a new method for model parameter identification

A NEW METHOD FOR MODEL PARAMETER IDENTIFICATION J. Ladvánszky, A. Hilt Innovation Company for Telecomm., H-1142 Budapest, Ungvár u. 64-66. Hungary Abstract: The problem is to determine parameter values of linear lumped element circuit model of microwave devices. An overdetermined set of linear equations for nominator and denominator coefficients of entries of the admittance matrix of the circuit model is built up and the least squares error solution is found. Advantages of our method are its speed and accuracy. 1. INTRODUCTION The problem is to find parameter values of linear lumped element circuit model of microwave devices. S-parameters at nfr values of frequency and the structure of the circuit model of the device are given. Known solutions of this problem are extraction [ACFET (5)], optimisation [Valtonen et al. (2)] or combination of the two methods. Approximate and exact kinds of extraction are known. The approximate extraction is as follows. Explicit expressions for S, Y, Z or other parameters of the circuit model are determined. These expressions are simplified substituting characteristic frequency values and/or omitting negligible terms at characteristic regions of frequency. Characteristic frequencies are resonant or antiresonant frequencies. Then explicit expressions for some of the model parameters are written if possible and solved. Applicability depends on the actual model parameter values. The number of model parameters that can be identified by approximate extraction is not greater than the sum of the resonant and the antiresonant frequencies and characteristic regions of frequency, multiplied by two, for every Sparameters. The exact extraction is explicit solution of circuit equations for model parameters. Model parameter values are calculated at all frequency values, then the arithmetic mean of the model parameter values are calculated over the set of frequencies of the S-parameters. The number of model parameters that can be identified applying exact extraction is not greater than the port number multiplied by two, if the explicit solution is obtained at one frequency value. First the approximate extraction is used and the model structure is simplified. Then, for the remaining part of the circuit model, the exact extraction is applied. Characteristics of model parameter extraction are that • specific measurements may be needed, • presently a fully automatic algorithm for extraction is not known. Application of extraction heavily depends on the kind and the type of the device to be modelled. A considerable amount of non-algorithmic work is needed for the preparation of the equations of model parameter extraction for the actual conditions, • results may depend on the kind of approximations, • calculations are non-iterative thus local minima cannot occur, • after the explicit expressions for the model parameter values are known, consumption of the computer time is smaller than any other methods of model parameter identification. Optimisation is a search for a global extreme value of an object function. Extreme value is searched over a domain of model parameter values. The object function is a weighted sum of the norm of the differences between the actual and the desired values of quantities characterising the circuit model. Initial values and the domain of model parameter values are given. When the object function is prescribed, requirements of the circuit design are taken into account. The optimisation algorithm is selected. Then actual values of model parameters are modified iteratively depending on the chosen algorithm of the optimisation. Characteristics of model parameter optimisation are that • specific measurements are not necessary, • the general form of the object function and the optimisation algorithm is known. At the beginning of the computations, actual form of object function and the actual optimisation algorithm is selected within limitations, • results may depend on the actual form of the object function and the optimisation algorithm, • there is a possibility of finding local minima, • consumption of computer time may be considerable. In this paper a method that differs from either extraction or optimisation is presented. The new method is based on the fact that entries of the admittance matrix of lumped element circuit models are rational fractions of frequency with common denominator. Nominator and denominator coefficients are polynomials of the model parameters. Thus, if admittance matrices of the circuit model are known at sufficiently large number of frequencies, then an overdetermined set of linear equations can be formulated for the coefficients. Finding the least square error solution of this set of equations, another set of equations is given for the model parameters that is independent of frequency. This set of equations can be solved by optimisation or selecting coefficients of lowest possible degree, closed form solution may be found. This method is compared to extraction and optimisation in examples. Modelled devices are a photodiode and a MESFET under illumination. Objectives of comparison are speed and accuracy. The method of comparison is that device models are simulated with random error added to port voltages and currents. Then model parameter values are determined applying the three methods. Speed and accuracy are compared for different maximum relative amplitudes of the random error. 2. THE NEW METHOD FOR MODEL PARAMETER IDENTIFICATION 1

