Patent Publication Number: US-6219629-B1

Title: Apparatus for combined simulation of electromagnetic wave analysis and circuit analysis, and computer-readable medium containing simulation program therefor

Description:
BACKGROUND OF THE INVENTION 
     1. Field of the Invention 
     The present invention relates to simulator apparatus for a combined simulation of electromagnetic wave analysis and circuit analysis, as well as to computer-readable media containing a simulation program designed therefor. More particularly, the present invention relates to a simulator apparatus for a combined simulation of electromagnetic wave analysis and circuit analysis, in which the time step size of the circuit analysis may vary in accordance with the circumstances, and a computer-readable medium containing a simulation program that performs such a simulation. 
     2. Description of the Related Art 
     In today&#39;s computational electromagnetics, the Finite-Difference Time-Domain (FD-TD) method is used as a practical technique to analyze the transitional behavior of electromagnetic waves by using a computer for numerical simulation. The FD-TD, which solves Maxwell&#39;s equations in the time and spatial domains with difference methods, is actually implemented to a variety of cases because of its wide scope of applications. 
     In the field of circuit analysis, on the other hand, there are different computer simulation tools which numerically analyze the transient behavior of an electronic circuit. Generally, they calculate the circuit&#39;s node voltages and branch currents on the basis of Kirchhoff&#39;s laws. The circuit analysis tools are further equipped with programs for modeling the characteristic equations of semiconductor devices and other circuit elements having nonlinear voltage-current characteristics, and to solve those nonlinear equations, they employ some numerical techniques such as Newton-Raphson methods. 
     Researchers have also proposed another numerical simulation method that performs both an electromagnetic wave analysis and a circuit analysis in a combined manner. In this proposed method, the simulation steps proceed along the time axis, keeping a close linkage between the electromagnetic field physics obtained by the electromagnetic wave analysis and the voltage-current physics obtained by the circuit analysis. With capabilities of both an electromagnetic wave analysis and a circuit analysis, this hybrid numerical simulation technique provides an advantage in that it solves the characteristics of circuit elements and the electromagnetic field behavior in their surrounding space in a unified manner. It is particularly useful for the analysis of high-frequency signals propagating in a circuit. 
     When using FD-TD techniques to numerically simulate the transient behavior of an electromagnetic wave, it is desirable that the time step size (Δt em ) be as large as possible within a range determined by the following condition for stability.                Δ                   t   em       ≤     1     v              (     1     Δ                   x   min         )     2     +       (     1     Δ                   y   min         )     2     +       (     1     Δ                   z   min         )     2                     (   1   )                         
     Here, v is the speed of electromagnetic waves; Δx min , Δy min , and Δz min  represent the minimum values of spatial discretization intervals defined in the x-, y-, and z-directions, respectively. The above constraint about the time step size derives from the following reasons: (a) the numerical solution would diverge if Δt em  was larger than the right side of the condition (1), and (b) the overall computation time of numerical simulation would increase if an unnecessarily small value was set to Δt em . Usually, the time step size Δt em  is determined within the following range.                0.90   ×     1     v              (     1     Δ                   x   min         )     2     +       (     1     Δ                   y   min         )     2     +       (     1     Δ                   z   min         )     2               ≤     Δ                   t   em       ≤     0.99   ×     1     v              (     1     Δ                   x   min         )     2     +       (     1     Δ                   y   min         )     2     +       (     1     Δ                   z   min         )     2                       (   2   )                         
     Once it is determined, the value of Δt em  is used without changes, throughout the simulation period. 
     As opposed to this, the time step size (Δt cs ) in the transient circuit simulation normally varies in accordance with the circumstances. For example, when numerically solving nonlinear circuit equations, the time step size should be reduced if the solution for potential differences does not converge. To allow the electromagnetic wave analysis and circuit analysis to exchange data with each other, it is therefore necessary to synchronize their simulation clocks in a reliable and efficient manner, so that they will transfer data to each other at the definite time points. 
     Now, the next paragraphs will discuss the timing of data transfer operations between the electromagnetic wave analysis and the circuit analysis. 
     FIG. 9 is a diagram which shows the timing of data transfer in a conventional simulator. The upper half of FIG. 9 shows a process of electromagnetic wave analysis, while the lower half shows a process of circuit analysis. In the equations appearing in the following section, the superscripts (e.g., n, n+½, . . . ) affixed to the variables indicate a specific time step k, or the k-th time step within a simulation process, meaning that a time of (k×Δt em ) has elapsed since the beginning of the process. 
     (S31) The simulator calculates magnetic field values at t em =(n+½)Δt em , where n is a natural number, and t em  is a simulation time in the electromagnetic wave analysis. Here, the magnetic field H is calculated by 
     
