Balanced-to-Balanced Gysel Filtering Power Divider With Arbitrary Power Division

This paper presents the balanced-to-balanced Gysel filtering power divider with arbitrary power division for first time. Based on the multicoupled series-resonator bandpass prototype network, a detailed theory analysis is presented and the corresponding design equations are concluded. Two filtering power dividers with the division ratio of 2 and 3 are realized using the half-wavelength resonator and the short-stub-loaded resonator. Finally, two example circuits are fabricated and measured for verification. The measured results show good agreement with the simulated and confirm the theory.


I. INTRODUCTION
The balanced RF circuit has been widely used in modern communication system due to its high immunity to noise compared with the single-ended circuit [1]. To construct the balanced RF receiver/transceiver circuit, the balanced filter and power divider, which present identical function with their single-ended counterparts for differential-mode (DM) signal and present suppression for common-mode (CM) signal, have been presented.
The balanced filter with single-band and dual-band bandpass response was presented in [2] and [3] respectively. In [4], a wideband balanced filter was reported by using coupled-line stubs/coupled-line stub-loaded resonators and the modified coupled feed-lines. A tunable balanced bandpass filter with constant bandwidth was proposed in [5]. Based on the balanced bandpass filter, the balanced-tobalanced diplexer was proposed in [6]. For the balancedto-balanced (BTB) power divider, several researches were presented [7]- [11]. In [7], a BTB power divider was presented but without good isolation between the output ports. A BTB Willkinson power divider with good isolation between output ports was reported in [8] by Bin Xia. Then, Bin Xia developed a BTB Gysel power divider with arbitrary power division The associate editor coordinating the review of this manuscript and approving it for publication was Yuhao Liu. in [9]. In [10], a wideband BTB Willkinson power divider was proposed. In [11], a balanced power divider with tunable power division was proposed. Moreover, the other balanced passive components were also reported, such as the balanced coupler [12] and the balanced crossover [13].
In order to future compact the RF system, the filtering power divider (FPD) which integrates the functions of filter and power divider has been extensively studied. In [14], both the single-and dual-band filtering power dividers were both reported. In [15], a filtering power divider with ultra-wideband bandpass response was designed using transversal filtering transformer. A detailed design method of FPD with arbitrary power division was presented in [16] using even-order filter and traditional Willkinson power divider. In [17]- [19], A synthesis method for filtering power divider with arbitrary power division and arbitrary order bandpass response was proposed but without considering the isolation. A Gysel filtering power divider with arbitrary power division and high isolation between two output ports was presented in [20]. Moreover, the ratrace hybrid with bandpass response was reported [21] using the electromagnetically-coupled resonator. The isolation was achieved by using the inherent phase-shift between electronic-and magnetic-coupling.
For the balanced-to-balanced filtering power divider, it was firstly reported in [22] using the same coupling scheme as presented in [21]. In [22], the FPD was realized by three full-wavelength resonators and a half-wavelength resonator. The full-wavelength resonators work at second-order resonance and the half-wavelength resonator works at the first-order resonance. By using the different phase-shift between first-order and second-order resonance, which was presented in [23], the equivalent electronic coupling was achieved. This configuration has good isolation between two output ports but also has a large size because the resonators work at second-order resonance. In our previously research [24], the BTB filtering power divider was realized by half-wavelength resonator and short-stub-loaded resonator (SSLR). Compared with the research proposed in [22], it achieved compacter size and wider suppression for the common-mode signal. Moreover, several other BTB filtering power dividers were presented in [25]- [27]. In [25], a BTB filtering power divider was designed with hish output isolation only using two square patch resonators. In [26] and [27], a wideband and dual-band BTB filtering power divider were presented respectively with high frequency selectivity using similar structure. For the BTB filtering power divider with unequal power division, it was firstly presented in [28] using the same structure presented in [22]. Although a series of design equations are provided, the corresponding theoretical analysis was insufficient in [28].
In this paper, the balanced-to-balanced Gysel filtering power divider (BTB-GFPD) with arbitrary power division is researched and designed. The research is presented with detailed analysis and conclude the corresponding design equations through a series of derivation and demonstration. The design topology of the proposed BTB-GFPD is extended from the design scheme in [21], [22], [24]. At first, the design topology is studied based on multicoupled series-resonator bandpass prototype network and the design equations are concluded. Then the BTB-GFPD realized using half-wavelength resonator and SSLR is also studied. And the detailed realization principle is proposed. Two BTB-GFPDs with the division ration of 2 and 3, and center-frequency of 2.4GHz are implemented and measured. The measured results are presented. Finally, a discussion about the limitation of division ration is concluded. Fig. 1 is the coupling scheme of the proposed BTB-GFPD. It consists of four resonators. When the input signal is excited at port 1, the output signal at port 2 and port 3 will have arbitrary power division, while port 4 is introduced as an insolation port. This coupling scheme is used in [22], [24] to realize the BTB Gysel filtering power divider with equal output power. In [22], [24], the FPDs have a symmetrical structure and can be analyzed by odd-and even-mode analysis method. To realize arbitrary power division, the proposed FPDs are unsymmetrical. So, the traditional odd-and even-mode analysis method could not be applied.

