Solid-State Electronics 52 (2008) 1092–1098

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Solid-State Electronics

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Evaluating MOSFET harmonic distortion by successive integrationof the I–V characteristics

Ramón Salazar a, Adelmo Ortiz-Conde a,*, Francisco J. García-Sánchez a, Ching-Sung Ho b, Juin J. Liou c,d

a Solid-State Electronics Laboratory, Simón Bolívar University, Caracas, Venezuelab ProMOS Technologies Inc., Hsinchu, Taiwanc Department of Electrical and Computer Engineering, University of Central Florida, Orlando, FL, USAd Department of ISEE, Zhejiang University, Hangzhou, China

a r t i c l e i n f o

Article history:Received 18 December 2007Received in revised form 26 March 2008Accepted 29 March 2008Available online 27 May 2008

The review of this paper was arranged byProf. S. Cristoloveanu

Keywords:Integral non-linearity functionHarmonic distortionTHDIP2IP3

0038-1101/$ - see front matter � 2008 Elsevier Ltd. Adoi:10.1016/j.sse.2008.03.018

* Corresponding author. Tel.: +58 212 9064010; faxE-mail address: ortizc@ieee.org (A. Ortiz-Conde).

a b s t r a c t

A new method, which we have named ‘‘Full Successive Integrals Method” (FSIM), is presented for evalu-ating harmonic distortion of semiconductor devices from their static I–V characteristics. To assess themethod’s applicability, it is used to calculate the harmonic distortion components (Hn) from the mea-sured output characteristics of several experimental n-MOSFETs with various channel lengths. The result-ing values of the harmonic components are compared to, and match very well, those obtained throughconventional Fourier analysis techniques. The proposed method’s main appeal is that its implementationis fast and straight forward, and inherently filters out measurement noise. It can be used to calculate dis-tortion harmonics for any desired input level, without having to deal with AC signals or having to performlengthy Fourier type analysis.

� 2008 Elsevier Ltd. All rights reserved.

0. Introduction

The evaluation of harmonic distortion in electron devices is animportant topic both in device [1–4] and circuit design [5]. Conse-quently, the measurement, characterization and study of distortionin MOSFET devices are relevant tasks in order to achieve optimal de-sign. The assessment of distortion has been traditionally carried outby calculating different figures of merit such as total harmonic dis-tortion (THD), nth-order harmonic distortion ratios (HDn), DynamicRange, IP2 and IP3, among others. All these quantification criteria re-quire performing Fourier analysis to AC signals, a relatively complexprocedure that consumes significant time and computationalresources.

Some integration-based methods have been proposed since2002 [6–13] to increase the computational efficiency when calcu-lating accurate values of distortion. Those methods, which are allclosely related and based on the idea of the Integral non-linearityfunction (INLF) [13], circumvent dealing with AC signals avoidingthe associated Fourier analysis. They only require performing rela-tively simple operations, comprising integration on the device’s

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static I–V characteristics. As a result, the computational efficiencyis highly improved over traditional Fourier analysis techniques. Itis important to point out that an underlying implicit assumptionwhen discussing equivalence between traditional figures of meritand these methods is the absence of dynamic processes that wouldselectively influence the AC response of the system. Therefore,some methods may not be accurate if the studied device (or sys-tem) is being operated at high enough frequencies that harmonicdistortion may arise as consequence of transient effects. As anadded advantage an integration-based method also serves to re-duce noise when experimental data is processed.

It was recently reported [7] that some integral methods areused to calculate harmonic components under conditions ofweakly non-linear behavior, and that the THD cannot be accuratelycalculated, unless the input amplitude to the system is sufficientlysmall so that, besides the fundamental, only one harmonic is dom-inant in the output.

An integral-based method, called the ‘‘Successive IntegralsMethod” (SIM), was recently proposed [6] to extract the coeffi-cients of a polynomial function. Once the coefficients are extracted,harmonic distortion can be calculated from them, and thereforeTHD, IP2 and IP3 corresponding to that polynomial functioncan be evaluated. This method was still not fully immune to

R. Salazar et al. / Solid-State Electronics 52 (2008) 1092–1098 1093

experimental noise because it involved operations which did notinclude integration of the device’s static I–V characteristics.

The presently proposed method extends the idea of the SIM inthat it improves the immunity to noise in the experimental data.In addition, it also allows calculation of harmonic distortion fig-ures, THD, IP2 and IP3, without having to restrict the input ampli-tude to the weakly non-linear regime.

In Section 1 of this paper we describe the FSIM and show how itcan be used to calculate harmonic coefficients for any input ampli-tude. A synthetic I–V characteristic is firstly used in Section 2 todemonstrate how applying the FSIM is completely equivalent tousing traditional Fourier analysis. In Section 3, the FSIM will be ap-plied to several measured ID–VDS triode region characteristics ofexperimental n-MOSFETs with 3.3 nm gate-oxide thickness, andchannel lengths ranging from 120 nm to 1 lm, in order to visualizeand compare their real distortion performance and its dependenceon channel downscaling.