       
         H n+½ =H n−½ −(Δt em /μ) rot E n   (3) 
       
     
     where μ is permeability, and “rot” represents the rotation (or curl) of a vector function. 
     (S32) The simulator then calculates current source data I on the basis of the magnetic field H obtained in step S31 as part of the electromagnetic wave analysis, and transfers the resultant value to the circuit analysis. This current source data I is obtained by the following. 
     
       
         I n+½ =ΣH n+½   (4) 
       
     
     (S33) Based on the current source data I supplied from the electromagnetic wave analysis, the simulator executes a circuit analysis at t cs =(n+{fraction ( 1 / 2 )})Δt em , where t cs  is a simulation time in the circuit analysis. Here, the behavior of the circuit is given by the following equation. 
     
       
         V n+½ =Z −1 I n+½   (5) 
       
     
     (S34) Based on the voltage V obtained through the circuit analysis, the simulator calculates the electric field E in a specific part (i.e., cell) of the computational domain in which the circuit is located. The electric field E, given by the following, is then passed to the electromagnetic wave analysis. 
      E n+1 =V n+½ /d,  (6) 
     where d is the dimension of a cell in which the circuit under analysis is located. 
     (S35) The simulator calculates the electric field at t em =(n+1)Δt em , according to the following equation. 
     
       
         E n+1 =E n +(Δt em /ε) rot H n+½ ,  (7) 
       
