Design and Implementation of a Parallel Algorithm for Computing Microwave Circuits
Time:2023-04-07
Views:1090
1、 Introduction
With the progress of computer technology, the finite difference Time Domain (FDTD) is a numerical calculation technique for solving electromagnetic problems, which was proposed by K.S.Yee in 1966. His basic idea is to use the differential equation of the field components of the time-domain Maxwell equation to replace the differential equation with difference and iterate each field component. However, this method will significantly increase the computational grid as the frequency increases, making it difficult for the performance of a PC to meet the needs, and relying solely on the improvement of computer performance is not practical. For example, when analyzing waveguide membrane filters, in order to correctly simulate the geometric structure of all the membranes, the grid size of the FDTD grid is selected very small, resulting in a very large number of grids describing the entire waveguide filter. Due to the uniform waveguide transmission line between each two membranes, it is obviously unnecessary to use the same grid as the membrane. People have used non-uniform FDTD grids to solve this problem. When the size of the grids varies greatly, not only is convergence difficult to control, but it still cannot guarantee the saving of calculation time. Applying the Diakoptics idea to the full wave analysis of microwave circuits, the circuit is divided into several independent parts, and different grids are used according to the specific structure of each part to independently perform full wave time domain analysis on each part. As the grids of each part are uniform, it is easy to ensure the convergence of the algorithm.
2、 The concept of Diakoptics
The concept of Diakotics comes from network theory. Its definition is: decomposing a network into several sub networks, solving the impact response of each sub network separately, and finally obtaining the total response of the network from the impact response of each sub network through certain connection conditions. The connection conditions can be divided into serial connection and parallel connection according to their different forms. Serial connection is a one-way connection from one end of the network to the other in a certain order, as shown in Figure 1. Its advantage is simplicity, but the biggest problem is that when the impact response of one of the subnetworks changes, it will have an impact on the subsequent networks. Parallel connections can overcome this drawback. Parallel connections can be made between any two adjacent subnetworks, and several parallel connections can be made independently at the same time. Parallel time-domain Diakoptics assumes that the subnetwork is an M+N port network, where M ports are connected to the previous subnetwork and N ports are connected to the subsequent subnetwork. The discrete Green‘s function of the sub network is the excitation at port t=0 for g (i, j, n ‘), i.e. j (j=1, M+N), and the impact response at port t=n‘ for i (i=1, M+N).
When studying microwave circuit problems, if microwave circuits can be equivalent to a linear network, it can be assumed that the Green‘s function describing the characteristics of microwave circuits can correspond to the impulse response function in circuit theory. From the perspective of electromagnetic field theory, the time-domain Green‘s function g (r, t; r0, t0) is the field of the unit impulse signal applied at time t0 of the point source located at point r0 at observation points r and t, and satisfies the equation
When two microwave sub circuits are connected, there is a complex coupling relationship on their connection reference surface, which can be vividly described by the reflection and transmission of electromagnetic waves in a medium with two discontinuous interfaces, as shown in Figure 1. So how can the Diakoptics algorithm be applied to the analysis of microwave circuit characteristics? Before introducing this point, this article first briefly introduces the mathematical description of the Diakoptics algorithm.
The reflection and transmission phenomena in Figure 1 medium can be used to vividly describe the coupling relationship between two microwave sub circuits
3、 Mathematical Description of Diakoptics Algorithm
Provide a mathematical description of the Diakoptics algorithm through the serial and parallel connections of two two-port networks. Figure 2 assumes that the impulse response functions of the reflected and transmitted waves of the two sub circuits are: gr1 (t), gr2 (t), gt1 (t), gt2 (t) and hr1 (t), hr2 (t), ht1 (t), ht2 (t). The superscript "r" represents the reflected wave, "t" represents the transmitted wave, subscript 1 represents the excitation from the input reference plane to the circuit, and subscript 2 represents the excitation from the output reference plane to the circuit. Let f be the impulse response function of the circuit after connecting two sub circuits. Using a serial algorithm, the impact response observed from the input reference surface of the f network is:
fr1（t）=gr1（t）+gt2（t）*hr1（t）*gt1（t）+gt2（t）*hr1（t）
*gr2（t）*hr1（t）*gt1（t）+…+gt2（t）*（hr1（t）
*gr2（t））n*hr1（t）*gt1（t）+…； （2）
Using parallel algorithms, the impact response functions fr1 (t), ft2 (t) viewed from the input port of the f circuit and fr2 (t), ft1 (t) viewed from the output port of the f circuit are:
fr1（t）=gr1（t）+gt2（t）*hr1（t）*gt1（t）+gt2（t）*hr1（t）
*gr2（t）*hr1（t）*gt1（t）+…+gt2（t）*（hr1（t）
*gr2（t））n*hr1（t）*gt1（t）+…
ft2（t）=gt2（t）*hr2（t）+gt2（t）*hr1（t）*gr2（t）*ht2（t）+…
+gr2（t）*（hr1（t）*gr2（t））n*hr2（t）+… （3）
fr2（t）=hr2（t）+ht1（t）*gr2（t）*ht2（t）+ht1（t）*gr2（t）
*hr1（t）*gt2（t）*ht2（t）+…+ht1（t）*（gr2（t）
*hr1（t））n*gr2（t）*ht2（t）+…
ft1（t）=ht1（t）*gt1（t）+ht1（t）*gr2（t）*hr1（t）*gt1（t）+…
+ht1（t）*（gr2（t）*hr1（t））n*gr1（t）+…
Among them, * represents time-domain convolution, and the meaning of superscripts and subscripts remains unchanged.
