Progress In Electromagnetics Research, Vol. 113, 111–126, 2011

SYNTHESIS OF MULTI-STEP COPLANAR WAVEGUIDE-TO-MICROSTRIP TRANSITION

S. Costanzo

Dipartimento di Elettronica, Informatica e SistemisticaUniversity of Calabria, Via P. Bucci, Rende (CS) 87036, Italy

Abstract—A synthesis procedure is developed in this paper for thedesign of N-step coplanar waveguide-to-microstrip transitions. Anequivalent circuit approach is adopted to model the structure in termsof N cascaded ABCD matrices relative to the N coplanar waveguidesections forming the transition. A constrained optimization problem isformulated as the minimum finding of a proper functional to accuratelydetermine the transition dimensions by imposing an upper bound tothe return loss within a prescribed frequency band. An iterative N-step procedure is developed to find the optimization problem solution.Numerical results on millimeter-wave transition configurations areprovided to demonstrate the effectiveness of the proposed synthesismethod. A back-to-back transition prototype with N = 3 sections isthen fabricated and characterized in terms of measured S-parametersto experimentally demonstrate a return loss better than 10 dB in thefrequency range from 1 GHz up to 40GHz.

1. INTRODUCTION

The coplanar waveguide (CPW) configuration offers significantadvantages and flexibility in the design of hybrid and monolithicmicrowave integrated circuits, especially when working at millimeterand sub-millimeter frequency ranges. As compared to standardmicrostrip lines, it provides small radiation loss, low dispersion,simple integration of active and passive devices, and also it eliminatesthe need for via holes [1]. The conventional CPW [1] consistsof a center strip conductor and two ground planes on both sides,which are made of finite extent in practical circuits. The CPWconfiguration with additional backside grounding, commonly called

Received 29 November 2010, Accepted 24 January 2011, Scheduled 28 January 2011Corresponding author: Sandra Costanzo (costanzo@deis.unical.it).

112 Costanzo

conductor-backed coplanar waveguide, provides further advantagesin terms of mechanical stability, thermal dissipation capability andintegration. The above features make CPW and its variant conductor-backed CPW ideally suited for the design of low-loss, compactand inexpensive uniplanar structures [2–4]. Applications involvingthe integration of CPWs and microstrip lines on the same circuitor adopting on-wafer measurement techniques with CPW probesrequire the flexible combination and compatibility of both microstripand CPW technologies. As a consequence of this, the design ofwideband, low-loss and reduced size transitions between CPWs andmicrostrip lines has become an essential point, subject to rigorousresearch since many years [5–10]. When working at millimeter-wave frequencies, the requirement of a transition structure is alsorelated to the dimensions limitations of standard connectors, oftenpreventing a direct connection to CPWs circuits. An interestingtransition structure which provides a gradual transformation of theelectric and magnetic fields between a CPW or conductor-backed CPWand a microstrip or a stripline was proposed in [11, 12]. It exploitsresults coming from the analysis of normal propagation modes inconductor-backed CPW structures [13, 14] to assume the characteristicimpedance of conductor-backed CPW as given by the combinationof two characteristic impedances associated to the CPW mode andthe microstrip mode, respectively. These impedances are stronglyinfluenced by the conductor-backed CPW parameters dimensions. Inparticular, for small values of the ratio between the center line-to-ground line separation and the substrate height, the CPW propagationmode is dominant. On the other hand, when this ratio increases,the conductor-backed CPW structure turns to a microstrip line andthe microstrip mode becomes more dominant. On the basis of thisconsideration, the configuration presented in [11, 12] realizes a smoothtransition, in terms of both impedance and field match, by graduallyenlarging the width of the center line of conductor-backed CPW tothe microstrip width, while keeping the characteristic impedance valueconstant along the structure. For simplicity of implementation, thetransition is divided into N sections of conductor-backed CPW havingequal length and impedance, with the geometrical parameters of eachsection derived from the structural expressions reported in [1]. Anumerical analysis based on the use of Ansoft HFSS simulator ispresented in [11, 12] to demonstrate the wideband behavior of thetransition structure, by examining the effect of the length as well as thenumber of sections on the operating bandwidth. Experimental resultson back to back transitions are also discussed in [11, 12] to validate theproposed transition configuration.