     
     where ε is a dielectric constant. The electric field data supplied in step S34 is then adapted to the electric field calculated in step S35, thereby yielding the entire electric field at t em =(n+1)Δt em . 
     The combined simulation for electromagnetic wave analysis and circuit analysis is accomplished in this way, by allowing the two processes to exchange the current source data and electric field data. The conventional simulation process, however, is likely to produce an unstable solution even if the time step size Δt em  satisfies the stability condition (2). This problem seems to be caused by a time lag of Δt em /2 between the magnetic field calculation (S31) and the electric field calculation (S35) as part the electromagnetic wave analysis. The conventional simulation process, however, does not take that time lag into consideration, unfortunately. That is, the simulator calculates the electric field at a specific time point (n+1)Δt em  on the basis of the voltage at a past time point (n+½)Δt em , which makes the solution unstable. It may be possible to avoid such instability of solutions by reducing Δt em  to about half the standard step size that is suggested by the stability condition (2). This choice, however, leads to two times longer computation time. Further, the value of t cs  is varied in accordance with the progress of the circuit analysis, and therefore, ten is not always equal to t cs . In a conventional simulation process, the analysis often fails due to the inconsistency in t cs  and t em . 
     SUMMARY OF THE INVENTION 
     Taking the above into consideration, an object of the present invention is to provide a simulator apparatus for a combined simulation of electromagnetic wave analysis and circuit analysis, which is configured to produce stable solutions with a smaller amount of processing loads. 
     Another object of the present invention is to provide a computer-readable medium containing a simulation program for a combined simulation of electromagnetic wave analysis and circuit analysis, which produces stable solutions with a smaller amount of processing loads. 
     To accomplish the above objects, according to the present invention, there is provided a simulator apparatus for performing a combined simulation in which a transient electromagnetic wave analysis for a prescribed space is linked in the time domain with a circuit analysis that analyzes a circuit placed in the prescribed space. This simulation apparatus comprises: an electromagnetic wave analyzer, a circuit analyzer, a current source data transfer unit, and an electric field data transfer unit. The electromagnetic wave analyzer performs an electromagnetic wave analysis by alternately executing a magnetic field calculation and an electric field calculation. The magnetic field calculation is based on a first simulation time that is advanced by a first time step size each time the magnetic field calculation is executed. The electric field calculation is based on a second simulation time that has a predetermined time offset with respect to the first simulation time and is advanced by the first time step size each time the electric field calculation is executed. In the electric field calculation, the electromagnetic wave analyzer waits for electric field data for a local space where the circuit is located, and upon receipt of the electric field data, it computes an entire electric field in the prescribed space by adapting the received electric field data. The circuit analyzer operates at a third simulation time that is advanced by a second time step size determined in accordance with circumstances. The circuit analyzer calculates a voltage developed across the circuit by solving a given circuit equation at the third simulation time. In this calculation of the voltage, the circuit analyzer waits for current source data for the circuit when a time difference between the first simulation time and the third simulation time falls below a first predetermined time difference threshold, and upon receipt of the current source data, it resumes the circuit analysis with the received current source data. The current source data transferring unit is activated when a time difference between the first simulation time and the third simulation time falls below the first predetermined time difference threshold. It calculates the current source data for the circuit on the basis of magnetic field obtained by the electromagnetic wave analyzer and transferring the calculated current source data to the circuit analyzer. The electric field data transfer unit is activated when the time difference between the second simulation time and the third simulation time falls below the second predetermined time difference threshold. It calculates the electric field data in the circuit&#39;s local space on the basis of the voltage obtained by the circuit analyzer, and transfers the calculated electric field data to the electromagnetic wave analyzer. 
     To accomplish the above objects, according to the present invention, there is provided another simulator apparatus for performing a first simulation process in combination with a second simulation process, allowing the first and second simulation processes to transfer simulation results to each other. Here, the first simulation process is based on a first simulation time that is advanced by a fixed time step size, while the second simulation process is based on a second simulation time that is advanced by a variable time step size. The simulation apparatus comprises: a first data transfer unit, an analyzer unit, and a second data transfer unit. Being activated when a time difference between the first simulation time and the second simulation time falls below a predetermined time difference threshold, the first data transfer unit transfers a simulation result of the first simulation process to the second simulation process. The analyzer unit executes the second simulation process on the basis of the transferred simulation result of the first simulation process, changing the variable time step size as required. The second data transfer unit, being activated when a time difference between the second simulation time and the first simulation time having been advanced by the fixed time step size falls below another predetermined time difference threshold, transfers a simulation result of the second simulation process to the first simulation process. 
     The above and other objects, features and advantages of the present invention will become apparent from the following description when taken in conjunction with the accompanying drawings which illustrate a preferred embodiment of the present invention by way of example. 
    
    
     BRIEF DESCRIPTION OF THE DRAWINGS 
     FIG. 1 is a conceptual view of the present invention; 
     FIG. 2 is a diagram which shows a procedure of data transfer between an electromagnetic wave analysis and a circuit analysis; 
     FIG. 3 is a flowchart which shows a procedure of simulation according to the present invention; 
     FIG. 4 is a side view of a microstrip transmission line; 
     FIG. 5 is a transverse cross section of the microstrip transmission line; 
     FIG. 6 is a diagram which shows the result of a conventional simulation with a time step size Δt em  of 0.315 ps; 
     FIG. 7 is a diagram which shows the result of a conventional simulation with a time step size Δt em  of 0.630 ps; 
     FIG. 8 is a diagram which shows the result of a simulation performed by the propose simulator, with a time step size Δt em  of 0.630 ps; and 
     FIG. 9 is a diagram which shows the timing of data transfer operations in a conventional simulator. 
    
