analysis of coupled microstrip lines for quad-band …

Progress In Electromagnetics Research B, Vol. 41, 187–211, 2012

ANALYSIS OF COUPLED MICROSTRIP LINES FORQUAD-BAND EQUAL POWER DIVIDERS/COMBINERS

A. M. El-Tager1, *, A. M. El-Akhdar2,and H. M. El-Hennawy2

1Electronic Engineering Department, M.T.C., Cairo, Egypt2Electronics and Communications Department, Ain Shams University,Cairo, Egypt

Abstract—This paper presents a novel quad-band power dividerwith equal power division ratio. The proposed power divider isrealized using two cascaded sections of dual-band transformers basedon coupled microstrip lines. Limitations of using dual-band quarter-wavelength transformers based on coupled lines are studied throughparametric analysis to obtain useful design guidelines related toavailable fabrication facilities. General closed-form expressions arededuced to calculate design parameters. To verify analysis anddesign methodologies, a prototype of quad-band equal power divider isproposed. Compared to conventional quad-band power dividers usingsections of transmission line transformers, the proposed power dividerrecords a size reduction of about 20% and reduced parasitic effects athigher frequencies according to the usage of only two resistors insteadof four with much smaller ohmic values. In addition, a quad-bandpower divider is proposed, fabricated and measured for 3G and 4Gapplications at 2.1, 2.5, 3.5, and 3.8GHz frequencies. Measured andsimulated data are in very good agreement which validates the noveldesign.

1. INTRODUCTION

Microstrip power dividers/combiners are widely used in RF/microwavefront-end systems such as power amplifiers, mixers, feed network of anantenna array system for distribution of low-power signals and labora-tory equipments. Many studies are made for device miniaturization [1–5], Ultra-wide band [6–8], and multiband operation [9–11]. They have

Received 8 March 2012, Accepted 21 May 2012, Scheduled 4 June 2012* Corresponding author: Ayman Mohamed Elsaye El-Tager ([email protected]).

188 El-Tager, El-Akhdar, and El-Hennawy

been subject of interest in recent years within the microwave engineer-ing community.

Modern wireless communication systems operate in dual-bandand multi-band frequencies. Therefore, many researchers developeddual-band RF components such as antennas [12] and powerdividers/combiners [13–16]. Power dividers and combiners play anessential role in microwave and millimetric wave systems. Severaldual-band Wilkinson power dividers have been developed basedon Monzon’s dual-band transformer using two cascaded sections oftransmission line transformers [17]. His theory was extended toprovide tri-band [18] and quad-band [19] transformers using cascadedtransmission lines, which are further more used in the design of multi-band power dividers/combiners [20–22]. Dual-band power dividers canalso be achieved using T-sections [23] and Pi structures [24–26] byconnecting open or short stubs to the conventional transmission line,but the main drawback of using such techniques is their large physicalsize. Another modification technique could be presented to providedual-band power dividers using artificial planar transmission lines [27–29], which need a very high resolution facility in fabrication.

On the other hand, dual-band power dividers can be achieved us-ing four cascaded sections of 1/6 wavelength transmission lines with dif-ferent characteristic impedances of transmission line transformers [30],as shown in Figure 1, which is limited to a frequency and its firstharmonic. To get rid of this limitation, a modification is done by con-necting series or parallel RLC circuits instead of the isolation resistor,according to any desired frequency ratio [31], but this is limited to lowfrequency applications. Another modification is applied by connectingtwo open stubs between the cascaded sections of transmission lines [32],as illustrated in Figure 2 which suffers a larger physical size. A thirdmodification is presented by replacing the conventional quarter wavetransmission line with a π-section [33] as shown in Figure 3. Finally, adual-band transformer based on coupled microstrip lines is presentedin [34], as shown in Figure 4. This technique has many advantages;first: size reduction over the presented techniques in Figure 2 and Fig-ure 3, second: there is no need to use any reactive components as inFigure 1. Therefore, this technique will be adopted throughout this

Z1 Z2 Z5 Z6

Z3 Z4 Z7 Z8

Z P1

P2

P3

Figure 1. Dual-band power divider using cascaded λ/6 sec-tions [30, 31].

