The ideal lossless symmetrical reciprocal network (ILSRN) is constructed and introduced to resolve the complex interconnections of two arbitrary microwave networks. By inserting the ILSRNs, the complex interconnections can be converted into the standard one-by-one case without changing the characteristics of the previous microwave networks. Based on the algorithm of the generalized cascade scattering matrix, a useful derivation on the excitation coefficients of antenna arrays is firstly proposed with consideration of the coupling effects. And then, the proposed techniques are applied on the microwave circuits and antenna arrays. Firstly, an improved magic-T is optimized, fabricated, and measured. Compared with the existing results, the prototype has a wider bandwidth, lower insertion loss, better return loss, isolation, and imbalances. Secondly, two typical linear waveguide slotted arrays are designed. Both the radiation patterns and scattering parameters at the input ports agree well with the desired goals. Finally, the feeding network of a two-element microstrip antenna array is optimized to decrease the mismatch at the input port, and a good impedance matching is successfully achieved.

Microwave system consists of microwave elements and transmission lines, and the microwave network theory is one of the most important analytical approaches to design and synthesize the microwave equipment. A field analysis using Maxwell’s equations is complete and rigorous. However, usually we are only interested in the terminal characteristics and the power flow through a device. And, sometimes, it is convenient to combine the elements together and find the responses without having to reanalyze the behavior of each element in combination with its neighbors. As a result, the complex and sensitive parts can be analyzed with a rigorous field analysis approach, while the characteristics of the entire network are obtained by microwave network theory.

Generally, a microwave network with an arbitrary number of ports can be characterized by the impedance, admittance, and scattering matrices. The transmission (

In this paper, the algorithm of the generalized cascade scattering matrix is studied and expanded to deal with the complex interconnections and the excitation coefficients of antenna arrays. The ideal lossless symmetrical reciprocal network (ILSRN) is firstly presented to facilitate the generalized microwave cascade network. It can be treated as the “virtual joint” to combine the surrounding networks together. As a result, the generalized cascade scattering matrix can be expediently operated with an intuitive idea. The only thing one should to do is just to offer the interconnection relationships, and then the operations of the generalized cascade scattering matrices can be automatically performed on the computer. Moreover, a useful derivation on the excitation coefficients of antenna arrays is firstly proposed with consideration of the coupling effects. The antenna array as a whole is considered to be a network load of the feeding network. The excitation coefficients of each unit can be obtained through the interconnection of the feeding network and antenna array. To demonstrate the accuracy of the proposed techniques, applications both in the fields of microwave circuits and antenna arrays will be presented in the following.

The framework of this paper is as follows. In Section

Considering two arbitrary microwave networks I and II, the scattering matrices can be written as

Combining (

Note that the derivation is under a hypothesis that each pair of interconnection ports is interconnected one-by-one. Although similar conclusions have been drawn in many researches and educational books, none clearly indicate how to deal with the complex situations, for example, the interconnections with more than two ports and self-loops with the ports coming from the same network. Here, the ILSRNs are introduced to resolve the problem. Assume a three-port ILSRN has a generalized form of

The above complex equation set contains four unknowns, and more conditions are needed to resolve it. At the three-port node, Kirchhoff’s voltage law and current law can be applied to give

To illuminate the decomposition of the complex interconnections, a sketch is shown in Figure

Decomposition of the complex interconnections.

Antennas radiate electromagnetic waves towards the surrounding space with the coupling effects existing between the elements. As depicted in Figure

Interconnection between the feeding network and antenna array.

It should be noted that all the input ports have been already considered, and

A home-made program is developed in the environment of MATLAB to realize the interconnections of the generalized cascade scattering matrices. Firstly, the complex network is segmented into a set of ILSRNs and simple subnetworks, for example, short end, open end, matched load, transmission line, and transformer. The scattering matrices of these subnetworks can be easily acquired with the characteristic parameters. As to the complex and sensitive parts, they can be analyzed with commercial EM tools. Sometimes, these parts include various elements, and it is more advisable to measure them. And then, the simulated or measured matrices can be introduced into the network theory. As a result, by combining the network theory and a flexible method for the complex and sensitive parts, we can deal with most of microwave networks by the cascade scattering matrices. Secondly, all the ports are sequentially numbered, and then the interconnection relationships can be uniquely represented by a set of vectors. For instance, the vector of

The calculation procedure has its merits in two aspects: (1) the complicated operations are performed on the computer, which sometimes is impossible by manual work, especially for the networks with bigger sizes; (2) the calculation procedure is more time-saving compared with the conventional simulation methods, which can be embodied in the optimization algorithms to seek the optimum parameters.

Magic-T is a kind of microwave equipment with four ports and allows incident signals to be combined or subtracted with well-defined relative phases. Generally, the typical

The explosive view of the magic-T.

The circuit diagrams of the magic-T. (a) Equivalent circuit. (b) Decomposition circuit.

The proposed techniques are adopted and the subnetworks are connected together. Numerical results indicate that the remaining ports will be the port-E (numbered by 63), port-H (numbered by 1), port-1 (numbered by 4), and port-2 (numbered by 7). In addition, the time-cost is only 0.1 seconds, which makes it possible to have the calculation procedure embodied in the hybrid genetic algorithm to find optimum parameters satisfying the wideband demands [

The optimized scattering parameters of the magic-T.