    
     DESCRIPTION OF THE PREFERRED EMBODIMENTS 
     An embodiment of the present invention will be described below with reference to the accompanying drawings. 
     FIG. 1 shows the concept of the present invention. The present invention proposes a simulator which performs an electromagnetic wave analysis in a predetermined space, in conjunction with a circuit analysis for a specific circuit located in the space. The proposed simulator comprises an electromagnetic wave analyzer  1 , a circuit analyzer  2 , a current source data transfer unit  3 , and an electric field data transfer unit  4 . Note that the term “circuit” denotes various electronics, ranging from a simple discrete device to an interconnected group of electrical and electronic components. It should also be noted that, in the following section, the term “simulation time” is used as a variable to indicate a virtual time point in a simulation process, but it has nothing to do with the time of day in real life. 
     The electromagnetic wave analyzer  1  divides a given three-dimensional space into a number of cells, and analyzes the transient behavior of electromagnetic waves by using FD-TD techniques. More specifically, the electromagnetic wave analyzer  1  calculates the electric field E on the edges of each cell and its surrounding magnetic field H to investigate their transient behavior. To represent specific time points in this process, the electromagnetic wave analyzer  1  has two time variables t em 01 (first simulation time) and t em 02 (second simulation time). During a simulation process, the first simulation time t em 01 advances by a fixed time step size Δt em  (first time step size). The second simulation time Δt em /2 has a fixed time offset, or lag, with respect to the first simulation time t em 01. 
     The electromagnetic wave analyzer  1  is configured to alternately calculate the magnetic field at t em 01 and the electric field at t em 02, thereby performing a transient analysis of electromagnetic waves. Concerning the electric field calculation, the electromagnetic wave analyzer  1  has to wait for electric field data for the circuit&#39;s “local space” to be supplied from the circuit analyzer  2 , where the term “local space” refers to a specific cell or cells in which the circuit of interest is placed. When this local electric field data is ready, the electromagnetic wave analyzer  1  calculates the electric field in the entire space so that it will reflect the supplied electric field data. In the present embodiment, the supplied local electric field data is introduced in the following way. First, the electromagnetic wave analyzer  1  obtains the electric field for the space other than the circuit&#39;s local space by solving Maxwell&#39;s equations in the time and spatial domains. Then it adapts the supplied local electric field data to the electric field obtained above. 
     The circuit analyzer  2  performs a circuit analysis on the basis of an equivalent circuit  2   a  that represents the circuit of interest in a current source approach. SPICE (Simulation Program with Integrated Circuit Emphasis), for example, is an appropriate tool for the circuit analysis. This analysis takes a current source approach, where a voltage V to be developed across the circuit of interest is calculated from a given current source. The equivalent circuit  2   a  contains a current source  2   ab  connected to the terminals of the circuit model  2   aa  and a capacitor C connected in parallel with the current source  2   ab . Here, another variable t cs  (third simulation time) is defined to indicate a specific time point at which the circuit analyzer  2  solves the given circuit equations. This simulation time t cs  advances in a stepwise manner, being incremented by a time step size Δt cs  (second time step size). Note that the time step size Δt cs  may vary as the circumstances require. The circuit analyzer  2  monitors the difference between t em 01 and t cs , and if the difference falls below a predetermined time difference threshold λ1 (first time difference threshold), it starts waiting for the next current source data to be supplied. When this data is ready, the circuit analyzer  2  updates the equivalent circuit with the supplied current source data to continue the circuit analysis. 
     When the simulation time t cs  is advanced by the time step size Δt cs , the circuit analyzer  2  further examines whether the resultant t cs  has exceeded t em 01, and whether their difference is not less than the predetermined time difference threshold λ1. If both criteria are met, the circuit analyzer  2  turns back t cs  by canceling the increment Δt cs  added above. Then it advances t cs  again with a new time step size Δt cs  that is now given a smaller value. 
     Similarly, the circuit analyzer  2  examines the value of t cs  after advancing it by Δt cs . If the resultant simulation time t cs  has exceeded t em 02, and if their difference is not less than a predetermined time difference threshold λ2 (second time difference threshold), the circuit analyzer  2  turns back t cs  by canceling the increment. Then it reduces the time step size Δt cs . and advances t cs  again by the reduced time step size Δt cs . Here, the predetermined time difference thresholds λ1 and λ2 have sufficiently small values, which is about one hundredth of Δt em , for instance. 
     The current source data transfer unit  3  compares t em 01 with t cs , and if their difference falls below the time difference threshold λ1, it starts to calculate current source data for use in the equivalent circuit, based on the magnetic field obtained by the electromagnetic wave analyzer  1 . The resultant current source data is then transferred to the circuit analyzer  2 . 
     The electric field data transfer unit  4  compares t em 02 with t cs , and if their difference falls below the time difference threshold λ2, it starts to calculate electric field data for the circuit&#39;s local space, based on the voltage obtained by the circuit analyzer  2 . The resultant electric field data is then transferred to the electromagnetic wave analyzer  1 . 
     Referring now to FIG. 2, the next few paragraphs will describe how the data is transferred between the electromagnetic wave analysis and the circuit analysis. In the equations appearing in the following section (and in FIG.  2 ), each superscript indicates a specific time step k, or the k-th step within a simulation process, meaning that a simulation time of (k×Δt em ) has elapsed since the beginning of the process. 
     (S1) The electromagnetic wave analyzer  1  calculates the magnetic field at a simulation time t em 01=(n+½)Δt em , where n is a natural number. This calculation is based on the following equation. 
     
