# An Aging Small-Signal Model for Degradation Prediction of Microwave Heterojunction Bipolar Transistor S-Parameters Based on Prior Knowledge Neural Network

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Device Structure and Experiment

_{CBO}to the BC junction can allow the device to operate properly without breakdown and cause non-negligible degradation [16]. We have chosen about 70–90% of the breakdown voltage for high-field stress aging. The breakdown voltage of the InP HBT device used is about 4.7 V; thus, we apply constant reverse bias voltages of 3.4–4.3 V to the BC junction of the device and select four stress biases in steps of 0.3 V to perform S-parameter tests after up to 300 minutes of stress application while keeping the BE junction open. To accurately assess the effect of high-field electrical stress on device RF characteristics, a Rohde & Schwarz ZVA 50 Network Analyzer controlled by IC-CAP 2018 software is employed for S-parameter testing, and the ground–signal–ground (GSG) microwave probe was used in the on-chip measuring process. In addition, to correct the error terms introduced by the test equipment itself, a system error calibration is required before testing with VNA. In this paper, the on-chip test of InP HBT S-parameters is a two-port network test system, and to obtain the actual S-parameters of the DUT, the TOSM (Through, Open, Short, Match) system error calibration method [17,18] is used to remove the parasitic elements introduced by the equipment components such as test cables and probe tips before the reliability test. It is worth noting that the calibration of the RF on-chip test platform is performed before the electrical stress is applied. Also, attention is paid to maintaining a stable test environment during the test to ensure that the calibration conditions do not deviate. All of the above on-chip tests are performed on the CASCADE MICROTECH Summit 11000 Prober, as shown in Figure 3.

## 3. Model Technique

#### 3.1. Prior Knowledge Construction

_{pbe}, C

_{pce}, and C

_{pbc}are the base-emitter, collector-emitter, and base-collector parasitic capacitances, respectively. L

_{b}, L

_{c}, and L

_{e}are the lead inductances associated with the base, collector, and emitter, respectively. The above pad parasitic parameters are extracted by the Open and Short Test Structure method described in [20]. R

_{b}, R

_{c}, and R

_{e}are the series resistances associated with the base, collector, and emitter, respectively, which can be determined in the cut-off condition measurements [21]. After peeling off the parasitic elements mentioned above, the periphery of the equivalent circuit features extrinsic distributed elements, in which C

_{bcx}is the base-collector distributed capacitance and C

_{bex}is the base-emitter distributed capacitance. Intrinsic model elements include the dynamic base resistance R

_{bi}, the dynamic base-emitter resistance R

_{be}, the intrinsic base-emitter capacitance C

_{be}, the intrinsic base-collector capacitance C

_{bc}, the DC transconductance G

_{m0}, and the delay time τ. We use the peeling algorithm from our earlier work [19] to obtain the values of these extrinsic and intrinsic elements.

_{aging}is the value of key model parameters after degradation, ΔP is the amount of degradation of key model parameters, and P

_{initial}is the value of key parameters in the fresh state. A

_{O}is the accelerated degradation saturation factor, a is the degradation acceleration factor, and μ is the degradation acceleration index factor. The degradation equations for the above key parameters are substituted back to the model topology shown in Figure 4 to obtain the degradation trend of the S-parameters of the device, and we use this rough degradation law as a priori degradation knowledge to improve the quality of machine learning-assisted aging modeling mapping.

#### 3.2. Dual-Extreme Learning Machine(D-ELM) Structure

_{CE}, I

_{b}), and NETWORK2 has additional reliability parameter inputs (V

_{CB,stress}, t

_{stress}). Thus, we use NETWORK1 and NETWORK2 to characterize the fresh S-parameters and the degradation of S-parameters in HBTs, respectively. Then, the two sets of results are summed to obtain the aging S-parameters.

#### 3.3. PKNN-Based Aging Modeling Method

_{ij}′ is the output of the coarse model degradation obtained from the aging equivalent circuit. The corresponding training target matrices of the two networks are shown in Equations (5) and (6) below:

_{ij}

^{init}is the value of S-parameters in the fresh state. S

_{ij}

^{degr}is the value of S-parameters after degradation.

