

equivalent LC parameters. The derived performance

of the DGS is fully unpredictable until the optimized

solutions are achieved through an iterative trial and

error process. Hence the conventional methods as

reported in the open literature are time consuming and

may not lead to the optimum design.

This paper overcomes this limitation by develop-

ing an equivalent circuit model, which is directly

derived from the physical dimensions of the DGS.

Figure 3 illustrates the flowchart of the method. As

can be seen from the figure, a generalized equivalent

circuit model of the DGS is developed first. The

design resonant frequency, dimensions and the

dielectric properties of the microwave laminate are

selected. The design parameter is varied in a do loop

until the required frequency is achieved. In the de-

velopment of the equivalent circuit model, the struc-

ture is assumed to be the combination of a pair of

u-shaped filaments of ground currents and the gap

and cross discontinuities. For the components, quasi-

static equivalent capacitances and inductances are

calculated and ABCD matrix parameters are ex-

tracted. Finally, the ABCD matrix parameters are

transformed to S-parameters vs frequency. This ap-

proach gives a comprehensive understanding of the

physical principle of DGS (how does the DGS create

bandstop and bandpass responses and which dimen-

sions are playing the most vital role to create the dis-

tinct performance.

The paper is organized as follows: Section II

presents the theory of DGS unit cell followed by the

design in Section III. Section IV deals with the analy-

sis of the equivalent circuit model obtained in Sec-

tion III. Results and discussion are presented in Sec-

tions V and VI respectively followed by conclusion

in Section VII.

II. QUASI-STATIC THEORY OF DGS

For a conventional microstrip transmission line, the

quasi-transverse electromagnetic (TEM) mode propa-

Figure 2. Conventional design and analysis methods of

DGS.

Figure 3. Proposed design and analysis method of DGS.

212 Mukul Roy et al.


gates under the microstrip filament and the infinite

ground plane. The electric and magnetic fields are

mostly confined under the microstrip-line. The return

current on the ground plane is the negative image of

the current on the microstrip line. As can be seen in

Figure 1 of the DGS perturbed microstrip transmis-

sion line, the return path of the current is fully dis-

turbed and this current is confined to the periphery of

the perturbation (Fig. 4) and returns to the under-

neath of the microstrip line once the perturbation is

over. On the basis of this observation, an equivalent

circuit model shown in Figure 5(a) is developed. The

ground plane has been truncated around the unit cell

DGS. The current distribution of the compact model

in Figure 5(a) is shown in Figure 5(b). This current

distribution in Figure 5(b) is a more regular version

of the current distribution of the actual ground plane

as shown in Figure 4. On the basis of the observation

of the maximum concentration of the return surface

current on the ground plane, the width of the side fil-

ament arms, which contribute to the inductance of

the DGS, is determined. Figure 6 shows the equiva-

lent filament model of the DGS. This equivalent cir-

cuit model comprises two crosses at the junction of a

dumbbell followed by a transmission line with the

arm length ‘‘b’’, the bend, arm length ‘‘a’’, the bend,

arm length ‘‘b’’, and return to the cross. The gap is

represented by the equivalent capacitances and is

connected vertically to the arms of the two crosses.

The power is impinged at one arm of the cross and

power is extracted from the opposite arm of another

cross. Now the equivalent circuit model of the cur-

rent filament can be extracted with the standard

equivalent inductances and capacitances of the

microstrip discontinuities. The inductances and

capacitances are derived from the physical dimen-

sions using quasi-static expressions for microstrip

crosses, lines and gaps available in the open litera-

tures [10–16].

The closed form expressions for various micro-

strip discontinuities considered here are based on

Quasi Static Analysis of a thin microstrip line. Note

that thickness of the line and ground plane does not

affect the behavior of this circuit in terms of location

of the attenuation pole. Only the bandwidth of the

stop band decreases [12]. In this approach, the mode

of propagation of wave is considered to be purely

TEM.