II. THEORY ANALYSIS
When the signal is excited at port 1, the proposed BTB-GFPD can be seen as a multicoupled series-resonator   Fig. 2. It is attained by normalization of center-frequency and bandwidth as follow [29]: where FBW is the fractional bandwidth. According to the Kirchhoff's law, the loop voltage can be calculated by loop current as follow:  where the s is the bandpass frequency variable(s = w − 1/w and w is the frequency variable).

A. ISOLATION
As a Gysel power divider, the isolation between two output ports is important. In the design of traditional VOLUME 8, 2020 Gysel power divider, the isolation is realized by loading a lumped resistor without producing extra signal loss. In the presented BTB-GFPD, the port 4 is introduced as an isolation port. An 50 matching load is connected to port 4 instead of using a lumped resistor to realize the isolation between two output ports. For avoiding the 50 matching load produces extra signal loss, the isolation between port 1 and port 4 must be achieved. This has been attained in the design and can be verified through measuring the transmission parameter between port 1 and port 4. The theory analysis about the isolation between port 1 and port 4, and the isolation between port 2 and port 3 are both provided in the following. When the input signal is added into port 1, the response of proposed BTB-GFPD can be attained by solving (2). Then, the following equation can be extracted as: By applying the Gaussian elimination method, the following equation can be attained: The equation (4) shows the relationship between i 1 and i 4 . If the i 4 is always zero whatever the i 1 is, port 4 will isolated from port 1. Therefore, for the isolation between port 1 and port 4, the following condition should be satisfied.
The equation (5) is the condition for realizing the isolation between port 1 and port 4. At the center frequency of bandpass (w = 1 rad/s), it can be simplified as follow: The transmission parameter between port 2 and port 3 can be researched by repeating the above analysis process. Similarly, the isolation between port 2 and port 3 can be concluded by as: If (6) and (7) are both satisfied, the isolation between two output ports will be achieved and the 50 matching load will not increase the insertion loss of the power divider.

B. BANDPASS RESPONSE AND POWER DIVISION
When the input signal is added into port 1, the output of port 2 and port 3 can be attained by solving (2). The following It can be expanded as follow: When the condition (6) is satisfied, i 4 will always be zero whatever i 1 is. Therefore, (9) can be simplified as follow: (10) is a simplification for (8) when the condition (6) is satisfied. Meanwhile, (10) can be seen as the mathematical representation of the prototype network in Fig. 3. This means that the prototype network in Fig. 2 and the prototype network in Fig. 3 will have same response when a signal is excited at port 1 and the condition (6) is satisfied. Based on (10), i 2 and i 3 can be calculated as follow: According to the calculation of power, the output power of port 2 and 3 at the center frequency can be calculated as follow: For achieving the power division of α 2 (P 2 /P 3 = α 2 ), the following condition should be satisfied.
According to (13), a simple formulas can be obtained.
Based on the simplified prototype network presented in Fig. 3, the coupling scheme can be further simplified as 36456 VOLUME 8, 2020    4. The simplified coupling scheme can be seen as a filtering power divider without isolation between two output ports, which has been detailed researched in [17], [18]. The Fig. 5 is the coupling prototype of traditional second-order two-port bandpass filter which can be synthesized according to the specifications of bandpass filtering response. The m 21 is its normalized internal coupling coefficient, and the m s1 and m 2L are both its normalized external coupling coefficients. For the proposed power divider, in order to realize the bandpass response, its coupling coefficient and external coupling parameters should satisfy the following condition as presented in [17], [18].

