CN107958129B - Algorithm for simulating microcosmic current distribution of zinc oxide piezoresistor - Google Patents
Algorithm for simulating microcosmic current distribution of zinc oxide piezoresistor Download PDFInfo
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- XLOMVQKBTHCTTD-UHFFFAOYSA-N Zinc monoxide Chemical compound [Zn]=O XLOMVQKBTHCTTD-UHFFFAOYSA-N 0.000 title claims abstract description 74
- 239000011787 zinc oxide Substances 0.000 title claims abstract description 37
- 238000004422 calculation algorithm Methods 0.000 title claims abstract description 25
- 238000004364 calculation method Methods 0.000 claims abstract description 37
- 238000005457 optimization Methods 0.000 claims abstract description 33
- 238000004088 simulation Methods 0.000 claims abstract description 14
- 238000012545 processing Methods 0.000 claims description 4
- 230000009471 action Effects 0.000 claims description 3
- 238000011217 control strategy Methods 0.000 claims description 3
- 230000000694 effects Effects 0.000 claims description 3
- 239000011159 matrix material Substances 0.000 claims description 3
- 230000011218 segmentation Effects 0.000 claims description 3
- 230000007246 mechanism Effects 0.000 abstract description 3
- 230000009286 beneficial effect Effects 0.000 abstract description 2
- 238000011160 research Methods 0.000 description 8
- 238000000034 method Methods 0.000 description 5
- 238000013461 design Methods 0.000 description 4
- 239000000463 material Substances 0.000 description 4
- 230000008569 process Effects 0.000 description 4
- 238000005094 computer simulation Methods 0.000 description 2
- 238000010586 diagram Methods 0.000 description 2
- 238000005315 distribution function Methods 0.000 description 2
- 238000012935 Averaging Methods 0.000 description 1
- 230000008859 change Effects 0.000 description 1
- 238000011156 evaluation Methods 0.000 description 1
- 238000013178 mathematical model Methods 0.000 description 1
- 238000012986 modification Methods 0.000 description 1
- 230000004048 modification Effects 0.000 description 1
- 238000005036 potential barrier Methods 0.000 description 1
- 239000002994 raw material Substances 0.000 description 1
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Abstract
An algorithm for simulating the microscopic current distribution of a zinc oxide piezoresistor comprises a formula establishing step, a sectional control step and an optimization step, wherein the formula establishing step, the sectional control step and the optimization step are sequentially carried out. The beneficial effects are as follows: the fast algorithm of the piecewise linearization enables the ZnO varistor calculation simulation model after the introduction of the grain boundary real conduction mechanism to be effectively solved, and the speed and efficiency of calculation simulation are greatly improved by fully optimizing the related algorithm.
Description
Technical Field
The invention relates to the field of research on novel high-performance ZnO piezoresistors with high voltage gradients, in particular to an algorithm for simulating the microcosmic current distribution of a zinc oxide piezoresistor.
Background
The most basic premise of material design is that the actual process of material experimental study and the various elements involved therein can be described by a suitable mathematical model; the most basic tool for material design is a calculation simulation model capable of truly reflecting the change rule of material characteristics.
Since the original optimization variables (raw material component formula and processing process conditions) and the final optimization target (macroscopic electrical performance parameters) in the optimization problem have a very complex correlation relationship, experimental research is almost the only tool and way for optimization solution for a long time. The important achievements in the aspect of optimizing the performance of the ZnO piezoresistor are basically established on the basis of a great amount of experimental research, and the important achievements are the main reasons that domestic manufacturers do not break through the novel high-performance ZnO valve plate in years of tracking research and related research achievements of domestic scientific research institutes and practical application requirements are different.
There is a very complex association between the optimization variables and the optimization objectives of the original optimization problem that is difficult to systematically generalize and describe. Compared with this, in each optimization problem divided into two steps, the original optimization variables and the intermediate optimization targets, and the association relationship between the intermediate optimization variables and the final optimization targets are definitely simpler and clearer. For the optimization variables and the optimization targets of each step in the optimization process divided into two steps, clear correlation relations between the optimization variables and the optimization targets can be established, and the correlation relations are basic conditions for further solving the original complete optimization problem.
For the calculation simulation model and algorithm of the ZnO varistor, some researchers have carried out related research and achieved certain results at present. However, the existing research results of calculating and simulating the ZnO piezoresistor still have certain problems and disadvantages, and it is difficult to obtain a direct correlation between the optimization variables and the target, that is, an intricate influence relationship and a correlation mechanism between the intrinsic microstructure and grain boundary characteristic parameters of the ZnO piezoresistor and macroscopic electrical performance parameters.