Progress In Electromagnetics Research, Vol. 113, 2011 113

In this paper, a synthesis procedure is developed to accuratelydetermine the dimensions of the conductor-backed CPW-to-microstriptransition configuration proposed in [11, 12]. Based on the approachoutlined in [11, 12], the fundamental mode propagation is assumed, andeach section of the conductor-backed CPW-to-microstrip transitionis properly dimensioned to prevent the excitation of higher-ordermodes [15, 16] within the operating frequency band. A gradual smoothtransformation is realized from the CPW mode to the microstripmode, while keeping the characteristic impedance constant along thetransition [12], so reflections due to impedance changes are avoidedand the assumption of single-mode operation gives accurate results.As a matter of fact, it is expected only that reactive phenomenaresponsible for resonances could be produced at the transition sectionsdue to dimensions changes. On the basis of single-mode assumption,a simplified equivalent circuit approach is adopted, with the transitionstructure modelled in terms of N cascaded ABCD matrices relative toits N sections, and a compact expression is derived for the reflectioncoefficient S11 at the transition input as a function of the propagationconstants of the N conductor-backed CPW transition sections. Byimposing a proper upper bound to the return loss within a frequencyband limited by the intersection frequency between the fundamentalmode and the first higher-order mode, a constrained optimizationproblem is formulated as the minimum finding of a N variablefunctional, and an iterative N-step procedure is developed to find thesolution in terms of the N phase constants relative to the N transitionsections. The structural formulas of conductor-backed CPWs arefinally applied to determine the N-step transition dimensions. Theeffectiveness of the synthesis procedure is demonstrated by discussingnumerical results on two millimeter-wave conductor-backed CPW-to-microstrip transition configurations with different number of sections,and validation with Ansoft Designer simulations are provided. Forboth the examined transition structures, a wideband behavior isnumerically predicted, with a return loss below −10 dB from 1 GHzup to 65 GHz. Furthermore, experimental validations are provided interms of measured S-parameters on a back-to-back 3-step transitionprototype, by confirming the correct operating behavior of the returnloss up to 40 GHz.

2. THEORY

The conductor-backed CPW-to-microstrip transition proposed in[11, 12] and examined in this paper is shown in Fig. 1. It is usedto connect a conductor-backed CPW of dimensions Wc, Sc, Gc

114 Costanzo

W c

h

W m

S c S c

W c

S i S i

W iL G Gc c

S c S c

(a) (b)

Figure 1. Layout of conductor-backed CPW-to-microstrip transition:(a) Top view and (b) Side view.

to a microstrip line of width Wm, and consists of N sections ofconductor-backed CPW with equal length L and parameters Wi, Si

(i = 1, 2, . . . , N). Each section is designed to have a characteristicimpedance Zc, while Zo is the impedance value of the connectedconductor-backed CPW and microstrip line. To guarantee the gradualchange of the electric and magnetic fields from the CPW mode tothe microstrip mode, the proper design of the transition in Fig. 1leads to increase the parameters Wi, Si (i = 1, 2, . . . , N) along thestructure, from the conductor-backed CPW side to the microstripline. In [11, 12], this design process is performed uniquely on thebasis of parametric simulations on Ansoft HFSS software, with nospecific criteria to determine the structure dimensions, so implyinga heavy repeatition procedure, with variable simulation costs whenchanging the design constraints in terms of operating frequency bandand dimensions. In this paper, a theoretical modelling of the structurein Fig. 1 is developed by assuming the equivalent circuit of Fig. 2, wherethe connected conductor-backed CPW and the microstrip line are bothdescribed by uniform transmission lines of characteristic impedance Zo

(Fig. 2(a)), and the transition is modelled in terms of N cascadedABCD matrices (Fig. 2(b)) relative to the N sections of uniformtransmission lines having characteristic impedance Zc.

