CN110516282B - Indium phosphide transistor modeling method based on Bayesian statistics - Google Patents
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Abstract
The invention discloses an indium phosphide transistor modeling method based on Bayesian statistics. Based on the Bayesian statistical technique, the characteristic prediction of the transistor is realized according to the change rule of the input and output states of the transistor under the condition that the internal characteristics of the transistor are not required to be known, and the high-precision transistor model establishment is realized. The invention breaks through the existing modeling technology of the InPDHBT device, can solve the problem of low prediction precision of the existing strong nonlinear device model, and has good feasibility and accuracy in developing the obtained device behavior model.
Description
Technical Field
The invention relates to the field of microelectronic device modeling, in particular to a modeling method of an indium phosphide (InP) double-heterojunction bipolar transistor (DHBT) small-signal behavior model based on a Bayesian statistical technique.
Background
InP is an important iii-v group compound semiconductor material, and has the advantages of extremely high mobility and good radiation resistance. The InPDHBT device has the characteristics of high frequency, low noise, high power, high efficiency, radiation resistance and the like, is one of strategic high points of the international high-technology development, can be widely applied to various amplifiers, oscillators, digital circuits and the like, and has important application value in the fields of radar, communication, remote sensing, detection, security imaging, biological medicine and the like which urgently need high power, low noise and high frequency characteristics.
However, as a basis of computer-aided design of a circuit, an accurate model of an InPDHBT device, which is applicable to large-scale Electronic Design Automation (EDA) simulation application, is still deficient, and the development of a modeling technology is relatively lagged. The model accuracy and feasibility of modeling techniques are critical to the success of computer aided design of integrated circuits. As the structure of the inpdh bt device becomes more complex, the power becomes larger, the frequency increases, and the nonlinearity is enhanced, new requirements for circuit design are continuously provided, which bring new problems and new challenges to the precise development of the model. The development of an accurate model of the behavior of the small signal of the InPDHBT becomes one of the recognized problems in the industry and the academic community, and is an area which needs to be broken through urgently.
The existing InPDHBT small-signal model mainly comprises a physical basis model and an equivalent circuit model, such as a MEXTRAM model, a HICUM model, a UCSD model, an Agilent HBT model and the like, most of models are developed aiming at a Si/SiGe HBT process, and the model fitting degree is not high because InPDHBT is not particularly targeted. In comparison, the UCSD and Agilent HBT models are more suitable for model development of an InP HBT, which corrects the time transfer function, but does not perform well in terms of the current blocking effect. With the development of measuring equipment and measuring technology, behavioral models have gained wide attention with their particular advantages. The behavior model has extremely high precision due to the simple principle, is suitable for transistors of different types made of all materials, and obtains a great deal of research in the working field of high-frequency and strong nonlinear transistors.
Therefore, if the technological advantages of the InP DHBT device are fully utilized and the CAD design level of the circuit is improved, a modeling method for describing the behavior of the transistor with high precision is required.
Disclosure of Invention
The invention overcomes the defects of the prior art, provides a modeling method of an InPDHBT device small signal behavior model based on a Bayesian statistical algorithm, solves the problem of low modeling precision of the existing InPDHBT device behavior model, and establishes a model capable of accurately predicting the nonlinear behavior of the InPDHBT device.
The technical scheme of the invention is as follows:
a modeling method of an InPDHBT device small signal behavior model based on Bayesian statistics specifically comprises the following steps:
step one, fitting a function initial step:
in the ultra-wideband frequency range, the bias state of the InP transistor and the corresponding S parameter characteristic sample are obtained through actual measurement, and the linear function of the bias state and the S parameter sample is obtained through fitting as follows:
y＝f(x)＝w T x + b (1) wherein w is a weight coefficient, x is a bias state, b represents a bias term, y is an S parameter sample, and T represents transposition;
the set of bias states and S-parameter samples is
n represents the total number of samples, i represents the sample number; x is a radical of a fluorine atom i ∈R d Representing bias conditions including bias voltage condition, bias current condition and frequency
y i E R, representing the parameter of the transistor S; r represents a real number set, and d represents the dimensionality of the real number set;
step two, bayesian inference derivation step:
in Bayesian inference, p (w) represents a prior probability, where w is referred to herein as a weight coefficient for the model. Bayesian inference formula notation is as follows:
p (w) in the formula represents prior probability, and represents the probability that the weight coefficient w is the weight coefficient required by the model; p (w | D) represents the probability of a weight coefficient of w given event D; p (D | w) is the likelihood probability, i.e., the probability that event D is determined given the model weight coefficient w; p (D) is a definite term, which remains unchanged for different weighting coefficients w.
