Normalized circuit parameters for each design frequency ratio.

## Abstract

With the remarkable progress in the use of Internet of Things (IoT) and 5G, there is a demand for higher performance such as miniaturization, broadband/multiband, low loss, and high integration for several microwave circuits. This chapter treats microwave power dividers/combiners used in amplifiers, mixers, phase shifters, antenna feeding networks, and so on. Here, the treated circuits are composed of LC-ladder circuits and an absorption resistor. It shows that multiband (dual-band and tri-band) and broadband can be achieved by changing the number of stages of the LC-ladder circuit. In addition, the effectiveness of this design method is demonstrated by electromagnetic simulations and prototype experiments.

### Keywords

- Wilkinson power divider
- lumped-element
- LC-ladder circuit
- compact
- broadband/multiband

## 1. Introduction

In recent years, wireless communication equipment has been rapidly researched and developed such as Internet of Things (IoT), wireless local area network (LAN), and 5G, and it is expected that the demand for wireless communication equipment will become more widespread in the future. Along with this, various microwave circuit elements mounted on wireless communication devices are also required to have higher performance, such as miniaturization, low loss, high integration, and wideband/multiband. The authors are paying attention to the power divider/combiner that divides/combines microwave signals among various microwave circuit elements. The reason is that the power divider/combiner is considered to be an important circuit element that is directly linked to its performance in microwave circuits.

As a power divider/combiner for a three-port network, the Wilkinson power divider (hereinafter referred to as a conventional circuit) composed of two quarter wavelength transmission lines at a design frequency and an absorption resistor connected between two output ports is widely used at several microwave/millimeter-wave circuit system such as a balanced amplifier, a mixer, a phase shifter, an antenna feeding network, and so on [1]. However, since the circuit size depends on the wavelength due to the distributed circuit configuration, there arises a problem that the area occupied by the circuit system becomes especially large in a low-frequency band. As a method for reducing the size of a microwave circuit, a method of replacing a transmission line with a Π-type/T-type circuit equivalent to that at the design frequency is often used, but the equivalence between the two circuits is guaranteed only at the design frequency [2]. Therefore, such a circuit generally has a narrow band characteristic. In addition to that, as a method of shortening the transmission line, methods of loading parallel capacitances or parallel open-circuited stubs at both ends or the center of the transmission line have been reported [3, 4, 5, 6]. In addition, some miniaturization design methods using composite right-/left-handed transmission lines and lumped elements have also been proposed [7, 8, 9, 10]. However, their operation bands are still narrower than that of the conventional circuit. Therefore, it is considered difficult to achieve both miniaturization and wide bandwidth of the circuit at the same time. On the other hand, our research group proposes a configuration using an LC-ladder circuit as a lumped-element circuit type Wilkinson power divider. It has been analytically and experimentally clarified that a configuration using a two-stage LC-ladder circuit on the input side can realize frequency characteristics equal to or higher than those of the conventional circuit. Furthermore, ultra-wideband power dividers, unequal power dividers, and * N*-way power dividers, etc., in a circuit configuration, using LC-ladder circuits have been also reported [11, 12, 13, 14, 15].

This chapter shows how to design a power divider that can be matched at arbitrary two frequencies with a simple circuit configuration with 9 lumped elements. The circuit is designed for application in IoT (920 MHz) and 5G (sub6 band: 3.7 GHz). The influence of the self-resonant frequency of the chip element used in the circuit configuration is considered in the SHF-band, so the inductance is realized using a meander line or a bent line. Electromagnetic field simulations and prototype experiments confirm the effectiveness of the two-frequency matching circuit with a quasi-lumped-element circuit configuration. It should be noted that this circuit also has a feature that high-pass or low-pass characteristics can be selected by replacing the inductance L and the capacitance C of the components.

In the circuit configuration described above, the frequency characteristic of either the high-frequency or the low-frequency band becomes a narrow band. Therefore, the number of stages of the LC-ladder circuit was increased, and a circuit with 15 elements in which an LC-ladder circuit and an LR/CR circuit were connected in parallel between the output ports enabled three-frequency matching. It was shown that by moving the matching frequency in the middle of the three matching frequencies closer to the low-frequency side or the high-frequency side, a divider having an absolute constant bandwidth in the low-frequency and high-frequency bands becomes possible.