In the high frequency region, small changes in in-

ductance, characteristic impedance and effective

dielectric constant (hence to capacitance) take place.

We have incorporated these small corrections by

curve fitting, as well as with interpolation from the

previous works and available data in [10–16].

When estimating the capacitance, unlike induct-

ance, frequency variation does not affect capacitance

[12]. The effective dielectric constant varies very lit-

tle with frequency [13] consequently the capacitance

varies a little with frequency.

On the contrary, the current distribution is sup-

posed to be a vector quantity and a function of fre-

quency. Hence the inductance decreases to a limited

extent with an increase in frequency. By curve fitting

and interpolation from the previous works, we have

made necessary correction of the DC inductances.

This results in a simplified form of calculation

that does not require using full-wave matrix solvers.

III. DESIGN OF DGS

The DGS unit cell is designed for the application of

GSM dual-band mobile communications where most

RF and microwave circuits are designed at L and S-

bands. Taking the attenuation pole at 2.4 GHz and

the cutoff frequency fC at 1.2 GHz, the length of aand b is kg/8, where kg is the guide wavelength of the

cutoff frequency fC. This resonant behavior of DGS

can be explained by an equivalent LC circuit model.

As mentioned above the circuit model is derived

from the quasi-static expressions for microstrip

bends, gaps and crosses [10–16]. The filament induc-

tances for bends and straight lines are calculated

using expressions found in [10–16]. For the calcula-

tions and practical prototyping TLXO Taconic ce-

ramic laminate of dielectric constant er ¼ 10 and

thickness h ¼ 0.63 mm is used for analysis and CST

Figure 4. Current distribution on the ground plane of a

unit cell DGS.



Microwave Studio for simulation of the S-parameters

vs frequency of the DGS for comparison.

IV. ANALYSIS

If we extend our view to the concentration of the

return current on the ground plane in Figures 5(b)

and 5(c), we observe an interesting fact. The current

is concentrated at all places along the length of the

microstrip line except at the defect on the ground. At

the gap on the ground, the current retraces a certain

amount of its path and goes along the side lobes

(being confined within a limited width only) and a

strong capacitance is introduced in the gap along the

length of the microstrip line. Additionally a small

amount of current goes directly along the side lobes

instead of traveling along the microstrip line.

The circuit model of this truncated structure was

then derived and its output behavior was calculated

with the help of Matlab 7 and then compared with

those of the actual unit cell DGS. Since the results,

are in good agreement over a wide range of parame-

ter variation, it can be rightly assumed to be an accu-

rate equivalent circuit model of a unit cell DGS.

The special feature of this circuit model is that it

incorporates the actual physical dimensions and

thereby predicts the outcomes and changes of a unit

cell DGS with the variation of the physical dimen-

sions. To the best of knowledge of the authors, this

type of analysis was not carried out before. The pre-

vious works [4] and [5] were based on the Butter-

worth Filter approximation of the DGS and there was

no relationship between the proposed design parame-

ters and the equivalent circuit components.

Length ‘‘b’’ is taken into account twice in calculat-

ing the inductance while the length ‘‘a’’ is taken into

account only once in calculating the inductance of

the side arms. So, length ‘‘b’’ of the side arm contrib-

utes twice to the inductance arising at the side lobes

in comparison to the length ‘‘a’’ of the side arm. As a

result, if the length of the other arm is varied while

the other is kept constant, the location of the attenua-

tion pole does not vary similarly for similar variance

Figure 5. (a) Truncated ground plane and substrate around unit cell DGS (b) Current distribution on the truncated ground

plane around unit cell DGS.

Figure 6. Direction of current on the truncated ground

plane around unit cell DGS.



of ‘‘a’’ or ‘‘b’’. Ideally a D� variation of arm ‘‘b’’should give similar result in terms of variation of

location of attenuation pole as of 2D� variation of

arm ‘‘a’’, but mutual inductances and other effects

make that effect small.