Progress In Electromagnetics Research B, Vol. 41, 2012 189

P2

P3

Z1 Z1 Z3 Z4

Z2 Z2 Z5 Z6

R1 P1

jYs1

jYs2

Figure 2. Dual-band power divider using T-sections [32].

P2

P3

Z2

Z4

Zs5

Zs3

Z1

Z3

R

P1

Zs2

Zs4

Zs1

Figure 3. Dual-band power divider using π-sections [33].

Z1e, Z1o

Z2e, Z2o

Z3a Z3b

Z4a Z4b

R

P2

P3

Z1b

Z2b

Z1a

Z2a

P1

Figure 4. Dual-band power divider based on coupled lines [34].

work. But it has some limitations, such as frequency ratio constraintswhile realizing feasible coupled microstrip lines. This will be studiedthroughout this paper to obtain some guidelines for other designers.Furthermore, a closed form expression is introduced to calculate designparameters.


In this paper, the technique of dual-band transformers based oncoupled microstrip lines is studied. Some parametric analyses areobtained through normalized curves to highlight the advantages aswell as the limitations of this technique. Then, a general closed-formexpression to calculate the design parameters is deduced. Furthermore,this technique is employed to propose a novel quad-band power dividerwith equal power split. The idea of this proposal is based on replacingeach single-band transmission line transformer used in conventionaldual-band Wilkinson power dividers [20] by its equivalent dual-bandstructure based on coupled microstrip lines studied in this paper.In other words; Monzon’s theory is extended to be realized usingcoupled microstrip lines with realistic impedance values for the sake ofachieving reduced size quad-band power dividers with flexible layoutdesign. This paper is organized in six Sections. Section 2 introducesgeneral expressions as well as parametric analyses used to calculatethe design parameters of dual-band transformers based on coupledlines. In Section 3, analysis of the quad-band power divider andthe combination of both techniques are proposed. Section 4 showsthe design, fabrication and measurements of the proposed prototype.Section 5 compares the proposed technique to other published two-wayquad-band power dividers/combiners. Finally, Section 6 presents theconclusions of this work.

2. STUDY OF DUAL BAND TRANSFORMER USINGCOUPLED MICROSTRIP LINES

Many studies were made to convert the conventional single bandtransmission lines to operate as dual-band ones. This section presentsdual-band transformers based on coupled microstrip lines illustratedin Figure 5(a). Such transformers are of great interest because ofthe current trend toward compact, more efficient, dual-band RF frontends. This section also presents the generalization of the closedform expressions used in the design of dual-band transformer basedon coupled microstrip lines. In addition, some design guidelines areproposed throughout further parametric analysis.

Dual-band transformer based on coupled microstrip lines shown inFigure 5(a) consists of two branch transmission lines with characteristicimpedances Zm and coupled microstrip lines connected between thetwo branch lines having characteristic impedances of Zne and Zno.Characteristic impedances are tuned to the average frequency fo =(f1 + f2)/2; where f1 is the center frequency of the first operatingband and f2 is the center frequency of the second operating band.θi represents the electrical length of dual-band transformer either at


(a) (b)

ZN

s Zm, i Zm, i

Zne, Zno i , θ

θθ θ,

Figure 5. (a) Dual-band transformer based on coupled lines and (b)its equivalent single-band transformer circuit.

frequency f1 which will be θ1 or at frequency f2 which will be θ2.On the other hand, Figure 5(b) shows the equivalent single-bandtransmission line transformer having characteristic impedance ZN andelectrical length θs.

2.1. Theory and Analysis

In this section; the general expressions used to design dual-bandtransformer based on coupled lines is presented. Analysis will takeplace by comparing dual-band transformer based on coupled lines withits equivalent circuit of single-band transformer. This is known asreplacing single-band elements with dual-band elements technique. Asshown in Figure 5, main design parameters are:

(i) Single-band transformer characteristic impedance (ZN ).(ii) Single-band transformer electrical length (θs).(iii) Dual-band transformer branch lines characteristic impedances

(Zm).(iv) Frequency ratio of the dual-band transformer (f2/f1).