At 1.5 GHz, the optimized characteristic parameters are transformed into the physical dimensions, and the corresponding prototype is fabricated on the two-layer substrates as shown in Figure

Photograph of the improved magic-T. (a) Top view. (b) Bottom view.

The results of the magic-T. (a) Return loss at port-E and port-H. (b) Return loss at port-1 and port-2. (c) Insertion loss at port-E-1 and port-E-2. (d) Insertion loss at port-H-1 and port-H-2. (e) Amplitude difference. (f) Phase difference. (g) Isolation at ports E-H and 1-2.

Waveguide slotted antennas are an important class of microwave antenna with numerous applications in radar and communication systems. Elliott derived three famous formulas, which are still widely adopted nowadays [

There are similarities and differences between our work and [

Two parallel dipoles.

In the following, two typical linear waveguide slotted arrays (resonant and traveling wave arrays) are presented to illuminate the principle of the proposed techniques. They are both operated at 10 GHz with 12 slots and based on the standard rectangular waveguide of WR-90, which has an inside dimension of 22.86 mm × 10.16 mm and a thickness of 1.27 mm. The longitudinal round-head slots are located in the broad wall with a fixed width of 1.6 mm. Once the offsets and lengths of the radiating slots are determined, the impedance matrix of the slotted array will be obtained in terms of the above-mentioned method. And then, the excitation distribution of the radiating slots can be acquired by connecting the scattering matrices of the feeding network and slotted array together. On the contrary, the offsets and lengths of the radiating slots can be synthesized with a desired excitation distribution, and they will be calculated according to the following steps.

Initialize the offset and length of the last radiating slot.

Find the offsets and lengths of the other radiating slots by minimizing the approximation errors of the calculated excitation distribution with respect to the desired one through optimization algorithms (e.g., genetic algorithm and hill-climbing algorithm).

Check the calculated scattering parameters at the remaining ports. If the demands are not met, modify the offset and length of the last radiating slot and repeat Steps

Stop.

The schemes of the resonant array (spaced at

Parameters of the resonant array.

Element | Offset (mm) | Length (mm) |
---|---|---|

1 | 0.8701 | 14.4029 |

2 | 1.2072 | 14.4239 |

3 | 1.8556 | 14.4810 |

4 | 2.5196 | 14.5580 |

5 | 3.0715 | 14.6303 |

6 | 3.3885 | 14.6727 |

7 | 3.3837 | 14.6721 |

8 | 3.0689 | 14.6285 |

9 | 2.5131 | 14.5564 |

10 | 1.8531 | 14.4807 |

11 | 1.2068 | 14.4241 |

12 | 0.8700 | 14.4029 |

The resonant array. (a) Configuration. (b) Equivalent circuit.

Radiation patterns of the resonant array.

The prototype is fabricated and connected with a coaxial-to-waveguide transducer, as shown in Figure

Photograph of the resonant waveguide slotted array.

Measured results of the resonant waveguide slotted array. (a) Reflection coefficients. (b) Radiation pattern.

The schemes of the traveling wave array (spaced at

Parameters of the traveling wave array.

Element | Offset (mm) | Length (mm) |
---|---|---|

1 | 0.9087 | 14.4148 |

2 | 1.1343 | 14.4255 |

3 | 1.5303 | 14.4579 |

4 | 2.0086 | 14.5151 |

5 | 2.4818 | 14.5847 |

6 | 2.8906 | 14.6473 |

7 | 3.1629 | 14.6845 |

8 | 3.1962 | 14.6832 |

9 | 2.9150 | 14.6402 |

10 | 2.3943 | 14.5678 |

11 | 1.8442 | 14.4934 |

12 | 1.5000 | 14.4473 |

The traveling wave array. (a) Configuration. (b) Equivalent circuit.

Radiation patterns of the traveling wave array.

Without losing its generality, a two-element microstrip antenna array (spaced at

The configuration of the two-element array.

Unlike the waveguide slotted arrays, the two-element array has a symmetrical configuration, and the excitation coefficients of antenna elements will be always equal. As a result, only the reflection coefficients are considered to decrease the mismatch at the input port. The equivalent circuit of the feeding network is constructed as shown in Figure

Equivalent circuit of the feeding network.

Photograph of the two-element array.

Reflection coefficients of the two-element array.

We present an approach to resolve the complex interconnections of two arbitrary microwave networks by introducing the ILSRNs. It greatly facilitates the operations of generalized microwave cascade network and is suitable for computer-aided procedures. The algorithm of the generalized cascade scattering matrix is expanded to the field of antenna arrays. The excitation coefficients of antenna arrays are calculated according to the incident waves, which are obtained by connecting the scattering matrices of the feeding network and antenna arrays through the proposed techniques. Quantitative analysis of the excitation coefficients is available for antenna arrays with consideration of the coupling effects. As a result, the characteristics of the microwave circuits and antenna arrays can be accurately predicted, and advanced modifications can be adopted to rectify the undesired characteristics. The proposed techniques are verified by the applications on the microwave circuits and antenna arrays. And they have a promising prospect for many similar applications.

The authors declare that there is no conflict of interests regarding the publication of this paper.