Frequency-Variation Based Method for Parameter Estimation of Transistor Ampliﬁer Akash Rathee and Harish Parthasarathy Abstract—In this paper, a frequency-variation based method has been proposed for transistor parameter estimation in a common- emitter transistor ampliﬁer circuit. We design an algorithm to estimate the transistor parameters, based on noisy measurements of the output voltage when the input voltage is a sine wave of variable frequency and constant amplitude. The common emitter ampliﬁer circuit has been modelled using the transistor Ebers-Moll equations and the perturbation technique has been used for separating the linear and nonlinear parts of the Ebers-Moll equations. This model of the ampli- ﬁer has been used to determine the amplitude of the output sinusoid as a function of the frequency and the parameter vector. Then, applying the proposed method to the frequency components, the transistor parameters have been estimated. As compared to the conventional time-domain least squares method, the proposed method requires much less data storage and it results in more accurate parameter estimation, as it exploits the information in the time and frequency domain, simultaneously. The proposed method can be utilized for parameter estimation of an analog device in its operating range of frequencies, as it uses data collected from different frequencies output signals for parameter estimation. Keywords—Perturbation Technique, Parameter estimation, frequency-variation based method. I. I NTRODUCTION A NALOG circuit designing involves equivalent modelling of various analog devices and simulations. The equiv- alent models require to perform as close as possible to the actual physical models of analog circuits, for various applied input signals of varying voltages, currents and frequencies, under changing environmental conditions. Variation of device parameters in view of applied signal variations and environ- mental conditions (such as variation of temperature) cannot be ruled out. If, the variation in devices parameters is ignored during the circuit design process, then the circuit model and simulation results would not match with the performance of actual physical circuit. Therefore analog devices parameter estimation is of great importance for analog circuit designing and simulations. Various techniques for analog devices pa- rameters estimation have been presented in existing literatures such as parameter estimation based on vectorial large-signal measurements [4], concurrent global optimization technique [2], waveform based parameter estimation [5], analytical ap- proach [6] , using wavelet transform [1] and so on. In [8] combinatorial algorithms for BJT (Bipolar Junction Transistor) Akash Rathee is a Teaching-Cum-Research Fellow in the Depart- ment of Electronics and Communication Engineering, Netaji Subhas Institute of Technology,Sector-3, Dwarka, New-Delhi, India. e-mail: akashrathee 7 @yahoo.co.in Harish Parthasarathy is a Professor in the Department of Electronics and Communication Engineering, Netaji Subhas Institute of Technology,Sector-3, Dwarka, New-Delhi, India. e-mail: harisignal@yahoo.com parameter extraction have been presented. V.B. vats et al. in [7] have presented BJT parameter estimation from state analysis and simulations. Sudipta Majumdar et al. in [3] have presented wavelet based transistor parameter estimation. In this paper we present a parameter estimation method which utilizes noisy measured variable frequency output voltages to estimate transistor parameters. As compared to least squares method of parameter estimation, our proposed method results in improved accuracy of parameter estimation as it takes into consideration the information in time and frequency domain simultaneously. Although, the transistor is represented by Ebers-Moll current equations in the model of common emitter ampliﬁer circuit yet only linear part of each equation is considered in order to obtain linearized transistor model. As the thermal voltage is more affected by thermal ﬂuctuations than the other parameters such as I C0 ,I E0 ,α r ,α f , we have estimated the thermal voltage V T . In the same way other parameters can also be estimated. The proposed method for pa- rameter estimation involves minimization of the squared error energy function calculated by varying the input frequencies. The proposed method applies K.A.M.(Kolmogorov Arnold Moser) technique, during the process of parameter estimation. A brief introduction of the K.A.M. technique is as follows: K.A.M. is the abbreviation of the famous theorem on averaging for quasi-periodic motion due to Kolmogorov Arnold and Moser. They used the technique to analyze the trajectories of ’n’ body gravitational problem. Averaging can be used to determine qualitative characteristics of the motion, like, the average time spent by the trajectories inside a set. In our paper, we use K.A.M. method to calculate the Fourier components averaging of the output waveform at any given frequency. If the output has the waveform: y(t) = ∑ m k=1 A k e jω k t , where, ω 1 ,ω 2 , ..., ω m are incommensurate, then y(t) is quasi- periodic signal and application of K.A.M. gives, A k = lim T −→∞ 1 T  T/2 −T/2 y(t)e −jω k t dt. This paper has been orga- nized in ﬁve sections. In section (2) we have developed frequency-variation based method for transistor parameters estimation. In section (3) this technique has been applied to the common emitter transistor ampliﬁer circuit, in order to estimate the transistor parameters. Section (4) presents simulations and discussions. In section (5) conclusions are presented. II. PARAMETER ESTIMATION USING FREQUENCY- VARIATION BASED METHOD An analog circuit can be represented by a system of multi- variable state equations. The system of equations consists of r World Academy of Science, Engineering and Technology International Journal of Electronics and Communication Engineering Vol:5, No:12, 2011 1823 International Scholarly and Scientific Research & Innovation 5(12) 2011 ISNI:0000000091950263 Open Science Index, Electronics and Communication Engineering Vol:5, No:12, 2011 publications.waset.org/8831/pdf