Due to the symmetric property of the dual frequency transformerthe even-odd decomposition method is used to obtain a generalexpression for calculating design parameters. Where

θo =θ1 + θ2

2= θs (1)

fo =f1 + f2

2(2)

Even or odd half circuit design parameters are calculated usingtransmission line theory as follows:


2.1.1. Even Mode Analysis

From the transmission line theory, all ports look open for even modeanalysis. According to Figure 6(a)

Zein = jZm

−Zne+Zm · tan2θi

(Zm + Zne) · tan θi(3)

While from Figure 6(b)

Zein = −jZN · cot

(θs

2

)(4)

By equating Equations (3) and (4) we get:

ZN = ZmZne−Zm · tan2θi

(Zm + Zne) · tan θi · cot(

θs2

) (5)

Or,

Zne = ZmZm · tan2θi − ZN · cot

(θs2

) · tan θi

Zm + ZN · cot(

θs2

) · tan θi

(6)

2.1.2. Odd Mode Analysis

From transmission line theory, all ports are shorted for odd modeanalysis. From Figure 7(a)

Zoin = jZm

tan θi · (Zno + Zm)Zm − Zno · tan2θi

(7)

Zine

Zine

(a) (b)

ZN , s /2Zm θi Zne

i θ θ, ,

Figure 6. (a) Even mode half circuit, (b) its equivalent circuit.

Zino Zin

o

(a) (b)

Zno i Zm i ZN , θs/2 θθ , ,

Figure 7. (a) Odd mode half circuit, (b) its equivalent circuit.


And from Figure 7(b)

Zoin = jZN · tan

(θs

2

)(8)

From Equations (7) and (8) we get

ZN = Zm(Zno + Zm) · tan θi · tan

(θs2

)

Zm − Zno · tan2θi(9)

Or

Zno = ZmZN · tan ( θs

2 ) + Zm · tan θi

ZN · tan ( θs2 ) · tan2θi−Zm · tan θi

(10)

2.1.3. General Expressions

Now after calculating the characteristic impedances of the coupledlines, an expression for the electrical length of the dual-bandtransformer is to be obtained.

By definition we have,

θ2

θ1=

f2

f1(11)

From Equation (1) we have

θ1 = 2θs − θ2 (12)

Substituting Equation (12) in Equation (13) leads to

θ1 = θs ·(

f1

fo

)(13)

And similarly

θ2 = θs ·(

f2

fo

)(14)

From Equations (13) or (14), Equations (6) and (10) can be rewrittenas

Zne = Zm

Zm · tan2[θs ·

(f1

fo

)]− ZN · cot

(θs2

) · tan[θs ·

(f1

fo

)]

Zm + ZN · cot ( θs2 ) · tan

[θs ·

(f1

fo

)] (15)

And

Zno = Zm

ZN · tan(

θs2

)+ Zm · tan

[θs ·

(f1

fo

)]

ZN · tan(

θs2

) · tan2[θs ·

(f1

fo

)]− Zm · tan

[θs ·

(f1

fo

)] (16)


Equations (15) and (16) are considered as general expressionscontaining all design variables of the dual-band transformer based oncoupled microstrip lines, which are; ZN , θs, (f2/f1) and Zm. Thissection studies the relations between the design variables and howthey could affect the choice of coupled lines characteristic impedancesZne and Zno which may lead to a non-realizable solution if misschosen. Therefore, to provide design guide, a closed form expression isintroduced in Appendix A to choose Zm. Moreover, further parametricanalyzes are studied in the next section to choose Zm graphicallyfor certain required frequency ratio with realizable values of coupledmicrostrip lines.

2.2. Parametric Analysis

Usually, ZN is given, while for quarter wave length transformers, θs

is chosen to be 90 degrees. But the frequency ratio (f2/f1) needsmore investigation. Based on the analysis introduced in [34], it shouldbe less than 1.64. By further insight thinking about this limit itis found to be non-realizable and requires more determination. Inorder to determine the limitation and to obtain a degree of freedom inrealizing the coupled microstrip lines, a new coefficient q′ is introduced.Assuming q′ is the difference between Zne and Zno. While, a variable qis defined as q′ normalized to Zo which is the geometric mean of Zne andZno(Zo = (Zev · Zod)1/2) [35, 36]. It is called a realization coefficient,which helps designers to obtain a feasible realizable solution as shownin this section.