In extracting the equivalent circuit model, closed

form expressions were used for calculating the circuit

parameters of certain microstrip discontinuities that

come into play in this truncated figure (Fig. 5(a)).

The following is a discussion of individual disconti-

nuities and their equivalent circuit parameters.

A. Microstrip Gap

The central discontinuity of the microstrip line of

Figure 6 can be represented as Microstrip Gap. The

equivalent circuit of the microstrip discontinuity is

shown in Figure 7. As can be seen, the gap is repre-

sented by two parallel capacitances to ground (Cp)

and a series capacitance Cgap. The values of these

capacitances are extracted from the even and odd

mode capacitances as given below. Interestingly, it

should be noted that all equivalent capacitances are

extracted from the physical dimensions of the gap

discontinuity and the dielectric constant.

Cp ¼ 1

2Ceven ð1Þ

Cgap ¼ 1

2Codd � 1

2Ceven

� �ð2Þ

CoddðpFÞ ¼ W � S

W

� �mo

eKo

CevenðpFÞ ¼ W � S

W

� �me

eKe

9>>=>>; for er ¼ 9:6 ð3Þ

Figure 7. Gap in a microstrip line with uniform ground

plane and its equivalent circuit.

Figure 8. Cross in a Microstrip line with uniform

ground plane and its equivalent circuit.



mo ¼ W

h0:619 log

W

h� 0:3853

� �

Ko ¼ 4:26� 1:453 logW

h

9>>=>>;

for 0:1 � S

W� 0:3 ð4Þ

me ¼ 0:8675

Ke ¼ 2:043W

h

� �0:12

9=;for 0:1 � S

W� 0:3

me ¼ 1:565

W

h

� �0:16� 1

Ke ¼ 1:97� 0:03Wh

9>>>>>=>>>>>;for 0:3 � S

W� 1 ð5Þ

CoddðpF=mÞ ¼ Codd 9:6ð Þ er9:6

� �0:8CevenðpF=mÞ ¼ Ceven 9:6ð Þ er

9:6

� �0:99>=>;

for 2:5 � er � 15 ð6Þ

‘‘S’’ in the above equation is the gap width.

B. Microstrip Cross

The two ends of the structure shown in Figure 6 are

represented by two microstrip crosses. The power is

impinged on one end and is extracted through the

other end. The equivalent circuit of such a disconti-

nuity is shown in Figure 8. As can be seen in the fig-

ure, the symmetric four arms of the microstrip cross

are represented by equivalent L1 and L2 and Cs. The

cross arms are inductively coupled by L3. Each end

of the circuit is terminated by width W1 and W2. The

equivalent capacitances and inductances are calcu-

lated using expressions (7)–(10).

CþðpFÞ ¼ W1

W1

h

� ��13

86:6W2

h� 30:9

ffiffiffiffiffiffiW2

h

rþ 367

!log

W1

h

� �

þ W2

h

� �3

þ74W2

hþ 130

266664

377775

� 1:5W1

h1�W2

h

� �þ 2h

W2

� 240

8>>>>>>>>>><>>>>>>>>>>:

9>>>>>>>>>>=>>>>>>>>>>;ð7Þ

The above expression is valid for er ¼ 9.9. For otherdielectric constants, as C ¼ e ðA=dÞ for rectangular

capacitances, using usual denotations, we have used

CþðerÞ ¼ Cþð9:9Þ er9:9. Inductance values are not af-

fected by change of dielectric constant.

L1ðnHÞ ¼ hW1

h

� ��3=2

�W1

h165:6

W2

hþ 31:2


h

r� 11:8

W2

h

� �2 !

� 32W2

hþ 3

26664

37775ð8Þ

L2ðnHÞ ¼ hW2

h

� ��3=2

�W2

h165:6

W1

hþ 31:2


h

r� 11:8

W1

h

� �2 !

