A Compact Six-Port Power Divider With Two Independent Out-of-Phase Dividing Operations

This paper presents a compact six-port power divider with simple structure that performs two independent out-of-phase divisions. The proposed power divider consists of a uniplanar ring crossover with four transmission lines for output and two feeding lines. The S-parameter matrix of the proposed power divider is derived analytically, and it is shown that the two independent out-of-phase divisions are realized when the circumferential length of the ring is one guided wavelength λg whereas conventional power dividers require a 2λg or 2.5λg ring. Two power dividers configured with striplines were designed to have -10-dB impedance bandwidths of 5% and 10% with a center frequency of 2.45 GHz. The designed power dividers were fabricated and two independent out-of-phase divisions having small amplitude imbalance were verified by measurements. Because of its compactness, the proposed power divider might be applied to the feeding network of dual-feed dual-polarized array antennas with narrow element spacing.

A compact feeding network might be realized by integrating two dividers into a single divider. A single power divider for dual-feed dual-polarized antennas needs to perform two independent out-of-phase dividing operations that correspond to the inputs from two separated ports, which cannot be realized by multi-way out-of-phase dividers [16]- [18], turnstile-based orthomode transducers [19], [20], sixport junctions [21], [22], or balanced-to-unbalanced power dividers [23], [24]. Up until now, few power dividers that perform two independent out-of-phase divisions have been reported [25]- [27]. The substrate integrated waveguide (SIW) power dividers [25], [26] are unsuitable for compact networks because the width of an SIW needs to be larger than half a wavelength in order to support the dominant TE 10 mode, and a long matching section is needed. The microstrip hybrid ring [27] does not require the matching section and is smaller than [25], [26]. However, [27] is still large for array antennas with narrow element spacing because the circumferential length of the ring is 2λ g , as with the power dividers with a 2.5λ g ring [23], [24]. Furthermore, the air bridges of microstrip lines in [27] increase the overall height and the complexity in structure. This paper presents a six-port out-of-phase power divider composed of a four-port uniplanar ring crossover [28], [29] and two feeding lines. Because the circumferential length of the ring is only λ g , the circuit area of proposed power divider is almost quarter of that of [27]. The proposed power divider configured with striplines is designed and constructed, and two independent out-of-phase divisions are verified.
The contributions of this paper are summarized as follows.
• A compact six-port power divider with simple structure is presented. The ring in the proposed power divider is only λ g in circumferential length whereas the conventional hybrid-ring power dividers are composed of a 2λ g or 2.5λ g ring. Because the proposed power divider does not have a long matching section or the air bridges of transmission lines, the overall height and structural complexity are reduced. • The closed-form expression of the S-parameter matrix of the proposed power divider is derived, which enables fast characterization. The characteristics of the proposed power divider are validated by both electromagnetic simulation and measurement. The remainder of this paper is organized as follows. In Section II, the S-parameter matrix of the proposed power divider is derived, and its characteristics are studied. In Section III, two design examples of the proposed power divider configured with striplines are shown. Section IV shows the experimental results of the fabricated power dividers, including comparisons with the conventional power dividers. The conclusion of this paper is presented in Section V. Fig. 1 shows the transmission line model of the proposed six-port out-of-phase power divider. The proposed power divider is composed of a ring crossover [28], [29] with four transmission lines for output and two feeding lines. The characteristic impedance of the four transmission lines, 2Z 0 , is twice that of the feeding lines (Z 0 ). The characteristic impedance of the ring is Z 1 , and the circumferential length of the ring is λ g at the center frequency f 0 .

A. OVERVIEW OF THE PROPOSED POWER DIVIDER
A wave input from port 1 is divided to ports 3 and 5 with opposite phases. Similarly, a wave input from port 2 is divided to ports 4 and 6 with opposite phases. The interport isolations, the amplitude imbalance, and so on change depending on Z 1 , as will be described in Section II-C.
As an application example, a dual-feed dual-polarized patch antenna fed by the proposed power divider is shown in Fig. 2. The patch antenna is excited by a differential pair of waves, and radiates yand xpolarized waves when ports 1 and 2 are excited, respectively. For array antennas with narrow element spacing (e.g. phased arrays with 0.5λ 0 spacing), a compact feeding network is indispensable. The proposed power divider may be a promising candidate as the feeding network of these array antennas. port 1 port 2 port 3 port 4 port 5 port 6 Transmission line model of the proposed six-port out-of-phase power divider that performs two independent out-of-phase divisions corresponding to the inputs from ports 1 and 2.