III. REALIZABTION
The realization structure of this BTB-GFPD is shown in Fig. 6. It is constructed by half-wavelength resonator (HR) and short-stub-loaded resonator (SSLR) as presented in Fig. 1. The resonator 1, resonator 2 and resonator 3 are all HRs, and the resonator 4 is a SSLR. The HR and SSLR have been researched in [24]. In this section, a thorough discussion is presented.

A. COMMON-MODE SUPPRESSION
The structure of the HR with corresponding feeding line structure is provided in Fig. 7. Although, the whole component is unsymmetrical in order to realize unequal power division. Each HR and the corresponding feeding line are symmetrical. The dominant resonant frequency of HR (f H ) can be calculated as follow: where, c is the velocity of light in free space and ε ef is the effective dielectric constant. The voltage distribution of dominant resonance is odd-symmetrical and also shown in Fig. 7. The electric coupling coefficient between HR and the feeding  line k e can be calculated as [30] where, CR represents the coupling region, V H (x) is voltage distribution of the dominant resonance of HR, V F (x) is voltage distribution of the feeding line and p represents a constant. As mentioned before, V H (x) is an odd function and the coupling region is symmetrical. When a CM signal is inserted, V F (x) will be a even function and k e will be zero as a result. Similarly, the magnetic coupling coefficient k m will be zero too when a CM signal is added. Therefore, CM input signal can not excite the dominant resonance of the HR. The HR is shut off for the CM signal. Thus, the suppression for CM signal is achieved.

B. EQUIVALENT MAGNETIC COUPLING
In the coupling scheme shown in Fig. 1, both electric and magnetic coupling are utilized to constructed the BTB-GFPD. The electric coupling and magnetic coupling have different phase shift in the transmission path which is VOLUME 8, 2020 used to realize the isolation of the BTB-GFPD. However, in the practical realization, only the electric coupling exists between resonators. When a pair of signal pass through SSLR and HR respectively, a difference of 180 • is existence between the phase shift of two transmission paths. This feature is used to equivalently realize magnetic coupling in the proposed BTB-GFPD. Fig. 8(a) and 8(b) depicts a signal passing through SSLR and HR respectively. The J 1 and J 2 are admittance inverters which represent the electric coupling between resonators. Usually, they are with difference value. Their ABCD matrix T 1 and T 2 are as follow: T a and T b is the ABCD matrix of transmission path a and b respectively. They can be calculated as: where T S and T H is the ABCD matrix of SSLR and HR respectively. The SSLR has a symmetric structure shown in Fig. 9(a). Fig. 9(b) and 9(c) is its odd-mode and even-mode equivalent circuit model respectively. The input admittances of odd-mode equivalent circuit Y ino S and even-mode equivalent circuit Y ine S can be obtained as: The ABCD matrix of SSLR T S can be attained as follow [31]: The dominant resonant frequency of SSLR (f S ) is its even-mode resonance and can be obtained by solving the following euqtion: Therefore, The T S can be simplified at the resonant frequency of f S .
The HR has been widely studied in many literatures and its ABCD matrix T H can be obtained in [29]. The dominant resonant frequency of HR (f H ) is its odd-mode resonance frequency. The T H at the resonant frequency of f H is as follow: In this design, the HR and SSLR are with same dominant resonant frequency. The f S and f H are equal to the center-frequency of the passband (f 0 ). Using (21), (22), (27) and (28), the T a and T b can be obtained.
Then, the transmission parameter at the center frequency f 0 of path a (S 21 a ) and path b (S 21 b ) can be calculated as follow: where, Y 0 is the characteristic impedance of the terminal. In this design, Y 2 is set equal to Y 0 . The second-order harmonic resonance of SSLR (f 2th S ) is approximately twice of f S . Therefore, For a bandpass filter with fractional bandwidth of 10%, the J 1 /Y 0 and J 2 /Y 0 are both with a very small value according [29]. So, The equation (35) indicates that the SSLR and HR have different phase shift in their transmission paths and the phase difference is 180 • . In this way, equivalently-magnetic coupling (negtive coupling) −M 31 can be realized. Therefore, the BTB-GFPD can be constructed based on the coupling scheme in Fig. 1 using SSLR and HR.

