**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 (*C*1–2, *L*1–2) connected between Port1 and Port2 (3), an L (*L*3), and an LRC circuit (*C*3, *L*4, *R*) connected between the output ports in parallel. Each parameter in the figure is normalized by the center angle frequency *ω*<sup>0</sup> and the characteristic impedance *Z*<sup>0</sup> of the input/output port, and each design value is obtained by *Rnor Z*0, *Lnor Z*0/*ω*0, and *Cnor*/(*Z*<sup>0</sup> *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

**Figure 1.**

*Circuit configuration. (a) Schematic of two-section LC-ladder divider and (b) its equivalent circuit with onefold symmetry.*

**Figure 2.** *Equivalent circuits at (a) even- and (b) odd-mode excitations.* and the input impedance seen from Port1 according to the theorem of maximum power supply.

$$\frac{1}{R\_1} = \frac{1}{j2L\_1} + \frac{1}{\frac{2}{jC\_1} + \frac{1}{\frac{1}{j2L\_2} + \frac{1}{\frac{1}{jC\_2} + R\_{(2,3)}}}} \tag{1}$$

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 *L*1,2, *C*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**.

$$\frac{1}{R\_{(2,3)}} = jC\_2 + \frac{2}{jL\_3} + \frac{1}{\frac{1}{j2C\_3} + \frac{jL\_4}{2} + \frac{R}{2}} \tag{2}$$

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 *L*3,4, *C*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.

#### *2.2.3 Frequency characteristics of scattering parameters*

The normalized circuit parameters obtained by the above procedure are *C*<sup>1</sup> = 1.23, *C*<sup>2</sup> = 1.20, *C*<sup>3</sup> = 0.54, *L*<sup>1</sup> = 1.20, *L*<sup>2</sup> = 0.61, *L*<sup>3</sup> = 1.74, *L*<sup>4</sup> = 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


**Table 1.**

*Normalized circuit parameters for each design frequency ratio.*

#### **Figure 3.**

*Frequency characteristics of scattering parameters for two-section LC-ladder dividers with several matching frequency ratios. (a) Input/output port reflection, (b) isolation, and (c) power division characteristics.*

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*<sup>2</sup> 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<sup>2</sup> . 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

#### **Figure 4.**

*Experimental results for broadband divider. (a) Simulation pattern, (b) its analysis result, (c) photograph of fabricated circuit, and (d) measured S-parameters.*

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 shortcircuited 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

#### **Figure 5.**

*Experimental results for dual-band divider. (a) Simulation pattern and its analysis result and (b)photograph of fabricated circuit and its measured S-parameters.*

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 lowfrequency 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.
