# A wideband quadrature power divider/combiner and its application to an improved balanced amplifier.

1. INTRODUCTIONMetamaterials are artificial electromagnetic materials that do not exist in nature and exhibit some unique properties. Successful applications have been proposed in dual-band components [1,2], enhanced-bandwidth devices [3], super-compact multilayer transmission lines [4], phase and impedance couplers with arbitrary coupling level [5,6] and negative/zeroth order resonators [7].

On the other hand, quadrature power dividers are key components for the implementation of balanced amplifiers, image rejection mixers, Doherty amplifiers, circularly polarized antennas, and others. Recently metamaterial quadrature power dividers and baluns have been proposed in [8-10].

In [8], the proposed broadband Wilkinson balun, the authors make use of a one-stage Wilkinson power divider in combination with a +90[degrees] composed right/left handed transmission line (CRLH-TL) and a -90[degrees] CRLH-TL. The +90 CRLH-TL consists of two series capacitor Clo, a shunt inductor LL0, and an initial host transmission line with characteristic impedance of 50 Q and length [d.sub.tL1]. The -90 CRLH-TL is composed by two series capacitor [C.sub.L2], a shunt inductor [L.sub.L2], and a host transmission line with characteristic impedance of 50 Q and length [d.sub.TL2]. The WPD is calculated at the central frequency [f.sub.0] and then the metamaterial phase-shifting lines are designed as follows: (a) to design the +90 CRLH -TL an initial host transmission line with characteristic impedance of 50 Q and length [d.sub.tL1] small enough in order for the effective medium equations presented in [12] remain unchanged is considered. With the purpose of achieve a phase advance for a positively traveling plane wave of the form [e.sup.-j[PHI]MM+], [[PHI].sub.MM+] must be chosen to be negative. Thus, letting [[PHI].sub.MM+] = - [pi]/2 in (1) listed in [8] and imposing the impedance matching condition of (2) listed in [8] the results of [C.sub.L0] and [L.sub.L0] are obtained. (b) The 50 [ohm] -90 MM-TL is designed to achieve a phase delay for a positively traveling plane wave of the form [e.sup.-j[PHI]MM-], therefore [[PHI].sub.MM-] must be chosen to be negative. Thus [C.sub.L2], [L.sub.L2], and [d.sub.TL2] are obtained letting [[PHI].sub.MM-] = [pi]/2 in (1) listed in [8], imposing the impedance matching condition and enforcing the phase-slope condition of (3) listed in [8]. The circuit was validated with a prototype at 1.5 GHz and it has achieved a FBW of 77%.

On the other, the modified Wilkinson power divider proposed in [9] makes use of a one-stage Wilkinson power divider in combination with a conventional microstrip transmission line and a CRLH-TL whose unit cell is composed by [L.sub.L], [C.sub.L], [C.sub.R], and [L.sub.R]. In order to design a broadband quadrature power divider, over the frequency range from [f.sub.1] to [f.sub.2], the CRLH-TL is chosen to have -90[degrees] and -180[degrees] at [f.sub.1] and [f.sub.2], respectively. Solving (1) and (2) listed in [9] the four equivalent circuit parameters LL, [C.sub.L], [C.sub.R], and [L.sub.R] are obtained. On the other hand, the MS-TL is designed to achieve the desired phase difference with the CRLH-TL therefore the microstrip line should be designed to have phase delay of -180[degrees] and -270[degrees] at [f.sub.1] and [f.sub.2] respectively. Proposed circuit has FBW of 94% and exhibits big size due to the electrical length of the microstrip line.

In [10], a broadband quadrature power divider with FBW of 104% was developed. The circuit uses the same circuit as [9] however they differ in the design method. In [10], the quadrature power divider is also designed at the center frequency [f.sub.0]. After, the length of the microstrip transmission line is properly chosen and simulated to determine the phase [[PHI].sub.MS1] and the phase [[PHI].sub.MS2] at [f.sub.1] and at [f.sub.2] respectively. The difference phase between the CRLH-TL and the microstrip transmission line must be +90 at [f.sub.1] and [f.sub.2], therefore the phase [[PHI].sub.MM1] and the phase [[PHI].sub.MM2] at [f.sub.1] and [f.sub.2] are obtained by means of (1) and (2) listed in [10]. Subsequently, two values P and Q must be obtained from Equations (6) and (7) listed in [10] and finally, the right-handed section of the CRLH-TL, [L.sub.L] and [C.sub.L] are obtained from P, Q and the matching condition. In [10], it is mentioned that the length of the microstrip transmission line is properly chosen however a criteria to choose the length of the microstrip transmission line is not clearly provided. On the other hand, because the solution depends directly of [f.sub.1] and [f.sub.2] it is worthy to mention that, a criterion to choose [f.sub.1] and [f.sub.2] is not mentioned.