       
         H n+½ =H n−½ −(Δt em /μ) rot E n   (8) 
       
     
     (S2) The circuit analyzer  2  simulates the circuit&#39;s behavior. When the condition |t cs −t em 01|&lt;λ1 is met during this circuit analysis, the circuit analyzer  2  stops the present simulation and waits the provision of new current source data. Also, when the condition |t cs −t em 01|&lt;λ1 is met, the current source data transfer unit  3  calculates the current source data according to the magnetic field obtained by the electromagnetic wave analyzer  1 . The resultant current source data is then transferred to the circuit analyzer  2 . The current source data I can be obtained by the following. 
      I n+½ =ΣH n+½   (9) 
     (S3) With the new current source data supplied, the circuit analyzer  2  resumes the circuit analysis from the simulation time t cs  according to the following equation. 
     
       
         V n+½ =Z 31 1 I n+½ ,  (10) 
       
     
     where Z is the impedance matrix of the circuit model. The circuit analyzer  2  repeats the calculation, incrementing t cs  by Δt cs . 
     (S4) When the condition |t cs −t em 02|&lt;λ2 is met, the electric field data transfer unit  4  calculates electric field data for the circuit&#39;s local space, based on the voltage obtained through the circuit analysis. The electric field data calculated by the following equation is then transferred to the electromagnetic wave analyzer  1 . 
     
       
         E n+1 =V n+½ /d  (11) 
       
     
     (S5) Upon receipt of the electric field data from the electric field data transfer unit  4 , the electromagnetic wave analyzer  1  calculates the electric field at t em 02=(n+1)Δt em  by 
     
       
         E n+1 =E n +(Δt em /ε) rot H n+½   (12) 
       