Based on bayesian inference, in order to obtain an optimal model equation, a Maximum A Posteriori (MAP) technique is used, and the obtained optimal model weight coefficients are as follows:
w MAP ＝arg max w p(D|w)p(w) (3)
i.e. the required model weight coefficients.
The weight coefficient w of the optimal model MAP The specific acquisition process of the calculation flow is as follows:
under the real number condition, based on the equation of the gaussian kernel function, the log-likelihood function equation can be obtained:
where β represents the inverse of the variance of the Gaussian distribution, w T ＝(w 1 ，w 2 ，w 3 ，...w k ) T Is the transposition of the weight coefficients, Φ (x) i )＝(Φ 1 ，Φ 2 ，Φ 3 ，...Φ k ) Is a basis function, k represents the number of coefficients, e.g. one-dimensionalPolynomial model, then its basis functions are: phi l (x i )＝x i l And 1 is an arbitrary number between 1 and k.
Selecting a Gaussian distribution with the mean value of 0 and the inverse variance of alpha as a prior probability, wherein the prior probability is as follows:
after negative logarithmic transformation we obtain:
thus, based on equations (2), (4), (6), the maximum a posteriori probability is the minimization equation:
the model coefficient obtained by the minimization (7) is the required optimal model equation coefficient w MAP 。
Step three, fitting the nonlinear behavior of the InP DHBT transistor:
for the relation between the small signal S parameter characteristic of the InP DHBT transistor and the bias state and frequency, the characteristic description equation can be used to represent:
wherein S is xx Representing four S parameters, S, corresponding to the transistor 11 、S 12 、S 21 And S 22 。I b Is the bias current of the base, V ce Is the collector and emitter bias voltage, and f is the corresponding operating frequency.
The input part is real number, but the output end is complex number, in order to adapt to Bayesian inference equation, the complex number of equation (8) is divided, and the divided equation is as follows:
wherein the content of the first and second substances,
and
the real part equation and the imaginary part equation of the Bayes S parameter model are adopted, and w represents the corresponding weight coefficient of the Bayes model.
Get S 11 For example, based on bayes theory, the equation of the model can be obtained as follows:
wherein w = (w) 1 ，w 2 ，...，w k ) For the model equation coefficients, k represents the weight coefficient index, Φ (I) b ，V ce ，f)＝(Φ，Φ 2 ，...，Φ k ) Representing the basis function equations.
Given a training set, the relationship between bias state and S-parameter can be expressed as: d = { I b ，V ce ，f；Real(S 11 )}。
The prior probability p (w) represents
Is the probability of the target optimum model; p (D | w) as likelihood probability represented in
The probability of the training data set is obtained under the equation of (2).
Based onEquation (7), S 11 And substituting variables corresponding to the real part to obtain:
obtaining the coefficient w of the real part corresponding to the optimal model by minimizing equation (12) MAP_real 。
Similarly, the modeling procedure for the imaginary part is the same as the real part, based on equation (7), and S is 11 And substituting variables corresponding to the imaginary part to obtain:
obtaining the coefficient w of the optimal model corresponding to the imaginary part by minimizing equation (13) MAP_imag 。
Obtaining S according to formula (14) 11 The behavioral model of the parameters is repeated by the equations (11) to (14) to obtain other S parameters S 12 、S 21 、S 22 。
S 11 ＝Real(S 11 )+Imag(S 11 )×j (14)
Where j represents the imaginary unit quantity.
Further, the ultra-wideband small signal behavior characteristic with the bandwidth from 200MHz to 325GHz.
Further, the actual measurements included S for the inpdh bt device under different bias conditions 11 、S 12 、S 21 And S 22 Ultra-wideband characteristic curve.
Compared with the prior art, the invention has the advantages that:
the invention breaks through the existing modeling technology of the InPDHBT device, can solve the problem of low prediction precision of the existing strong nonlinear device model, and has good feasibility and accuracy in developing the behavior model of the device.
