InP-based heterojunction bipolar transistors(HBTs)are gradually applied for the field of microwave and millimeter wave integrated circuits(ICs)with the growing frequencies [
Journal of Infrared and Millimeter Waves, Volume. 41, Issue 2, 430(2022)
Accuracy estimation of microwave performance for InP HBTs based on Monte-Carlo analysis
In order to verify the reliability of the circuit, the microwave performance for InP HBTs is analyzed based on Monte-Carlo method. The larger the fluctuation range is, the higher the accuracy of extracting intrinsic parameters is required. The allowable accuracy range can be inferred from the output performance. The microwave characteristics are composed of modeled S-parameters, current gain cut-off frequency and maximum oscillation frequency. The standard deviation of Monte-Carlo numerical analysis can be derived from the uncertainty curve of the intrinsic parameters of the π-topology small signal model. The results of Monte-Carlo analysis show that the requirements for the accuracy of measurement parameters are quite different under different frequencies and bias conditions, which verifies the reliability of the circuit under different conditions.
Introduction
InP-based heterojunction bipolar transistors(HBTs)are gradually applied for the field of microwave and millimeter wave integrated circuits(ICs)with the growing frequencies [
In the process of device modeling,the main reason for the accuracy of the output microwave characteristics of the model is the measurement uncertainty. The S-parameters measured by the vector network analyzer(VNA)are used to establish the small-signal models of the devices. The following uncertainty contributions from measurement should be taken into account:vector network analyzer noise and uncertainty of calibration standard definitions [
The uncertainty of measured parameters of each frequency is included in the factory instructions in vector network analyzer. The uncertainty of measured S-parameters will be transferred to the microwave performance accuracy of the HBT model through the de-embedding process and the small-signal circuit parameters extraction. The process of stripping parasitic components(de-embedding)will transfer the uncertainty of measurement parameters. However,many experiments show that the uncertainty transitivity of parasitic components is insignificant,especially for on-chip testing [
Monte-Carlo method is a common numerical tolerance analysis method,which is often used in statistics. In the equivalent circuit,the Monte-Carlo analysis method can be used to estimate the accuracy range of the whole model in combination with the uncertainty of all parameters. The uncertainty of each component has an impact on the accuracy of the HBT small-signal model. The Monte-Carlo method arranges and combines the possible values of all the components in the accuracy range,and carries out the circuit analysis after random sampling.
In the process of electronic circuit design,Monte-Carlo method is one of the important means to verify the results of circuit reliability analysis. It can analyze the impact on electronic components deviating from the standard value on the output performance,and predict the pass rate before the product is putting into production. On the contrary,the accuracy range of components can be deduced when a certain qualification rate is required.
Electronic circuit tolerance analysis can improve the reliability of electronic circuit designs,shorten the design time and save the design cost. Sun has analyzed the tolerance to the circuit accuracy by using the method of moments before[
This paper is organized as follows. After describing the small-signal model and uncertainties of the intrinsic elements in Sect. 1,the cutoff frequency,maximum oscillation frequency and S-parameters analysis based on Monte-Carlo methods is derived in Sect. 2. Sect. 3 presents the results and discussion. Finally,conclusion is drawn in Sect. 4.
1 Uncertainty of small-signal model
The uncertainty estimation is based on the small-signal model after de-embedding for HBT devices.
Figure 1.The intrinsic part of π-topology small-signal equivalent circuit model of HBT devices
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Where
where
Direct extraction method is the simplest and most efficient way to build an equivalent model. In the previous paper [
Ignoring the effect of de-embedding,the transfer of uncertainty of the intrinsic part is mainly by the extraction of small-signal elements. Just consider the part in the dotted line of
The sensitivity analysis of the extracted values of the elements should be considered when estimating the uncertainty of the intrinsic elements. To illustrate the uncertainty estimation of model parameters with sensitivity directly is difficult. Therefore,combined with sensitivity analysis and VNA instrument uncertainty,the uncertainty of each element can be expressed concretely [
It is worth noting that the uncertainty of each element varies with frequency. But when extracting parameters,the author refers to the point of the minimum uncertainty as the best extraction frequency point. So we only need to consider the minimum uncertainty of each element when we estimate the accuracy of the model circuit performance.
