INTRODUCTION

With a decrease in the size of crystals and an increase in their operating frequencies, negative effects related to the signal’s integrity begin to appear, for example, crosstalk, resonances caused by the simultaneous switching of digital blocks, and noise. These effects complicate the design process for high-frequency ICs on-chip topology, IC package requirements, and PCB layout requirements. If the board is the final product or functional unit of the system, then the same package can be used to package different microcircuits. The development of circuitry, taking into account the parameters of the used package, allows us to save on creating a new package and minimize its negative impact on the operation of microcircuits. Hence, two tasks arise: (1) selection of the equivalent circuit of the package and creation of a simulation model that will take into account the parameters of the package; (2) determination of the parameters of the model in a wide range of operating frequencies, including the frequencies of the proposed microcircuit.

In this paper, we consider the optimization algorithm by the stochastic method of the classical model, which uses the concepts of the self-capacitance of the output of the conductor’s inductance and the communication capacitance in order to achieve a better match of the model to the investigated package with the minimum increase in its computational complexity.

BUILDING THE PACKAGE OF THE MODEL FOR A CIRCUIT SIMULATION PROGRAM

In the works [1, 2], the package was studied to determine the parameters of its interconnections. In the work [3], the package interconnections and their influence on the quality of signal transmission at the PCB level were investigated. The process of preparing the package of the model for a schematic editor is conventionally divided into two large stages:

(1) getting a matrix of the S-parameters of the outputs of the package by the calculation method or by measuring a real device. It is advisable to carry out both the calculation and measurements so that the results obtained by two independent methods confirm and complement each other;

(2) selection of the appropriate model of interconnections, which will take into account the internal conductors of the package, the wire of the classical technology of welding crystals or elements of the welding of the flip-chip technology, and the external conductors with which the microcircuit is attached to the board; and transformation of the matrix obtained at the first stage of S-parameters, i.e., matrices of complex numbers of size 2N × 2N, where N is the number of outputs of the package, for each frequency value to the parameters of the model used for simulation. It should be borne in mind that each value of the frequency of the matrix of S-parameters will correspond to their values of capacitances, inductances, and resistances of the model.

In this study, 3D electromagnetic modeling in the frequency domain is carried out to extract the parameters of the package. The 3D model of the enclosure is constructed from the existing package drawings and internal wiring. Based on the results obtained in the electromagnetic modeling of the S-parameters, an electrical circuit suitable for the simulation program is constructed. For simplicity, only the two central outputs on one side of the package are considered.

The simplest option is to simulate the electrical circuit, taking into account the obtained S-parameters in the electromagnetic simulation. However, not all simulators work directly with S-parameters. In addition, the size of the matrix of S-parameters is directly proportional to the number of ports of the model and the frequency range; therefore, the calculation using such a model in the time domain (analysis of transient processes) significantly increases the simulation time. During the development of blocks, an increase in the simulation time should be avoided, but this may be acceptable when validating the final solution. Thus, a simplified model is needed, the behavior of which will correspond to the behavior described by the S-parameters, but the computational complexity will be an order of magnitude lower. The most common model of the transmission line path, which is the lead conductor of the package, is the RLGC transmission line model [4, 5]. The advantage of this model is that it can work with both distributed and lumped parameters. The lumped model is suitable for simulation and has a sufficient degree of reliability for low and medium frequencies. Using the RLGC model as a base, we can get simpler or more complex models by removing or adding new elements.

Figure 1 shows the equivalent RLC model of the output of the package. This is a simplified RLGC model where the Inner and Outer outputs correspond to the inner and outer outputs of the package, Rs and Ls are the resistance and inductance of the conductor of the output of the package, CGND is the output capacity with respect to the ground range of the package, and Cp is the capacity of the connection to the adjacent output. There is no resistance that simulates the leakage of capacitance between the output and ground due to the high rating and low degree of influence on the dynamic processes.

Fig. 1.
figure 1

Standard equivalent RLC model of output of package.

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The coupling inductance in the RLC model is absent due to the practical complexity of its measurement, and the parameters of the welding wire are not taken into account, since the position of the wire in the housing and its parameters must be determined for each specific welding option separately.

Figure 2 shows the frequency dependences of the impedances (matrix coefficients of the Z- and Y-parameters) in the frequency range 1–10 GHz for the central output of one of the sides of the package under study. Based on the presented parameters, an equivalent RLC output model is built using the following formulas:

$$C = {\text{imag}}\left( {1{\text{/}}{{Z}_{{22}}}} \right){\kern 1pt} {\text{/}}{\kern 1pt} \left( {2\pi f} \right),$$
(1)
$$L = {\text{imag}}\left( {{\text{1/}}{{Y}_{{{\text{11}}}}}} \right){\kern 1pt} {\text{/}}{\kern 1pt} \left( {2\pi f} \right),$$
(2)
$$R\left( {DC} \right) = {\text{real}}\left( {1{\text{/}}{{Y}_{{11}}}} \right),$$
(3)

where Yij and Zij are the parameter matrices of the Z- and Y-parameters, respectively (the Y-parameter is presented in the minus first degree minus for ease of perception).

Fig. 2.
figure 2

Z-parameters of central output on one side of the package.

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However, these dependences are not similar to the frequency dependences of inductance or capacitance in their pure form. Thus, in the frequency range 1–10 GHz, in the classical RLC model, the output inductance changes in the range 1.8–2.4 nH, and its own capacitance changes in the range of 0.4 to 0.8 pF. Upon reaching a certain frequency, the RLC model ceases to correspond to the physical object. Also, the impedance of the output under study in the frequency range 4.5–7.5 GHz is not optimized (see Fig. 2).