q =q′

Zo(17)

q′ − qZo = 0 (17a)

where

q′ = Zne − Zno (17b)

Zo =√

ZneZno (17c)

2.2.1. Variation of Frequency Ratio Fr

Figure 8(a) shows the coupled lines characteristic impedances atdifferent values of Zm. Figure 8(a) shows the curves presented inFigure 6 of Ref. [34] at different values of Zm. While Figure 8(b)shows how the introduced realization coefficient (q) is varying versusthe frequency ratio at different values of branch line characteristic


(a) (b)

0

10

20

30

40

50

60

70

80

1.4 1.45 1.5 1.55 1.6 1.65 1.7

Frequency ratio (f2/f1)

Z a

nd Z

ev

od

Zm=30 Ω

Zm=70 Ω

Zm=60 Ω

Zm=50 Ω

Zm=40 Ω

0

1

2

3

4

5

6

7

8

9

10

1 1.1 1.2 1.3 1.4 1.5 1.6 1.7

Rea

liza

tio

n c

oef

fici

ent

(q)


Zm = 30 Zm = 70

Figure 8. (a) Coupled lines characteristic impedances (Zev and Zod),(b) realization coefficient q versus frequency ratio Fr at different valuesof Zm.

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

1.4 1.5 1.6 1.7


Rea

liza

tio

n c

oef

fici

ent

(q) Zm=30Ω

Zm=70Ω

Zm=50Ω

Zm=40Ω

Figure 9. Realization coefficient versus frequency ratio at differentvalues of Zm.

impedance Zm. In Figure 8(b), the solid line presents the curve ofZm = 30 Ω. While, the dashed line shows the curve of Zm = 70Ω.

From Figure 8, it is observed that as frequency ratio increases,Zm increases (but it shouldn’t exceed the value of ZN , otherwise itwill lead to unacceptable solution). Although this curve plots q withlarge values, for feasible realization of coupled microstrip lines, therealization coefficient q is usually small (may be less than 2) as shownin Figure 9, because as q decreases Fr increases. On the other hand,Figure 9 can be used to determine the maximum allowed q at certainZm and certain Fr. Therefore, further investigation of this realizationcoefficient is presented as follows.


0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 2 4 6 8 10

Normalized separation (S')

Real

izat

ion c

oe

ffic

ien

t(q)

Zm /ZN= 0.67

q max2

q min

S' min2

S' maxS' min1

q ma x1

Zm /ZN = 1

Re f. [34]

Figure 10. Realization coefficient (q) versus separation betweencoupled lines (S′).

2.2.2. Separation between Coupled Lines

Figure 10 introduces the relation between coupled lines characteristicimpedances (Zev and Zod) presented in the realization coefficient (q)and the separation between coupled lines with respect to substratethickness (S′ = S/H). This normalized curve is important torealize feasible dual-band transformer based on coupled microstriplines. Usually, the available fabrication process has certain resolutionwhich limits (S′) to a certain minimum value S′min which determinesqmax 1. On the other hand, the maximum separation limit S′max whichcorresponds to qmin is determined by the condition (Zev = Zod)at which (Zm/ZN = 0.67). One can observe that the region ofnormalized separation is (S′min < S/H < 8), which is different fromthat of conventional coupled lines (0.1 < S/H < 10). However, formore practically realizable dual-band transformer based on coupledmicrostrip lines, a region of solution is provided starting from (ZN =Zm) which is qmax 2, that dominates the maximum limit of q. Therefore,the minimum frequency ratio Fr is to be obtained applying themaximum q from Figure 10 into Figure 9. It is clear from Figure 10that the realization coefficient q can vary in (0.04 < q < 0.72), whilethe design example of reference [34] is plotted, giving q = 0.32.

2.2.3. Choice of Zm

Figure 11 shows the variation of realization coefficient (q) versusthe normalized branch line characteristic impedance Zm at different


0

1

2

3

4

5

6

0.4 0.5 0.6 0.7 0.8 0.9 1

Normalized Zm (Zm/ ZN)

Rea

liza

tio

n c

oe

ffic

ien

t (q

)

Fr(

f2/f

1)

Fr= 1.6 Fr= 1.5 Fr= 1.4 Fr= 1.3

Figure 11. Realization coefficient (q) versus normalized branchimpedance Zm at different frequency ratios Fr.