The frequency values fi, i=1,2, ... nfr, where nfr is the number of frequency values, and the complex reflection coefficients corresponding to these frequency values are given. Admittance values y(ωi) are calculated from the reflection coefficient values at fi (ωi=2πfi). The admittance of a linear, lumped element one-port is expressed as n = n max

∑a

y( jω i ) =

n

( jω i ) n

d

( jω i )

n = n min

(2.1)

d = d max

∑β

d

d = d min

where an and βd are real. We assume that

d min = 0

(2.2)

Then we can rearrange (2.1) to obtain

β0 = 1

(2.3)

A further rearrangement of (2.1) yields

[

y ( j ω i ) = a n min ( j ω i ) n min +...+a n max ( j ω i ) n max − y ( j ω i ) β 1 j ω i +...+β d max ( j ω i ) d max

]

(2.4)

According to (2.4), a set of linear equations is written for the nominator and denominator coefficients as unknowns:

Ax = b

(2.5)

where entries of the matrix A are given in Appendix 1,

x T = α nmin

... α nmαx

β1 ... β dmαx

(2.6)

bT = Re( y ( j ω 1 )) Im( y ( j ω 1 )) ... Re( y ( j ω nfr )) Im( y ( j ω nfr )) (2.7) where T denotes matrix transpose. The number of frequencies is large in order to provide that the set of equations (2.5) be overdetermined. The number of equations is greater than the number of unknowns. Thus a least squares error solution can be found by minimising

Ax − b in the following way

[Peters and Wilkinson (1)]:

x = ( A T A ) −1 A T b

(2.8)

where Ax − b denotes the L2 norm of the column matrix Ax − b . The nominator and denominator coefficients that are expressed as polynomial of the model parameters form another set of equation for the model parameters. This set of equations is solved for the model parameter values. Entries of the admittance matrix of a multiport circuit model have common denominator. Thus the number of columns of the admittance matrix can be reduced (Appendix 2). 3. COMPARISON OF OUR METHOD TO EXTRACTION AND OPTIMISATION IN EXAMPLES In the examples circuit models of microwave devices are identified applying extraction, optimisation and our method. The objective of comparison between them is speed and accuracy. Speed of the methods is characterised by the total time consumption that is needed for the actual computed solution of the identification problem and the related input-output operations. Accuracies of the methods are compared in the following way. Random error is added to port voltages and currents in the analysis. Two different relative amplitudes of the error are used. Identification methods are applied to the analysis results. Relative difference between circuit parameter value in the analysis and in the optimisation characterises accuracy. Some details of implementation are described in Appendix 3. Example 1: A 5 element circuit model of a photodiode is shown in Fig. 1 [Liang and Aitchison (4)]. The photodiode is under modulated laser illumination. Frequencies of laser modulation and port excitation signals are identical. Small signal reflection coefficient val2

