I. Introduction
With the rapid development of computer technology, the Finite Difference Time Domain (FDTD) method has become a powerful tool for analyzing microwave circuits. Initially used for passive and linear circuits, it has now expanded to active and nonlinear systems, as well as from TEM (Transverse Electromagnetic) systems to dispersive ones. Despite its widespread application, FDTD faces challenges when dealing with complex geometries or large electrical sizes. In such cases, the computational resources required—both in terms of memory and time—can be extremely high, sometimes making it impossible to achieve the desired accuracy. For example, in simulating a waveguide bandpass filter, the need to accurately model all diaphragms leads to a very fine grid, resulting in an excessive number of grid points. Since there are uniform sections between the diaphragms, using the same fine grid throughout is inefficient. To address this, non-uniform FDTD grids have been introduced, but they still face issues with convergence and computational efficiency. This is where the Diakoptics approach comes into play, offering a more efficient way to analyze microwave circuits by dividing them into sub-circuits, each with its own mesh, and coupling them through appropriate boundary conditions.
II. The Concept of Diakoptics
Diakoptics is a technique that decomposes a complex circuit into simpler sub-circuits. Each sub-circuit is analyzed independently, and their interactions are determined based on connection conditions. In linear circuit theory, the characteristics of a sub-circuit are represented by impulse response functions, and the coupling between sub-circuits can be handled through either serial or parallel algorithms. The serial algorithm processes the sub-circuits sequentially, which makes it straightforward to implement but limits flexibility. If a sub-circuit needs adjustment, the entire sequence must be recalculated, leading to inefficiency. On the other hand, the parallel algorithm allows simultaneous processing of multiple sub-circuits, enabling independent adjustments without affecting the rest of the system. This makes it more efficient, especially for repeated modifications.
From an electromagnetic perspective, the time-domain Green’s function describes the field response at a given point due to a unit impulse applied at another point. When two sub-circuits are connected, the interaction is complex and can be visualized using reflection and transmission phenomena in a medium with discontinuities. This graphical representation helps understand how the Diakoptics algorithm works in practice.
III. Mathematical Description of the Diakoptics Algorithm
To illustrate the Diakoptics algorithm, consider two two-port networks connected in series. The impulse response functions for reflected and transmitted waves are denoted as $ g_r1(t), g_t1(t), h_r1(t), h_t1(t) $, etc., where the superscript denotes whether it's a reflected or transmitted wave, and the subscript indicates the excitation direction. Using the serial algorithm, the overall impulse response seen from the input reference plane is:
$$
F_{r1}(t) = g_{r1}(t) + g_{t2}(t) * h_{r1}(t) * g_{t1}(t) + \dots
$$
Similarly, the parallel algorithm calculates responses from both input and output ports, allowing for more flexible and efficient computation. When dealing with multi-port sub-circuits, these expressions are extended to matrix forms, where each impulse function becomes a sub-matrix. This approach enables the analysis of complex networks while maintaining accuracy and efficiency.
IV. Implementation of Diakoptics Algorithm in Microwave Circuit Analysis
To apply Diakoptics to microwave circuits, an equivalent time-domain network model must first be established. This involves representing the electromagnetic fields at the reference planes using orthogonal basis functions. These functions allow the circuit to be modeled as a multi-mode network, which can then be treated as a multi-port network. The choice of basis functions affects the accuracy and efficiency of the simulation. While rectangular pulse functions can be used, they require many terms for high precision. Orthogonal functions, such as sine and cosine, are often more efficient, especially for regular structures like waveguides.
The impact response function of the circuit is obtained using a Gaussian pulse modulated by the center frequency of the operating band. This pulse is then processed via windowed Fourier transform to extract the frequency response. The key is ensuring that the spectral characteristics of the excitation pulses match across all sub-circuits to maintain accuracy during the Diakoptics coupling process.
V. Application Examples and Discussion
As an example, a five-diaphragm rectangular waveguide bandpass filter (WR34) was analyzed using the FDTD-Diakoptics method. The filter was divided into five parts, each simulated using FDTD. The field distribution at the interface planes was extracted and used in the Diakoptics algorithm. A Gaussian pulse with a center frequency of 26.0535 GHz was used as the excitation, and the results showed excellent agreement with traditional FDTD simulations. This demonstrates the effectiveness of the Diakoptics approach in reducing computational load while maintaining accuracy.
Conclusion
This paper presents the FDTD-Diakoptics method as a novel approach for analyzing complex microwave circuits. It overcomes the limitations of traditional FDTD methods, such as high memory usage and long computation times, by efficiently breaking down the problem into manageable sub-circuits. The method has been validated through practical examples, showing both flexibility and high accuracy. It offers a promising alternative for the simulation and analysis of microwave circuits, particularly those with complex geometries.
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