0 20 40 60 80 100 120 140 1600

50

100

150

X: 70

Y: 36.72

ZN (Ohms)

Zne a

nd Z

no (

Oh

ms)

zne

zno

Zm=30

Zm=10

Zm=50Zm=70

Figure 12. Variation of Zne and Zno with ZN at different values ofZm.

frequency ratios (Fr). It is recognized that the realization coefficient(q) increases, as the frequency ratio decreases. On the other hand,for certain frequency ratio, as Zm increases, the realization coefficient(q) slightly increases. This observation is matched with the resultsobtained in Figure 9. If the designer needs higher frequency ratio, Zm

should be chosen with higher value. But, the maximum allowed Zm islimited by the value of ZN to satisfy (Zm/ZN = 1).

Moreover, Zm can also be determined graphically throughFigure 12. It shows the relation between single band characteristicimpedance ZN and coupled lines characteristic impedances Zne andZno at different values of branch lines characteristic impedances Zm.Figure 12 shows the acceptable and unacceptable solutions of Zm. Forexample if we pick Zm = 30 Ω, that leads to unacceptable solutionbecause Zno > Zne. On the other hand, if we choose Zm = 70 Ω,


that will lead to an acceptable solution but with large value of q′ thatyields to small separation between microstrip coupled lines throughrealization. Based on certain microstrip fabrication process this smallseparation might not be realizable. For simplicity an expression tocalculate the optimum value of branch line characteristic impedanceZm is introduced in Appendix A.

From the previous studies, we can define the unacceptable regionof solutions in the following conditions:

a) Zne < Zno, which is unacceptable solution.b) q′ > Zo, that obtains tight separation between coupled lines which

might not be reached in the available fabrication process.c) If the difference q′ is very small, a large separation between coupled

microstrip lines occurs leading to extra radiation losses.

As a conclusion, this section presented the variation of designparameters and how they affect the coupled lines characteristicimpedances using both analytical and graphical solutions. Thefollowing section presents the application of these dual-band coupledlines to design a novel quad-band power divider with equal power split.

3. DESIGN OF PROPOSED QUAD-BAND POWERDIVIDER

This section proposes a novel design methodology of quad-band powerdivider/combiner with equal power division ratio. The proposedpower divider consists of two sections of dual-band transformersbased on coupled microstrip lines. Each single-band transmissionline transformer used in dual-band Wilkinson power divider shown inFigure 13 is replaced by its equivalent dual-band structure based onthe previously studied coupled microstrip lines technique.

Figure 13 presents dual-band Wilkinson power divider. The powerdivider consists of two sections of transmission line transformers andtwo isolation resistors for port isolation. This conventional technique

R1 R2

Z1 θ1 Z2 θ2

P3 Z1 θ1 Z2 θ2

P2

P1

, ,

,,

Figure 13. Dual-band Wilkinson power divider based on Monzon’stheory for cascaded transmission line transformers [20].


f1

f01

f2

f0

f3

f02

f4

Figure 14. Frequency line indicator.

is utilized to provide tri-band (using three sections of transmission linetransformers) [21] and quad-band (using four sections of transmissionline transformer) [22]. The former conventional technique is to becompared to the novel design of this work in Section 5. The proposeddesign methodology is presented as follows:

1) Starting with a given four frequencies f1, f2, f3, and f4, andexploiting the symmetric property of the dual-band transformeras shown in Figure 14, the intermediate frequencies fo1 and fo2

are calculated using Equations (18) and (19) as follows:

f01 =f1 + f2

2(18)

f02 =f3 + f4

2(19)

While, the center frequency fo is calculated using Equation (20)as follows:

f0 =fo1 + fo2

2(20)

2) The design parameters Z1 and Z2 in the conventional Wilkinsonpower divider shown in Figure 13 are calculated using Monzon’sexpression shown in Equations (21) and (22)

Z2 = Zo

√12α

+

√1

4α2 + 2 (21)

Z1 =2Z2

o

Z2(22)

where;α = tan2 (βl) = P 2 and P =

nπ

1 + m

m presents the frequency ratio (f2/f1).