The two-port ABCD matrix elements of the ith section are given

Progress In Electromagnetics Research, Vol. 113, 2011 115

Microstrip line :W CBCPW line: W ,S , G

CBCPW-to-MicrostripTransitionABCD-T

CBCPW-to-MicrostripTransition: ,

ABCD-2W S

CBCPW-to-MicrostripTransition: ,

ABCD-1W S

CBCPW-to-MicrostripTransition: ,

ABCD-NW SN N

(a)

(b)

m

Z o

c c c

Z o

2 21 1

Figure 2. Equivalent circuit topology of conductor-backed CPW-to-microstrip transition: (a) Full cascaded circuit and (b) Cascadedcircuit of N-step transition.

by [17]:

ABCDi =[

cosh γiL Zc sinh γiL1

Zc· sinh γiL cosh γiL

](1)

where γi = αi + jβi is the complex propagation constant, with:

βi = βo

√εieff (2)

βo being the free-space propagation constant.The attenuation constant αi and the effective dielectric constant

εieff of the ith section are determined by the quasi-static formulas based

on the conformal mapping technique [1].The ABCD matrix of the entire transition is derived by cascading

the ABCD matrices relative to all sections, so giving:

ABCDT =[AT BT

CT DT

]=

N∏

i=1

[Ai Bi

Ci Di

](3)

Some manipulations lead to obtain:

AT = DT = cosh

[(γ1 −

N∑

i=2

γi

)L

]

BT = Zc sinh

[(γ1 +

N∑

i=2

γi

)L

]

116 Costanzo

CT =1Zc· sinh

[(γ1 +

N∑

i=2

γi

)L

](4)

The reflection coefficient S11 at the transition input can be expressedin terms of ABCDT parameters as follows [17]:

S11 =AT + BT

Zo− CT Zo −DT

AT + BTZo

+ CT Zo + DT

(5)

The substitution of expressions (4) into relation (5) gives:

S11 =

(Z2

c−Z2o

Z2c +Z2

o

)· sinh

[(γ1 +

∑Ni=2 γi

)L

]

sinh[(

γ1+∑N

i=2 γi

)L

]+ 2ZcZo

Z2c +Z2

ocosh

[(γ1 −

∑Ni=2 γi

)L

] (6)

A compact form of expression (6) can be written as:

S11 = P ·sinh

(θ1 +

∑Ni=2 θi

)

sinh(θ1 +

∑Ni=2 θi

)+ Q cosh

(θ1 −

∑Ni=2 θi

) (7)

where:

P =Z2

c − Z2o

Z2c + Z2

o

, Q =2ZcZo

Z2c + Z2

o

, θi = γiL, i = 1, 2, . . . , N (8)

Following Fano’s considerations on the theoretical limitations ofbroadband matching [18], we can impose the magnitude of thereflection coefficient S11 to be smaller than or equal to a specifiedvalue ρm within a prescribed frequency band B, i.e.:

|S11(f)| ≤ ρm, ∀f ∈ B (9)

Due to the assumption of fundamental mode propagation, the reliableband B of the approach adopted in the paper is upper limited by theintersection frequency between the fundamental mode and the firsthigher-order mode [15, 16]. However, this upper bound does not givestrong limitations, as a proper selection of CPW dimensions can inhibithigher-order modes, so providing a very wideband behavior.

By substituting expression (7) into relation (9), we have:∣∣∣∣∣

sinh(θ1 +∑N

i=2 θi)

sinh(θ1 +∑N

i=2 θi) + Q cosh(θ1 −∑N

i=2 θi)

∣∣∣∣∣ ≤ρm

|P | (10)

The application of Schwarz’s inequality [19] gives:

Progress In Electromagnetics Research, Vol. 113, 2011 117

∣∣∣∣∣sinh

(θ1 +

N∑

i=2

θi

)+ Q cosh

(θ1 −

N∑

i=2

θi

)∣∣∣∣∣

≤∣∣∣∣∣sinh

(θ1 +

N∑

i=2

θi

)∣∣∣∣∣ + Q

∣∣∣∣∣cosh

(θ1 −

N∑

i=2

θi

)∣∣∣∣∣ (11)

from which: ∣∣∣sinh(θ1 +

∑Ni=2 θi

)∣∣∣∣∣∣sinh

(θ1 +

∑Ni=2 θi

)∣∣∣ + Q∣∣∣cosh

(θ1 −

∑Ni=2 θi

)∣∣∣≤ ρm

|P | (12)

