I. Introduction
With the rapid development of computer technology, the Finite Difference Time Domain (FDTD) method has become a widely used tool for analyzing microwave circuits. Initially applied to passive circuits, it has now extended to active, linear, and nonlinear systems. Furthermore, FDTD has been successfully implemented in both TEM (Transverse Electromagnetic) and dispersive systems. However, when dealing with complex geometries or large electrical sizes, FDTD can require excessive memory and computation time, sometimes making it impractical due to precision limitations. For example, in simulating waveguide filters with multiple diaphragms, the grid size must be very fine to capture the detailed geometry, leading to a massive number of grid points. Between each pair of diaphragms, there is usually a uniform transmission line, which does not need such a fine grid. To address this issue, non-uniform FDTD grids have been introduced. However, these grids can still cause convergence issues and may not significantly reduce computation time. The Diakoptics approach offers an alternative by decomposing the circuit into sub-circuits, allowing different meshing strategies for each part. This makes it easier to control convergence and improves computational efficiency.
II. Concept of Diakoptics
Diakoptics is a technique that divides a complex circuit into simpler sub-circuits, computes their individual characteristics, and then couples them through connection conditions. In linear circuits, the characteristics of each sub-circuit are represented by impulse response functions, and coupling is achieved through serial or parallel algorithms. The serial algorithm connects sub-circuits sequentially from one end to the other, treating each as a load for the previous one. While simple to implement, it suffers from inefficiency when adjustments are needed, as the entire sequence must be recalculated. The parallel algorithm, on the other hand, allows simultaneous computation of adjacent sub-circuits, making it more efficient and flexible, especially when sub-circuits are frequently modified.
When analyzing microwave circuits, if the circuit can be modeled as a linear network, its Green’s function corresponds to the impulse response function in circuit theory. From an electromagnetic perspective, the time-domain Green’s function describes the field at a point in space and time due to a unit impulse at another point. When two sub-circuits are connected, they interact through a complex coupling relationship, visualized by the reflection and transmission of electromagnetic waves across discontinuities. Understanding how to apply the Diakoptics algorithm to microwave circuits requires a clear mathematical framework, which is explored next.
III. Mathematical Description of the Diakoptics Algorithm
The Diakoptics algorithm can be mathematically described using the concept of two-port networks. Suppose the impulse responses of reflected and transmitted waves from two sub-circuits are gr1(t), gr2(t), gt1(t), gt2(t) and hr1(t), hr2(t), ht1(t), ht2(t). Using the serial algorithm, the overall impulse response seen from the input reference plane is calculated by convolving the individual responses. Similarly, the parallel algorithm calculates the responses from both input and output ports. These calculations involve convolution operations, where * denotes time-domain convolution. The formulas illustrate how the coupling between sub-circuits affects the overall system response. When multi-port sub-circuits are involved, the equations generalize to matrix form, with each impulse response becoming a sub-matrix. This approach ensures accurate representation of complex interactions between sub-circuits.
IV. Implementation of Diakoptics Algorithm in Microwave Circuit Analysis
To apply Diakoptics to microwave circuits, an equivalent time-domain network model is first established. This involves representing the fields at reference planes using orthogonal basis functions, effectively transforming the circuit into a multi-mode network. The choice of basis functions depends on the circuit geometry, with rectangular pulses often used for complex structures. These functions allow the circuit to be treated as a multi-port network, enabling the application of Diakoptics. The impact response function of the system is obtained using Gaussian pulse excitation and windowed Fourier transform techniques, ensuring accuracy while reducing computational burden. Each sub-circuit is excited with the same spectral characteristics to maintain consistency in the analysis.
V. Application Examples and Discussion
A practical example of the Diakoptics-FDTD method is demonstrated through the analysis of a rectangular waveguide bandpass filter. The filter is divided into sub-circuits, each analyzed using FDTD to obtain field distributions at connecting reference planes. A Gaussian pulse with a modulation frequency centered within the operating band is used as the excitation source. The results show excellent agreement with traditional FDTD simulations, validating the effectiveness of the Diakoptics approach. The method proves to be both efficient and accurate, making it a valuable tool for analyzing complex microwave circuits.
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