ues in the 2-18 GHz frequency band are computed. Model parameter values are determined using extraction, optimisation and our method. Model parameter values are extracted from the reflection coefficient values at 200 MHz, 5 GHz and 15 GHz. Model parameters at three illumination intensities that generate photocurent amplitude of Iph = 10, 100 and 1000 µA are identified. Random error of three different levels is added to port voltage and current. Extraction, optimisation and our method are compared in Tables 1, 2 and 3. Novelty in this example is that our method can be applied for several illumination intensities simultaneously by separating nominator coefficients in (2.4) corresponding to different illumination intensities. Minimum, maximum and R.M.S. relative errors of the model parameter values are compared in Tables 4, 5 and 6. Best values are written in bold letters. In Tables 4, 5 and 6 the numbers of smallest error values are 1, 5 and 21 for extraction, optimisation and our method, respectively. Extraction is the quickest and optimisation is the slowest from the three methods. Example 2: Circuit model of a MESFET under illumination, shown in Fig. 2, is investigated. The three methods are compared in the 200 MHz-18 GHz frequency range. Frequency step of 200 MHz is chosen, thus the number of frequency values equals to 90. Three levels of laser power are applied. Random error of three different levels is added to the port voltages and currents. Table 7 shows the comparison between element values. In Table 8 minimum, maximum and R.M.S. relative differences between the model parameter values determined by the three methods and that in the simulation are compared. In Table 8 the numbers of the smallest error values are 0, 2, 7 for extraction, optimisation and our method, respectively. S-parameters in the case of 5% error are plotted in Fig. 3. 4. CONCLUSIONS A method is given for identification of linear lumped element circuit model parameters of one- and two-port electrical devices. The speed of the method is between the speed of extraction and optimisation. Advantage of the method is that R.M.S. relative differences between model parameter values in identification and in simulation are, in most cases, smaller than that in the case of extraction and optimisation. Refinements of the method are intended to be included in the oral presentation. 5. ACKNOWLEDGEMENTS This work has been done with the support of the Innovation Company for Telecomm. and the Department of Microwave Telecommunications of the Technical University of Budapest. Special thanks to Dr. A. Baranyi for his scientific supervision during common research work with the authors. REFERENCES (1) G. Peters and J. H. Wilkinson: „The least squares problem and pseudo-inverses”, 1970, The Computer Journal, Volume 13, Number 3, pp. 309-316 (2) M. Valtonen et al.: „APLAC - An Object Oriented Analog Circuit Simulator and Design Tool”, 1992, Reference Manual, User´s Manual, Systems Simulation Manual, Helsinki University of Technology (3) DesignSoft Inc.: „TINA 3.0 - Toolkit for Interactive Network Analysis”, 1995, User′s Guide (4) P. J. Y. Liang, C. S. Aitchison: „Improvement in Microwave to Optical Communication Systems Interfaces”, 1995, Progress Report, Brunel University (5) Technical University of Bologna, Department of Electronics, Informatics and Systems Sciences: „ACFET program” APPENDIX 1 Taking the real and imaginary parts of (2.4) two equations from the set of equations (2.5) are obtained:

Re( y ( j ω i )) = α 0 − α 2 ω i2 + α 4 ω i4 − +... + Im( y ( j ω i ))β 1ω i + Re( y ( j ω i ))β 2 ω 2i − Im( y ( j ω i ))β 3 ω 3i − Re( y ( j ω i ))β 4 ω i4 + + − −... Im( y ( j ω i )) = α 1ω − α 3 ω 3i + α 5ω 5i − +... − Re( y ( j ω i ))β 1ω i + Im( y ( j ω i ))β 2 ω 2i + Re( y ( j ω i ))β 3 ω 3i − Im( y ( j ω i ))β 4 ω i4 − + + −...

(A1.1)

(A1.2)

Therefore entries of the matrix A can be expressed as follows: 1

A 2i −1, n − n min +1 = ( −1) 2 1

A 2i , n − n min +1 = ( −1) 2

3 1 sgn(mod( n, 4) − ) + 2 2

3 1 sgn(mod( n, 4) − ) + 2 2

* (ω i ) n mod( n + 1,2)

(A1.3)

* (ω i ) n * mod( n,2)

(A1.4) 3

A 2i −1,d − d min + n max − n min +1 = ( −1)

1 3 1 sgn(mod( d −1, 4) − ) + 2 2 2

* (ω i ) d * (Re( y ( j ω i )) mod(d + 1,2) + Im( y ( j ω i )) mod(d,2))

(A1.5)

A 2i ,d − d min + n max − n min +1 = ( −1)

1 3 1 sgn(mod( d + 2, 4) − ) + 2 2 2

* (ω i ) d * (Re( y ( j ω i )) mod(d,2) + Im( y ( j ω i )) mod(d + 1,2)) (A1.6)

where

i = 1.... n fr

n = n min ... n max

d = d min + 1... d max

(A1.7)

