Dual-Band Compact Wilkinson Power Divider Using Common Inductor and Complex Load

A 1.8/2.4-GHz dual-band compact Wilkinson power divider was designed and implemented using an integrated passive device process. The design used a common inductor to match impedance in the even mode. The common inductor replaced two shunt inductors in the circuit so the circuit size was reduced. An additional complex load was then used for matching impedance in the odd mode. The circuit measurements showed reflection at all ports better than −11.9 dB, isolation at the output ports better than −14 dB and transmission coefficient better than −4.3 dB, at both operating frequencies. The circuit occupied only $0.036\lambda _{\mathrm {o}}\times 0.018\lambda _{\mathrm {o}}$ at the higher operating frequency.


I. INTRODUCTION
Power dividers were used in many circuit blocks such as low noise amplifiers, power splitters, array antennas, etc. Dual-band and Multi-band operations are also of great benefits in various applications. Among power dividers, Wilkinson power dividers (WPD) have been studied extensively in both single and dual-band operations. They were implemented in many forms such as printed-circuit board (PCB), CMOS or in an Integrated Passive Device (IPD). On PCB, microstrip transmission lines were normally used in WPD circuits. With larger transmission line traces, this type of WPD can handle the largest power of the three forms mentioned. There are several techniques to design a dual-band Wilkinson power divider (WPD). Distributed transmission lines with periodically loaded slow wave structure were used by Rawat and Ghannouchi to make dual-band WPD [1]. Multi-section transmission lines were also used in some works. Additional lumped components connected in the shunt path were used to make dual-band WPDs [2]- [4]. Stubs instead of lumped components have been used [5]- [8]. Cascaded multi-section transmission lines without shunt components were also implemented [9]- [12].
The associate editor coordinating the review of this manuscript and approving it for publication was Yong Chen .
Composite right-/left-handed transmission lines were also used in dual-band WPDs [13], [14]. All these PCB designs had similar performance with some size differences, which were usually greater than 0.1λ o ×0.1λ o .
Beside WPD on PCB forms, some works were also implemented in CMOS process. Chiang, et al. used complementary-conducting-strip transmission lines in designing a WPD for Ka-band application [15]. A π-network [16] and T-network [22] lumped circuits were used to replace distributed transmission lines to reduce the circuit sizes in single-band designs but the insertion losses in both designs were relatively high due to low Q inductors. Active inductors were also used in CMOS to reduce the circuit sizes [17], [18]. Since the CMOS substrate was quite lossy, lower performances was expected in passive WPDs. The active ones also suffered from power consumption, noise and nonlinearity issues. Nevertheless, the circuit sizes were much smaller than those of PCB implementations.
WPDs implemented in an IPD process were as small as those of CMOS but had better performance since the inductors implemented had better quality factors. Normally, π-network right-handed transmission lines were common in implementing transmission lines in an IPD. The 900-MHz and 2.4-GHz Wilkinson power dividers based on π-network right-handed transmission lines implemented on VOLUME 8, 2020 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ GaAs-IPD technology have been shown [19], [20]. In this paper, a dual-band WPD was designed and implemented on IPD process to operate at 1.8 and 2.4 GHz. In addition to π-network right-handed transmission lines, the design included a common inductor and a complex load connecting the output ports. The design used even and odd mode analysis. Combining a common inductor, complex load and two section transmission lines, a compact dual-band WPD was implemented. Note that since there is a trade-off between insertion loss and isolation in practice, design optimization had low insertion loss as the first priority. The detailed analysis and design are discussed in the following sections.

II. COMPACT POWER DIVIDER WITH COMMON INDUCTOR AND COMPLEX LOAD
Conventional power dividers, are usually implemented in distributed transmission lines. For a compact form in integrated circuits, LC-based lumped circuits were used. For a single operating frequency circuit, the transmission lines used can be either λ/4 [17] or 3λ/4 [19]. For a λ/4 line, a π-network lumped transmission line circuit was normally usedwhile a T-network was used for 3λ/4 line. In both cases, the circuits had at least one inductor. Since the size of the inductor greatly impacts the size of the overall circuit, a minimal number of inductors were used, or smaller size inductor were all preferred.
To make the circuit operate at dual-frequency bands, shorter transmission lines with a common inductor and complex load were used. Figure 1 shows the circuit topology. The transmission lines were divided into two sections, T.L.1 and T.L.2. T.L.1 was next to Port 1, while the common inductor was placed in the middle between T.L.1 and T.L.2. The complex load was placed between T.L.2 and Port 2 and Port 3. The circuit was analyzed using the even and odd mode technique. The even and odd mode equivalent circuits are shown in Fig. 2.
The benefits of the common inductor are even mode matching and size reduction. In the even mode, a shunt element along the transmission lines must connect to ground and therefore two elements are needed for the shunt branch between T.L.1 and T.L.2. A smaller common inductor replaced two shunt inductors, since it represented a larger equivalent shunt inductor in the even mode due to the mutual inductance. Therefore, both even matching and circuit size reduction can be achieved. The complex load was designed In Fig. 2, the common inductor L com was considered as L com,e and L com,o for the equivalent even-and odd-mode circuit. The L com,e value has been increased in the even mode due to the mutual inductance of the common inductor. The detailed analysis of the common inductor, even and odd modes are considered next.