B. DERIVATION OF THE S-PARAMETER MATRIX
In this subsection, the S-parameter matrix of the proposed power divider is derived analytically in order to find the condition to realize two independent out-of-phase power divisions with equal-amplitude division, no reflection, and perfect inter-port isolations. Because the even-odd mode analysis [30] of a six-port network is complicated, a simple and straightforward technique based on an admittance matrix [31] was employed. To avoid several elements of the admittance matrix from diverging, nodes 1 and 2 were shifted by the phase ∆ψ as shown in Fig. 3. The limit as ∆ψ → 0 was taken after deriving the S-parameter matrix.
The element of the admittance matrix [Y ], Y mn = I m /V n | V k =0 (k =n) , is found by driving port n with voltage V n , short-circuiting the other ports, and then calculating the current I m at port m [14]. When nodes 1-6 on the ring shown in Fig. 3 are used as the references for ports 1-6, the admittance matrix [Y ] of the power divider shown in Fig. 3 is expressed as (1) in the next page. Then, the S-parameter matrix [S] is obtained by as described in [32]. By taking the limit as ∆ψ → 0, the S-parameter matrix of the power divider shown in Fig. 1 is obtained. The matrix operation in (2) and the limit operation ∆ψ → 0 were implemented with MATLAB®Symbolic Math Toolbox™ [33]. The closed-form expression of the Sparameter matrix is so long that it is not shown in this paper due to length restrictions. When the circumferential length of the ring is λ g (i.e. ψ = π/2), the S parameters for port 1 input are reduced to where (12) As can be derived from (8) and (10), S 31 and S 51 are out of phase regardless of Z 1 or Z 2 . This is obvious because the difference in the path lengths from port 1 to ports 3 and 5 is . S 11 becomes zero when Z 2 = 2Z 0 . The remaining elements of the S-parameter matrix are obtained by substituting Z 1 = 0 and Z 2 = 2Z 0 , and finally, the S-parameter matrix of the proposed power divider for Z 1 = 0 is given by Because S 31 = −S 51 = 1/ √ 2, a wave input to port 1 is divided to ports 3 and 5 with opposite phases and equal amplitudes with no reflection and perfect inter-port isolations.  Similarly, a wave input to port 2 is divided to ports 4 and 6 with opposite phases and equal amplitudes as S 42 = −S 62 .

C. PARAMETRIC STUDY OF THE POWER DIVIDER
In Section II-B, it was shown that two independent out-ofphase divisions are realized when the circumferential length of the ring is λ g as in Fig. 1. It was also shown that equal-amplitude division, no reflection, and perfect interport isolations are realized when Z 1 = 0 and Z 2 = 2Z 0 . In actual transmission lines, however, Z 1 = 0 cannot be realized. To study the characteristics of the proposed power divider with a realistic value of Z 1 , the S parameters were calculated by sweeping Z 1 (> 0) with Z 2 fixed to 2Z 0 . Fig.  4 shows |S m1 | as a function of the normalized characteristic impedance Z 1 /Z 0 . |S 11 |, |S 21 |, |S 41 |, and |S 61 |, as well as the amplitude imbalance between |S 31 | and |S 51 | increase with Z 1 /Z 0 , whereas the phase difference between S 31 and S 51 remains constant at −180 • regardless of Z 1 /Z 0 , as explained in Section II-B. The S parameters for the port 2 input, S m2 , are not shown because the tendency of S m2 is the same as that of S m1 due to the symmetrical structure. The frequency characteristics can be obtained by substituting ψ = (π/2)·(f /f 0 ) into (1), where f is the frequency. The −10-dB impedance bandwidth, the fractional bandwidth for the reflection |S 11 | below −10 dB, as a function of Z 1 /Z 0 is shown in Fig. 5. The bandwidth increases with Z 1 /Z 0 , and reaches the maximum of 118.6% when Z 1 /Z 0 = 5.89. However, the amplitude imbalance between |S 31 | and |S 51 | is as large as 20.2 dB in this case. The reflection and the interport isolations are also high. To reduce the amplitude imbalance, the reflection, and the inter-port isolations, smaller value of Z 1 /Z 0 needs to be chosen. When Z 1 /Z 0 = 0.450, VOLUME   for instance, a 10% bandwidth is obtained as shown in Fig. 5.
The amplitude imbalance at f 0 is 0.32 dB in this case. When Z 1 /Z 0 = 0.233, the amplitude imbalance at f 0 is reduced to 0.09 dB. Meanwhile, the bandwidth is also reduced to 5%. As explained above, there is a trade-off between the impedance bandwidth and the other parameters, including the inter-port isolations and the amplitude imbalance. Based on Figs. 4 and 5, Z 1 /Z 0 can be determined by considering the impedance bandwidth, amplitude imbalance, reflection, and inter-port isolations that are acceptable for the applications of the proposed power divider.