^{N1}and β

^{N2}are the connection weight matrices between the hidden and output layers to be solved during the neural network training, respectively. The optimal weight between the hidden and output layers can be found by obtaining the Moore–Penrose generalized inverse [23] of the output matrix of the hidden layer to complete the training of the ELM network. In summary, we can obtain the objective function of the proposed PKNN-based D-ELM network training as shown in Equation (16), and its least squares solution can be obtained from Equation (17).

#### 3.4. Model Optimization

_{mea}and y

_{pre}represent the measured output and the predicted output of the neural network, respectively. N is the total number of training data.

_{x}and mutation probabilities p

_{m}are set to be adaptive, as shown in Equations (19) and (20):

_{max}is the maximum fitness in the population. f

_{avg}is the average fitness of the population. f

_{max}–f

_{avg}reflects the convergence state of optimization. f is the larger fitness of the two individuals to be reorganized. f

^{′}is the fitness of the individual to be mutated. k

_{1}, k

_{2}, k

_{3,}and k

_{4}are constants less than 1.

_{x}and p

_{m}are reduced to preserve them; for low-quality individuals with low fitness, p

_{x}and p

_{m}are increased to eliminate them. In addition, in the early stage of iterative optimization, the population needs a large reorganization and mutation probability to achieve a fast search for the optimal solution, while in the late stage of convergence, the population needs a small reorganization and mutation probability to enable the population to converge quickly after searching for the optimal solution. Then, the superior individuals in the offspring were reinserted into the paternal generation to form a new population. The above process will not stop until the number of training cycles satisfies the number of evolutionary generations set in the experiment. Finally, we decode to obtain the optimal values of the weight between the input layer and the hidden layer and the bias of the hidden neurons. At this point, we can obtain the trained AGA-D-ELM neural network model to predict the aging S-parameters.

## 4. Results and Discussion

_{mea}and y

_{pre}represent the measured output and the predicted output of the neural network, respectively. N is the number of data.

^{2}InP DHBT devices under the bias of (V

_{CE}= 1.6 V, I

_{b}= 200 μA) and subjected to two stress conditions: (V

_{CB,stress}= 3.4 V, t

_{stress}= 250 min) and (V

_{CB,stress}= 4.3 V, t

_{stress}= 310 min). The model predictions of the degradation of the real and imaginary parts of each component of S-parameters are compared with the corresponding test results using the NETWORK2’ and NETWORK2 trained in the second and third sets of experiments, respectively. In addition, to further complement the predictive effectiveness of the proposed model, we present the simulation results of the model in Figure 12b and Figure 13b for the magnitude and phase of the degradation of each S-parameter component in the Advanced Design System (ADS). The simulation accuracy of NETWORK2’ on the degradation of S-parameters is still able to reach or even be better than that of NETWORK2 after the dependence on reliability training samples is reduced. It can be seen that the proposed PKNN network structure is able to predict the degradation trend of the S-parameters of the HBT devices after the accelerated aging well, which provides guidance for the establishment of a highly efficient and accurate aging model for microwave devices in practical applications.

## 5. Conclusions

^{2}InGaAs/InP DHBTs in the frequency range of 0.1–40 GHz.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 10.**Evolution of fitness during optimization for (

**a**) NETWORK1, (

**b**) NETWORK2, and (

**c**) NETWORK2’.

**Figure 11.**Comparison between predicted and measured results of the S-parameters in the fresh state from 0.1 to 40 GHz.

**Figure 12.**Comparison between predicted and measured results of (

**a**) real and imaginary parts and (

**b**) magnitude and phase of degradation of S-parameters from 0.1 to 40 GHz at V

_{CB,stress}= 3.4 V, t

_{stress}= 250 min.

**Figure 13.**Comparison between predicted and measured results of (

**a**) real and imaginary parts and (

**b**) magnitude and phase of degradation of S-parameters from 0.1 to 40 GHz at V

_{CB,stress}= 4.3 V, t

_{stress}= 310 min.