Furthermore, ultra-wideband characteristics are possible by increasing the number of stages in the LC-ladder circuit. As a method for widening the bandwidth of impedance transformers, quarter wavelength multistage transformers are also described in Pozer’s book and are often used. By using the concept of this multistage impedance transformer [16] and L-type matching circuit [17], a circuit with a relative bandwidth exceeding 100% in the UHF band was realized. Specifically, we have experimentally confirmed an ultra-wideband divider with a relative bandwidth of 100% or more, which covers the 80 MHz–370 MHz band used for public radio in Japan, with a lumped-element circuit configuration.

## 2. Two-stage LC-ladder divider

This section shows a lumped-element power divider that realizes the same frequency characteristics as the conventional Wilkinson power divider. Furthermore, a two-frequency matching divider operating in the UHF and the SHF band will be described.

### 2.1 Circuit configuration

Figure 1a shows the dual-band power divider with arbitrary two matching frequencies treated in this section [18, 19]. This circuit consists of an LC-ladder circuit (_{1–2}, _{1–2}) connected between Port1 and Port2 (3), an L (_{3}), and an LRC circuit (_{3}, _{4}, * R*) connected between the output ports in parallel. Each parameter in the figure is normalized by the center angle frequency

ω

_{0}and the characteristic impedance

Z

_{0}of the input/output port, and each design value is obtained by

R

_{nor}Z

_{0},

L

_{nor}Z

_{0}/

ω

_{0}, and

*/(*C

_{nor}

Z

_{0}

ω

_{0}). In order to apply the even/odd mode excitation method to the proposed circuit, Figure 1b shows an equivalent circuit symmetric with respect to the plane AA’. At that time, each input/output port is represented by terminal resistors

R

_{1},

R

_{2}, and

R

_{3}, and since Port1 is parallelized, it is twice as large as the output port.

### 2.2 Design method and scattering matrix

#### 2.2.1 Even mode

When a signal of the same phase and amplitude is applied to each output port Port2/3 of the equivalent circuit shown in Figure 1b, the plane AA’ becomes a magnetic wall. It is not necessary to consider the inflow of current to the L and RLC parallel circuits, and the signal applied to the output port propagates to the input port side while maintaining its potential. Therefore, the equivalent circuit can be simplified as shown in Figure 2a. The following equation expresses the signal non-reflection condition at the input end for conjugate matching of the terminal resistance of Port1 and the input impedance seen from Port1 according to the theorem of maximum power supply.

For each of the real and imaginary parts, by determining the parameters, so that Eq. (1) is satisfied with two matching frequencies, the circuit parameters on the input side _{1,2}, _{1,2} can be derived.

#### 2.2.2 Odd mode

In the odd-mode excitation in which a signal of opposite phase and the same amplitude is applied to the output port of the circuit shown in Figure 1b, the plane AA’ becomes an electric wall, and when the potential becomes 0 on the plane AA’. Therefore, the inflow of current to the input side can be ignored. Therefore, in this case, the equivalent circuit can be simplified as shown in Figure 2b.

By satisfying Eq. (2) for the real and imaginary parts and designing it to operate as an impedance transformer at the design frequency, the circuit parameters on the output port side _{3,4}, _{3}, and * R* can be calculated.

The above operation can derive all parameters, and it is possible to design an equal power divider that matches at arbitrary two frequencies. Table 1 shows the normalized circuit parameters for some design frequency ratios.

Frequency ratio | 0.8/1.2 | 0.6/1.4 | 0.4/1.6 |
---|---|---|---|

_{1} | 1.23 | 1.52 | 2.46 |

_{2} | 1.20 | 1.11 | 0.90 |

_{1} | 1.20 | 1.11 | 0.90 |

_{2} | 0.61 | 0.76 | 1.23 |

_{3} | 0.54 | 0.96 | 1.85 |

_{3} | 1.74 | 2.14 | 3.46 |

_{4} | 1.93 | 1.24 | 0.83 |

1.62 | 1.12 | 0.92 |

#### 2.2.3 Frequency characteristics of scattering parameters

The normalized circuit parameters obtained by the above procedure are _{1} = 1.23, _{2} = 1.20, _{3} = 0.54, _{1} = 1.20, _{2} = 0.61, _{3} = 1.74, _{4} = 1.93, * R* = 1.62 when the matching frequency ratio (