Remainder of the division of x by y, where x and y are integer numbers, is denoted by mod(x,y).

sgn( x ) = 1 if x > 0, 0 if x = 0, − 1 if x < 0

(A1.8)

APPENDIX 2 In the case of models of two-port devices (2.5) can be rewritten for the four entries of the admittance matrix. In this case size of the matrix A is 8nfr by (n11max-n11min+1+ n21max-n21min+1+ n12max-n12min+1+ n22max-n22min+1+4dmax+4). Entries of the admittance matrix have common denominators, thus the size of A can be reduced to 8nfr by (n11max-n11min+1+n21maxn21min+1+n12max-n12min+1+ n22max-n22min+1+dmax+1). APPENDIX 3 In Example 1, if the illumination is modulated and the frequency of modulation is not equal to the frequency of excitation, then the model is a time-varying circuit. To avoid this situation, modulation and excitation frequencies are chosen equal. Our algorithm is implemented as a problem for APLAC [Valtonen et al.(2)]. In Example 2, expressions of entries of the admittance matrix are computed by a symbolic analysis program [DesignSoft Inc. (3)]. Illumination is not modulated. Effect of illumination on the device is modelled by a DC voltage shift of the gate voltage. Other effects are neglected. Our algorithm is implemented as a problem for APLAC. Rs ∆

Ls o

Cd

Cp

y(jω)

Iph o

Fig. 1. A 5 element photodiode model (3). Notations: Iph-photocurrent, Cd-diode capacitance, Rs, Ls-series resistance and inductance, Cp-parallel capacitance Table 1. Comparison between three methods for model parameter identification of the photodiode under illumination without error on the port voltage and current Model parameter Iph1 (µA) Iph2 (µA) Cd (fF) Rs (mΩ) Ls (nH) Cp (fF) Time consumption (sec)

Assumed value 10 100

Extraction

Our method

9.972

Optimisation 15.151

Extraction

10.001

Assumed value 100

100.002

104.658

200 500

199.956 499.642

1 100 *

1.000 100.067 3.13

Our method

100.002

Optimisation 180.966

100.004

1000

199.775 765.838

200.000 499.237

1.001 99.705 1 min 50 sec

0.998351 99.997 22.41

Extraction

100.001

Assumed value 1000

1000

Optimisation 2011

1000

1208

200 500

200.022 500.016

1 100

0.999995 99.967 3.13

Our method

999.998

10

9.972

14.435

10.001

199.864 703.745

200.000 500.004

200 500

199.503 497.003

200.104 785.113

199.995 500.013

1.001 99.843 1 min 50 sec

1.000 100.000 22.52

1 100

1.000 100.757 3.13

599.482 100.162 3 min 16 sec

1.000 100.001 22.63

1000

* Analyses of the model representing different illumination intensities are performed separately. Table 2. Comparison between three methods for model parameter identification of the photodiode under illumination with 1% error on the port voltage and current Model parameter Iph1 (µA) Iph2 (µA) Cd (fF) Rs (mΩ) Ls (nH) Cp (fF) Time consumption (sec)


Extraction 114.895

Optimisation 23.446

Our method 5.000

Assumed value 100

-158.070

208.082

69.063

1000

200 500

240.100 999.999

199.449 498.795

181.451 586.408

1 100

1.029 40.000 5.88

1.002 99.675 2 min 18 sec

1.009 116.847 20.32

Extraction

Our method 73.466

Assumed value 1000

Extraction

-158.070


1020

2602

978.777

10

114.895

16.812

5.000

200 500

169.093 100.000

199.907 711.390

188.540 556.618

200 500

240.049 100.015

199.424 556.551

188.195 527.445

1 100

1.053 135.429 5.22

1.001 99.740 2 min 35 sec

0.995064 112.561 22.57

1 100

1.045 40.001 4.72

1.002 99.647 3 min 10 sec

1.071 99.434 24.00

4

1020

Optimisation 2183

Our method 990.167

Table 3. Comparison between three methods for model parameter identification of the photodiode under illumination with 5% error on the port voltage and current Model parameter Iph1 (µA) Iph2 (µA) Cd (fF) Rs (mΩ) Ls (nH) Cp (fF) Time consumption (sec)


