# Theory and Design of a Flexible Two-Stage Wideband Wilkinson Power Divider

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## Abstract

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## 1. Introduction

- Simple and generalized mathematical analysis.
- Fully Resistive isolation network.
- Closed-form equations for determining the value of isolation resistor without involving optimization.
- Flexible transmission bandwidth.
- Good isolation between two ports along with excellent port matching.
- Ready scalability for DC blocking application.

## 2. Analysis of Proposed Power Divider

#### 2.1. Even-Mode Analysis

#### 2.2. Odd-Mode Analysis

#### 2.3. Design Steps

- Choose the center frequency ${f}_{c}$.
- Divide the network designed using transmission lines and short-circuit stubs into half-circuit because of the symmetrical structure.
- Perform the even-mode analysis to determine the impedance values, ${Z}_{A}$, ${Z}_{B}$, ${Z}_{S1}$ and ${Z}_{S2}$ according to (12) at centre frequency for distinct ${\theta}_{c}$.
- Perform the odd-mode analysis to determine the values of the isolation resistors from (14).
- For EM simulation and eventually layout, model the junction discontinuities properly using tapers and bends.

## 3. Case Studies, Results and Analysis

## 4. Wideband DC Isolated WPD

#### 4.1. Even-Mode Analysis

#### 4.2. Odd-Mode Analysis

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 12.**Measurement Setup for (

**A**): ${S}_{11}$; (

**B**): ${S}_{21}$ for ${\theta}_{c}$ = 55${}^{\circ}$.

**Figure 13.**EM vs. Measured S-parameters for (

**A**) ${S}_{11}$; (

**B**) ${S}_{23},{S}_{22}$ for ${\theta}_{c}$ = 55${}^{\circ}$.

**Figure 14.**EM vs. Measured S-parameters for (

**A**) ${S}_{21}$; (

**B**) ${S}_{31}$ for ${\theta}_{c}$ = 55${}^{\circ}$.

**Figure 18.**Variation of ${S}_{23}$(dB) for ${\theta}_{c}$ = 55${}^{\circ}$ with different isolation resistors.

**Figure 23.**Measurement Setup for (

**A**): ${S}_{23}$; (

**B**): ${S}_{21}$ for ${\theta}_{c}$ = 55${}^{\circ}$.

**Figure 24.**EM vs. Measured S-parameters for (

**A**) ${S}_{21}$; (

**B**) ${S}_{31}$ for ${\theta}_{c}$ = 55${}^{\circ}$.

**Figure 25.**EM vs. Measured S-parameters for (

**A**) ${S}_{11}$; (

**B**) ${S}_{23},{S}_{22}$ for ${\theta}_{c}$ = 55${}^{\circ}$.

Parameters | Cases | ||
---|---|---|---|

${\mathit{\theta}}_{\mathit{c}}$ | 55${}^{\circ}$ | 60${}^{\circ}$ | 65${}^{\circ}$ |

${Z}_{A}$($\Omega $) | 115 | 127 | 120 |

${Z}_{B}$ ($\Omega $) | 81 | 90 | 94 |

${Z}_{S1}$ ($\Omega $) | 110 | 55 | 35 |

${Z}_{S2}$($\Omega $) | 40 | 31 | 24 |

${R}_{1}$ ($\Omega $) | 131 | 94 | 162 |

${R}_{2}$ ($\Omega $) | 200 | 150 | 200 |

Parameters | Cases | ||
---|---|---|---|

${\mathit{\theta}}_{\mathit{c}}$ | 55 ${}^{\circ}$ | 60${}^{\circ}$ | 65${}^{\circ}$ |

${Z}_{A}$ ($\Omega $) | 110 | 50.82 | 35 |

${Z}_{B}$ ($\Omega $) | 88.63 | 127 | 92 |

${Z}_{S1}$ ($\Omega $) | 56.5 | 90 | 68 |

${Z}_{S2}$ ($\Omega $) | 70 | 47.4 | 48 |

${Z}_{e}$ ($\Omega $) | 131 | 160 | 129 |

${Z}_{o}$($\Omega $) | 30 | 69 | 42 |

${R}_{1}$ ($\Omega $) | 38 | 100 | 62 |

${R}_{2}$($\Omega $) | 150 | 150 | 150 |

Reference | Frequency Range (GHz) | Topology | FBW * (|${\mathit{S}}_{21}$ = ${\mathit{S}}_{31}$|) | DC Isolation | Isolation Network | Size (${\mathit{\lambda}}_{\mathit{g}}\times {\mathit{\lambda}}_{\mathit{g}}$) |
---|---|---|---|---|---|---|

MWCL [40] | 2.7–4.7 | Dual Resonant Modes in single resonator | 53.5% (3 dB) | No | 1 Resistor | 1.06 × 0.89 |

MWCL [54] | 1.25–2.5 | Coupled-lines at Input/Output port | 51% (3 dB) | Yes | 1 Resistor | 0.89 × 0.57 |

TMTT [45] | 1.45–4.6 | Quasi-coupled Lines | 77% (3 dB) | No | 1 Resistor 1 Capacitor | 0.2 × 0.15 |

MWCL [43] | 1.55–4.24 | Embedded Transversal Signal Interference | 84% (3 dB) | No | 1 Resistor 1 Inductor 1 Capacitor | 0.8 × 0.5 |

Access [50] | 1.42–3.42 | Three Line coupled structure | 58% (3 dB) | Yes | 1 Resistor | 0.62 × 0.5 |

TCPMT [47] | 1.5–3.55 | Three Line Coupled structure | 64% (3 dB) | No | 1 Resistor | 0.05 × 0.32 |

This Work | 1.3–2.9 | Two-stage Configuration | 80% (3 dB) | No | 2 Resistors | 1.03 × 0.62 |

1.01–3.02 | Two-stage with Coupled Lines | 82.5% (3 dB) | Yes | 2 Resistors | 1.2 × 0.86 |

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**MDPI and ACS Style**

Saxena, A.; Hashmi, M.; Banerjee, D.; Chaudhary, M.A.
Theory and Design of a Flexible Two-Stage Wideband Wilkinson Power Divider. *Electronics* **2021**, *10*, 2168.
https://doi.org/10.3390/electronics10172168

**AMA Style**

Saxena A, Hashmi M, Banerjee D, Chaudhary MA.
Theory and Design of a Flexible Two-Stage Wideband Wilkinson Power Divider. *Electronics*. 2021; 10(17):2168.
https://doi.org/10.3390/electronics10172168

**Chicago/Turabian Style**

Saxena, Antra, Mohammad Hashmi, Deepayan Banerjee, and Muhammad Akmal Chaudhary.
2021. "Theory and Design of a Flexible Two-Stage Wideband Wilkinson Power Divider" *Electronics* 10, no. 17: 2168.
https://doi.org/10.3390/electronics10172168