Let us impose:

θ1 −N∑

i=2

θi = a1 + jb1, θ1 +N∑

i=2

θi = a2 + jb2 (13)

where:

a1 =

(α1 −

N∑

i=2

αi

)L, b1 =

(β1 −

N∑

i=2

βi

)L (14)

a2 =

(α1 +

N∑

i=2

αi

)L, b2 =

(β1 +

N∑

i=2

βi

)L (15)

In the case of small losses (αi ¿ βi,∀i = 1, 2, . . . , N), we canassume a1 ' 0 and a2 ' 0, so a compact form of relation (12) can beeasily derived as:

C1| sin b2| ≤ C2| cos b1| (16)

where

C1 =(

1− ρm

|P |)

, C2 =ρmQ

|P | (17)

The unknowns βi(f) (f ∈ B, i = 1, 2, . . . , N) into Eq. (16) aredetermined as solutions of a nonlinearly constrained optimizationproblem [20], formulated as the finding of the least value of thefunctional:

ψ (β1, β2, . . . , βN ) = C1| sin b2| − C2| cos b1| (18)

subject to the constraint given by Eq. (2).

118 Costanzo

From the knowledge of the phase constant βi relative to the ithsection, the effective dielectric constant εi

eff is derived, which in turnsis related to the section dimensions Wi, Si by the expression [1]:

εieff =

1 + εr · K(k′)K(k) · K(k1)

K(k′1)

1 + K(k′)K(k) ·

K(k1)K(k′1)

(19)

where k = ab , k1 = tanh(πa

4h)

tanh( πb4h

), k′ =

√1− k2, k′1 =

√1− k2

1, a = Wi,

b = Wi + 2Si, K(. . .) being the complete elliptic integral of the firstkind.

To solve the problem, an iterative N-step optimization procedureis performed, for a given number N of sections. Due to the smallchanges between adjacent sections, we assume, as initial guess, all βi

are equal, so that:

b1 = −(N − 2)β1, b2 = Nβ1 (20)

By substituting relations (20) into Eq. (18), the functional to beminimized at the first step with respect to the single variable β1 canbe expressed as:

ψ1 (β1) = C1 |sinβ1 · UN−1(cosβ1)| − C2 |TN−2(cosβ1)| (21)

where Tj(. . .) and Uj(. . .) are the Chebyshev polynomials of the firstand the second kind, respectively, and order j [19].

At the second step, the terms b1, b2 are expressed as:

b1 = β1 − (N − 1)β2, b2 = β1 + (N − 1)β2 (22)

Relations (22) are substituted into Eq. (18) for defining thesecond-step functional to be minimized with respect to the unknownβ2, namely:

ψ2 (β2) = C1 |sinβ1 · [TN−1(cosβ2) + cosβ1 · UN−2(cosβ2)]|−C2

∣∣cosβ1 · TN−1(cosβ2) + sin2 β2 · UN−2(cosβ2)∣∣(23)

At the generic nth step, the optimization procedure involves the finding

Progress In Electromagnetics Research, Vol. 113, 2011 119

of the least value of the functional:

ψn(βn) = C1

∣∣∣∣∣sin(

n−1∑

i=1

βi

)· TN−n+1(cosβn)

+ cos

(n−1∑

i=1

βi

)· sinβn · UN−n(cosβn)

∣∣∣∣∣

−C2

∣∣∣∣∣cos

(β1 −

n−1∑

i=2

βi

)· TN−n+1(cosβn)

+ sin

(β1 −

n−1∑

i=2

βi

)· UN−n(cosβn)

∣∣∣∣∣ (24)

Finally, at the Nth step, the functional to be minimized isexpressed as:

ψN (βN )

