Small-Signal Model Parameter Optimization Process

Figure 7 summarizes the optimization flow of the IGWO algorithm for small-signal model parameters.

Figure 7

Figure 7 Flowchart of the IGWO small-signal model parameters optimization process.

Step 1. Initialization and setting related parameters.

The number of dependent-bias model parameters including intrinsic parameters and Cbcx constitute the solution space D of the IGWO algorithm. The location of the grey wolf population is initialized by Equation (38).

Where rand represents the random value in the interval [0,1]. (Xi)U and (Xi)L represent the lower bound (LB) and upper bound (UB) of dependent-bias model parameters. The total number of iterations Titeration and the population size N are set to constant values. The control parameters, such as A, C and a1, are randomly generated by the algorithm.

Step 2. Calculation of the fitness value of each grey wolf and saving the positions of the three best grey wolves.

The IGWO algorithm is based on the fitness value of individuals, so a reasonable fitness function can reflect the relative superiority and inferiority of each. The top three best grey wolves are selected according to the fitness value and their position vectors Xα, Xβ and Xδ are saved.

Step 3. Updating the positions of other individuals.

Equations (34) through (36) are used to update the positions of other individuals except the top three best grey wolves.

Step 4. Updating parameters.

Equations (31), (33) and (37) are used to update A, C and a1, respectively.

Step 5. Determination of termination conditions.

If the predetermined maximum number of iterations Titeration is reached, the calculation stops and the optimal position Xα and the optimal function value is output; otherwise, the executions of Steps 2 through 4 are repeated.

MODEL VALIDATION

InP HBTs with emitter areas of 1×15 μm2 are used to verify the effectiveness of the proposed small-signal model. S-parameters are measured with an Agilent 8510C vector network analyzer and an Agilent B1500A semiconductor device analyzer provides DC bias for the device under test. The testing process is controlled by IC-CAP software. In this work, the dependent-bias model parameters of the InP HBT are optimized in the frequency range of 0.1 to 40 GHz using the GA, GWO and IGWO under two bias points (VCE = 0.7 V, IC = 4 mA and VCE = 2.0 V, IC = 7.5 mA). The used LB and UB with their optimized results are shown in Tables II and III.

TABLE II - OPTIMIZED PARAMETER VALUES FOR VCE=0.7 V, IC=4 mA

TABLE III - OPTIMIZED PARAMETER VALUES FOR VCE=2.0 V, IC=7.5 mA

The comparisons of measured S-parameters with simulated results adopting the GA, GWO and IGWO from 0.1 to 40 GHz are shown in Figures 8 and 9. It is apparent that IGWO-modeled S-parameters exhibit better agreement with measured results over the entire frequency range than GA and GWO-modeled S-parameters.

Figure 8a

 

Figure 8b

Figure 8 Comparison of measured and simulated S-parameters for VCE = 0.7 V, IC = 4 mA: real (a) and imaginary (b).

Figure 9a

 

Figure 9b

Figure 9 Comparison of measured and simulated S-parameters for VCE = 2.0 V, IC = 7.5 mA: real (a) and imaginary (b).

The accuracy of each method can be obtained more intuitively by calculating the S-parameter errors between the measured and the simulated values. The expression for the calculated error is given in Equation (39).

where Sm and Sc are the measured and simulated S-parameters, respectively. N is the total number of simulated points. The error results of different algorithms are shown in Table IV. The mean error obtained from IGWO is the smallest, verifying its superiority over the other approaches.

TABLE IV - ERROR RESULTS FOR DIFFERENT BIAS POINTS

CONCLUSION

The GWO algorithm is improved and successfully applied to optimize small-signal model parameters of HBT devices. The IGWO algorithm uses a nonlinear convergence factor based on the tangent function description instead of the linear decreasing convergence factor of the GWO algorithm. This effectively balances the global and local search ability.

The validity of the method is verified with an InP HBT in the frequency range of 0.1 to 40 GHz for two different bias conditions. It is proved that the IGWO-based optimization approach can yield more accurate small-signal model parameters than the GA and GWO algorithms. The algorithm is implemented in MATLAB and can be easily used to optimize small-signal model parameters of HEMT, SiC and MOSFET devices.

ACKNOWLEDGMENT

This work was supported by the Foundation of the Department of Science and Technology of Henan Province (Grant No. 222102210172, 212102210286).

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