## 1. **Why Design a Dual-Band Power Divider?**

With the rapid development of wireless communication technology, new mobile communication standards are emerging, requiring RF systems to support multiple frequency bands simultaneously.

To reduce the complexity of RF systems and meet the high performance and miniaturization requirements of modern communication systems, various microwave devices capable of operating in multiple frequency bands are needed.

A dual-band power divider designed using dual-line K-transformers has a simple structure, high-frequency band isolation, and certain bandpass filter characteristics.

This makes it widely applicable in various communication circuits and dual-band RF systems.

## 2. **Achieving the Circuit and Design Method of a Dual-Line K-Transformer**

**Parallel Structure for Dual-Band K-Transformer**

To achieve a dual-band K-transformer, a parallel structure as shown in Figure 1(a) can be used.

This structure has the symmetry of a K-transformer and four design freedoms: Z_{1}, Z_{2}, *θ*_{1}, and *θ*_{2}.

By reasonably selecting these parameters, this structure can achieve K-transformation at two frequency points.

**Equivalent K-Transformer at Two Frequencies**

Considering that the parallel dual-line structure in Figure 1(a) is equivalent to the K-transformer in Figure 1(b) at f_{1} and f_{2}, with a K value of Z_{T}, the admittance matrix of this network at the two frequency points is:

**Deriving the Admittance Matrix**

For the two-port network with symmetrical characteristics in Figure 1(a), we use the odd-even mode analysis method to derive its admittance matrix.

We can calculate and plot the characteristic impedance values Z_{1} and Z_{2} of the parallel dual-line relative to the frequency ratio, as shown in Figure 2.

**Frequency Ratio Range and Characteristic Impedance**

If *θ*_{2}=2*θ*_{1}=2*θ*, the applicable frequency ratio range of the parallel dual-line transformer is between 1.8-1.9 and 2.1-2.3.

At this range, the characteristic impedance of the dual lines is between 25-150Ω, which can be realized using microstrip transmission lines.

**Adjusting Frequency Ratios**

If other frequency ratios of dual-band impedance transformers are needed, the ratio between *θ*_{1} and *θ*_{2} can be changed, and parameters can be optimized using methods such as genetic algorithms.

## 3. **Design Example and Simulation Test Results**

**Circuit Structure**

Figure 3 shows the circuit structure of the dual-line dual-band power divider.

The common end 1 is connected to output ends 2 and 3, each with a dual-line dual-band K-transformer, and there is an isolation resistor between output ends 2 and 3 with a resistance value of 2Z_{0}.

**Operating Frequencies and Design Parameters**

If the designed dual-band power divider's operating frequencies are 2GHz and 4.4GHz with a corresponding frequency ratio m=2.2, a dual-band K-transformer with Z_{T}=70.7Ω and *θ*_{2}=2*θ*_{1}=2*θ* is needed.

**Microstrip Implementation**

From the admittance matrix, the characteristic impedance and electrical length parameters of the dual-line dual-band K-transformer transmission lines can be obtained.

This microstrip dual-band power divider is implemented using a substrate with a thickness of 0.8mm and a dielectric constant of 2.55.

The circuit area is approximately 0.22λ_{g}×0.18λ_{g}, where λ_{g} is the guided wavelength of the microstrip line at the frequency f_{1} (2GHz).

**Testing with Network Analyzer**

Using a Keysight N5230A network analyzer, the S-parameters of the dual-band power divider were tested.

At the frequency points 2GHz and 4.4GHz, the return loss S11 was 24.2dB and 26.5dB, respectively.

The return loss of S22 and S33 reached 35dB and 28dB, respectively.

The insertion loss of S21 and S31 from port 1 to ports 2 and 3 was 3.2dB at both frequency points.

The isolation between the two output ports (ports 2 and 3) was greater than 25dB at both frequency points.

**Achieving Design Objectives**

The parallel dual-line impedance transformer not only achieved impedance matching and power division at the two frequency points but also acted as a filter at other frequency points, meeting the design objectives.