Extraction

Our method

-302.354

Optimisation 6.821

5.000

Assumed value 100

117.615

226.159

158.787

1000

200 500

50.001 100.037

203.262 567.295

239.087 285.095

1 100

2.451 382.767 6.92

0.983262 105.273 4 min 32 sec

0.991106 57.786 20.98

Extraction 117.615


3.053

1922

200 500

279.980 999.998

1 100

0.930309 40.001 4.67

Our method 151.923

Assumed value 1000

1050

10

198.718 798.871

227.670 369.221

1.006 97.469 2 min 21 sec

0.992457 71.288 19.44

Extraction 3053

Optimisation 2859

Our method 990.167

-302.354

9.956

5.000

200 500

50.030 999.747

202.976 725.414

188.195 527.445

1 100

2.440 396.734 4.61

0.984758 104.959 2 min 17 sec

1.071 99.434 23.84

Table 4. Minimum relative error of the model parameter values Error on port voltage and current Iph1 (µA) Iph2 (µA) Extraction Optimisation Our method

0%

10 100 9.081 µ 1.125 m 31.079 µ

1%

100 1000 4.531 µ 681.532 µ 702.806 n

1000 10 105.269 µ 518.079 µ 1.077 µ

5%

10 100 29.359 m 2.290 m 8.922m

100 1000 19.512 m 465.622 µ 4.936 m

1000 10 19.512 m 2.436 m 5.657 m

10 100 176.147 m 16.309 m 8.594 m

100 1000 69.691 m 5.660 m 7.543 m

1000 10 749.850 m 4.357 m 5.657 m

100 1000 2.053 921.997 m 519.227 m

1000 10 31.235 1.859 500.000 m

100 1000 400.438 m 236.213 m 110.823 m

1000 10 5.250 318.977 m 85.249 m

Table 5. Maximum relative error of the model parameter values Error on port voltage and current Iph1 (µA) Iph2 (µA) Extraction Optimisation Our method

0%

10 100 2.785 m 531.677 m 1.778 m

1%

100 1000 330.961 µ 809.660 m 8.819 µ

1000 10 7.573 m 1.011 84.139 µ

5%

10 100 10.490 1.345 500.000 m

100 1000 2.581 1.602 265.345 m

1000 10 10.490 1.183 500.000 m

10 100 31.235 1.262 587.868 m

Table 6. R.M.S. relative error of the model parameter values Error on port voltage and current Iph1 (µA) Iph2 (µA) Extraction Optimisation Our method

0%

10 100 493.383 µ 123.621 m 477.616 µ

1%

100 1000 74.992 µ 155.004 m 2.091 µ

1000 10 1.725 m 207.051 m 15.517 µ

5%

10 100 1.811 287.527 m 107.062 m

100 1000 454.992 m 287.779 m 53.429 m

1000 10 1.757 228.244m 85.249 m

10 100 5.236 218.208 m 166.400 m

Vph Cgd

G Cgs

D

V Rds

Ri

Cds

gm*V VGS

VDS

S Fig. 2. Linear lumped element circuit model of a MESFET under illumination, without bonding parasitics. Vph is the gate voltage shift caused by illumination. Only values of Cgs and gm change with illumination intensity Table 7. Comparison between three methods for model parameter identification of the MESFET under illumination Error on port voltages and currents Model parameter Cgs1 (fF) Cgs2 (fF) Cgs3 (fF) Ri (Ω) Cgd (fF) gm1 (mS) gm2 (mS) gm3 (mS)