In this paper, a novel metamaterial broadband quadrature power divider and a simply but robust design methodology based on a closed set of equations are proposed. Also, the proposed quadrature power divider has been utilized to design a balanced amplifier whose measured results show that a balanced amplifier with FBW of 124.4% has been successfully implemented.

2. PROPOSED QUADRATURE POWER DIVIDER AND ITS DESIGN METHOD

The proposed quadrature power divider consists of a two-section power divider in combination with a 90[degrees] [+ or -] 10[degrees] metamaterial differential phase shifter as is shown in Figure 1(a). The two-section power divider has been calculated with Seymour's method [11], to provide equal power split, high output port isolation, and good return losses at all three ports within a fractional bandwidth of 120%. All the arms of the power divider have electrical length of -90[degrees] at the center frequency and characteristic impedance [Z.sub.1] and [Z.sub.2] of 64.77 [ohm] and 77.19[ohm] respectively. Also [R.sub.1] and [R.sub.2] were calculated as 165.22 [ohm] and 137.73 [ohm] respectively.

The proposed differential phase shifter consists of a CRLH-TL and PRH-TL named CRLHu-TL and PRHd-TL respectively. The CRLHu-TL is composed by a Pure-Left-Handed section whose unit-cells elements are [C.sub.L] and [L.sub.L] and a pure right handed section, named PRHp. PRHp is due to the parasitic pad effect of [C.sub.L] and [L.sub.L]. The PRHd-TL is composed by a pure right handed section, named PRHb, and two sections equivalents to PRHp. Let [[empty set].sub.PLH] and [[empty set].sub.PRH] be the phase shift of the pure left handed section and the phase shift of the PRHb respectively, then the differential phase shift between top branch and the bottom branch of the quadrature power divider shown in Figure 1(a) is given by (1) [12].

[PHI]([omega]) = [[PHI].sub.PLH]([omega]) - [[PHI].sub.PRH]([omega]) (1)

where

[[PHI].sub.PLH]([omega]) = N[1/[omega][square root of([L.sub.L][C.sub.L])] (2)

[[PHI].sub.PRH]([omega]) = -N[omega][square root ([C.sub.R][L.sub.R])] (3)

It can be shown that the derivative and second derivate of (1) are given by (4) and (5) respectively. Since N > 0, w > 0, LL > 0 and [C.sub.L] > 0 then [empty set]" > 0 thus the differential phase shift is a convex function because it is a twice differentiable function and its second derivative is positive. In particular, if [empty set]'([omega] = [[omega].sub.min]) = 0, then [empty set]([[omega].sub.min]) is a global minimum. From this condition we calculated (6) which represent the frequency in which the differential phase shift is minimal.

[PHI]'([omega]) = [d/d[omega]][PHI]([omega]) = N ([-1/[[omega].sup.2][square root of ([L.sub.L][C.sub.L])]) + [square root ([C.sub.R][L.sub.R])]) (4)

[PHI]"([omega]) = N(2/[[omega].sup.3][square root of ([L.sub.L][C.sub.L])]) (5)

[[omega].sup.2.sub.min] = 1/[square root of ([L.sub.L][C.sub.L][C.sub.R][L.sub.R])] (6)

In this work, the accepted differential phase shift is considered as 90[degrees] [+ or -] 10[degrees] however an analog procedure can be performed for other tolerances. Because the accepted differential phase shift is 90[degrees] [+ or -] 10[degrees] then the minimum accepted differential phase shift is 80[degrees] (= 4[pi]/9 radians) and maximum accepted differential phase shift is 100[degrees] (= 5[pi]/9 radians). Now, let [[omega].sub.1] be the frequency in which the differential phase shift is maximum and [[omega].sub.min] the frequency in which the differential phase shift is minimum, then

[PHI]([[omega].sub.min]) = N(1/[[omega].sub.min][square root ([L.sub.L][C.sub.L])]] + [[omega].sub.min][square root of ([C.sub.R][L.sub.R])]) = 4[pi]/9 (7)

and

[PHI]([[omega].sub.1]) = N(1/[[omega].sub.1][square root ([L.sub.L][C.sub.L])]] + [[omega].sub.min] [square root of ([C.sub.R][L.sub.R])]) = 5[pi]/9 (8)

Now, from (7) and (8) and after some straightforward we have obtained the polynomial equation of the second degree given by (9) which solution is given in (10).