IV. IMPLEMENT AND MEASUREMENT
Using ANSYS's HFSS, two BTB filtering powewr divider with the division ratio of 2 and 3 are designed on Rogers RO4350B with a permittivity of 3.  Fig. 10(a) presents the transmission parameters of DM signal and CM signal. The simulated insertion loss for DM signal between port 2 and port 1, and between port 3 and port 1 is 2.78dB and 5.89dB respectively at the center-frequency of 2.4GHz with 3dB absolutely bandwidth of 214MHz and 3dB fractional bandwidth of 8.91%. The measured S dd21 is centered at 2.403GHz with the insertion loss of 3.12dB, 3dB absolutely bandwidth of 202MHz and 3dB fractional bandwidth of 8.41%. The center-frequency, insertion loss, 3dB absolutely bandwidth and 3dB fractional bandwidth of measured S dd31 is 2.404GHz, 6.21dB, 194MHz and 8.08% respectively. The S cc21 is <-50dB up to 4GHz in the simulation and is <-45dB up to 4GHz in the measurement. The simulated and measured S cc31 is <-51dB up to 4GHz and <-48dB up to 4GHz respectively. This BTB-GFPD is designed with the power division of 2. Thus, the difference between S dd21 and S dd31 should be 3dB in theory. In simulation, this value is 3.11dB. In measurement, it is 3.09dB. The imbalance of magnitude of output power excluded the ideal difference of 3dB is presented in Fig. 10(e). The simulated imbalance of output power magnitude in passband is <0.16dB but the measured is <0.47dB. The deterioration is mainly caused by the manufacturing tolerance. The manufacturing tolerance generate the difference deviation both in center-frequency and bandwidth and also aggravates the imbalance of output power magnitude.
The Fig. 10(b) presents the isolation between port 2 and port 3 for the DM and CM signal. For the DM signal, the simulated isolation is >32dB and the measured isolation is >35dB. For the CM signal, the isolation is >64dB in simulation and >55dB in measurement. When a DM signal is added into port 1, the port 4 is isolated. The isolation performance is also presented in Fig. 10b which is >45dB in simulation and >34dB in measurement. For the common application, the 50 matched load can be replaced by a 50 lumped resistor. The isolation between port 4 and port 1 for DM signal guarantee the isolation resistor could not produce extra insertion loss. In the measurement, the isolation between port 2 and port 3 for the DM signal is improved but the isolation between port 4 and port 1 for the DM signal is deteriorated. This is mainly caused by the manufacturing tolerance.
The Fig. 10(c) shows the mode conversion between port 2 and port 1, and between port 3 and port 1. The simulated S dc21 , S cd21 , S dc31 and S cd31 are all <-50dB. But all of them are degraded in measurement. The measured S dc21 is <-31dB, the measured S cd21 is <-26dB, The measured S dc31 is <-36dB and the measured S cd31 is <-28dB. The mode conversion between port 2 and port 3 is shown in the Fig. 10(d). In simulation, the S dc32 and S cd32 are both <-60dB. In measurement, the S dc32 and S cd32 are both <-45dB.
The This BTB-GFPD is designed with the power division of 3. Its response and paragraph are all shown in Fig. 11. The transmission parameters of the DM and the CM signal is presented in the Fig. 11(a). The insertion loss for DM signal between port 2 and port 1 is 2.33dB in simulation and 2.66dB in measurement. The insertion loss for DM signal between port 3 and port 1 is 7.16dB in simulation and 7.46dB in measurement. The simulated S dd21 and S dd31 have the center-frequency of 2.4GHz, 3dB absolutely bandwidth of 203MHz and 3dB fractional bandwidth of 8.45%. For the measured S dd21 , the center-frequency, 3dB absolutely bandwidth and 3dB fractional bandwidth is 2.402GHz, 194MHz and 8.08% respectively. For the measured S dd31 , the centerfrequency, 3dB absolutely bandwidth and 3dB fractional bandwidth is 2.404GHz, 194MHz and 8.07% respectively. The simulated S cc21 and S cc31 are both <-50dB up to 4GHz. In measurement, the S cc21 and S cc31 are both <-47dB up to 4GHz.
The ideal difference of S dd21 and S dd31 is 4.77dB when the division ratio is 3. The magnitude imbalance of output power excluded the ideal difference of 4.77dB is shown in Fig. 11(e). The simulated magnitude imbalance in passband is <0.1dB but the measured is <0.26dB. VOLUME 8, 2020 The isolation between port 2 and port 3 and the isolation between port 4 and port 1 for the DM signal are presented in the Fig. 11(b). The simulated S dd32 and S cc32 is >30dB and >60dB respectively. But the measured S dd32 and S cc32 is >35dB and >55dB respectively. The isolation between port 4 and port 1 for DM signal S ds14 is >48dB in the simulation and is >40dB in the measurement. Fig. 11(c) shows the mode conversion between port 2 and port 1, and between port 3 and port 1. The simulated S dc21 , S cd21 , S dc31 and S cd31 are all <-50dB. But the measured S dc21 , S cd21 , S dc31 and S cd31 is <-33dB, <-25dB, <-37dB and <-37dB respectively. The mode conversion between port 2 and port 3 is shown in the Fig. 11(d). In the simulation, the S dc32 and S cd32 are both <-59dB. In the measurement, the S dc32 is <-50dB and S cd32 is <-40dB.  0.35mm. The effective size of this FPD is 23.1mm×30.6mm and 0.3λ g ×0.4λ g , where λ g denotes the guided wavelength at 2.4 GHz.
The measured isolation between two output ports of the designed FPDs are both >35dB and better than the isolation of the FPD presented in [28] which is only >16dB. This depends on the precise control for the external quality factor of resonator 4 (Q e4 ) by adjusting the coupling strength between resonator 4 and the load. Furthermore, the FPD proposed in [28] was using the same structure presented in [22]. The structure of designed FPD in this paper is similar with the structure proposed in our previously research [24]. Thus, the designed FPDs in this paper keep the advantages in the suppression for common-mode signal and size compared with the FPD presented in [28].