Take
Figure 2.Estimated relative uncertainty of intrinsic parameters Cπ versus frequency
The extracted values and the minimum uncertainties
As can be seen from
2 Theoretical analysis
2.1 Monte-Carlo methods
Monte-Carlo method is a numerical statistics method used to analyze the fluctuation range of device characteristics,which is a mathematical analysis method based on probability. It can not only estimate the accuracy of microwave performance,but also analyze the tolerance in the opposite direction based on the design requirements. By constructing or describing the random distribution of each model parameter in the range of uncertainty and sampling from the random distribution of all parameters,the aggregation of output functions can be established [
Simple mathematical expressions are used to indicate this abstract description. For the random distribution of each variable,Gaussian distribution is selected to express independent variables according to the possible distribution of the elements in the circuit. Suppose there are N numbers of elements in an equivalent circuit,and the expression of each variable can be written as:
where
here M is the number of samples. As long as M is large enough,the aggregation of
In the HBT small-signal model described in
2.2 Calculation of microwave characteristics
In order to show the accuracy more intuitively,it is necessary to quantify the microwave characteristics. As the important frequency characteristics of devices,
The physical meanings of
where
There are also two ways to define
All the above parameters expressed by the formulas are based on the small-signal circuit model of HBT and its modeled S-parameters. The accuracy range of the microwave characteristic
3 Results and analysis
3.1 S-parameters analysis
The comparison between modeled S-parameters and measured S-parameters is one of the criteria to test the correctness of small-signal model in the process of parameter extraction. On the basis of the accuracy of the whole model obtained by Monte-Carlo method,it must impact the magnitude and phase of the four S-parameters.
The experimental observations show that the uncertainties of elements have little effect on the phase of each S-parameter,which hardly fluctuates.
Figure 3.Magnitude accuracy of S-parameters(45MHz∼40GHz)
It can be observed from
The main parameters affecting the microwave performances of the device are
Figure 4.Positive and negative deviation rates of S-parameters(45 MHz∼40 GHz)
In order to quantify the fluctuation range,a concept called deviation rate is proposed. The definition of deviation rate is as follows:
Both positive and negative deviation rates change with frequencies,which are also shown in
3.2 Current gain cutoff frequency analysis
From the definition of characteristic frequency(current gain cutoff frequency)
Figure 5.H21 versus frequency and accuracy range of current gain cutoff frequency
It is worth noting that
Informed of
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3.3 Maximum oscillation frequency analysis
Figure 6.U and
It can be seen from
And the accuracy ranges of
It can be seen from
4 Conclusion
This paper presents an accuracy estimation method for microwave performance of InP HBTs based on Monte-Carlo analysis. Based on the π-topology small-signal model,the intrinsic elements transmit the uncertainty of measurement parameters. Through the derivation and analysis of Monte-Carlo method,an approach to estimate the overall microwave performance of the model is obtained. Accuracy ranges of the modeled S-parameters,current gain cutoff frequency and maximum oscillation frequency are presented in details. The results reflect the uncertainty of VNA measurement S-parameters affects the microwave performance in different degrees. It is necessary to evaluate the performance of devices before the circuit design,and Monte-Carlo analysis is an important index. Thus,some EDA simulation tools have the function of Monte-Carlo simulation to improve reliability. This paper mainly from the statistical point of view on the Monte-Carlo analysis,combined with sensitivity and uncertainty,analyzed the accuracy performance of microwave performance.
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Ke-Jing CAO, Ao ZHANG, Jian-Jun GAO. Accuracy estimation of microwave performance for InP HBTs based on Monte-Carlo analysis[J]. Journal of Infrared and Millimeter Waves, 2022, 41(2): 430
Category: Research Articles
Received: Aug. 9, 2021
Accepted: --
Published Online: Jul. 8, 2022
The Author Email: Jian-Jun GAO (jjgao@ee.ecnu.edu.cn)