f

_{1}/

f

_{2}) is 0.8/1.2. Figure 3a–c shows the frequency characteristics of the scattering matrix of a dual-band power divider with several matching frequency ratios. Here, the arbitrary matching frequency ratios

f

_{1}/

f

_{2}are set to 0.8/1.2, 0.6/1.4, and 0.4/1.6, respectively. Due to the symmetry of the circuit, the scattering matrix elements shown in the figure show the reflection characteristics (

S

_{11},

S

_{22}) in (a), the isolation characteristics (

S

_{32}) in (b), and the power division characteristics (

S

_{21}) in (c). The relative bandwidth is defined as the value obtained by dividing the band where the reflection characteristics and isolation characteristics are −20 dB or less by the matching frequency. The relative bandwidths of the two matching frequency bands are 51.8%/77.7%, 6.6%/23.5%, and 1.9%/25.0% for each matching frequency ratio described above, and good power division characteristics can be confirmed within each bandwidth. It can also be seen that the relative bandwidth is wide on the high-frequency side of two matching frequencies and narrow on the low-frequency side. Therefore, in order to realize a wideband divider, a prototype experiment is conducted with a matching frequency of 0.8/1.2, which has a common operating frequency band.

### 2.3 Simulation and experimental results

In order to confirm the validity of the circuit design method, we designed a broadband power divider in the 920 MHz band using a commercial electromagnetic simulator (Sonnet em). The design conditions are a dielectric substrate with a relative permittivity of 2.2, a thickness of 0.787 mm, and each port has a microstrip line configuration with a characteristic impedance of 50 Ω. Since the commercially available 1005 size chip inductor has a self-resonant frequency in the UHF/SHF bands, it is difficult to use it in circuit design above the UHF band. Therefore, as a lumped element model, the circuit pattern is designed using a spiral inductor that directly reproduces the metal pattern on the dielectric substrate and a commercially available chip capacitor. In addition, the circuit pattern was determined by trial and error to reduce the influence of the land pattern on the characteristics while securing the land pattern for soldering required for the chip element. In addition, * S*-parameter data related to GRM series capacitors are used for the simulation. Figure 4a shows a circuit pattern with a board area of 8.8 × 14.8 mm

^{2}. From the figure, the short-circuited part of the inductor is connected to the ground conductor by a via hole. Figure 4b shows the frequency characteristics of the scattering matrix obtained from the circuit pattern using the electromagnetic simulator. The figure shows the input/output reflection characteristics (

S

_{11},

S

_{22},

S

_{33}), isolation characteristics (

S

_{32}), power division characteristics (

S

_{21},

S

_{31}), and output phase difference characteristics (arg(

S

_{21}/

S

_{31})). The relative bandwidth for -20 dB reflection/isolation characteristics was 45.8%. This value is larger than that of the conventional circuit based on the distributed circuit theory. Furthermore, the maximum output phase difference in the band was 2.9°.

Considering the practical application of the proposed circuit, a prototype experiment was conducted under the same conditions using the circuit pattern shown in Figure 4a. A conductor pattern was formed on the dielectric substrate Rogers/Duroid 5880 using a substrate processing machine (ProtoMat S63) made by LPKF. The chip elements used are the same 1005 size commercially available chip capacitors (GRM series) and thick film chip resistors (MCR series) used in the simulation and soldered to the conductor pattern. In addition, the via hole part of the simulation pattern is short-circuited with the ground conductor by making a hole with a diameter of 0.3 mm at the desired position, inserting silver paste, and sintering it. Figure 4c shows a prototype circuit photograph. Figure 4d shows the frequency characteristics of the scattering matrix of the prototype circuit measured using a vector network analyzer. From the figure, the measured results are almost the same as the electromagnetic simulation results, but the relative bandwidth with reflection characteristics and isolation characteristics of −18 dB or less is about 45.9%, and some deterioration can be seen. This is thought to be due to the tolerance of each chip element and manufacturing error of the spiral inductor. However, in the actual measurement, it was confirmed that the two frequencies were matched, and the power division characteristics were flat around the matching frequency band, and the maximum output phase difference was 2.6°.