3) The isolation resistors R1 and R2 are calculated usingEquations (23) and (24) as follows:

R1 =

√BE

AD(23)

R2 =E

C −DR1(24)

where

A = 2Z2

(1 +

Z2T2

Z1T1

)(a)

B = 2Z2 (b)

C =2Z2

2T2

Zo(c)

D =(

T2 − Z2

Z1T1

)(d)

E = 2ZoC (e)T1 = tanB1l1 (f)T2 = tanB2l2 (g)

Next step is to replace each single band transmission line by itsequivalent dual-band structure based on coupled microstrip lines asshown in Figure 15. The design procedure of this step is obtainedgraphically suing parametric analysis of Section 2.2 and analyticallyusing Equations (15), (16), (17) and Appendix A.

As a conclusion, this section presented the design guidelines toprovide quad-band power divider using two cascaded sections of dual-band transformers based on coupled microstrip lines. This design

Figure 15. Novel quad-band power divider based on coupledmicrostrip lines.


procedure is summarized in the flowchart of Appendix B. Section 4verifies this design methodology through a fabricated prototype. Onthe other hand, Section 5 compares this novel technique to conventionalquad-band power divider techniques.

4. PROPOSED QUAD-BAND POWER DIVIDER FOR 3GAND 4G APPLICATIONS

This section presents design, fabrication and measurements of a quad-band power divider with equal power division based on coupledmicrostrip lines. The proposed power divider is useful for 3G and 4G,such as WiMax and UMTS applications. It operates at frequencies2.1GHz, 2.5 GHz, 3.5GHz, and 3.8GHz. It is interesting to mentionthat the proposed design satisfies 3G and 4G application bandwidths.

4.1. Realization Using Coupled Microstrip Lines

Following the design process proposed in the flowchart of Appendix B,a quad-band power divider with equal power division is designed asfollows:

• Using Equations (19) and (20) we obtain ZN = ZN1 = ZN2 =70.7Ω for θs = θs1 = θs2 = θs3 = θs4 = 90 degrees.

• Graphically, using Figures 8 through 12 or analytically, using thededuced expression of Appendix A, the branch line characteristicimpedance can be calculated as follows: Using the general expression from Equations (15), (16), and

the closed form expression presented. The branch linescharacteristic impedance Zm is calculated using the closedform expression presented in Appendix A, and graphicallyusing the design curves presented in Figures 10 through 12 asfollows:

¦ Zm = Zm1 = Zm2 = Zm3 = Zm4 = 50Ω¦ Ze = Ze1 = Ze2 = Ze3 = Ze4 = 36 Ω¦ Zo = Zo1 = Zo2 = Zo3 = Zo4 = 27 Ω

• The isolation resistors R1 and R2 are calculated usingEquations (23) and (24).

After calculating the design parameters of the quad-band powerdivider, quad-band power divider is realized as shown in Figure 15using low loss Teflon substrate RT/Duroid6010 with relative dielectricconstant εr = 10.5, height H = 50 mil, metal thickness T = 17µm, andloss tangent tan δ = 0.001. Table 1 presents the final realized physicalparameters of the proposed microstrip transformer.


Table 1. Physical parameters of the proposed quad-band powerdivider.

Parameters Coupled lines Branch linesIsolation

resistor Ω

Electrical

parameters

Z1e = Z2e

= Z3e = Z4e

Z1o = Z2o

= Z3o = Z4o

Zm1 = Zm2

= Zm3 = Zm4

R1 R2

Impedances

(Ω)36 27 50 100 200

Physical

values (mm)

Width Separation Length Width Length100 200

2.4 0.75 9.12 1.1 9.5

Figure 16. Schematic of the proposed quad-band equal power divider.

4.2. Design Simulations and Layout Generation

This section presents the simulation of the proposed quad-band powerdivider based on coupled microstrip lines to include all microstripdiscontinuities and to prepare the design for layout generation process.Figure 16 shows the schematic of quad-band equal power divider. Thesimulation is done using Agilent ADS. Resultant Scattering parametersof the proposed structure are plotted when compared to measuredparameters in Section 4.3.

4.3. Fabrication and Measurements

The designed proposed prototype is fabricated, which gives promisingmeasurements. Figure 17 shows the fabricated prototype of theproposed quad-band equal power divider based on coupled microstrip


Figure 17. Quad-band power divider connected to the HP8510c VNA.