= C1

∣∣∣∣∣sin(

N−1∑

i=1

βi

)· cosβN + cos

(N−1∑

i=1

βi

)· sinβN

−C2

∣∣∣∣∣cos

(β1−

N−1∑

i=2

βi

)·cosβN+sin

(β1−

N−1∑

i=2

βi

)·sinβN

∣∣∣∣∣ (25)

Once solved the optimization problem with respect to theunknowns βi (i = 1, 2, . . . , N), the effective dielectric constant εi

eff

of each section is obtained from Eq. (2) and the dimensions Wi, Si

are determined as solutions of Eq. (19) which satisfy the physicalrealizability constraints of the structure, namely:

Wc ≤ Wi ≤ Wm

Si > Sc

Wi > Wi−1, i = 2, 3, . . . , N

Si > Si−1, i = 2, 3, . . . , N (26)

The optimization procedure described above can be summarizedas follows:

1. assign the dimensions of the conductor-backed CPW andthe microstrip line to be connected, both having the samecharacteristic impedance Zo;

2. determine the reliable frequency band B by performing thecharacterization of higher-order modes relative to the conductor-backed CPW;

120 Costanzo

3. fix the number N of transition sections, all having the samecharacteristic impedance Z1, and a total transition length equalto λg/4, λg being the guided wavelength;

4. impose a suitable upper bound ρm on the magnitude of thereflection coefficient at the transition input within the reliableband B determined from stage 2;

5. apply the N-step optimization procedure to determine the phaseconstant of each transition section;

6. from the phase constants obtained in stage 5, determine theeffective dielectric constant and the dimensions of the N transitionsections which satisfy the physical realizability constraints.

The proposed synthesis approach can have practical applicationsin the design of CPW-to-microstrip transitions useful to the packagingof compact integrated circuits [20–29].

Even if formulated for the connection between a conductor-backedCPW and a microstrip line, the optimization procedure can be alsoapplied to the case of CPW without back plane, as originally consideredin [11, 12], by simply adopting the proper structural formulas [1]. Inthis case, it is expected that the absence of the back plate will reducethe occurrences of unwanted higher-order modes [15].

3. NUMERICAL RESULTS

The synthesis procedure outlined in the previous section isapplied to accurately design two millimeter-wave transitions betweena conductor-backed CPW and a microstrip line, both havingcharacteristic impedance Zo = 50 Ω on a dielectric substrate ArlonDiClad 880 (εr = 2.2 and tan δ = 0.0009) of height h = 0.254mm.On the basis of quasi-static formulas reported in [1], the dimensionsof the conductor-backed CPW are selected as Wc = 0.623mm, Sc =0.1mm and Gc = 3 mm, at a central design frequency fo = 30GHz.The microstrip line has width Wm = 0.773mm. To determine thereliable frequency band B, a characterization of the CPW higher-ordermodes is performed by following the approach outlined in [16], wherethe resonance effects in both the substrate thickness and the lateraldirections are modelled in terms of an effective permittivity εmn

eff foreach higher-order mode HMmn. In Table 1, the intersection frequencybetween the εeff curves of the fundamental mode and the generichigher-order mode HMmn is reported for the first three propagatingmodes in the conductor-backed CPW. It can be observed that thefirst higher-order mode HM01 intersects the fundamental mode at a

Progress In Electromagnetics Research, Vol. 113, 2011 121

frequency equal to 67 GHz, so an operating frequency band B goingfrom 1 GHz up to 65 GHz is imposed.

The synthesis method is applied to design two different transitions,with N = 3 and N = 5 conductor-backed CPW sections, respectively,all having the same characteristic impedance Zc = 51 Ω slightlydifferent from Zo. In both cases, an upper bound ρm = −10 dBis imposed for the magnitude of the reflection coefficient S11 atthe transition input, and a total length equal to 1.68 mm (λg/4,λg being the guided wavelength) is assumed for the transition. ASequential Quadratic Programming (SQP) procedure based on aconstrained quasi-Newton method [30] is applied to solve the N-stepoptimization problem formulated in Section 2 with respect to theunknowns βi(f) (i = 1, 2, . . . , N). The effective dielectric constant

Table 1. Intersection frequency fc between fundamental mode andhigher-order modes HMmn for conductor-backed CPW.