On the other hand, by separating the matching two frequencies, it is possible to realize a divider that operates in two bands. Here, the results of electromagnetic field simulations and prototype experiments for the divider shown by the green line in Figure 3 are introduced. The two matching frequencies are selected as 920 MHz and 3.68GHz used in IoT and 5G (sub6 band). Figure 5a shows the simulation pattern and its analysis results, and Figure 5b shows the prototype circuit photograph and measurement results. It can be confirmed that the matching frequency on the low-frequency side is slightly shifted to the higher side, and the matching frequency on the high-frequency side is slightly different. However, the measurement results and the simulation results are in good agreement.

### 2.4 Summary

This section has proposed a design method for a Wilkinson-type dual-band power divider with a new configuration using an LC-ladder circuit. It is known that a power divider using lumped elements in order to reduce the circuit area in a relatively low-frequency band generally has a narrow band frequency characteristic. We conducted a trial experiment of a power divider with lumped elements design in the 920 MHz band and showed that a wide operating frequency band with a relative bandwidth of about 45.9% could be obtained. Furthermore, it was shown that a divider operating in two separate bands (920 MHz/3.68GHz) could be realized. The proposed circuit is useful in reducing the circuit area in the UHF/SHF band. Next, we will conduct a prototype experiment in the 5G (Sub6) band to confirm its usefulness further.

## 3. Multiband divider

Since research on circuits and devices that support multiband systems is also actively conducted [20, 21], this section describes power dividers that can be matched at arbitrary three frequencies. Further, as an application thereof, it is shown that a dual-band power divider having an absolute constant bandwidth can be realized by moving the middle frequency closer to the low-frequency side among any three matching frequencies.

### 3.1 Circuit construction and design method

Figure 6a shows the circuit configuration of a power divider with three sections of LC-ladder circuits at the input port side [22]. The circuit parameters are normalized as described above. In designing a circuit using the even-/odd-mode excitation methods, consider an equivalent circuit having a onefold symmetry with respect to the plane AA’ in Figure 6b. Each input/output port is represented as a terminal resistor. When the circuit structure is vertically symmetrical, _{1,2,3,6}, _{1,2,4,5}, and * R* are considered as two elements.

#### 3.1.1 Even-/odd-mode analysis

Since the circuit in Figure 6b is also symmetric with respect to the plane AA’, the even-/odd-mode analysis can be applied as in 2.2.1 and 2.2.2. Figure 7 shows equivalent circuits at even-/odd-mode excitations. The conditional equations for obtaining the circuit parameters are as follows.

All parameters can be derived by the above operation, and an equal power divider with arbitrary three matching frequencies can be designed.

#### 3.1.2 Scattering parameters

If the three normalized matching frequencies _{1}/_{2}/_{3} are set to 0.4/0.6/1.6, 0.4/0.8/1.6, and 0.4/1.0/1.6 according to the above design procedure, the circuit parameters of the equal power divider are shown in Table 2. The frequency characteristics of the scattering matrix for each power divider are shown in Figure 8a-c. The scattering matrix elements shown in this figure refer to reflection characteristics (_{11}, _{22}), division characteristics (_{21}), and isolation characteristics (_{32}). Good power division characteristics and reflection/isolation characteristics at the desired frequency can be confirmed for each circuit. Next, we are studying a dual-band power divider with absolute constant bandwidth in the UHF/SHF band. Here, the bandwidth of the UHF band is expanded by bringing two of the abovementioned three matching frequencies closer to each other. Figure 9 shows the frequency characteristics when the design frequency ratio is 0.4/0.45/1.6. As shown in this graph, it can be seen that the band of the 920 MHz band is expanded. In the next section, when designing a dual-band power divider in the UHF/SHF band, the circuit pattern is examined by referring to each element value when the matching frequency ratio is 0.4/0.45/1.6 in Table 2.