0% Assumed value 346.755 349.960 355.983 10

Extraction

30 14.252 32.408 41.588

1%

5%

Our method

Extraction

346.748 349.953 355.976 10.000

Optimisation 347.568 350.504 356.574 10.112

Our method

Extraction

333.784 354.471 341.290 9.967

Optimisation 350.591 349.622 348.132 11.142

346.755 349.960 355.983 10.000

30.000 14.251 32.408 41.588

30.057 14.286 32.440 41.738

30.000 14.252 32.408 41.588

334.180 332.346 336.981 5.243

30.889 16.397 36.236 36.392

29.890 14.570 32.459 41.401

29.824 14.511 32.779 41.717

5

Our method

333.784 354.471 341.290 9.967

Optimisation 350.591 349.622 348.132 11.142

30.889 16.397 36.236 36.392

29.890 14.570 32.459 41.401

29.824 14.511 32.779 41.717

334.180 332.346 336.981 5.243

Rds (Ω) Cds (fF) Time consumption (sec)

100

100.000

99.826

100.000

96.298

98.121

98.832

96.298

98.121

98.832

200

200.000 6.04

200.204 8 min 26 s

200.000 2. min 33 s

213.366 8.24

207.391 3 min 57 s

212.710 2. min 34 s

213.366 8.24

207.391 3 min 57 s

212.710 2. min 34 s

Table 8. Error of the model parameter values at different error on port voltages and currents Minimum relative error of the model parameter values 0% 1% 5%

Error on port voltages and currents Extraction Optimisation Our method

163.765*µ 973.190µ 4.447n

771.485µ 685.484µ 196.714µ

3.255m 964.954µ 3.110m

Maximum relative error of the model parameter values 0% 1% 5%

48.229µ 11.187m 29.809µ

30.394m 84.977m 9.789m

150.528m 114.227m 475.675m

R.M.S. error of the model parameter values 0% 1% 5%

6.025µ 1.277m 3.338µ

APLAC 6.20 User: TKI, Hungary Jan 28 1996 0.00 FI11 -45.00 [deg ] -90.00

800.0m

-135.0

6.000G

10.000G Freq [Hz]

14.000G

18.000G

-180.0 2.000G

APLAC 6.20 User: TKI, Hungary Jan 28 1996 180.0 FI21 135.0 [deg ] 90.00

1.00

45.00

6.000G

10.000G Freq [Hz]

14.000G

18.000G

0.00 2.000G

APLAC 6.20 User: TKI, Hungary Jan 28 1996 90.00 FI12 45.00 [deg ] 0.00

35.00m

-45.00

6.000G

10.000G Freq [Hz]

14.000G

18.000G

-90.00 2.000G

APLAC 6.20 User: TKI, Hungary Jan 28 1996 0.00 FI22 -45.00 [deg ] -90.00

200.0m

-135.0

6.000G

10.000G Freq [Hz]

14.000G

18.000G

14.000G

18.000G

6.000G

10.000G Freq [Hz]

14.000G

18.000G

6.000G

10.000G Freq [Hz]

14.000G

18.000G

APLAC 6.20 User: TKI, Hungary Jan 28 1996

800.0m S22 600.0m [] 400.0m

0.00 2.000G

10.000G Freq [Hz]


80.00m S12 65.00m [] 50.00m

20.00m 2.000G

6.000G


4.00 S21 3.00 [] 2.00

0.00 2.000G

24.944m 12.614m 48.749m


1.10 S11 1.00 [] 900.0m

700.0m 2.000G

5.7m 8.632m 1.203m

-180.0 2.000G

6.000G

10.000G Freq [Hz]

14.000G

18.000G

Fig. 3. Comparison between extraction, optimisation and our method applied to simulated S-parameters of a six element MESFET model when 5% error was added to port voltages and currents. Markers: simulation: diamonds, extraction: crosses, optimisation: squares, our method: triangles 6