The invention is based on Bayesian statistical technology, realizes the characteristic prediction of the transistor and the establishment of a high-precision transistor model according to the input and output state change rule of the transistor without knowing the internal characteristics of the transistor, reduces the complexity of the model and improves the modeling efficiency and the modeling precision compared with the existing model established based on an equivalent circuit method.
The invention develops a model for behavior prediction of an InPDHBT device under strong nonlinearity and ultra-wideband conditions based on Bayesian statistics, and obtains higher precision. The model obtained by the Bayesian statistical method is well fitted with the measured data.
Drawings
FIG. 1 is a topological structure diagram of a model of the present invention;
FIG. 2 shows a test small signal S according to the present invention 11 A parameter real part broadband characteristic diagram; wherein (a) is a characteristic curve; (b) is an error;
FIG. 3 is a test small signal S of the present invention 11 A parameter imaginary part broadband characteristic diagram; wherein (a) is a characteristic curve; (b) is an error;
fig. 4 is a real-part versus imaginary-part characteristic curve of all four S-parameters tested for InP DHBT transistors in bias state Ib =400ua, vce =1.0 v; wherein (a) is S 11 And (b) is S 12 (c) is S 21 And (d) is S 22 ；
FIG. 5 shows a test small signal S according to the present invention 11 And S 22 Smith artwork characteristics;
FIG. 6 shows a test small signal S according to the present invention 12 And S 21 Smith artwork properties.
Detailed Description
The invention is further described with reference to the following figures and detailed description.
As shown in fig. 1 to 5, a modeling method of an infiniband hbt device small signal behavior model based on bayes statistics specifically includes the following steps:
a modeling method of an InPDHBT device small signal behavior model based on Bayesian statistics specifically comprises the following steps:
step one, fitting a function initial step:
in the ultra-wideband frequency range, actually measured InP transistor bias state and S parameter characteristic samples represent bias voltage current state and frequency, and a linear function of the bias state and S parameter samples is obtained by fitting as follows:
y＝f(x)＝w T x + b (1) wherein w is a weight coefficient, x is a bias state, b represents a bias term, y is an S parameter sample, and T represents transposition;
the set of bias states and S parameter samples is
n represents the total number of samples, i represents the sample number; x is the number of i ∈R d Representing bias conditions including bias voltage condition, bias current condition and frequency
y i The method comprises the following steps that (1) an element belongs to R, represents a transistor S parameter, R represents a real number set, and d represents the dimensionality of the real number set;
step two, bayesian inference derivation step:
in Bayesian inference, p (w) represents a prior probability, where w is referred to herein as a weight coefficient for the model. Bayesian inference formula notation is as follows:
p (w) in the formula represents prior probability, and represents the probability that the weight coefficient w is the weight coefficient required by the model; p (w | D) represents the probability of a weight coefficient of w given event D; p (D | w) is the likelihood probability, i.e., the probability that event D is determined given the model weight coefficient w; p (D) is a definite term, which remains unchanged for different weighting coefficients w. Based on bayesian inference, in order to obtain an optimal model equation, a Maximum A Posteriori (MAP) technique is used, and the obtained optimal model weight coefficients are as follows:
w MAP ＝arg max w p(D|w)p(w) (3)
i.e. the required model weight coefficients.
The optimal model weight coefficient w MAP The specific acquisition process of the calculation flow is as follows:
under the real number condition, based on the equation of the gaussian kernel function, the log-likelihood function equation can be obtained:
where β represents the inverse of the variance of the Gaussian distribution, w T ＝(w 1 ，w 2 ，w 3 ，...w k ) T Is the transposition of the weight coefficients, Φ (x) i )＝(Φ 1 ，Φ 2 ，Φ 3 ，...Φ k ) Is a basis function, and k represents the number of coefficients, e.g., a one-dimensional polynomial model, then its basis function is: phi (phi) of l (x i )＝x i l And 1 is an arbitrary number between 1 and k.
Selecting Gaussian distribution with the mean value of 0 and the inverse variance of alpha as prior probability, and adopting the following formula:
after negative logarithmic transformation we obtain:
thus, based on equations (2), (4), (6), the maximum a posteriori probability is the minimization equation:
the model coefficient obtained by the minimization (7) is the required optimal model equation coefficient w MAP 。
Step three, fitting the nonlinear behavior of the InPDHBT transistor:
for the relation between the small signal S parameter characteristic of the InP DHBT transistor and the bias state and frequency, the characteristic description equation can be used for expressing the relation between the small signal S parameter characteristic of the InP DHBT transistor and the bias state and frequency:
wherein S is xx Representing four S parameters, S, corresponding to the transistor 11 、S 12 、S 21 And S 22 。I b Is the bias current of the base, V ce Is the collector and emitter bias voltage, and f is the corresponding operating frequency.