Frequency ratio | 0.4/0.6/1.6 | 0.4/0.8/1.6 | 0.4/1.0/1.6 | 0.4/0.45/1.6 |
---|---|---|---|---|

_{1} | 2.07 | 1.75 | 1.42 | 1.51 |

_{2} | 1.11 | 4.62 | 2.41 | 0.93 |

_{3} | 0.97 | 1.40 | 1.83 | 1.24 |

_{1} | 1.92 | 2.74 | 2.55 | 2.49 |

_{2} | 2.12 | 7.61 | 8.13 | 1.87 |

_{3} | 2.02 | 1.64 | 1.43 | 1.52 |

_{4} | 1.26 | 1.02 | 0.81 | 0.64 |

_{5} | 0.65 | 0.63 | 0.75 | 0.09 |

_{6} | 1.49 | 1.43 | 1.38 | 1.49 |

_{4} | 0.57 | 0.70 | 0.77 | 0.46 |

_{5} | 1.49 | 1.15 | 0.84 | 0.21 |

1.05 | 0.89 | 1.01 | 1.04 |

### 3.2 Simulation and experiment

Based on the circuit analysis using the even-/odd-mode excitation method mentioned above, we are studying the circuit pattern on the dielectric substrate by an electromagnetic analysis as a preliminary step to the trial production. The circuit pattern is shown in Figure 10a. The design frequency is 0.4/0.45/1.6 with a frequency ratio to the center frequency of 2.3 GHz, that is, 920 MHz/1.03 GHz/3.68 GHz, and the conditions for the electromagnetic simulation are the same as in Section 2.3. Assume the use of GRM Series and MCR series for capacitors and resistors, respectively. Assuming the influence of the self-resonant frequency of the element on the circuit characteristics, the inductor is arranged by the bending pattern of the line instead of the chip element. In Figure 10a, the circuit size is 7.0 × 9.3 mm^{2} excluding the input/output ports. The frequency characteristics of the scattering matrix obtained by the electromagnetic simulation of the circuit pattern in Figure 10a are shown in Figure 10b. In addition to good reflection/isolation characteristics and division characteristics at the desired design frequency, the absolute constant bandwidth based on the center frequency in the UHF/SHF band is about 8.6%/7.4%. A prototype experiment is being conducted under the same conditions for the circuit pattern examined by the electromagnetic simulation above. A circuit is realized by forming a conductor pattern on a dielectric substrate using a commercially available substrate processing machine (ProtoMat S63/LPKF) and soldering each element to a predetermined position. A conductor pin is inserted into the via hole with a diameter of 0.3 mm and sintered to connect to the ground conductor. Figure 10c shows a photograph of the prototype circuit. The frequency characteristics of this circuit were measured using a vector network analyzer. The results are shown in Figure 10d. Good characteristics are almost identical to the analysis results, and the absolute bandwidth of 5.6%/4.8% can be confirmed in both operating bands.

### 3.3 Summary

As mentioned above, the usefulness of the circuit that can be designed by arbitrarily determining the three matching frequencies using the LC-ladder type configuration was examined. It was analytically and experimentally shown that a dual-band power divider with an absolute constant bandwidth in the UHF/SHF (sub6) band can be realized by closing the two matching frequencies to each other on the low-frequency side. The design method in this study is considered to be very useful in the situation where the use of the SHF band becomes more active due to social factors such as the spread of 5G. In the future, we plan to conduct an experimental study on the design of power dividers in the higher frequency range, such as the X-Band.

## 4. Multistage LC-ladder divider

### 4.1 Circuit construction and design results

By increasing the number of stages of the LC-ladder circuit, the operating band can be expanded, corresponding to the number of stages. In this section, the number of stages of the LC-ladder circuit on the input side is set to 8, and the ultra-wideband (relative bandwidth 100% over), an equal power division circuit with characteristics, will be described.