� 32W1

hþ 3

26664

37775ð9Þ

L3ðnHÞ ¼ �h 337:5þ h

W2

1þ 7h

W1

� ��

� 5W2

hcos

p2

1:5�W1

h

� �� ð10Þ

C. Inductance Calculations

The two U-shaped filaments of the truncated Ground

Plane of lengths ‘‘2b þ a’’ are represented as induc-

tances and are discussed as follows.

The inductances here have been calculated on the

basis of [10–16]. Since the filaments that we are re-

ferring to are actually rectangular cross-sections of

wire, a correction must be made to account for the

effect of the extra conductor. This is done by consid-

ering the rectangular cross-sections of filaments sepa-

rated by a distance ‘‘R’’ known as the geometric

mean distance (GMD). The GMD is found by calcu-

lating 1/n of the sum of the logarithm of the distance

between pairs of points.

To start calculating the GMD, initially the wire

must be broken up into ‘‘n’’ points (as shown in Fig-

ure 9) of equal volume that are small compared with

the width and depth of the wire. Then the GMD is

calculated by summing the logarithms of the distan-

ces between the ‘‘n’’ pairs of points and then taking

‘‘1/n’’ of it. This method of GMD allowed us to cal-

culate the inductance created by the side arms of the

dumbbell.



The concept of the GMD can also be used to cal-

culate the self-inductance of a filament. The self in-

ductance of a single conductor can be computed by

assuming that the self inductance of the conductor is

simply the sum of all the partial mutual inductance

of each pair of filaments within the single conductor.

This gives rise to the GMD of a filament from itself.

It can be calculated by [16]

R ¼ limN!1

YNm¼1

YNn¼1n 6¼m

rmn

0B@

1CA1=m0

B@1CA1=m

ð11Þ

If two or more very closely placed inductors are

considered in parallel and the net inductance is calcu-

lated using two-port circuit parameters, the results

are incorrect if the mutual inductances are not taken

into consideration. So the following relation is used

that includes the mutual inductances.

Leffective ¼ LSelf �X

MMutual

In our calculation, the side arms are purely induc-

tive and the two arms are combined into a single in-

ductance by using the above mentioned relation and

usual circuit theory. Thus we have ensured that all

the inductive effects of the side arms are included.

At DC, the current is distributed uniformly

throughout the conductor. However as the frequency

rises there will be a tendency for the current to con-

centrate at the surface of the filament. It is assumed

that the current decreases exponentially inside the fil-

ament with depth (skin effect), and that the current is

same at the top and bottom of the conductor i.e. side

lobe of the ground plane as shown from the surface

current distribution diagram Figure 5(b) and Figure 6.

As a result, the outer portions of the conductor con-

tribute less than the inner parts to the overall self in-

ductance (current has more difficulty passing through

the inner parts because of skin effect). If current is

concentrating on the surface, the inductance will

decrease. With the increase of frequency, this con-

centration of current on the surface increases. Thus

with an increase of frequency, there is a decrease in

the inductance. Using the expressions of eqs. (7)–(15)

and by curve fitting [10–16], we have done our cal-

culations for the inductances.

If two conductors meet at an angle, mutual induct-

ance has also to be taken into consideration at the

bend, which we represented as Lbend in the circuit

diagram (Fig. 10). When two strips of lengths m1 and

l1 meet at an angle e, then the junction gives a mutual

inductance given by MJunction [10].

MJunctionðnHÞ ¼ 200 cos e

� l1 tanh�1 m1

l1 þ Rþ m1 tanh

�1 l1m1 þ R

� � ð12Þ

As e ¼ 908 here, hence MJunction is zero, which

implies Lbend in Figure 10 is 0.

The self and mutual inductance can be calculated

with the Magnetic Flux method for Mutual Induct-

ance and Energy Method for Effective Inductance

calculation [10–16] taking into consideration the

high frequency of operation [10].