Among them, * represents time-domain convolution, and the meaning of superscripts and subscripts remains unchanged.
Figure 2 illustrates the connection diagram of the two sub circuits of the Diakoptics algorithm
When multi port sub circuits are connected, the above algorithm is still valid, except that each bump function in the formula should be replaced with a corresponding sub matrix. For example, let the g network be an M+N port network with M input ports and N output ports, and the h network be an N+L port network with N input ports and L output ports (the number of ports adjacent to g and h should be the same). The reflection and transmission submatrixes at the input reference surface of the g network are:
gentle
The subscript in the equation represents the reference plane, where i ← j means: i is the reference plane where the response is located, and j is the reference plane where the excitation is located; The superscript represents the port, and m ← n means: n is the input port, and m is the output port. Similarly, the reflection and transmission sub matrices at the output reference surface of the g network are:
gentle
The corresponding sub matrices of the h network can be obtained using the same method. The impact response function [f] of the connected network is:
[fr1（t）]=[gr1（t）]+[gt2（t）]*[hr1（t）]*[gt1（t）]+[gt2（t）]
*[hr1（t）]*[gr2（t）]*[hr1（t）]*[gt1（t）]+…
[ft2（t）]=[gt2（t）]*[ht2（t）]+[gt2（t）]*[hr1（t）]*[gr2（t）]*[ht2（t）]+…
[fr2（t）]=[hr2（t）]+[ht1（t）]*[gr2（t）]*[ht2（t）]+[ht1（t）]
*[gr2（t）]*[hr1（t）]*[gr2（t）]*[ht2（t）]+…
[ft1（t）]=[ht1（t）]*[gt1（t）]+[ht1（t）]*[gr2（t）]*[hr1（t）]*[gt1（t）]+… （4）
Where [fr1 (t)], [ft1 (t)], [fr2 (t)], and [ft2 (t)] are M, respectively × M、L × M、L × L and M × L-order submatrix. The following is an example of how to calculate matrix convolution using [gt2 (t)] * [ht2 (t)], and the first element of [gt2 (t)] * [ht2 (t)] is used as an example to illustrate its physical meaning:
Where [fr1 (t)], [ft1 (t)], [fr2 (t)], and [ft2 (t)] are M, respectively × M、L × M、L × L and M × L-order submatrix. The following is an example of how to calculate matrix convolution using [gt2 (t)] * [ht2 (t)], and the first element of [gt2 (t)] * [ht2 (t)] is used as an example to illustrate its physical meaning:
G1 ← 11 ← 2 * h1 ← 11 ← 2: The input of the first port on the output reference plane of the h sub network is generated through the coupling of the first port on the gh connection plane at port 1 on the input reference plane of the g sub network; G1 ← 21 ← 2 * h2 ← 11 ← 2: The input of the first port on the output reference plane of the h sub network is generated through the coupling of the second port on the gh interface at port 1 on the input reference plane of the g sub network; G1 ← N1 ← 2 * hN ← 11 ← 2: The input of the first port on the output reference plane of the h sub network is coupled through the nth port of the gh interface, and the output generated by port 1 on the input reference plane of the g sub network; So the first element of [gt2 (t)] * [ht2 (t)] describes the coupling of the input on the first port of the output reference plane of the h network to the output of the first port of the input reference plane of the g network.
4、 Implementation of Diakoptics Algorithm in Microwave Circuit Analysis
Diakoptics originates from network theory. In order to apply it to the analysis of microwave circuits, it is first necessary to establish an equivalent circuit model suitable for using the Diakoptics method in microwave circuits.
1. Equivalent time-domain network model for microwave circuits
The basic method for establishing an equivalent time-domain network model of microwave circuits is to use basis function technology (also known as time-domain mode technology) to represent the field at the reference surface as a linear combination of selected orthogonal basis functions, to equivalent a microwave network to a multimode circuit, and then to equivalent a multimode circuit to a multi-port network.
