# A Procedure to Design Feasible Dual Band Matching Networks with Minimum Complexity

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Matching Procedure

#### 2.1. Third Line Elements

## 3. Numerical Results and Experimental

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Circuit diagram of the matching network. ${Z}_{L}$ is the load to be matched at two frequencies ${f}_{a}$ and ${f}_{b}$; ${Z}_{1\left(2\right)}$ and ${L}_{1\left(2\right)}$ are parameters of transmission lines to be determined to match ${Z}_{L}$ to ${Z}_{0}$. ${Z}_{3}$ and ${L}_{3}$ parameters of a possible additional transmission line: it can be placed in front of the load as a stub (open or short) (

**a**) or alternatively in series (

**b**).

**Figure 2.**Reflection coefficient of the matching networks corresponding to the first case in Table 1 (i.e. ${Z}_{L,a}=52-j456\mathsf{\Omega}@1\mathrm{G}\mathrm{H}\mathrm{z}$, ${Z}_{L,b}=28-j180\mathsf{\Omega}@2.5\mathrm{G}\mathrm{H}\mathrm{z}$). The numbering of the legend corresponds to the presenting order in Table 1.

**Figure 5.**Schematics of the realized circuit with relevant parameters: (

**a**) circuit of the load; (

**b**) transmission line (T.L.) model of the matching network; (

**c**) compensated microstrip line (M.L.) model of the matching network.

**Figure 6.**Comparison of simulative results for four circuit models of the experimental matching circuit.

**Figure 7.**Simulative reflection coefficient of the compensated microstrip line matching network compared to the measurement of the realized circuit. The inset shows a picture of the realized circuit while the enlargements show details of the circuit (

**left**) and connector (

**right**).

${\mathit{Z}}_{\mathit{L},\mathit{a}}$ ${\mathit{Z}}_{\mathit{L},\mathit{b}}$ | ${\mathit{f}}_{\mathit{a}}$ ${\mathit{f}}_{\mathit{b}}$ | ${\mathit{Z}}_{1}$$/{\mathit{\theta}}_{1,\mathit{a}}$ | ${\mathit{Z}}_{2}/{\mathit{\theta}}_{2,\mathit{a}}$ | ${\mathit{Z}}_{3}$$/{\mathit{\theta}}_{3,\mathit{a}}$ |
---|---|---|---|---|

52 − j456 28 − j180 | 1.0 2.5 | 43.8/90.9 | 75.7/86 (o) | 70/27.5 (l) |

22.5/76.7 (o) | ||||

124.7/67.3 | 13.2/139 (o) | |||

34/24 (s) | ||||

35.1/48.2 | 18.6/99.6 (o) | 70/53.1 (o) | ||

56.7/43.8 | 11.7/160.8 (s) | |||

21.5/112.7 | 12.2/99.9 (o) | 70/27.7 (s) | ||

22 − j 403 16 − j 386 | 0.867 0.920 | 15.7/74.2 | 40.8 (o)/94.2 | 40/80 (l) |

34.2/89.7 | 110.5 (s)/176.7 | |||

22.5 − j22 5 + j66.5 | 1.0 2.4 | 51/57.6 | 25.8/28.5 (o) | n.n. |

72.2/146.6 (s) | ||||

57.6/50.3 | 44/43.1 (o) | |||

91.2/152.8 (s) | ||||

25/115.8 | 49/40.2 (o) | |||

105.8/151.3 (s) | ||||

22.2/125.7 | 44/34.6 (o) | |||

105.3/148.8 (s) | ||||

[14] a 400 + j30 400 + j30 | 1 2 | 80/122.4 | 40/119 (o) | 50/58.5 (s) |

13.5/59 (s) | ||||

[14] b 400 + j40 400 + j40 | 1 3.5 | 120/80.8 | 32/160 (o) | n.n. |

[19] a 200 + j60 250 + j100 | 2.4 5.8 | 100/109 | 36/102 (s) | n.n. |

[19] b 100 − j 35.4 150 − j6.4 | 0.9 2.4 | 44/45.2 | 37.7/51.2 (s) | n.n. |

27.9/149.2 (o) | ||||

[20] 10 − j66.3 10 − j44.2 | 2.4 3.6 | 78.3/28.7 | 14.4/142.3 (o) | n.n. |

[22] a 100 + j80 150 + j100 | 1 1.8 | 33.4/68.5 | 27.3/130.5 (o) 11/65.9 (s) | n.n. |

[22] b 50 + j80 60 + j 120 | 1 1.8 | 15/66.4 | 11/131.9 (o) | n.n. |

[22] c 100 + j0 120 + j0 | 1 2 | 44/52 | 40/54.7 (s) | n.n. |

20.4/160.1 (o) | ||||

68.6/103.6 | 59.6/101.9 (s) | |||

[25] a 92.8 − j15.8 78.8 − j21.5 | 1 2 | 57.5/50.8 | 64.1/58.5 (s) | n.n. |

57.6/111.1 | 64.4/121.5 (s) | |||

[25] b 10.3 − j11.9 11.1 + j24.4 | 1 2 | 21.8/85.6 | 20.5/66.4 (s) | n.n. |

65.5/125.7 (o) | ||||

[25] c 155.9 − j27 130.4 − j54.8 | 1 2.5 | 84.8/95.5 | 54.2/108 (s) | n.n. |

37.4/102.6 (s) | ||||

[25] d 86.5 + j13.4 90.1 + j21.4 | 1.5 2.5 | 29.6/138.8 | 36.2/134 (s) | n.n. |

87.5/66.8 (o) | ||||

61.8/77.9 | 64.3/69.6 (s) | |||

[26] 101 − j35.4 101 − j15.2 | 0.9 2.1 | 59.6/47.5 | 62.4/52.2 (s) | n.n. |

27.8/160.9 (o) | ||||

62.5/101.3 | 31.7/109.3 (s) |

^{1}Units: impedance: ohm; frequency: GHz; electric length: degrees: Symbols in the table: (s) short circuit stub; (o) open circuit stub; (l) line to ground; n.n. not necessary.

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**MDPI and ACS Style**

DiCarlofelice, A.; DiGiampaolo, E.; Tognolatti, P.
A Procedure to Design Feasible Dual Band Matching Networks with Minimum Complexity. *Electronics* **2022**, *11*, 3569.
https://doi.org/10.3390/electronics11213569

**AMA Style**

DiCarlofelice A, DiGiampaolo E, Tognolatti P.
A Procedure to Design Feasible Dual Band Matching Networks with Minimum Complexity. *Electronics*. 2022; 11(21):3569.
https://doi.org/10.3390/electronics11213569

**Chicago/Turabian Style**

DiCarlofelice, Alessandro, Emidio DiGiampaolo, and Piero Tognolatti.
2022. "A Procedure to Design Feasible Dual Band Matching Networks with Minimum Complexity" *Electronics* 11, no. 21: 3569.
https://doi.org/10.3390/electronics11213569