MJunctionðnHÞ ¼ 200 cos e

� l1 tanh�1 m1

l1 þ Rþ m1 tanh

�1 l1m1 þ R

� � ð13Þ

LselfDC ðnHÞ ¼ 200 llength

� 2llengthlwidth þ lthickness

þ 1

2� 0:00211

� � ð14Þ

MijðnHÞ ¼ 200 l

� lnl

dþ

ffiffiffiffiffiffiffiffiffiffiffiffiffi1þ l2

d2

r !�

ffiffiffiffiffiffiffiffiffiffiffiffiffi1þ d2

l2

rþ d

l

" #ð15Þ

In eq. (13), a single turn of the coil is considered

and ‘‘d’’ is the skin depth.

Taking into consideration, the aforesaid closed

form expressions and circuit parameters, we model

the equivalent circuit of the truncated DGS (Fig. 10)

Figure 9. Rectangular filament of wire broken into m � n

points.



and then carry on the following conversions using

two port circuit parameters as shown in Figure 11.

According to Figures 1 and 10, for sections of the

microstrip line following the microstrip cross up to

the gap and then proceeding the next cross, the

ground plane is merely a parallel strip instead of the

theoretical infinite one. For such a finite ground

plane, the characteristic impedance and effective

dielectric constant respectively are given by Eqs.

(16) and (17) [17]

eeffective ¼ er þ 1

2þ er � 1

2: 1þ 6h

w

� ��1=2

ð16Þ

Figure 10. Equationuivalent circuit model of unit cell DGS.



Z0 ¼ 120pffiffiffiffiffiffiffiffiffiffiffiffiffiffieeffective

p 2wh þ 1:393þ 0:667 ln 2h

w þ 1:444� �

forw

h� 1 ð17Þ

Figure 10 is then folded into a single block using

two port circuit theories and considering the ABCD

parameters of each block to combine them in series

or parallel, resulting in one final block. The S-param-

eters of that final block is extracted from its ABCD

parameters and plotted against frequency. The whole

folding of the circuit is shown in steps from 1 to 6 in

Figure 11.

As can be seen in Figure 11, the complete equiva-

lent circuit model in terms of crosses, bent lines and

Figure 11. Folding of the entire circuit into a single Block.



the gap capacitances is fully characterized by expres-

sions (1–17). All these expressions take care of the

dimensions of the DGS and the dielectric properties of

the substrate. Therefore from the equivalent circuit

model, a direct correlation between the design para-

meters and the design specification is established.

In the following section (Section V Results),

results of different parametric studies of the unit cell

DGS are shown. The results are followed by a dis-

cussion in the next section (Section VI Discussion).

The basis of selecting the lwidth is discussed in Dis-

cussions.

V. RESULTS

This section presents the parametric study of the pro-

posed DGS and the influence of these parameters on

the attenuation pole and the cut-off frequency. This

parametric study will give rise to the frequency

behavior of a DGS assisted 50-Ohm transmission

line. The parametric study leads to the design curves

for the generic dumbbell-shaped DGS circuit. There-

fore, this study is very useful for the designer com-

munity. Also this study gives insight into the physical

properties of the DGS in the frequency behavior.

Every set of theoretically calculated results is com-

pared with those obtained from commercially avail-

able EM solver CST Microwave Studio. Good agree-

ments between the two theoretical results validate the

proposed theory. Finally the theoretical results are

compared with the measured results of the fabricated

prototyped DGS circuit on Taconic Substrate. The

agreement is in general good showing on the spot

attenuation pole. The DGS assisted microstrip trans-

mission line is measured on an Agilent 8510C Vector

Network Analyzer (VNA).

A. Parametric Study of DGS

In all the figures from Figures 12–17, the lengths of

‘‘a’’ and ‘‘b’’ are taken to be 5 mm each with Gap

Figure 12. Variation of resonant frequency with arm

length ‘‘a.’’

Figure 13. Variation of resonant frequency with arm

length ‘‘b.’’

Figure 14. Variation of resonant frequency with both

side arm lengths simultaneously.