For solving a large-scale nonlinear equation system involved in a ZnO varistor calculation simulation model, the most common numerical calculation method which is also commonly adopted by researchers in the past is a traditional Newton iteration method. Because the resources and the solving time required by calculation are increased in geometric progression along with the increase of the equation and the unknown quantity in the equation set, researchers generally carry out numerical calculation simulation on the ZnO piezoresistor in the order of hundreds of grains, and each calculation usually takes more than ten hours.
Disclosure of Invention
The invention aims to solve the problems and designs an algorithm for simulating the microcosmic current distribution of the zinc oxide piezoresistor. The specific design scheme is as follows:
an algorithm for simulating the microcosmic current distribution of a zinc oxide piezoresistor comprises a formula establishing step, a sectional control step and an optimization step which are sequentially carried out,
in the formula establishing step, the effect of the added voltage increment on the equivalent circuit is solved, and the nonlinear circuit is subjected to approximate linearization processing;
in the step of sectional control, controlling the current increment of the external power supply and determining a control strategy that the external power supply gradually increases from zero voltage to the maximum value;
in the optimization step, the equation set established in the segmentation control step is evolved to simplify calculation.
In the formula establishing step, in the k and k +1 steps, the voltage and current data of each node and branch in the equivalent circuit need to satisfy a first nonlinear equation set and a second linear equation set,
the first nonlinear equation is:
the second linear equation set is:
defining the voltage and current data difference value of each node and branch in the k step and the k +1 step to obtain a third equation set of voltage and current:
and (3) substituting the linear equation set I and the linear equation set II into an equation set III to obtain a deformed equation set IV:
performing Taylor series expansion on the deformed equation set four to obtain an equation set five:
in the step of the sectional control, necessary changes are carried out on the constraint conditions of the external power supply branches in the equation group five to obtain an equation group six:
and for the increment of the current of the external power supply, controlling based on the current density value flowing through the whole ZnO piezoresistor to obtain a seventh equation set:
in the formula establishing step, voltage and current data of nodes and branches generated by the increment of the voltage of the external power supply are calculated, actual voltage and current data of each node and branch in the equivalent circuit under the action of the voltage of the current external power supply are obtained by three-step superposition of an equation set on the calculation result obtained in the previous step, various macroscopic electrical performance parameters of the ZnO piezoresistor are comprehensively analyzed, and the piecewise linearization algorithm of calculation simulation of the ZnO piezoresistor is realized.
In the optimization step, data corresponding to the relevant matrix in the equation set and the external power supply branch in the vector are segmented with other data to obtain the following equation:
A＝[AGB,AS]；
substituting the equation into equation set six to obtain equation set eight:
in the eighth equation set, the 2 nd equation in the equation set is substituted into the 4 th equation, the 4 th equation is substituted into the 1 st equation in sequence to obtain the voltage increment of each node in the direct solution equivalent circuit
Equation one of (1):
by
Obtaining the voltage and current increment delta U of each branch in the equivalent circuit through direct calculation(k+1)、
The system of equations of (a) is:
in seven of the equation set, SBulkIn order to calculate the estimated value of the overall cross-sectional area of the simulated ZnO piezoresistor, J _ exp _ min is the minimum value of the current density actually flowing through the ZnO piezoresistor, the value is-9, J _ exp _ step is the incremental value of the corresponding current density, the value is 0.01, and the calculation is iterated until the current density reaches the maximum value J _ exp _ max, and the value is 5.
The algorithm for simulating the microscopic current distribution of the zinc oxide piezoresistor obtained by the technical scheme has the beneficial effects that:
the fast algorithm of the piecewise linearization enables the ZnO varistor calculation simulation model after the introduction of the grain boundary real conduction mechanism to be effectively solved, and the speed and efficiency of calculation simulation are greatly improved by fully optimizing the related algorithm.
Drawings
FIG. 1 is a graph of a piecewise linearization algorithm of a ZnO varistor calculation simulation according to the present invention;
fig. 2 is an error analysis diagram of the computational simulation algorithm of the present invention.
Detailed Description
The present invention will be described in detail below with reference to the accompanying drawings.