The input part is real number, but the output end is complex number, in order to adapt to Bayesian inference equation, the complex number of equation (8) is divided, and the divided equation is as follows:
wherein, the first and the second end of the pipe are connected with each other,
and
the real part equation and the imaginary part equation of the Bayes S parameter model are adopted, and w represents the corresponding weight coefficient of the Bayes model.
Get S 11 For example, based on bayes theory, the equation of the model can be obtained as follows:
wherein w = (w) 1 ，w 2 ，...，w k ) For the model equation coefficients, k represents the weight coefficient index, Φ (I) b ，V ce ，f)＝(Φ，Φ 2 ，...，Φ k ) Representing the basis function equations.
Given a training set, the relationship between bias state and S-parameter can be expressed as: d = { I b ，V ce ，f；Real(S 11 )}。
The prior probability p (w) represents
Is the probability of the target optimum model; p (D | w) as likelihood probability, represented in
The probability of the training data set is obtained under the equation of (2).
Based on equation (7), S 11 And substituting variables corresponding to the real part to obtain:
obtaining the coefficient w of the real part corresponding to the optimal model by minimizing equation (12) MAP_real 。
Similarly, the modeling procedure for the imaginary part is the same as for the real part, and based on equation (7), S is set 11 And substituting variables corresponding to the imaginary part to obtain:
obtaining the coefficient w of the optimal model with the imaginary part corresponding to the imaginary part by minimizing equation (13) MAP_imag 。
And finally obtaining a behavior model of the complete S parameter according to the following formula.
S xx ＝Real(S xx )+Imag(S xx )×j (14)
Where j represents the imaginary unit quantity.
In summary, the actual InPDHBT transistor is tested to obtain the S parameter characteristics of the device in the ultra wide band range, including S 11、 S 12 、S 21 And S 22 Characteristic curves at different biases; and extracting model parameters of the test data according to the modeling method of the invention to obtain the model parameters of the test transistor.
And (3) fitting and comparing the test data with the model simulation data, wherein the results are compared by using characteristic curves with the working frequency of 200MHz to 325GHz in FIGS. 2 to 6. Referring to FIG. 2 (a), FIG. 2 (b), FIG. 3 (a) and FIG. 3 (b), the transistor is in a bias state I b ＝200uA，V ce S of =1.0V 11 The real part and imaginary part characteristic curves and errors of the parameters, and fig. 2 and fig. 3 respectively comprise measured data, traditional equivalent circuit prediction data and a Bayes model provided in the patent, and the results show that the provided model can better predict the characteristics of the transistor than the traditional equivalent circuit function, even if the transistor works in a strong nonlinear area (the curve has obvious bending), the provided model still provides high prediction precision, the output characteristic of the test piece is well reflected, and the superiority and effectiveness of the device behavior modeling technology provided by the scheme are verified. FIG. 4 shows the testing of InP DHBT transistors in bias state I b ＝400uA，V ce The proposed model also gives high accuracy results in ultra-wideband characteristic prediction for real and imaginary characteristic curves for all four S-parameters of = 1.0V. Meanwhile, the S parameter of the transistor is shown in I in FIG. 5 b ＝400uA，V ce The characterization characteristics of the S parameter of =1.5V on the Smith original graph, and the model based on the Bayesian statistical method still provides good modeling precision.
The foregoing is only a preferred embodiment of the present invention, and it should be noted that, for those skilled in the art, several modifications and decorations can be made without departing from the spirit of the present invention, and these modifications and decorations should also be regarded as being within the scope of the present invention.