Figure 15. Variation of resonant frequency with gap

dimension.



dimension 0.5 mm. The dielectric constant of the

substrate is 2.45 and height is 0.7874 mm. The

microstrip transmission line has a width of 2.28 mm

(50 � characteristic impedance). When any of the

above mentioned parameters are varied, the other

parameters are kept fixed as mentioned above.

Figure 12 shows the variation of the attenuation

pole or the resonant frequency of the DGS unit cell

with the arm length ‘‘a’’. It can be seen in the figure

that, the frequency decreases with the arm length.

The agreement between the CST Microwave Studio

simulation and the theory is very good.

Figure 13 shows the variation of attenuation pole

or resonant frequency of the DGS unit cell with the

arm length ‘‘b’’.Figure 14 shows the variation of the resonant fre-

quency when both ‘‘a’’ and ‘‘b’’ vary together. It can

be seen from the figures, the frequency decreases

with an increase in the arm length. Again the agree-

ment between the CST Microwave Studio simulation

and the theory is very good.

Figure 15 shows the variation of the resonant fre-

quency with the gap distance ‘‘g’’. As can be seen in

the figure, the gap capacitance diminishes with the

Figure 16. Variation of resonant frequency with width

‘‘w’’ of the unit cell DGS (Microstrip line width is kept

constant at characteristic impedance of 50 �).

Figure 18. Comparison of S parameter vs frequency of

CST microwave studio simulation and theory. The arm

lengths are 5 mm each and the gap dimension is 0.5 mm

and the substrate considered is Taconic of er ¼ 2.45 and

thickness 0.7874 mm. Microstrip line is characteristic im-

pedance of 50 �.

Figure 17. Variation of resonant frequency with dielec-

tric constant of the substrate (Microstrip line width is var-

ied according to the characteristic impedance of 50 �depending on the dielectric constant of the substrate).

Figure 19. Comparison of S11 parameters between CST

microwave studio simulation and theory. The arm lengths

are 5 mm each and the gap dimension is 0.5 mm and the

substrate considered is Taconic of er ¼ 2.45 and thickness

0.7874 mm. Microstrip line is characteristic impedance of

50 �.



gap distance of the DGS unit cell. As a result, the

resonant frequency increases with the increase of the

gap distance.

Figure 16 shows the variation of the resonant fre-

quency with the width ‘‘w’’ (Refer Figs. 1(b) and

5(a)) of the unit cell DGS keeping the width of the

microstrip line constant. As can be seen in the figure,

the inductance of the side arms increases with the

increase of the width ‘‘w’’ of the cell. As a result, the

resonant frequency decreases with the increase of

the width ‘‘w’’ of the DGS.Finally, in the parametric study the dielectric con-

stant of the substrate material is varied. It can be

seen in Figure 17, the resonant frequency decreases

with the dielectric constant of the substrate. Here

also the agreement between the CST Microwave Stu-

dio and the theory is very good.

B. Simulation and Measured Resultsof DGS

After the satisfactory agreement of the comprehen-

sive parametric study of the unit cell DGS between

the CST Microwave Studio and the proposed theory,

the complete S-parameter vs frequency plots are

shown in Figures 18–20. As can be seen in Figure 18,

the attenuation poles for the CST and the theory are

in very good agreement at 7.785 and 7.8 GHz, re-

spectively. There is a deviation in the stop bandwidth

of the two calculations. Similar discrepancies can be

observed in Figures 19 and 20. The discrepancies can

be attributed to the simple equivalent circuit model

of the proposed theory. In the proposed theory, the

microstrip discontinuities and the dielectric substrate

are assumed loss less and whereas in the full-wave

analysis, these losses are considered.

Finally the theoretical calculation of the DGS is

compared with the measured results on a Agilent HP

8510C VNA. Figure 21 shows the measured, simu-

lated and calculated S-parameters of the unit cell

DGS vs frequency. As can be seen for the figures,

similar magnitude of agreement as shown in Figure

17 for the case of CST Microwave Studio simulation

and the proposed theory is achieved. The discrepan-

cies may be attributed to the simplified circuit model

considered, which is more frequency dependent/sen-

sitive compared with those of CST full wave analysis

and measured results.