     
     The electromagnetic wave analyzer  1  obtains the electric field for the predetermined space by applying the received electric field data to the circuit&#39;s local space and calculating the electric field for the remaining space according to Equation (12). 
     In the simulator described above, the transfer operation of current source data occurs when the circuit analyzer&#39;s simulation time t cs  has approached the electromagnetic wave analyzer&#39;s simulation time t em 01 for the magnetic field. Similarly, the transfer operation of electric field data occurs when the circuit analyzer&#39;s simulation time t cs  has approached the electromagnetic wave analyzer&#39;s simulation time t em 02 for the electric field. As a result, every data transfer operation is synchronized with the internal calculation timing of the electromagnetic wave analysis, despite the timing offset between magnetic field calculation and electric field calculation. 
     Further, the present invention controls the simulation time t cs  not to advance too far beyond the simulation time t em 01 or t em 02, thus preventing the simulation from failure. The following section will explain the simulation procedure in more detail, including the control of t cs . FIG. 3 is a flowchart showing the simulation procedure according to the present invention. Note here that the steps described in FIG. 3 are executed by the elements shown in FIG. 1, with the exception that the first step  11  is the responsibility of a system controller (not illustrated) which totally supervises the simulation process. 
     (S11) The system controller allocates an appropriate data area within the main memory for storage of coefficients and other data, as well as initializing variables and parameters to be used in the simulation process. The time variables indicative of the circuit analyzer&#39;s simulation times (t cs ) and the electromagnetic wave analyzer&#39;s simulation times (t em 01 and t em 02) are all initialized to 0.0 in this step S11. 
     (S12) The circuit analyzer  2  calculates time-independent coefficients and the like. 
     (S13) The electromagnetic wave analyzer  1  adds Δt em /2 to t em 01, and Δt em  to t em 02. 
     (S14) The circuit analyzer  2  increments t cs  by Δt cs . 
     (S15) The circuit analyzer  2  examines whether the difference between t cs  and t em 01 falls below λ1 (i.e., |t cs −t em 01|&lt;λ1 is true) or not. If this condition is true, the process advances to step S16, and otherwise, the process proceeds to step S19. The above criterion can alternatively be |t cs −t em 01|≦λ1. 
     (S16) The electromagnetic wave analyzer  1  calculates the magnetic field. 
     (S17) The current source data transfer unit  3  transfers the current source data obtained through the electromagnetic field analysis to the circuit analyzer  2 . 
     (S18) The electromagnetic wave analyzer  1  increments t em 01 by Δt em . That is, the simulation time is advanced by the time step size Δt em  for the electromagnetic wave analysis. The process then proceeds to step S20. 
     (S19) The circuit analyzer  2  examines whether the condition t cs &gt;t em 01 is true or false. That is, the circuit analyzer  2  checks that the simulation time t cs  of the circuit analysis has not exceeded the simulation time t em 01 of the electromagnetic wave analysis. If the simulation time t cs  is beyond t em 01, the process branches to step S27. Otherwise, the process advances to step S20. 
     (S20) The circuit analyzer  2  solves a given circuit equation, thereby obtaining a voltage developed across the circuit at the simulation time t cs . 
     (S21) The circuit analyzer  2  checks whether the solution has successfully converged or not. If converged, the process advances to step S22, and otherwise, the process goes to step S27. 
     (S22) The circuit analyzer  2  examines whether the difference between t cs  and t em 02 falls below λ2 (i.e., the condition |t cs −t em 02|&lt;λ2 is true) or not. If this condition is met, the process advances to step S23, and otherwise, the process proceeds to step S26. The above criterion can alternatively be |t cs −t em 02|≦λ2. 
     (S23) The electromagnetic wave analyzer  1  calculates the electric field. In this step S23, the electric field in the space excluding the circuit&#39;s local space is calculated. 
     (S24) The electric field data transfer unit  4  supplies the electromagnetic wave analyzer  1  with the electric field data obtained as a result of the circuit analysis. The electromagnetic wave analyzer  1  applies the received electric field data to the circuit&#39;s local space. 
     (S25) The electromagnetic wave analyzer  1  increments t em 02 by Δt em . That is, the time step for the electric field calculation is advanced by the time step size Δt em , which is a predetermined parameter for the electromagnetic wave analysis. The process then advances to step S28. 