OPTIMIZATION OF THE CLASSICAL APPROACH

The classical RLC model can be used at low and medium frequencies; it is not advisable to use it at high frequencies. By replacing the elementary elements of this model with more complex resonant chains, it is possible to achieve a better correspondence of the impedance of the model to the impedance of the real output. However, the problem of determining the parameters of the model arises, since there are more unknown elements in the model than there are possible equations. A chain of the simplest elements, the total impedance of which will exactly correspond to the impedance of the output of the package over the entire operating frequency range, can be quite large and consist of a thousand or tens of thousands of elements or more. The model built based on such circuits will have computational complexity comparable to that of a VLSI, i.e., the transfer from the S-parameters to the equivalent model increases the level of computational complexity of the model and increases the simulation time. A simple circuit in terms of computational complexity and at the same time exactly matching the original S-parameters of the circuit may not exist, and its search is an ill-posed problem [6], which has no solution.

The following groups of methods exist to solve such problems [7]: iterative, stochastic [8], and evolutionary algorithms [9]. To find the parameters of the circuit (Fig. 3), we will use the stochastic method [10], based on the algorithm shown in Fig. 4. At the beginning of the algorithm, the parameters of the circuit elements are randomly selected; at each iteration, the parameters of the elements are randomly changed according to the formula

$${{X}_{k}} = {{X}_{{k - 1}}}(1 + CM),$$

where X is the element parameter (resistance, inductance, or capacitance); k is the iteration number; C is the coefficient of the change; and M is a random number in the range [–1:1].

Fig. 3.
figure 3

Optimized circuit cell (a) and the optimized circuit (b).

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Fig. 4.
figure 4

Stages of solving the optimization problem by the stochastic method.

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Next, for the newly obtained solution, the penalty function is calculated by the formula

$${{X}_{i}} = \sqrt {\frac{1}{N}\mathop \sum \limits_{n = 1}^N \left( {{{{\left( {R\left( {{{{{\omega }}}_{n}}} \right) - R} \right)}}^{2}} + {{{\left( {{{\chi }}\left( {{{\omega }_{n}}} \right) - {{\chi }}} \right)}}^{2}}} \right)} ,$$

where N is the number of points (used frequency values); R is the real part of the impedance; χ is the imaginary part of the impedance; and \({{\omega }}\) is the frequency.

If the new solution turns out to be better, it goes to the next iteration. Otherwise, the old solution goes to a new iteration, and the algorithm parameters are optimized if necessary. If for many iterations the solution remains unchanged or the penalty function of the solution at iteration i + 1 differs little from the iteration solution’s penalty function i, then the algorithm is close to reaching the optimum. In this case, additional optimizations are required:

—reduction of coefficient C, which will narrow the set of feasible solutions; i.e., a new solution is sought near the current one;

—removal of elements with excessively large or small parameters;

—removal of elements, whose contribution of which is compensated by the neighboring elements of the chain;

—unification of the circuit elements that allow us to do this.

The result of this algorithm as applied to the circuit in Fig. 3 with parameters i = j = 5 and C = 1 is the circuit shown in Fig. 5, where block Zself is the equivalent of its own output capacity, block Zmutual is the impedance of the connection of the investigated output with the neighboring output, and block Zlead is the impedance of the output conductor. Of the 200 initial elements simulating the connection capacity and their own output capacity, 10 and 11 elements remain, respectively. The number of elements that mimic the lead conductor is 3. Table 1 shows the parameters of models with the reference: the result of the transformation of the S-parameters at a frequency of 10 GHz. For the calculation, we used formulas (1)–(3); the RLC model more closely corresponds to the S-parameters in the active resistance, but at the same time it loses significantly in inductance and capacitance.

Fig. 5.
figure 5

Final model of a single package output.

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Table 1.   Model parameters at 10 GHz
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COMPARISON OF RESULTS

Figure 6 shows the graphs of the frequency dependence of the output impedance at different switching on instances. In fact, the presented parameters are parasitic for a microcircuit placed in this package, respectively, and taking them into account in the development of circuitry will improve the parameters of the circuit.

Fig. 6.
figure 6

Graphs of the frequency dependence of the output impedance at open circuit (a), with a matched load (b), and between two adjacent output (c): ZS is the result of matrix transformation of the S-parameters (standard); ZRLC is the classical model (see Fig. 1); Zopt is the impedance of the optimized circuit.

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The behavior of the optimized model (see Fig. 5) is more consistent with the S-parameters than the behavior of the model based on the classical approach. Table 2 shows the values of the mean relative deviation of the impedances of both models from the standard and the standard deviation (RMS). The standard in this case are the S-parameters and comparison is carried out in the frequency range 1–10 GHz.

Table 2.   Comparison of model accuracy
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Thus, the classical model with 4 elements, supplemented by 20 elements, made it possible to increase the compliance of the model with the standard model by 3–5 times (it is only valid for this case).

CONCLUSIONS

The reliability of the electrical model of the package’s output, built based on the calculated circuits, at high frequencies is 3–5 times greater than the reliability of the classical RLC model. The similarity of the resulting model with the reference model turned out to be imperfect, since it was impossible to replace the microwave path with an equivalent electrical circuit, and it remains possible to search for a better configuration of elements in the circuit. If we build a model based on the S-parameters of the package with welding elements (wires, solder balls, etc.) obtained by measurement or simulation, we can take into account the welding elements.