VI. DISCUSSION

The width of the microstrip line (W1) considered here

corresponds to 50 � characteristic impedance. The

width of the side arms (W2) is chosen here to be 0.2

mm. and it is one of the key parameters of this

model. W2 has been arrived after a comprehensive

investigation of a wide range of the lengths of the

side arms of a dumbbell-shaped DGS. It has been

Figure 20. Comparison of S21 parameters between CST

microwave studio simulation and theory. The arm lengths

are 5 mm each and the gap dimension is 0.5 mm and the

substrate considered is Taconic of er ¼ 2.45 and thickness

0.7874 mm. Microstrip line is characteristic impedance of

50 �.

Figure 21. Comparison of S parameter vs frequency of

Agilent HP 8010C VNA measurement, CST microwave

studio simulation and theory for a unit cell DGS using

Taconic substrate with er ¼ 10 and height 0.63 mm. Both

the arm lengths ‘‘a’’ and ‘‘b’’ are 10 mm and gap dimen-

sion is 0.3 mm. Characteristic impedance of the microstrip

line is 50 �.



seen during the computer aided investigation process

using CST Microwave Studio1 that with changes in

arm lengths consequently, the changes in return cur-

rent path on the perturbed ground plane, the width of

the return current sheet remains around 0.2 mm.

Therefore an approximation of W2 ¼ 0.2 mm. is used

throughout the model.

In earlier related literature, the unit cell DGS has

been described as a one pole Butterworth filter [1, 5]

where the capacitance comes only from the gap and

the inductance comes only from the loop. After doing

this analysis of the unit cell DGS, we can say that the

variance of the inductance and the capacitance does

not follow any linear rule and also we can explain

why they don’t follow.

If we observe Figures 4(b) and 5(b), it can be seen

clearly that the density of current is higher at the

bends. It is because there is no mutual inductance at

a right angle bend and hence current flows with

much more ease through this region.

The cross capacitances are similar in magnitude

with the gap capacitance. Although due to different

connectivities they are not directly connected in par-

allel or series with the gap capacitance, they still play

a big role in determination of the location of the

attenuation pole. Assuming cross capacitance as zero

leads to significant deviation of the computed results

from the measured or simulated one. Additionally,

like distributed inductance, there is distributed capac-

itance of the ground plane along the length of the

microstrip line.

VII. CONCLUSION

The dumbbell-shaped DGS is one of the most popu-

lar meta materials. It generates distinct pass-bands

and wide stop-bands. It has been used in many

microwave and mm-wave active and passive circuits

for their performance improvement and modifica-

tions. In this paper, we have presented a novel equiv-

alent circuit model of a unit cell DGS. We have pre-

sented a physics based model for the dumbbell-

shaped DGS. The DGS is treated as a discontinuity

and analyzed in terms of various other discontinu-

ities. Equivalent inductances and capacitances have

been included leading to the characterization. The

approach used could lead to similar work for

EBG. The equivalent circuit model is derived from

the equivalent inductance and capacitance, which

develop due to the perturbed returned current path on

the ground and the narrow gap, respectively. The fila-

ment current path is modeled as a current sheet on

the ground plane. The current tightly coupled to the

periphery of the dumbbell-shaped DGS. Hence it is

logical to model this unit cell DGS as a combination

of various microstrip line discontinues such as

crosses, bends and the gap capacitances. On the basis

of the developed theory a comprehensive parametric

study is performed and compared with the simulated

results of CST Microwave Studio. Excellent agree-

ment regarding the location of the attenuation poles

between the proposed theory and CST Microwave

Studio simulation has been obtained. The theory is

validated fully against the s-parameter vs frequency

plot for both the commercial full-wave solvers CST

Microwave Studio and the theory. Finally, the calcu-

lated results are compared with the measured results.