Claims (2)
1. An indium phosphide transistor modeling method based on Bayesian statistics is characterized by comprising the following steps:
step one, initializing a fitting function:
actually measuring the bias state of the InP transistor and the corresponding S parameter characteristic sample in the ultra-wideband frequency band range, wherein the bias state comprises a bias voltage state, a bias current state and a frequency
The S parameter includes S 11 、S 12 、S 21 And S 22 And fitting to obtain a linear function of the bias state x and the S parameter y as follows:
y＝f(x)＝w T x+b (1)
wherein w is a weight coefficient, b represents a bias term, and T represents transposition;
the set of bias states and S parameter samples is
n represents the total number of samples, i represents the sample number; x is the number of i ∈R d ，y i E is an element R, R represents a real number set, and d represents the dimensionality of the real number set;
step two, bayesian inference derivation step:
in Bayesian inference, p (w) represents the prior probability, where w denotes the weight coefficient of the model; bayesian inference formulas are labeled as follows:
p (w) in the formula represents prior probability, and represents the probability that the weight coefficient w is the weight coefficient required by the model; p (w | D) represents the probability of a weight coefficient of w given event D; p (D | w) is the likelihood probability, i.e., the probability that event D is determined given the model weight coefficient w; p (D) is a definite term, and is kept unchanged for different weight coefficients w;
based on bayesian inference, in order to obtain an optimal model equation, a Maximum A Posteriori (MAP) technique is used, and the obtained optimal model weight coefficients are as follows:
w MAP ＝arg max w p(D|w)p(w) (3)
the model weight coefficient is the required model weight coefficient;
the optimal model weight coefficient w MAP The specific acquisition process of the calculation flow is as follows:
under the real number condition, based on the equation of the gaussian kernel function, the log-likelihood function equation can be obtained:
where β represents the inverse of the variance of the Gaussian distribution, w T ＝(w 1 ，w 2 ，w 3 ，...w k ) T Is the transposition of the weight coefficients, Φ (x) i )＝(Φ 1 ，Φ 2 ，Φ 3 ，...Φ k ) Is a basis function, k represents the number of coefficients, where l (x i )＝x i l L is any number between 1 and k;
selecting Gaussian distribution with the mean value of 0 and the inverse variance of alpha as prior probability, and adopting the following formula:
after negative logarithmic transformation we obtain:
thus, based on equations (2), (4), (6), the maximum a posteriori probability is the minimization equation:
the model coefficient obtained by the minimization (7) is the required optimal model equation coefficient w MAP ；
Step three, fitting the nonlinear behavior of the InP DHBT transistor:
for the relationship between the InP DHBT transistor small signal S-parameter and the bias state, the characterization equation can be used to represent:
wherein S is xx Representing four S parameters S corresponding to the transistor 11 、S 12 、S 21 And S 22 ；I b Is the bias current, V, of the corresponding base of the transistor ce Is the bias voltage of the corresponding collector and emitter of the transistor, f is the corresponding working frequency of the transistor;
since the input part is real number but the output end is complex number, in order to adapt to the bayesian inference equation, the complex number of equation (8) is divided, and the divided equation is as follows:
wherein the content of the first and second substances,
and
the real part equation and the imaginary part equation of the Bayes S parameter model are adopted, and w represents a corresponding weight coefficient of the Bayes model;
wherein S 11 For example, based on bayes theory, the equation of the model can be obtained as follows:
wherein w = (w) 1 ，w 2 ，...，w k ) For the model equation coefficients, k represents the weighting coefficient index, Φ (I) b ，V ce ，f)＝(Φ 1 ，Φ 2 ，...，Φ k ) Representing a basis function equation;
given a training set, the relationship between bias state and S-parameter can be expressed as: d = { I b ，V ce ，f；Real(S 11 )}；
The prior probability p (w) represents
Is the probability of the target optimum model; p (D | w) as likelihood probability represented in
Obtaining the probability of the training data set under the equation condition of (3);
based on equation (7), S 11 And substituting variables corresponding to the real part to obtain:
obtaining the coefficient w of the real part corresponding to the optimal model by minimizing equation (12) MAP_real ；
Similarly, the modeling procedure for the imaginary part is the same as for the real part, and based on equation (7), S is set 11 And substituting variables corresponding to the imaginary part to obtain:
obtaining the coefficient w of the optimal model corresponding to the imaginary part by minimizing equation (13) MAP_imag ；
Obtaining S according to formula (14) 11 Behavior of a parameterModel, repeating equations (11) - (14) to obtain other S parameters S 12 、S 21 、S 22 ；
S 11 ＝Real(S 11 )+Imag(S 11 )×j (14)
Where j represents the imaginary unit quantity.
2. The modeling method for indium phosphide transistors based on bayesian statistics as defined in claim 1, wherein the ultra-wideband frequency band in step one is in the range of 200MHz to 325GHz.
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