A. COMMON INDUCTOR
The common inductor can be represented by two-port Z-parameters as shown in Fig. 3. The input signals of both input ports rotate in the same direction and meet each other at the middle of the structure. Physically they can be represented by two inductors with a mutual coupling factor. As a two-port structure, its Z-parameters at desired frequencies can be extracted by electromagnetic simulation.  To be used in even-and odd-mode analysis, an equivalent T-network shown in Fig. 4 was used to represent the network.
where the Z-parameters of the inductor is The complex load consists of two capacitors, one inductor and one resistor. Its equivalent circuits of R p -C p parallel network were used in the odd mode analysis. Figure 5 shows the complex load and its equivalent circuit for the odd mode analysis. The component values can be found as R p = R m 2 (3a) The half circuit of Fig. 5(b) was then used in the analysis of the odd mode. In the odd mode analysis, port 1 impedance is replaced with short circuit. The components in the equivalent circuit are T.L.1, T.L.2, common inductor and complex loadas shown in Fig. 2(b). The input impedance at Port 2 is where, (7b)-(7c) as shown at the bottom of the next page. The matching conditions in the odd mode analysis at both frequencies are  With the conditions in (6) and (8), the required specification in (1) can be achieved.

III. TRANSMISSION LINE CIRCUIT WITH LUMPED COMPONENTS
The circuit was implemented in an IPD process with size limitation. Therefore, the conventional distributed transmission lines were replaced by lumped transmission line circuits. A simple lumped transmission line circuit is a π-network right-handed transmission line shown in Fig. 6. The circuit consists of one inductor and two capacitors. At a given electrical length, its characteristic impedance is proportional to the inductor and inversely proportional to the capacitor. Proper values of these components must be considered, since an inductor occupies a large area. This structure has a limitation since its transmission line behavior degrades with its electrical length [21]. In addition, lower Q components contribute to higher insertion loss on the overall circuit. Transmission lines 1 and 2 are considered next.

A. TRANSMISSION LINE 2
Transmission line 2 was chosen to have smaller characteristic impedance than that of the ports. Its electrical length was set to be relatively small so that a simple π-network righthanded transmission line can be used. The components and transmission line properties are related by If its electrical length at 2.4 GHz is limited at θ 2x , then the inductor-capacitor multiplication factor is where ω 2 is the angular frequency at 2.4 GHz. Pakasiri, et al. showed that with a low quality factor for the inductor on a lossy substrate, the electrical length of a π-network right-handed transmission line should be less than 60 • [21]. Here, transmission line 2 was limited to 50 • at the upper frequency band. Since the T.L.2 implementation was a single-section lumped transmission line, its transmission line performance degradation made the overall circuit bandwidth narrower. Wider bandwidth could be achieved with a multisection T.L.2 [21], but the circuit size would be larger.

B. TRANSMISSION LINE 1
The electrical length of this line was quite long compared to the second line. The modified circuit, shown in Fig. 7, was used so that it acted as a transmission line at both operating frequencies. The electrical length and characteristic impedance of the line are To set a limitation on the electrical length θ 1x , a transcendental equation below was applied for determining the capacitors and inductors.
In this work, θ 1x was set at 90 • at the upper frequency band.

IV. IMPLEMENTATION AND MEASUREMENTS
The circuit was implemented in an IPD. There were three metal layers on a silicon substrate as illustrated in Fig. 8. Dielectrics 1 and 2 had dielectric constant values of 6.7 and 2.65. The metal layers and vias were copper. In the layout, all components, especially inductors, possessed resistance, which effected the transmission line behaviors. The capacitor layouts were parallel plates using the M1 and M2 layers. The inductors were rectangular spirals. The resistor was in meander trace form. A chip image is in Fig. 9 Fig. 10, which shows good agreement. Table 3 summarizes some simulated and measured S-parameters at both operating frequencies. At 1.8 GHz, the measured circuit performance differed a little from the simulation. However, at 2.4 GHz, the difference was greater, as the trends of measured S-parameter graphs deviated from those of the simulation. Overall, at both operating frequencies, the reflection at all ports were better than −11.9 dB, the isolations at the output ports were better than −14 dB and the transmission coefficients were better than −4.3 dB. Table 4 summarizes previously reported power dividers. The |S 21 | or |S 31 | values from [12], [7], [8] and [9] were high because they were off-chip dual-band power dividers, where high Q components could be realized. However, the circuit sizes were large. Some other works were single-band power  dividers so their sizes were smaller and performances better [19], [20], [22]. For dual-band dividers, in addition to dual-band circuit design criteria, there was a compromise between each frequency optimization, since component value variation and parasitics were added into a practical circuit. Therefore, the dual-band designs were generally larger. It is noted that this work generated smaller size than previous dual-band designs-see Table 4.

V. CONCLUSION
A compact 1.8/2.4-GHz dual-band Wilkinson power divider was designed and implemented in an integrated passive device process. To meet the circuit conditions at both frequencies, a common inductor and complex load were used with lumped transmission lines. The circuit parameters were determined by even and odd mode analysis of the circuit. For the even mode, the common inductor was used with lumped transmission lines to reduce the size of the circuit, while the input impedances of both ports were matched. In the odd mode, the complex load was used in addition to the lumped transmission lines and common inductor to match both ports. Overall, the simulated and measured results agreed well. For both frequencies, measured reflections at all ports were better than −11.9 dB, isolations at the output ports were better than −14 dB and transmission coefficients were better than −4.3 dB.