2[[omega].sup.2.sub.min] - 5[[omega].sub.1][[omega].sub.min] + 2[[omega].sup.2.sub.1] = 0 (9)

[[omega].sub.1] = [[omega].sub.min]/2 (10)

Thus replacing (10) in (7), a new equation in terms of -1 can be set as follows:

N(1/2[[omega].sub.1][square root of ([L.sub.L][C.sub.L])] + 2[[omega].sub.1] [square root of ([C.sub.R][L.sub.R])]) = 4[pi]/9 (11)

Furthermore, if the matching conditions listed in (12) and (13) are imposed in the analysis, then (8), (11), (12) and (13) define a linear system of four equations with the four unknowns ([C.sub.L], [L.sub.L], [C.sub.R] and [L.sub.R]). The solution of this set of equations is given by (14)-(17).

[Z.sub.C] = [square root of ([L.sub.R]/[C.sub.R])] (12)

[Z.sub.C] = [square root of ([L.sub.L]/[C.sub.L])] (13)

[C.sub.R] = [pi]/9N[Z.sub.C][[omega].sub.1] (14)

[L.sub.R] = [pi][Z.sub.C]/9N[[omega].sub.1] (15)

[C.sub.L] = 9N/4[pi][Z.sub.C][[omega].sub.1] (16)

[L.sub.L] = 9N[Z.sub.C]/4[pi][[omega].sub.1] (17)

Replacing (14)-(15) in (1) it can be found that [empty set]([omega] = [[omega].sub.1]) = [empty set]([omega] = 4[[omega].sub.1]) = 100[degrees], thus the lowest frequency and the highest frequency where the phase difference is accepted are [omega] = [[omega].sub.1] and [omega] = 4[[omega].sub.1]. Therefore the fractional bandwidth of the proposed circuit is 120% and the central frequency is [[omega].sub.0] = 5[[omega].sub.1]/2.

On the other hand, replacing [[omega].sub.1] = 2[[omega].sub.0]/5 into (14)-(17) one arrives at (18)-(21). These equations are very important because they establish [C.sub.L], [L.sub.L], [C.sub.R] and [L.sub.R] in terms of central frequency [[omega].sub.0] which facilitates the design; in this context the proposed methodology is simpler to implement than those proposed in [8-10].

[C.sub.R] = 5[pi]/18N[Z.sub.C][[omega].sub.0] (18)

[L.sub.R] = 5[pi][Z.sub.C]/18N[[omega].sub.0] (19)

[C.sub.L] = 45N/8[pi][Z.sub.C][[omega].sub.0] (20)

[L.sub.L] = 45N[Z.sub.C]/8[pi][[omega].sub.0] (21)

If the PRHb section is considered as a conventional transmission with characteristic impedance [Z.sub.PRH] (= [Z.sub.c] = 50 [ohm]) and electrical length [[THETA].sub.PRH] then replacing (18) and (19) into (3) ones arrives to the electrical length of the bottom branch is [[THETA].sub.PRH] = 50[degrees] at the central frequency. In this context proposed differential phase shifter exhibits smaller size than those proposed in [8-10].

It is important to mention that, if the differential phase shift is considered as [[PHI].sub.PRH]([omega])-[[PHI].sub.PLH]([omega]) then the differential phase difference is a concave function and the analysis is slightly modified. However, the equivalent circuit proposed in Figure 1(a) is valid for the design.

To validate the schematic of the proposed metamaterial quadrature power divider/combiner, the simulated response of a design at central frequency of 2 GHz with 3 unit cells is shown in Figure 2. This figure shows that from 804 MHz to 3.202 GHz, the differential phase shift is 90[degrees] [+ or -] 10[degrees], insertion loss are 3dB, return loss and isolation are better than 10dB.

3. IMPLEMENTATION AND RESULTS OF A BALANCED AMPLIFIER BY USING THE PROPOSED QUADRATURE POWER DIVIDER

The use of balanced amplifiers is a practical method for implementing the broadband amplifier that has flat gain and good input and output return loss [13]. Figure 1(b) shows the most popular arrangement of a balanced amplifier. It is well known, from the S-parameters of a balanced amplifier, that if the two amplifiers are identical, then the best return loss are achieved ([S.sub.11] = 0 and [S.sub.22] = 0) and therefore the return loss depend on the combiner/divider [13]. In this context several balanced amplifiers have been proposed in [14-22]. From Table 1, given in [14], it can be found that the maximum FBW of published balanced amplifiers is 97.9%. This FBW was achieved, in [14], by using metamaterial quadrature power dividers/combiners.

Based on the schematic given in Figure 1(b) a balanced amplifier at central frequency of 2 GHz was developed by using two identical ATF-36077 transistors. The balanced amplifier was designed and fabricated on a RT-duroid substrate with [[epsilon].sub.r] = 3.38 and h = 1.524 mm.