Mode Frequency fc [GHz]

HM01 67

HM02 133

HM10 890

Table 2. Effective dielectric constant and dimensions of conductor-backed CPW-to-microstrip transition of Fig. 1 (results from synthesisprocedure of Section 2 — Case N = 3).

Section number εieff L (mm) Wi (mm) Si (mm)

i = 1 1.78 0.56 0.65 0.12

i = 2 1.81 0.56 0.7 0.17

i = 3 1.83 0.56 0.73 0.23

Table 3. Effective dielectric constant and dimensions of conductor-backed CPW-to-microstrip transition of Fig. 1 (results from synthesisprocedure of Section 2 — Case N = 5).

Section number εieff L (mm) Wi (mm) Si (mm)

i = 1 1.77 0.336 0.64 0.11

i = 2 1.79 0.336 0.67 0.14

i = 3 1.81 0.336 0.7 0.17

i = 4 1.83 0.336 0.73 0.23

i = 5 1.86 0.336 0.76 0.3

122 Costanzo

εieff and the dimensions Wi, Si retrieved from the synthesis procedure

are summarized in Tables 2 and 3 for each section of the two designedtransitions with N = 3 and N = 5, respectively. Each step of thesynthesis procedure has required 3 iterations for the minimization ofthe relative functional.

To demonstrate the effectiveness of the optimization method,the analytical values of the phase constants βi (i = 1, 2, . . . , N)are computed from the synthesized parameters of Tables 2 and 3by adopting the conformal-mapping expressions reported in [1]. Asuccessful comparison between analytical and synthesized results canbe observed under Fig. 3 for the phase constants β1 (Fig. 3(a)), β2

(Fig. 3(b)) and β3 (Fig. 3(c)) relative to the transition with N = 3sections. Analogous results are obtained for the case with N = 5sections.

To verify the satisfaction of constraint (9), the return loss S11

is computed from the synthesized phase constants βi(i = 1, 2, . . . , N)by adopting Eq. (7) under the hyphotesis of small losses, as discussedin Section 2. For both transition configurations, with N = 3 andN = 5 sections, respectively, the magnitude of the reflection coefficientis correctly below the fixed upper bound ρm = −10 dB within the

(a) (b)

(c)

Figure 3. Analytical and synthesized phase constants values for thetransition with N = 3 sections: (a) β1, (b) β2, (c) β3.

Progress In Electromagnetics Research, Vol. 113, 2011 123

10 20 30 40 50 60-60

-50

-40

-30

-20

-10

0

10

Frequency (GHz)

(dB

)

upper bound ρm

N = 5 sections

N = 3 sections

No matching structure

Figure 4. Return loss ofconductor-backed CPW-to-microstrip transitions obtainedfrom the synthesis procedure.

0.77

1.68

unit: mm

Figure 5. Photograph of back-to-back transition with N = 3sections.

prescribed frequency band going from 1 GHz up to 65 GHz (Fig. 4). Inthe same Fig. 4, the return loss for the connection with no matchingstructure is reported for reference.

4. EXPERIMENTAL RESULTS

The experimental validation of the proposed synthesis techniqueis performed by examining the S-parameters behavior of a back-to-back transition obtained from two cascaded conductor-backedCPW-to-microstrip transitions, with N = 3 sections, connectedthrough a uniform conductor-backed CPW line. The two transitionsdimensions are those summarized in Table 2, obtained from the N-stepoptimization procedure discussed in Section 2. A 10 mm length is fixedfor the central conductor-backed CPW and the input microstrip lines.A photograph of the back-to-back prototype, realized by conventionalphotolitography, is illustrated in Fig. 5, where a particular of theN = 3 steps transition is also reported. S-parameters measurementsare performed by using the Anritsu 37269C network analyzer togetherwith the Anritsu text fixture. Due to the frequency limitations ofavailable instrumentation equipments, well working up to 40 GHz,higher frequencies are excluded from the measurement range. Theexperimental S-parameters of the back-to-back transition, shown inFigs. 6–7, are found to be in a satisfactory agreement with resultsobtained from Ansoft Designer simulations (Planar EM simulator),over the frequency range between 1 GHz and 40GHz, thus confirmingthe broadband transmission behavior of the designed conductor-backedCPW-to-microstrip transition. In particular, the measured return losscorrectly remains below the limit of−10 dB all over the frequency rangefixed in the synthesis procedure.