     (S26) The circuit analyzer  2  examines whether the condition t cs &gt;t em 02 is true or false. That is, the circuit analyzer  2  checks that the time t cs  of the circuit analysis has not outstripped the time t em 02 of the electromagnetic wave analysis. If t cs  has outstripped t em 02, the process proceeds to step S27. Otherwise, the process advances to step S28. 
     (S27) The circuit analyzer  2  cancels the preceding increment by subtracting Δt cs  from t cs , and it then reduces the time step size by multiplying Δt cs  by χ, where χ is a predetermined real number within the range of 0&lt;χ&lt;1. The process then returns to step S14. 
     (S28) The circuit analyzer  2  examines whether t cs  has exceeded t max  or not. If this condition is met, the process is terminated. Otherwise, the process proceeds to step S29. Here, t max  is a predetermined parameter that indicates the maximum simulated time. 
     (S29) The circuit analyzer  2  determines the next Δt cs , and returns to step S14. 
     In this way, the simulator of the present invention accomplishes a combined simulation of electromagnetic wave analysis and circuit analysis. The proposed simulation process never fails, since the circuit analyzer&#39;s simulation time t cs  is controlled not to exceed that of the electromagnetic wave analyzer, t em 01 or t em 02. 
     The simulator of the present invention transfers current source data and electric field data, considering the presence of a time offset of Δt em /2 between the magnetic field calculation and electric field calculation in the electromagnetic wave analysis. This contributes to the stability of solution. Furthermore, the proposed simulator allows the electromagnetic wave analyzer to use the condition ( 2 ) to determine its time step size Δt em , and requires no further reduction in its magnitude. Therefore, the computational loads does not increase in the present embodiment, unlike the conventional methods. 
     In the present embodiment, the circuit analyzer&#39;s time step size Δt cs  is controlled not to exceed Δt em /2, although it is allowed to be changed according to the circumstances. With this constraint, the simulation time t cs  is less likely to go beyond the simulation time t em 01 or t em 02, thus reducing the chances of executing extra process steps (i.e., the loop including step S27 in FIG.  3 ). 
     The next section will now present an example of a microstrip line simulation performed by the proposed simulator. 
     FIG. 4 is a side view of a microstrip transmission line. The computational domain, or calculation space, is divided into cells each having a volume of dx×dy×dz=0.265×0.389×0.400 [mm 3 ]. The illustrated microstrip transmission line is formed on a substrate  21  with dielectric constant ε=2.2, having a thickness of three cell distances in the X-axis direction. A conducting strip  31  and a ground plane conductor  32  are thin film conductors formed on the top side and the bottom side of the substrate  21 , respectively. As FIG. 4 shows, a diode  33  is placed at the right-hand end of the conducting strip  31 . The anode of this diode  33  is connected to the conducting strip  31  by a thin wire  34 , while its cathode is connected to the ground plane conductor  32  by another thin wire  35 . Further, to feed a high-frequency signal, an AC power source  36  is disposed between the left-hand end of the conducting strip  31  and the ground plane conductor  32 . 
     The conducting strip  31  formed on the top surface of the substrate  21  has a length of 25 cell intervals in the Z-axis direction, while the ground plane conductor  32  formed on the bottom surface of the substrate  21  has a length of 30 cell intervals in the Z-axis direction. The space surrounding the substrate  21  is filled with air  22 . There is provided an absorbing boundary  41  as indicated by a dotted line in FIG. 4, whose dimensions are  16  cell intervals in the X-axis direction by 30 cell intervals in the Z-axis direction. 
     FIG. 5 is a transverse cross section of the same microstrip transmission line. The conducting strip  31  has a width of six cell intervals in the Y-axis direction, and the anode-side wiring from the diode  33  is done on its center line. Likewise, the ground plane conductor  32  has a width of 20 cell intervals in the Y-axis direction, and the cathode-side wiring from the diode  33  is done on its center line. The absorbing boundary  41 &#39;s width in the Y-axis direction is 20 cell intervals. 
     For comparison, the above microstrip line is evaluated with two simulators: a conventional simulator and a new simulator proposed in the present invention. The following paragraphs will show the simulation results with reference to FIGS. 6 to  8 . 