FIG. 1 is a sectional linearization algorithm chart of the ZnO varistor calculation simulation of the invention, as shown in FIG. 1, an algorithm for simulating the microscopic current distribution of the zinc oxide varistor comprises a formula establishing step, a sectional control step and an optimization step, wherein the formula establishing step, the sectional control step and the optimization step are sequentially carried out,
in the formula establishing step, the effect of the added voltage increment on the equivalent circuit is solved, and the nonlinear circuit is subjected to approximate linearization processing;
in the step of sectional control, controlling the current increment of the external power supply and determining a control strategy that the external power supply gradually increases from zero voltage to the maximum value;
in the optimization step, the equation set established in the segmentation control step is evolved to simplify calculation.
In the formula establishing step, in the k and k +1 steps, the voltage and current data of each node and branch in the equivalent circuit need to satisfy a first nonlinear equation set and a second linear equation set,
the first nonlinear equation is:
the second linear equation set is:
defining the voltage and current data difference value of each node and branch in the k step and the k +1 step to obtain a third equation set of voltage and current:
and (3) substituting the linear equation set I and the linear equation set II into an equation set III to obtain a deformed equation set IV:
performing Taylor series expansion on the deformed equation set four to obtain an equation set five:
in the step of the sectional control, necessary changes are carried out on the constraint conditions of the external power supply branches in the equation group five to obtain an equation group six:
and for the increment of the current of the external power supply, controlling based on the current density value flowing through the whole ZnO piezoresistor to obtain a seventh equation set:
in the formula establishing step, voltage and current data of nodes and branches generated by the increment of the voltage of the external power supply are calculated, actual voltage and current data of each node and branch in the equivalent circuit under the action of the voltage of the current external power supply are obtained by three-step superposition of an equation set on the calculation result obtained in the previous step, various macroscopic electrical performance parameters of the ZnO piezoresistor are comprehensively analyzed, and the piecewise linearization algorithm of calculation simulation of the ZnO piezoresistor is realized.
In the optimization step, data corresponding to the relevant matrix in the equation set and the external power supply branch in the vector are segmented with other data to obtain the following equation:
A＝[AGB,AS]；
substituting the equation into equation set six to obtain equation set eight:
in the eighth equation set, the 2 nd equation in the equation set is substituted into the 4 th equation, the 4 th equation is substituted into the 1 st equation in sequence to obtain the voltage increment of each node in the direct solution equivalent circuit
Equation one of (1):
by
Obtaining the voltage and current increment delta U of each branch in the equivalent circuit through direct calculation(k+1)、
The system of equations of (a) is:
in seven of the equation set, SBulkIn order to calculate the estimated value of the overall cross-sectional area of the simulated ZnO piezoresistor, J _ exp _ min is the minimum value of the current density actually flowing through the ZnO piezoresistor, the value is-9, J _ exp _ step is the incremental value of the corresponding current density, the value is 0.01, and the calculation is iterated until the current density reaches the maximum value J _ exp _ max, and the value is 5.
Example 1
And according to the sixth equation set, voltage and current data of the nodes and the branches generated by the additional power supply increment in the step (k +1) can be obtained, and the voltage and current data of each node and branch in the equivalent circuit in the step (k +1) can be obtained by adding the voltage and current data to the calculation result obtained in the step (k) through the third equation set.
FIG. 2 is an error analysis diagram of the computational simulation algorithm of the present invention, as shown in FIG. 2, the data actually corresponds to a certain data point on the tangent extension of the kth step data point on the nonlinear curve, i.e., point A, rather than being directly located on the nonlinear curve; for the solving process of the (k + 2) th step, an equation is adopted, which is actually equivalent to a point B serving as a tangent extension line, so that the linearization of a nonlinear curve is realized.
Example 2
On the basis of embodiment 1, for the point B, the voltage data of each node branch in the equivalent circuit is completely the same as the point a, so that the kirchhoff-related law is still satisfied, and the current data of the branch is no longer the same as the point a, so that a certain deviation exists from the requirements of the kirchhoff-related law. And obtaining branch current data corresponding to the B point through a complete equation set:
wherein V is the voltage applied to two sides of the potential barrier;
Qiis the charge density of the surface state filling;
Ni(E) is a surface state energy distribution function;
fi(E) is a fermi distribution function;
kBboltzmann (Boltzmann) constants;
t is the absolute temperature;
a is the Richardson constant;
kBboltzmann constant;
t is the absolute temperature;
ξenergy level difference ξ between conduction band energy level Ec and Fermi level ξiIs a quasi-fermi level.