Table 2. Measured S-parameters of quad-band power divider.

Parameters 2.1GHz 2.5GHz 3.5GHz 3.8GHz Notes

Input

return loss15 13 11 15 > 11 dB

Output

return loss21 15 15 22 > 15 dB

Insertion

loss0.05 0.4 0.6 0.6 < 0.6 dB

Isolation

between

P1 & P2

21 26 15 15 > 15 dB

lines. The proposed quad-band equal power divider based on coupledmicrostrip lines has a physical size of 5.5 cm × 3 cm. Figure 18 showsthe simulated and measured return losses |S11|, |S22| and |S33| ofthe designed quad-band power divider. While Figure 19 presents themeasured insertion |S12|, |S13| and isolation |S23|.

Figure 18 shows that measured input return loss |S11| is betterthan 11 dB, while output return losses |S 22| and |S 33| are better than15 dB. Figure 19 shows that measured insertion losses |S 12| and |S 13|are better than 0.6 dB compared to ideal 3 dB level, while measuredisolation is better than 15 dB at the four operating frequencies. Themeasured S-parameters are recorded in Table 2.

Measurement process is done using HP8510c VNA as shown inFigure 17. There are some slight discrepancies between measurementsand simulations which may be due to any uncounted microstripdiscontinuities and/or tolerances in the fabrication process. The quad-band equal power divider based on coupled microstrip lines recordspromising measurements at the four operating frequencies.


(a)

(b) (c)

-35

-30

-25

-20

-15

-10

-5

0

1500 2000 2500 3000 3500 4000 4500

|S 11| measured

|S 11| simulate d

Inp

ut r

etu

rn

Frequency in MHz

-35

-30

-25

-20

-15

-10

-5

0

1500 2000 2500 3000 3500 4000 4500

|S 22| measured

|S 22| simulate d

Ou

tpu

t ret

urn

-35

-30

-25

-20

-15

-10

-5

0

1500 2000 2500 3000 3500 4000 4500

|S 33|m easured

|S 33|sim ulated

Frequency in MHz

Ou

tpu

t ret

urn

Frequency in MHz

Figure 18. Simulated and measured return losses (a) Magnitude of|S11|. (b) Magnitude of |S22|. (c) Magnitude of |S33|.

Next section presents a comparison between quad-band powerdivider using other techniques, and also presents the features of theprovided design.

5. COMPARISON TO OTHER CONVENTIONALTECHNIQUES

Figure 20 shows quad-band equal power split power divider usingcascaded sections of transmission line transformers, and four resistorsconnected at the end of each section to provide matching and isolationpresented in [22]. It operates at harmonic frequencies 0.5, 1, 1.5, and2GHz. The presented quad-band power divider based on cascadedsections of transmission line transformers has some disadvantages:1) long physical length of the divider due to the usage of fourcascaded sections of transmission line transformers which have beenreduced by using curved transmission lines instead of the conventionalstraight ones as shown in Figure 21. 2) Parasitic effects causedby the four resistors. 3) The need of an optimization CAD tool(swarm particle optimization). 4) Applicable only for low frequencyapplications according to the high values of the used isolation resistors


(a)

-10

-9

-8

-7

-6

-5

-4

-3

-2

1500 2000 2500 3000 3500 4000 4500

|S 12|m easured

|S 12|sim ulated

Frequency in MHz

Inse

rtio

n L

oss

-35

-30

-25

-20

-15

-10

-5

0

1500 2000 2500 3000 3500 4000 4500

|S 23|m easured

|S 23|sim ulated

Iso

lati

on

Frequency in MHz

(b)

Figure 19. Measured and simulated (a) insertion |S12| and (b)isolation |S23|.

R2

P2

P3

Z3 Z4

Z3 Z4

R3 R4

Z1

P1

Z2

Z1 Z2

R1

Figure 20. Quad-band power divider presented in [22].

Figure 21. Quad-band powerdivider using cascaded sections oftransmission line transformers aspresented in [22].

Figure 22. Quad-band equalpower divider using curved trans-mission lines based on this work.