The proposed prototype was implemented in three steps. The first step is to develop two identical amplifiers, the second step is to develop a metamaterial differential phase shifter and the third step is to implement the two-stage PD. In the first step, each amplifier is formed by an input bias tee, an ATF-36077 transistor (biased at [V.sub.ds] = 1.5 V and [I.sub.ds] = 10mA), and an output bias tee. The response of a single fabricated amplifier is shown in Figure 3. It can be seen in Figure 3(b) that return losses are very bad. It is due to input matching and output matching networks are not included in the design. However, this was done deliberately to demonstrate the ability of proposed balanced amplifier to improve the return losses.

In the second step, the pure-left handed section of the quadrature power divider is designed by considering one unit cell. The unit-cell values, [C.sub.L] (= 2.85pF) and [L.sub.L] (= 7.124nH), are calculated from (2) and (3) respectively. Each unit cell is formed by two commercial chip capacitors (ATC-600S5R6) with capacitance of 5.6 pF 2Cl) and a microstrip stub inductor w = 0.2 mm and length l = 7.555 mm) shorted to the ground plane by a via is utilized to implement [L.sub.L].

On the other hand, the PRHb is designed to have characteristic impedance of 50 [ohm] and electrical length of 50[degrees]. Thus, the PRH-TL has length of 12.688 mm and width of 3.512 mm respectively.

Up to this point, the design of the QPD is almost finished. However when the differential phase shift is measured, it can be found that the differential phase shift is lower than the desired valued due to the parasitic pad effect of [C.sub.L] and [L.sub.L]. The problem is easily solved by adding a 50 microstrip transmission line, on the bottom branch, whose length increases from zero degrees until the required differential phase shift is achieved. For our design, a transmission line with length l = 2.95 mm and width w = 3.512 mm is added to the PRHb section to take into account the parasitic pad effects. Figure 4 shows the differential phase shift before and after compensation. The other parameters are not presented because the changes are only performed on the phase of the pure-right handed transmission line.

In the third step, the dimensions of the two-stage power divider were calculated and arranged as is shown in Figure 5. The width and length of the transmission line with impedance of 64.770 [ohm] and electrical length of 90[degrees] are w = 2.259 mm and l = 23.247mm respectively. Also, the width and length of the transmission line with impedance of 77.190 [ohm] and electrical length of 90[degrees] are w = 1.607 mm and l = 23.530 mm.

Figure 5 and Figure 6 show the photograph of the fabricated wideband balanced amplifier and its measured results respectively. From Figure 6, it is clear that the return losses are better than 10 dB through the 841.6 MHz to 3.610 GHz frequency range. This frequency range represents a FBW of 124.4% and central frequency at 2.226 GHz. Furthermore the gain of the amplifier is 12.333 dB [+ or -] 1.6dB. From Figure 3 and Figure 6, it can be seen that the balanced amplifier has very good response despite that each individual amplifier has very bad return loss. These results show that with the proposed circuit, we were able to successfully design a broadband balanced amplifier with good return losses and flat gain.

4. CONCLUSIONS

In this work, a novel broadband metamaterial quadrature power divider with fractional bandwidth up to 120% was presented. The design is based on a straightforward, but robust methodology which is based on a set of closed exact equations. The proposed circuit was utilized to develop a broadband balanced amplifier with measured fractional bandwidth of 124.4%.

ACKNOWLEDGMENT

The authors would like to thank to INAOE, CICESE and CONACyT-Mexico (Project No. 180061), for partially funding this project.

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J.-L. Olvera-Cervantes (1), *, A. Corona-Chavez (1), R. Chavez-Perez (2), H. Lobato-Morales (1), J.-R. Ortega-Solis (2), and J.-L. Medina-Monroy (2)

(1) National Institute for Astrophysics, Optics and Electronics (INAOE), Puebla, Mexico

(2) Investigacion Cientifica y de Educacion Superior de Ensenada, Ensenada, Baja California, Mexico

* Corresponding author: Jose-Luis Olvera-Cervantes (jolvera@inaoep.mx).

Received 13 June 2012, Accepted 19 October 2012, Scheduled 24 October 2012

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Author: | Olvera-Cervantes, J.-L.; Corona-Chavez, A.; Chavez-Perez, R.; Lobato-Morales, H.; Ortega-Solis, J.-R |
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Publication: | Progress In Electromagnetics Research C |

Article Type: | Abstract |

Geographic Code: | 1USA |

Date: | Jan 1, 2013 |

Words: | 3730 |

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