Example 3
On the basis of example 2, the following calculation errors are defined on this basis:
in the above definition of the calculation error Ierr, the essence of the numerator portion is the sum of all the currents flowing into and out of each node of the equivalent circuit, and the essence of the denominator portion is the total current value flowing through each node.
According to kirchhoff's correlation law, the numerical value of the sum of the currents flowing into and out of any node in the circuit is supposed to be zero, the requirement of the law can be met for branch current data I (k +1) at the point A, and the branch current data corresponding to the point B does not meet the requirement of the law, namely a certain calculation error is generated.
And dividing the error data of the sum of the currents of each node by the total current value flowing through each node to obtain error ratio data which can better reflect the severity of the calculated current error of each node. And accumulating and averaging the current error ratio data of all the nodes to serve as an error evaluation index of the whole calculation simulation algorithm.
For the practical calculation simulation example, the calculation result obtained by adopting the piecewise linearization fast algorithm proposed by the paper and various related optimization measures has the calculation error Ierr only in the order of magnitude of 10-3 according to the definition, and the error value is very small, thereby showing that the related calculation simulation algorithm is completely reasonable and reliable.
The technical solutions described above only represent the preferred technical solutions of the present invention, and some possible modifications to some parts of the technical solutions by those skilled in the art all represent the principles of the present invention, and fall within the protection scope of the present invention.
Claims (2)
1. An algorithm for simulating the microcosmic current distribution of a zinc oxide piezoresistor comprises a formula establishing step, a sectional control step and an optimization step, and is characterized in that the formula establishing step, the sectional control step and the optimization step are sequentially carried out,
in the formula establishing step, the effect of the added voltage increment on the equivalent circuit is solved, and the nonlinear circuit is subjected to approximate linearization processing;
in the step of sectional control, controlling the current increment of the external power supply and determining a control strategy that the external power supply gradually increases from zero voltage to the maximum value;
in the optimization step, the equation set established in the segmentation control step is evolved to simplify calculation,
in the formula establishing step, in the k and k +1 steps, the voltage and current data of each node and branch in the equivalent circuit need to satisfy a first nonlinear equation set and a second linear equation set,
the first nonlinear equation is:
the second linear equation set is:
defining the voltage and current data difference value of each node and branch in the k step and the k +1 step to obtain a third equation set of voltage and current:
and (3) substituting the linear equation set I and the linear equation set II into an equation set III to obtain a deformed equation set IV:
performing Taylor series expansion on the deformed equation set four to obtain an equation set five:
in the step of the sectional control, necessary changes are carried out on the constraint conditions of the external power supply branches in the equation group five to obtain an equation group six:
and for the increment of the current of the external power supply, controlling based on the current density value flowing through the whole ZnO piezoresistor to obtain a seventh equation set:
in the formula establishing step, voltage and current data of nodes and branches generated by the increment of the voltage of the external power supply are calculated, actual voltage and current data of each node and branch in the equivalent circuit under the action of the voltage of the current external power supply are obtained by three-step superposition on the calculation result obtained in the previous step according to an equation set, various macroscopic electrical performance parameters of the ZnO piezoresistor are comprehensively analyzed, and a piecewise linearization algorithm of calculation simulation of the ZnO piezoresistor is realized,
in the optimization step, data corresponding to the relevant matrix in the equation set and the external power supply branch in the vector are segmented with other data to obtain the following equation:
substituting the equation into equation set six to obtain equation set eight:
in the eighth equation set, the 2 nd equation in the equation set is substituted into the 4 th equation, the 4 th equation is substituted into the 1 st equation in sequence to obtain the voltage increment of each node in the direct solution equivalent circuit
Equation one of (1):
by
Obtaining the voltage and current increment delta U of each branch in the equivalent circuit through direct calculation(k+1)、
The system of equations of (a) is:
2. the algorithm for modeling the micro-current distribution of a zinc oxide varistor as claimed in claim 1, wherein S in the seventh equation setBulkIn order to calculate the estimated value of the overall cross-sectional area of the simulated ZnO piezoresistor, J _ exp _ min is the minimum value of the current density actually flowing through the ZnO piezoresistor, the value is-9, J _ exp _ step is the incremental value of the corresponding current density, the value is 0.01, and the calculation is iterated until the current density reaches the maximum value J _ exp _ max, and the value is 5.
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