V. FURTHER DISCUSSION ON THE DIVISION RATIO
According to (14), (15), (16) and (17), it can be concluded that the division ratio is determined by the internal coupling coefficient M 21 and M 31 , and the external coupling coefficient VOLUME 8, 2020 Q e1 , Q e2 and Q e3 are all independent of the division ratio. For the designed two FPDs, they have different division ratio, but they have identical external coupling coefficients.
In the design, the division ratio α will be always larger than 1 if the output power P 2 is always larger than P 3 . Then, the following equations can be obtained based on (13), (14) and (15).
When M 21 is equal to M 31 , the output power P 2 is equal to P 3 and the division ratio α is 1. When M 31 is zero, the output power P 3 is zero too and the division ratio α is infinity. So, the following equations can be attained obviously: The M 31 can be regarded as zero if the g 2 is big enough. Therefore, the infinity of division ratio is achievable. Under this condition, the isolation between port 3 and port 2 and the isolation between port 4 and port 1 can be both achieved if the M 24 is zero. So, it is concluded that the arbitrary power division is achievable.

VI. CONCLUSION
This paper presents the balanced-to-balanced Gysel filtering power dividerwith arbitrary power division ratio. The coupling scheme is proposed and investigated based on multicoupled series-resonator bandpass prototype network. The corresponding design equations are also derived. The proposed FPDs are constructed by half-wavelength resonator and short-stub-loaded resonator. The detailed realization principle is also presented. For demonstration, two BTB filtering power dividers working at 2.4GHz with the power division of 2 and 3 are designed and measured. The results in measurement show good agreement with the simulation and verify the theory.