A. CONFIGURATION
A power divider configured with striplines is shown in Fig.  6. Two 0.75-mm-thick polyphenylene ether (PPE) substrates (dielectric constant ε r = 3.64 and loss tangent tan δ = 0.002) are stacked and glued with a 5-µm-thick bonding film (ε r = 2.50 and tan δ = 0.002). To facilitate its application as the feeding network of antenna arrays, two SMA connectors are mounted on the bottom layer of the substrate. Ports 1 and 2 are assigned at the ends of the SMA connectors with Z 0 = 50Ω. The reference planes of ports 3-6 are shown as red dotted lines in Fig. 6. To prevent the parallel-plate mode from traveling, metallic vias are placed along the striplines.

B. DESIGN PROCEDURES
As examples, two power dividers were designed to have 5% and 10% bandwidths at 2.45 GHz, in the following steps. Electromagnetic simulations were performed using Ansys®HFSS™ [34].
1) The width of the four striplines for output, w 2 , was chosen to be 0.14 mm in order to have the characteristic impedance of 100Ω (= 2Z 0 ). 2) Based on the relation between Z 1 /Z 0 and the bandwidth described in Section II-C, the widths of the ring w 1 were chosen to be 5.78 mm and 2.67 mm in order to have the characteristic impedances 11.5Ω (= 0.233Z 0 ) and 22.5Ω (= 0.450Z 0 ) for 5% and 10% bandwidths, respectively.
3) The radius of the ring r 1 was determined to minimize the reflection at 2.45 GHz. r 1 was determined based on the reflection (not on the circumferential length) because it is difficult to know the guided wavelength in the ring.

C. SIMULATED RESULTS
The S parameters of the power dividers designed to have 5% and 10% bandwidths are shown in Figs. 7 and 8, respectively. The S parameters obtained by the electromagnetic simulation are similar to those obtained by the equations. The −10-dB impedance bandwidths obtained by the simulation are 5.01% and 9.92%, which are in a good agreement with the desired ones (5% and 10%). The simulated amplitude imbalances of the power dividers (5% and 10% bandwidths) are 0.02 dB and 0.13 dB at 2.45 GHz, respectively. The simulated S 31 and S 51 are almost out of phase as shown in Figs. 7(b) and 8(b). The losses in the power dividers (5% and 10% bandwidths) are 0.26 dB and 0.16 dB, respectively. The discrepancy in the S parameters obtained by the equations and the simulation may come from the SMA connectors and the structural discontinuities that are not considered in the equations.