124 Costanzo

5 10 15 20 25 30 35 40-80

-70

-60

-50

-40

-30

-20

-10

0

10

Frequency [GHz]

Ref

lect

ion [

dB

]

Simulated

Measured

Figure 6. Reflection parameterof back-to-back N = 3 sectionstransition prototype: comparisonbetween simulations and measure-ments.

5 10 15 20 25 30 35 40

Frequency [GHz]

Tra

nsm

issi

on

[d

B]

SimulatedMeasured

10

5

0

-5

-10

Figure 7. Transmission parame-ter of back-to-back N = 3 setionstransition prototype: comparisonbetween simulations and measure-ments.

REFERENCES

1. Simons, R. N., Coplanar Waveguide Circuits, Components, andSystems, Wiley-Interscience, New York, 2001.

2. Zhang, G. M, J. S. Hong, B. Z. Wang, Q. Y. Qin, J. B. Mo, andD.-M. Wan, “A novel multi-folded UWB antenna FED by CPW,”Journal Electromagnetice Waves Applications, Vol. 21, No. 14,2109–2119, 2007.

3. Chen, N.-W. and Y.-C. Liang, “An ultra-broadband, coplanar-waveguide FED circular monopole antenna with improvedradiation characteristics,” Progress In Electromagnetics ResearchC, Vol. 9, 193–207, 2009.

4. Dastranj, A. and M. Biguesh, “Broadband coplanar waveguide-FED wide-slot antenna,” Progress In Electromagnetics ResearchC, Vol. 15, 89–101, 2010.

5. Simons, R. N. and R. Q. Lee, “Coplanar-waveguide/Microstripprobe coupler and applications to antennas,” IEE ElectronicsLetters, Vol. 26, 1998–2000, 1990.

6. Mirshekar-Syahkal, D., D. J. Newson, D. Wake, and I. D. Henning,“Wide-band transitions for applications in MMIC’s and OEIC’s,”IEEE Microwave Guided Wave Lett., Vol. 4, 299–300, 1994.

7. Dib, N. I. R., N. Simons, and L. P. B. Katehi, “New uniplanartransitions for circuit and antenna applications,” IEEE Trans.Microwave Theory Tech., Vol. 43, 2868–2873, 1995.

8. Gauthier, G. P., L. P. Katehi, and G. M. Rebeiz, “W-bandfinite ground coplanar waveguide (FGCPW) to microstrip linetransition,” 1998 IEEE MTT-S Int. Microwave Symp. Digest,

Progress In Electromagnetics Research, Vol. 113, 2011 125

107–109, 1998.9. Ellis, T. J., J.-P. Raskin, L. P. Katehi, and G. M. Rebeiz,

“A wideband CPW-to-microstrip transition for millimeter-wavepackaging,” 1999 IEEE MTT-S Int. Microwave Symp. Digest,629–632, 1999.

10. Raskin, J.-P., G. Gauthier, L. P. Katehi, and G. M. Rebeiz, “Modeconversion at GCPW-to-microstrip-line transitions,” IEEE Trans.Microwave Theory Tech., Vol. 48, 158–161, 2000.

11. Safwat, A. M. E., K. A. Zaki, W. Johnson, and C. H. Lee,“Novel design for coplanar waveguide to microstrip transition,”2001 IEEE MTT-S Int. Microwave Symp. Digest, 607–610, 2001.

12. Safwat, A. M. E., K. A. Zaki, W. Johnson, and C. H. Lee, “Noveltransition between different configurations of planar transmissionlines,” IEEE Microwave Wireless Comp. Lett., Vol. 12, 128–130,2002.