As a design method that uses lumped elements and realizes wideband characteristics while avoiding an increase in circuit size, there is a method that uses an LC-ladder impedance transformer. Here, we show the design method of the multiband LC-ladder divider proposed by Okada et al. and focus on the viewpoint of wideband and multiband. Figure 11a shows a multiband power divider consisting of multiple LC-ladder circuits and an RL series circuit on the output side. The figure shows an * N*-stage LC-ladder circuit (

L

_{1 − N},

C

_{1 − N}) connected between Port1 and Port 2/3 and a ladder circuit with

*and*L

*interchanged between Port2/3 (*C

C

_{(N + 1) − (2N − 1)},

L

_{(N + 1) − (2N − 1)}),

RL

_{2N}series circuit. Since the number of matching frequencies in this circuit corresponds to the number of stages

*of the LC-ladder circuit, it can operate as a power divider with arbitrary*N

*frequency matching according to the number of stages. In applying the even-/odd-mode excitation method to the circuit in Figure 11a, consider an equivalent circuit having a mirror image symmetry structure in the vertical direction as described above. Figure 11b shows the equivalent circuit of a multiband circuit with the plane AA’ as the plane of symmetry. Considering this figure, each input/output port has terminal resistors*N

R

_{1},

R

_{2}, and

R

_{3}, the input side element is doubled in parallel, and the output side element is divided, and the value is halved. The circuit design is possible using even-/odd-mode excitation methods as in Sections 2 and 3. As a result of designing the circuit, it can be seen that the relative bandwidth is expanded by increasing the number of stages of the LC-ladder divider. In Figure 12a-c are the cases where the number of stages is 4, 6, and 8, and the specific bandwidths are 98%, 108%, and 115%, respectively. By increasing the number of stages, the band becomes a wide band, but it becomes saturated to some extent when the number of stages is about 8.

### 4.2 Experiments

Figure 13 is a photograph of the prototype circuit of the circuit shown in Figure 12 and the measurement result of its * S*-parameter. The chip elements used in each circuit are 19, 31, and 37, respectively, and the circuit areas excluding the input/output ports are 51.5mm

^{2}, 138.4mm

^{2}, and 182.4mm

^{2}. Comparing the measurement results with the abovementioned analysis results, it can be seen that they are in good agreement. In manufacturing the prototype circuit, the circuit pattern is determined by electromagnetic simulations, as in the case of the two-section LC-ladder divider.

### 4.3 Summary

As mentioned above, it has been shown that the operating band of the power divider is expanded by increasing the number of stages of the LC-ladder circuit. Setting eight stages, an ultra-wideband circuit with a relative bandwidth exceeding 100% becomes possible. The effectiveness was also shown experimentally in a prototype experiment in the VHF band. With a wideband characteristic of over 100%, it is available for public radio in Japan.

## 5. Conclusions

By using a multistage impedance transformer and an L-type matching circuit design method to downsize the microwave power divider, the characteristics are equal to or higher than those of the conventional Wilkinson power divider designed based on the distributed circuit theory. First, it was shown that in a circuit capable of dual-band operation, a wideband circuit or two-frequency operation is possible by moving the operating two frequencies closer to or further away from each other. Next, a circuit configuration that enables three-frequency matching is shown, and by utilizing the feature that the frequency at the center of the three matching frequencies can be arbitrarily selected, a divider having an absolute constant bandwidth in the UHF/SHF band was made possible. Finally, it was shown that the operating band can be expanded by increasing the number of stages of the LC-ladder circuit and that an ultra-wideband circuit exceeding 100% can be realized if the number of stages is 8. These results have been clarified analytically and experimentally.