     FIG. 6 is a diagram showing the result of a conventional simulation with a time step size of Δt em =0.315 ps. The horizontal axis of FIG. 6 represents simulation time in picoseconds (ps), and the vertical axis shows the voltage developed across the diode  33 . The same convention applies to other diagrams, FIGS. 7 and 8. Although this small time step size, Δt em =0.315 ps allows the conventional simulator to provide stable solutions, it is so shorter than usual that a fair amount of processing loads are imposed on the electromagnetic wave analyzer. 
     To reduce the loads, the microstrip line simulation has been conducted again by using the same conventional simulator, but now with a time step size doubled. FIG. 7 shows the result of this simulation with a time step size of Δt em =0.630 ps. With this doubled time step size, however, the voltage solution exhibits apparent instability, including oscillations that increases with time. 
     Then, for comparison purposes, the same simulation has been done with the same time step size as in FIG. 7, but using the simulator of the present invention. FIG. 8 shows the result of this simulation performed by the proposed simulator, with a time step size of Δt em =0.630 ps. As FIG. 8 shows, the simulator of the present invention yields stable solutions even if a larger time step size is used, and therefore, it is no longer necessary for the practitioners to set a reduced time step size when conducting an electromagnetic wave analysis in combination with circuit analysis techniques. As such, the present invention enables a stable simulation with a smaller amount of processing loads. 
     The above-described processing mechanisms will actually be implemented as software functions of a computer system. The functionalities of the proposed simulator are encoded in a computer program, which will be stored in a computer-readable storage medium. A computer executes this program to provide the above simulation processes and functions. The suitable computer-readable storage media include magnetic storage media and solid state memory devices. Some portable storage media, such as CD-ROMs and floppy disks, are also suitable for circulation purposes. Further, it is possible to distribute the program through an appropriate server computer deployed on a network. The program file delivered to a user is normally installed in his/her computer&#39;s hard drive or other local mass storage devices, which will be executed after being loaded to the main memory. 
     The above discussion will now be summarized as follows. According to the present invention, the simulator performs both an electromagnetic wave analysis and a circuit analysis with appropriate linkage mechanisms, in which the electric field data obtained in the circuit analysis is transferred to the electromagnetic wave analysis when the circuit analyzer&#39;s time step has approached the timing for the electric field calculation in the electromagnetic wave analysis. This configuration minimizes the time difference between the calculation of electric field data in the circuit analysis and its application to the other electric field, thus improving the stability of solutions. 
     The concept of the present invention can be paraphrased in more generic form as a simulator apparatus which executes a first simulation process in combination with a second simulation process, where the two simulation processes are allowed to transfer their simulation results to each other only when the difference between the first simulation process&#39;s time and the second simulation process&#39;s time falls below a predetermined threshold. This configuration permits one simulation process to use the other&#39;s simulation result with a minimized time difference, which contributes to the stability of numerical solutions. 
     Further, the present invention provides a computer-readable medium containing a simulation program which performs both an electromagnetic wave analysis and a circuit analysis with appropriate linkage mechanisms. This simulation program runs on a computer system, allowing a stable simulation for an electromagnetic wave analysis combined with a circuit analysis. 
     The above concept of the present invention can be paraphrased in more generic form as a computer-readable medium containing a simulation program which performs both a first simulation and a second simulation, where an appropriate linkage mechanism allows the two simulations to exchange their simulation results. The simulation program runs on a computer system, enabling a stable simulation for an electromagnetic wave analysis linked with circuit analysis techniques in the time domain. 
     The foregoing is considered as illustrative only of the principles of the present invention. Further, since numerous modifications and changes will readily occur to those skilled in the art, it is not desired to limit the invention to the exact construction and applications shown and described, and accordingly, all suitable modifications and equivalents may be regarded as falling within the scope of the invention in the appended claims and their equivalents.