In general, good agreements between the theory, the

commercially available numerical analyses and the

experimental results have validated the developed

theoretical model. However this simplified model has

a discrepancy of the bandwidth calculation. The dis-

crepancy can be attributed to the quasi static equiva-

lent circuit model itself where the simplified models

of discontinuities are more frequency sensitive when

compared with full wave analysis and measurement.

ACKNOWLEDGMENTS

The work was supported by ARC Discovery grant No:

DP0665523, Chipless RFID for Barcode Replacement.

Software supported by CST is also acknowledged.

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BIOGRAPHIES

Sushim Mukul Roy obtained his Bachelor

of Engineering in Electronics and Telecom-

munication Engineering from Bengal Engi-

neering College (Deemed University), India

in 2003. At present he is pursuing a PhD at

Monash University, Clayton, Australia. He

worked at Samtel Color, Ghaziabad, India

in the Process Engineering Department

from August 2003 to July 2004. His areas

of interest include passive microwave devices and his present

research is Chipless Radio Frequency Identification Systems.

Nemai Chandra Karmakar (S‘91-M’91-

SM’99) obtained the B.Sc (EEE) and M.Sc

(EEE) from Bangladesh University of Engi-

neering and Technology, Dhaka in 1987

and 1989 respectively, the M.Sc. degree in

Electrical Engineering from the University

of Saskatchewan, Saskatoon, Canada in

1991 and the PhD degree from the Univer-

sity of Queensland, Brisbane, Australia in

1999. His PhD thesis work concerned the

area of switched beam and phased array antennas for mobile sat-

ellite communications. His PhD work was one of the most signifi-

cant findings at The University of Queensland in 1998 and pub-

lished in national media such as ABC Radio and Canberra Times.

His PhD work was elected the third best student paper in 1997

Asia Pacific Microwave Conference held in Hong Kong. From

1989 to 1990, he worked as an Assistant Engineer in Electronics

Institute, Atomic Energy Research Establishment, Dhaka, Bangla-

desh. In August 1990, he was a Research Assistant at the Com-

munications Research Group, University of Saskatchewan, Can-

ada. From 1992 to 1995 he worked as a Microwave Design Engi-

neer at Mitec, Brisbane, Australia where he contributed to the

development of land mobile satellite antennas for the Australian

Mobilesat. From 1995 to 1996 he taught final year courses on

Microwaves and Antenna Technologies at Queensland University

of Technology, Brisbane, Australia. From September 1998 to

March 1999 he worked as a research engineer within the Radar Lab-

oratory, Nanyang Technological University, Singapore. From

March 1999 to July 2004 he was an Assistant Professor and Gradu-

ate Advisor in the Division of Communication Engineering, the

School of Electrical and Electronic Engineering, Nanyang Techno-

logical University, Singapore. From July 2004 to date he is a Senior

Lecturer in the Department of Electrical and Computer Systems En-

gineering, Monash University, Clayton, Australia. Dr. Karmakar’s

research interests cover areas such as smart antennas for mobile and

satellite communications, EBG assisted RF devices, planar phased

array antennas, broadband microstrip antennas and arrays, beam-

forming networks, near-field/far-field antenna measurements,

microwave device modeling, monostatic and bistatic radars. He has

published more than 130 referred journal and conference papers,

and five book chapters. His biography has been included (by invita-

tion) in Marquis Who’s Who in Science and Technology 2002–

2006 edition as a pioneer in planar phased arrays.

Isaac Balbin obtained a Bachelor of Engi-

neering (BE) and a Bachelor of Science

(BSc) from Monash University, Australia in

2005. At the present point in time he is

pursuing a Masters of Engineering by

research at the same university. The areas

of interest include passive microstrip tech-

nology, particularly with regard to NRI

metamaterials and also to development of

chipless RFID systems.



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Dumbbell-shaped defected ground structure