Network Types | Time Required Per Training Session(s) | Residual Error (%) |
---|---|---|

MLP | 11.72 | 8.22 |

D-ELM | 9.64 | 3.93 |

Group | Training Networks | Knowledge Samples | Knowledge Injection Share (%) | Total Sample Size | Training Data Share (%) | Training Sample Sets | Test Sample Sets |
---|---|---|---|---|---|---|---|

1 | ELM | 0 | 0 | 2412 | 50 | 1206 | 1206 |

2 | PKNN | 4824 | 50 | 2412 | 50 | 1206 | 1206 |

3 | PKNN | 9648 | 100 | 2412 | 50 | 1206 | 1206 |

4 | BP | 0 | 0 | 2412 | 50 | 1206 | 1206 |

5 | RBF | 0 | 0 | 2412 | 50 | 1206 | 1206 |

6 | ELM | 0 | 0 | 2412 | 80 | 1930 | 482 |

7 | PKNN | 9328 | 100 | 1458 | 80 | 1166 | 292 |

Group | MRE for Test Sets (%) | Total Error (%) | |||||||
---|---|---|---|---|---|---|---|---|---|

ΔRe(S11) | ΔIm(S11) | ΔRe(S12) | ΔIm(S12) | ΔRe(S21) | ΔIm(S21) | ΔRe(S22) | ΔIm(S22) | ||

1 | 1.27 | 2.88 | 1.22 | 3.84 | 4.17 | 4.01 | 4.25 | 5.13 | 3.35 |

2 | 0.96 | 1.39 | 1.12 | 2.98 | 1.82 | 1.37 | 2.59 | 1.97 | 1.78 |

3 | 0.16 | 0.62 | 0.38 | 1.86 | 1.28 | 0.40 | 1.10 | 1.04 | 0.86 |

4 | 2.59 | 8.07 | 9.69 | 13.36 | 0.95 | 1.31 | 3.84 | 2.25 | 5.26 |

5 | 2.55 | 3.27 | 10.43 | 19.52 | 0.23 | 1.17 | 1.96 | 6.24 | 5.67 |

6 | 1.04 | 1.40 | 0.76 | 2.73 | 1.94 | 2.29 | 3.72 | 2.38 | 2.01 |

7 | 0.51 | 0.38 | 0.32 | 1.47 | 0.67 | 0.47 | 0.98 | 1.09 | 0.74 |

Comparison | NETWORK1 | NETWORK2 | NETWORK2’ |
---|---|---|---|

Fitness range | 2.03~2.38 | 2.11~2.84 | 7.30~9.55 |

Total duration of optimization (s) | 1956 | 5845 | 3728 |

Number of iterations to reach convergence | 864 | 845 | 583 |

Convergence time (s) | 1690 | 4939 | 2173 |

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## Share and Cite

**MDPI and ACS Style**

Cheng, L.; Lu, H.; Yan, S.; Liu, C.; Qiao, J.; Qi, J.; Cheng, W.; Zhang, Y.; Zhang, Y.
An Aging Small-Signal Model for Degradation Prediction of Microwave Heterojunction Bipolar Transistor *S*-Parameters Based on Prior Knowledge Neural Network. *Micromachines* **2023**, *14*, 2023.
https://doi.org/10.3390/mi14112023

**AMA Style**

Cheng L, Lu H, Yan S, Liu C, Qiao J, Qi J, Cheng W, Zhang Y, Zhang Y.
An Aging Small-Signal Model for Degradation Prediction of Microwave Heterojunction Bipolar Transistor *S*-Parameters Based on Prior Knowledge Neural Network. *Micromachines*. 2023; 14(11):2023.
https://doi.org/10.3390/mi14112023

**Chicago/Turabian Style**

Cheng, Lin, Hongliang Lu, Silu Yan, Chen Liu, Jiantao Qiao, Junjun Qi, Wei Cheng, Yimen Zhang, and Yuming Zhang.
2023. "An Aging Small-Signal Model for Degradation Prediction of Microwave Heterojunction Bipolar Transistor *S*-Parameters Based on Prior Knowledge Neural Network" *Micromachines* 14, no. 11: 2023.
https://doi.org/10.3390/mi14112023