(maximum value is 400 Ω). On the other hand, a proposed quad-bandequal power divider based on coupled microstrip lines based on thiswork and utilizing curved transmission lines is presented as shown inFigure 22. The proposed power divider based on coupled microstriplines operates at frequencies 1.1, 1.4, 1.8, 2.1GHz. Table 3 showsthe comparison between the two quad-band power dividers. Quad-band power divider presented in this work illustrated in Figure 22features reduced size, reduced parasitic effect, and simplicity in designparameters calculation.


Table 3. Comparison between quad-band equal power dividers.

Parameters This work Ref. [22]Enhancement

[%]Comments

Need of Particle

swarm

optimization

method

No Yes

Analytical

solution

is presented

Simplicity in

direct

substitution

Size

[cm × cm]10× 5 12× 5 20 %

Length

reduction

No. of

Isolation

Resistors

Only 2 4 50%

Less

parasitic

effects

Max. Resistor

Value (Ω)200 400 50%

Applicable

for higher

frequency

applications

Table 3 shows a size reduction of about 20%. The presented designuses only two isolation resistors with reduced values, which allowsoperation in higher frequencies with reduced parasitic effects. Theproposed design has a bit degraded performance compared to the onepresented in [22] but, it fulfills the power dividers performance. Therecorded input return loss is better than 14 dB at the four operatingfrequencies. Output return losses are better than 15 dB. The recordedinsertion losses are better than 0.3 dB, while the recorded isolationbetween the output ports is better than 19 dB at the four designfrequencies.

6. CONCLUSION

This paper presents a novel design of Quad-band power di-vider/combiner with equal power division ratio; the divider uses twocascaded sections of dual-band transformers based on coupled mi-crostrip lines. Moreover, the used dual-band transformer based on cou-pled microstrip lines are studied and analysed to obtain a closed formexpression for the determination of the design parameters. Furtherparametric analyses are conducted using general normalized curves tohelp designers to obtain a realizable solution based on their availablefabrication process resolution. The proposed power divider featuresreduced size and uses fewer resistors with lower values than other con-


ventional techniques which make it applicable to higher frequencies. Aproposed prototype is designed, simulated and fabricated at 2.1, 2.5,3.5 and 3.8 GHz for 3G and 4G applications. The achieved agreementbetween simulations and measurements verifies the design methodologyand make the proposed structure very promising. The proposed de-sign could be easily tailored to be applied in package level such as Lowtemperature co-fired ceramics (LTCC) and chip level such as CMOSand GaAs Monolithic microwave integrated circuits (MMIC).

APPENDIX A.

The branch line characteristic impedance can be a design freedom.The following expression helps the designer to calculate the branchline impedance which will provide an optimum solution for the coupledlines characteristic impedances Zev and Zod. For simplicity the designparameters used in the coupled lines characteristic impedances arestated as follows:

a = Zm (A1)b = ZN (A2)

c = tan[θs ·

(f1

fo

)](A3)

d = tan(

θs

2

)(A4)

The following expressions are used to calculate the branch lineimpedance Zm:

Zm =α1 +

√β1

δ1(A5)

where

α1 = qd2 + 2d2 − 2d2c4 + qc4d2 + 2qc2 (A6)β1 = 4d4 + 2q2d4c4 − 4d4c8q + 8d2c6q + q2c8d4

−8qc2d2 + q2d4 + 4qd4 − 8d4c4 + 4d4c8 + 4q2c2

−16c6d2 − 32d2c4 + 8q2c4d2 − 16d2c2 (A7)

δ1 =b

2cd (−2c2 − 2 + qc2 − q)(A8)

APPENDIX B.

Proposed design steps used to calculate the design parameters of quad-band equal power divider based on coupled microstrip lines.


f1, f2, f3, f4, and θs

ZN1 and ZN2 are calculated through Equations (21) and

(22)

fo1, fo2, and fo are calculated through Equations (18),

(19), and (20)

R1 and R2 are calculated through Equations (23) and

(24)

Zm1 and Zm2 are calculated through Equation (A)

Zne and Zno are calculated

through Equations (15) and (16)

START

END

Equations

(17)

Choose appropriate q

from the design curves

in Figures 9 , 10, 11

No

Yes

ZN, Zm, Fr, Zne, Zno

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