## References

- 1.
Wilkinson EJ. An N-way hybrid power divider. IEEE Transactions on Microwave Theory and Techniques. 1960; 8 (1):116-118 - 2.
Noriega F, Gonzales PJ. Designing LC Wilkinson Power Splitters. RF Interconnects/Interfaces. 2002; 48 :18-24 - 3.
Hirota T, Minakawa A, Muraguchi M. Reduced-size branch-line and rat-race hybrids for uniplanar MMIC’s. IEEE Transactions on Microwave Theory and Techniques. 1990; 38 (3):270-275 - 4.
Tang C-W, Chen M-G. Realization of ultra-compact planar microstrip branch-line couplers with high-impedance open stubs. In: IEEE MTT-S International Microwave Symposium Digest. London: IEEE; 2007. pp. 995-998 - 5.
Pu T, Cui J, Chang H, Lu J. “A Novel ultra compact Luzzatto power divider using miniaturized stubs and folded structure,” 2020 IEEE MTT-S International Wireless Symposium, 2020. IEEE London - 6.
Scarddelletti MC, Ponchak GE, Weller TM. Miniaturized Wilkinson power dividers utilizing capacitive loading. IEEE Microwave and Wireless Components Letters. 2002; 12 (1):6-8 - 7.
Deutschmann B, Jacob A. Compact ultra-broadband power dividers with integrated resistors. In: Proceedings of the 48th European Microwave Conference. London: IEEE; 2018. pp. 620-623 - 8.
Lu LH, Bhattacharya P, Katehi LPB. X-Band and K-Band Lumped Wilkinson power dividers with a micromachined technology. In: IEEE MTT-S International Microwave Symposium Digest. London: IEEE; 2000. pp. 287-290 - 9.
Song A, Wang X, Ma Z, Ohira M. Design theory of dual-band Wilkinson power divider with different frequency ratio ranges. In: Proceedings of 2018 Asia-Pacific European Microwave Conference. London: IEEE; 2018. pp. 1486-1491 - 10.
I. Chen, W. Lee, and Y. Pang, “A Compact Power divider using dual composite right−/left-handed resonators (D-CRLH) with filtering response,” Proceedings of 2018 Asia-Pacific European Microwave Conference, pp. 1366–1368, 2018. IEEE London - 11.
Kawai T, Nakashima Y, Kokubo Y. Okada, and Ohta Lumped-element wilkinson power dividers using LC-ladder circuits. In: Proceedings of International Symposium RAD System Space Place, C1-C1407. London: IEEE; 2007. pp. 35-38 - 12.
Kawai T, Nagano K, Enokihara A. A 920MHz lumped-element Wilkinson power divider utilizing LC-ladder circuits. IEICE Transactions on Electronics. 2018; E101-C (10):801-804 - 13.
Okada Y, Kawai T, Enokihara A. Wideband lumped-element wilkinson power dividers using LC-ladder circuits. In: Proceedings of the 45th European Microwave Conference. London: IEEE; 2015. pp. 115-118 - 14.
Okada Y, Kawai T, Enokihara A. Design method for multiband WPDs using multisection LC-ladder circuits. IEEE Microwave and Wireless Components Letters. 2017; 27 (10):894-896 - 15.
Okada Y, Kawai T, Enokihara A. Design of multi-way LC-ladder dividers with multi-band operation. IEICE Transmission Electron. 2017; E100-C (10):893-900 - 16.
Pozar DM. Microwave Engineering. 4th ed. Hoboken, New Jersey, United States: John Wiley and Sons, Inc.; 2011. pp. 229-234 - 17.
Pozar DM. Microwave Engineering. 4th ed. Hoboken, New Jersey, United States: John Wiley and Sons, Inc.; 2011. pp. 246-254 - 18.
Nagano K, Tsuchiya A, Kawai T, Enokihara A. “Experimental study of quasi-lumped-element power divider utilizing LC-ladder circuits at SHF-Band,” IEEE Transaction of Electronic Information and Systems, vol. 141, no. 2, pp. 94–98, Feb. 2021 (Japanese ed.) - 19.
Kawai T, Nagano K, Enokihara A. Dual-band semi-lumped-element power dividers at UHF/SHF Bands. In: Proceedings of the 50th European Microwave Conference. London: IEEE; 2021. pp. 844-847 - 20.
Yang X, Wang Z, Xie J, Zhang X, Liao Z, Yang M, et al. A tri-band gysel power divider based on tri-band phase inverter. In: 2020 Cross Strait Radio Science & Wireless Technology Conference (CSRSWTC). London: IEEE; 2020 - 21.
Abdelrahman BM, Ahmed HN, Nashed AI. A novel tri- band Wilkinson power divider for multiband wireless applications. IEEE Microwave and Wireless Components Letters. 2017; 27 (10):891-893 - 22.
Tsuchiya A, Okuda T, Kawai T, Enokihara A. Experimental study of UHF/SHF dual-band semi-lumped-element power dividers. In: Proceedings of 2021 IEEE International Symposium on Radio-Frequency Integration Technology (RFIT)|IEEE Conference, 210773. London: IEEE; 2021