13. Riaziat, M., I. J. Feng, R. Majidi-Ahy, and B. A. Auld, “Single-mode operation of coplanar waveguides,” IEE Electronics Letters,Vol. 23, 1281–1283, 1987.

14. Safwat, A. M. E., K. A. Zaki, W. Johnson, and C. H. Lee,“Mode-matching analysis of conductor backed coplanar waveguidewith surface etching,” Journal of Electromagnetics Waves andApplications, Vol. 15, No. 5, 627–641, 2001.

15. Riaziat, M., R. Majidi-Ahy, and I.-J. Feng, “Propagation modesand dispersion characteristics of coplanar waveguides,” IEEETrans. Microwave Theory Tech., Vol. 38, 245–251, 1990.

16. Heinrich, W, F. Schnieder, and T. Tischler, “Dispersion andradiation characteristics of conductor-backed CPW with finiteground width,” 2000 IEEE MTT-S Int. Microwave Symp. Digest,1663–1666, 2000.

17. Pozar, D. M., Microwave Engineering, John Wiley and Sons, NewYork, 2005.

18. Fano, R. M., “Theoretical limitations of the broadband matchingof arbitrary impedances,” J. Franklin Institute, Vol. 429, 57–85,139–154, 1950.

19. Abramowitz, M. and I. A. Stegun, Handbook of MathematicalFunctions, Dover, New York, 1972.

20. Wang, S.-N. and N.-W Chen, “Compact, ultra-broadbandcoplanar-waveguide bandpass filter with excellent stopbandrejection,” Progress In Electromagnetics Research B, Vol. 17, 15–28, 2009.

21. Vinoy, K. J. and P. U. Reddy, “Design of narrowband bandpass

126 Costanzo

filter on coplanar waveguide using spiral slots,” Progress InElectromagnetics Research Letters, Vol. 6, 139–148, 2009.

22. Chen, J., G. Fu, G.-D. Wu, and S.-X. Gong, “Compact gradedcentral feeder line CPW-fed broadband antenna,” Journal ofElectromagnetics Waves and Applications, Vol. 23, Nos. 14–15,2089–2097, 2009.

23. Chen, H., Y. H. Wu, Y. M. Yang, and Y. X. Zhang, “A noveland compact bandstop filter with folded microstrip/CPW hybridstructure,” Journal of Electromagnetics Waves and Applications,Vol. 24, No. 1, 103–112, 2010.

24. Sze, J.-Y., T.-H. Hu, and T.-J. Chen, “Compact dual-bandannular-ring slot antenna with meandered grounded strip,”Progress In Electromagnetics Research, Vol. 95, 299–308, 2009.

25. Alkanhal, M. A. S., “Composite compact triple-band microstripantennas,” Progress In Electromagnetics Research, Vol. 93, 221–236, 2009.

26. Liao, W.-J., S.-H. Chang, and L.-K. Li, “A compact planarmultiband antenna for integrated mobile devices,” Progress InElectromagnetics Research, Vol. 109, 1–16, 2010.

27. Sze, J.-Y. and Y.-F. Wu, “A compact planar hexa-bandinternal antenna for mobile phone,” Progress In ElectromagneticsResearch, Vol. 107, 413–425, 2010.

28. Malekabadi, S. A., A. R. Attari, and M. M. Mirsalehi, “Design ofcompact broadband microstrip antennas using coplanar coupledresonators,” Journal of Electromagnetic Waves and Applications,Vol. 23, No. 13, 1755–1762, 2009.

29. Jaw, J.-L., F.-S. Chen, and D.-F. Chen, “Compact dualbandCPW-fed slotted patch antenna for 2.4/5GHz Wlan operation,”Journal of Electromagnetic Waves and Applications, Vol. 23,No. 14–15, 1947–1955, 2009.

30. Powell, M. J. D. “A fast algorithm for nonlinearly constrained op-timization calculations,” Lecture Notes in Mathematics, Vol. 630,144–157, 1978.