IV. MEASURED RESULTS
The power dividers designed in Section III were fabricated (Fig. 9). To show the striplines printed on the inner layer, non-laminated substrates were also fabricated in another lot. The inner layers of the power dividers are shown in Figs. 9(a) and (b). To connect the striplines with the SMA connectors having different characteristic impedances, quarterwavelength impedance transformers were inserted between them, as shown in Fig. 9(d).
The S parameters of the fabricated power divider were measured. The effect of the SMA connectors and the impedance transformers were eliminated by the TRL (Thru-Reflect-Line) calibration [14], using the Thru, Reflect, and Line patterns shown in Fig. 10. The reference planes for ports 3-6 are shown as red dotted lines in Figs. 9(a) and (b). The measured S parameters are shown in Figs. 11 and 12. The measured S parameters have a similar tendency to the simulated ones except for a slight shift in frequency. The measured −10-dB impedance bandwidths of the power dividers are 5.01% and 10.0%, which are close to the desired ones (5% and 10%, respectively). The amplitude imbalances of the power dividers (5% and 10% bandwidths) measured at 2.45 GHz are 0.04 dB and 0.01 dB, respectively. The losses in the power dividers (5% and 10% bandwidths) measured at 2.45 GHz are 0.37 dB and 0.04 dB, respectively. The mea-  Although not shown in this paper, similar tendencies of the measured S parameters for the port 2 input, S m2 , were also observed. Given that S 31 −S 51 and S 42 −S 62 , two independent out-of-phase divisions with a small amplitude imbalance have been verified by measurements.
A shift in the frequency is often caused by the difference between the dielectric constant of substrates considered in the design process and that of the actual substrates. The actual dielectric constant may differ from that on the datasheet as explained in [35]. To estimate the effective dielectric constant of the fabricated substrates, the Line-Line (LL) method [36] was applied to the measured results of the Thru and Line patterns shown in Fig. 10. The effective dielectric constant ε r , including the effect of the bonding film, was found to be 3.66 at 2.45 GHz. The layer configuration of the substrates was modified for re-simulation as shown in Fig. 13. The value of tan δ was kept to 0.002 because the estimated tan δ (0.014 at 2.45 GHz) includes the effects of both the dielectric and conductor losses that are difficult to decompose. Figs. 14 and 15 show the S parameters measured and simulated with the estimated ε r . The frequency shift is mitigated by using the estimated ε r . The error in the phase is still observed in Fig.  15(b), which might come from the solder and the misalignment of the SMA connectors that were not considered in the  re-simulation.
Six-port power dividers having two input ports are compared in Table 1. The six-port junctions [21], [22] are large in terms of wavelength because they are composed of multiple dividers and couplers. Furthermore, additional baluns are needed in order to perform out-of-phase division, which leads to even larger size. Although smaller size and out-ofphase division without additional baluns are realized with hybrid rings [23], [24], they cannot perform two independent divisions required for dual-feed dual-polarized antennas. The SIW power dividers [25], [26] can perform two independent out-of-phase divisions. However, the matching section increases overall circuit size. The air bridges in [27] increase the number of substrates and structural complexity as well. Whereas the conventional hybrid rings are 2λ g [27] or 2.5λ g [23], [24] in circumferential length, the proposed ones are based on a λ g ring without air bridges or matching sections. Because of the compact ring, the circuit area of the proposed power divider is about quarter of that of [27]. Although the bandwidths of the presented power dividers (5% and 10%) are relatively narrow, the bandwidth can be enhanced by choosing larger Z 1 /Z 0 as described in II-C. Because of the compactness, the proposed power divider could be integrated in the feeding network of array antenna with narrow element spacing.

V. CONCLUSION
This paper presented a compact six-port power divider with simple structure that performs two independent out-of-phase divisions. The S-parameter matrix of the proposed power divider was derived analytically, and it was shown that two independent out-of-phase divisions are realized with a λ g ring, whereas the conventional hybrid-ring power dividers are based on a 2λ g or 2.5λ g ring. In addition, the proposed power divider does not require the air bridges of transmission lines [27] that increase overall size and structural complexity, which could lead to low-cost and stable fabrication. Two power dividers configured with striplines were designed to have −10-dB impedance bandwidths of 5% and 10% with a center frequency of 2.45 GHz. The designed   by the difference in the effective dielectric constants.
Whereas the proposed power divider has a narrower bandwidth compared with the conventional ones, the proposed power divider is much smaller in terms of wavelength. Because of its compactness, the proposed power divider could be adopted to the feeding network of dual-feed dualpolarized antenna arrays with narrow element spacing. Design of dual-feed dual-polarized antenna arrays equipped with the proposed power divider is a future research topic.