The selected basis function satisfies the following conditions: it is only a function of spatial coordinates; Independent of time; Form a complete orthogonal set. And for a given microwave circuit, the selected basis function should be able to effectively describe the distribution of electromagnetic fields in the circuit. Assuming that the X-Y plane is the plane where the circuit cross-section is located, Z is the propagation direction, and the circuit is in Dirac- δ The electric field distribution at z=z0 under functional excitation is Ei (x, y, z0, t){ φ Mn (x, y)} is the family of basis functions, using φ Mm (x, y) can represent the field at t=t0, z=z0 in a microwave circuit as:
Among them, amn (z0, t0) is the coefficient, i.e. amplitude, of the (m, n) th basis function, so that the microwave circuit seen from the reference plane z=z0 can be equivalent to an equivalent time-domain multimode circuit based on the basis function. The functional form of the basis function can be either an orthogonal function suitable for general circuits or a special orthogonal function particularly suitable for certain types of circuits. Generally speaking, when the geometric structure of a circuit is complex and it is difficult to select special orthogonal functions as basis functions based on circuit characteristics, a rectangular pulse function can be selected (taking the values of grid nodes as the average value of the entire grid, so the pulse width is the width of one grid). However, due to the fact that the pulse function only describes local information of the system, in order to achieve sufficient accuracy, the number of expansion terms of the function is relatively large. When orthogonal functions can effectively express the global information of a circuit, they usually only require a few or a dozen items to achieve the required accuracy. For example, for a uniformly filled rectangular waveguide problem, if the basis function is selected as the {sin, cos} orthogonal function set based on the distribution characteristics of the field inside the waveguide, usually only 5 terms are needed to meet the requirements. By comparison, at least 60 pulses or 60 nodes are required to accurately describe the spatial field distribution on the cross-section of the waveguide system.
The orthogonality of the basis function enables each basis function to be regarded as an independent port, so the above equivalent time-domain multi-mode circuit based on the basis function can be further regarded as a multi port network.
2. Calculation of equivalent multi-port network characteristics
The spectrum of the bump function is infinite, so the FDTD algorithm cannot be directly used to solve the shock response function of the system. FDTD Diabopics uses Gaussian pulse modulated waves as excitation and obtains the impulse response function of a finite bandwidth microwave circuit using windowed Fourier transform technology. Among them, the modulation frequency of Gaussian pulse excitation is the center frequency of the circuit operating frequency band, and the pulse width and pulse time sampling interval depend on the frequency resolution and bandwidth. Although the excitation pulse has a limited bandwidth, the impact response function obtained by FDTD Diabetics includes the influence of windowing (referred to as the quasi impact response function at this time), as long as the following conditions are met: when using FDTD Diabetics to analyze the characteristics of the entire circuit, each sub circuit uses excitation pulses with the same spectral characteristics, taking into account the broadening effect of windowing on time-domain pulses, The frequency domain response of the resulting impact response function is sufficiently accurate.
5、 Application Examples and Discussion of FDTD Diabopics
The finite difference Time Domain (FDTD for short) method differentiates Maxwell equations in time and space. Using the Leap frog algorithm - alternating calculations of electric and magnetic fields in the spatial domain, simulating changes in the electromagnetic field through updates in the temporal domain, achieving numerical calculations. When using this method to analyze problems, it is necessary to consider various issues such as the geometric parameters, material parameters, computational accuracy, computational complexity, and computational stability of the research object. Its advantage is that it can directly simulate the distribution of the field, with high accuracy, and is currently one of the commonly used numerical simulation methods.
This article takes the characteristic analysis of a waveguide bandpass filter as an example to illustrate the application of this algorithm. Figure 3 shows a rectangular waveguide bandpass filter (WR34) with 5 diaphragms. According to this method, the filter is first divided into 5 parts and calculated using FDTD to obtain the field distribution at all connected reference surfaces (as shown by the dashed line in the figure). In FDTD calculation, the modulation function of Gaussian pulse is: f (t)=AmaxA (x, y) exp [- ((t - t0)/T) 2]. sin (2 π f0t), where the modulation frequency f0 is the center frequency of WR34-TE10 mode single-mode operating band; A (x, y) represents the spatial distribution of excitation amplitude, and the Diakoptics algorithm needs to calculate the response of all possible basis functions under a single excitation. Therefore, A (x, y) is selected as each basis function in sequence. The amplitude of the activation function Amax depends on its attenuation along the propagation direction and the calculation accuracy. The basic principle is that the field at the discontinuity and observation surface still has enough amplitude. The value of T should ensure that the energy at the cutoff frequency point on the excitation pulse spectrum is sufficiently small. In this example, the single mode operating frequency band of WR34 is: fTE10=17.369GHz, fTE20=34.738GHz, f0=26.0535GHz, T=200 (ps), t0=3T, Amax=10, and the basis function is φ N (x)=sin, and the corresponding coefficient an (z0, t) is shown in Figure 4 (due to the length of the article, only one result is given). Figure 5 shows the frequency characteristics of the filter obtained using the method presented in this paper, and it can be seen that the results are in good agreement with the FDTD results.
Figure 3 Schematic diagram of five diaphragm WR34 waveguide bandpass filter
The coefficients of the reflected wave basis function of the first subcircuit in Figure 3 obtained by the method in this article are shown in Figure 4
The frequency characteristics of the WR34 waveguide filter shown in Figure 5 and Figure 3
6、 Conclusion
This article introduces a new method for analyzing complex microwave circuits: the FDTD Diakoptics method. It can overcome the drawbacks of traditional FDTD methods that require large memory and long calculation time, and fully leverage the advantages of FDTD in making it easy to study complex geometric structure circuits. Through several analysis and design examples by the author, it has been proven that this method is not only flexible but also has high accuracy, making it a relatively effective simulation and analysis method for microwave circuits.
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