1. The Full Successive Integrals Method (FSIM)

1.1. Definition of FSIM

Let y be a function represented by an nth-order polynomial:

y ¼Xn

j¼1

ajxj; ð1Þ

where y and x indicate the output and input (respectively) of a device,and aj are the coefficients of the defining polynomial. In order to keepthe illustration of the proposed method simple, let us assume, with-out loss of generality, that y may be adequately represented by a thirdorder polynomial (n = 3). To use the FSIM on such a function will re-quire applying an operator comprised of the addition of three (in gen-eral n) terms containing successive integrals of the function. Theoperator is defined as follows for this case of n = 3:

Ep � ax2Z x

0ydxþ bx

Z x

0

Z x

0ydxdxþ c

Z x

0

Z x

0

Z x

0ydxdxdx; ð2Þ

where the values of a, b and c are constants chosen (calculated) asdescribed latter. Inspection of (2) indicates that Ep acts as a low-pass filter for any possible high frequency noise that might be pres-ent in the signal, since y is always contained under the integral sign.From this feature comes the word ‘‘Full” in the name of this method.Of course, higher-order polynomials (n > 3) would require applyingan operator similar to (2) but comprised of the corresponding addi-tion of n terms containing up to n successive integrals of the func-tion. Although the FSIM can be used in general to extract thecoefficients of arbitrary non-linear functions represented by poly-nomials, in the present work the emphasis will be focused on theuse of FSIM to calculate harmonic distortion from semiconductordevice’s static I–V characteristics.

It is convenient to normalize (2) by dividing it by x2 (in generalxn�1)

ENP �

Ep

x2 ¼ aZ x

0ydxþ b

x

Z x

0

Z x

0ydxdxþ c

x2

Z x

0

Z x

0

Z x

0ydxdxdx;

ð3Þ

where the superscript N means that this is a normalized operator.Substituting (1) (with n = 3) in (3) leads to

ENP ¼

X3

j¼1

ajxjþ1Fj; ð4Þ

where

Fj ¼a

ðjþ 1Þ þb

ðjþ 1Þðjþ 2Þ þc

ðjþ 1Þðjþ 2Þðjþ 3Þ

� �; ð5Þ

which can be evaluated at j = 1, 2 or 3 to yield F1, F2 and F3,respectively.

A possible way to select the values of a, b and c could be definedwhenever we wish to isolate the contribution of a single coeffi-cient. Let p, m and k represent, in any arbitrary order, the numbers1, 2 and 3. Then, the values of a, b and c, needed to isolate ak andeliminate the other two terms ap and am, are given by the followingsystem of three linear equations:

aðkþ 1Þ þ

bðkþ 1Þðkþ 2Þ þ

cðkþ 1Þðkþ 2Þðkþ 3Þ

� �¼ 1 ð6Þ

aðpþ 1Þ þ

bðpþ 1Þðpþ 2Þ þ

cðpþ 1Þðpþ 2Þðpþ 3Þ

� �¼ 0 ð7Þ

and

aðmþ 1Þ þ

bðmþ 1Þðmþ 2Þ þ

cðmþ 1Þðmþ 2Þðmþ 3Þ

� �¼ 0 ð8Þ

The solution to this linear system of equation yields the values of a,b and c

a ¼ þ6þ 11kþ 6k2 þ k3

pm� pk�mkþ k2 ; ð9Þ

b ¼ �ðk3 þ 6k2 þ 11kþ 6Þðpþ 5þmÞ

pm� pkþ k2 �mk; ð10Þ

c ¼ ð6þ 11kþ 6k2 þ k3Þð3pþ 3mþ pmþ 9Þpm� pk�mkþ k2 : ð11Þ

Keeping in mind that, for simplicity’s sake, we have decided toassume that in this example y is adequately represented by a thirdorder polynomial, it can be easily shown that the three Fouriercoefficients for this example can be written in terms of a1, a2 anda3, as described in Appendix A.

The practical implementation of our procedure is as follows: Toobtain a given coefficient ak, (where k = 1, 2 or 3), a, b and c are tobe evaluated from (9)–(11), using the corresponding values of k, pand m. Then, Hk may be evaluated directly by substituting ak, intothe formulas shown in Appendix A.

In general, distortion analysis needs not refer to a sinusoidal in-put signal. However, harmonic distortion analysis does imply asinusoidal signal, which may or may not have a dc component. Inthe case illustrated here we assume that the input is in general asinusoidal signal with a dc component, xdc, and a peak amplitude,xp. The distortion analysis may therefore be implemented by ana-lyzing the I–V characteristic in the range xdc � xp < x < xdc + xp. Arigorous proof is presented in Appendix B to show that this is in-deed the case. It is obvious that the FSIM method could also be ap-plied to any particular case, including that of a signal with zero DC,by performing a translation of the origin of coordinates as de-scribed in Appendix B.

2. Comparison of FSIM to Fourier analysis using synthetic data

For comparison purposes, a synthetic I–V characteristic wassimulated using a third order polynomial (I = a1V + a2V2 + a3V3 witha1 = 0.1, a2 = 0.05 and a3 = 0.01), and a range of 0 < V < Vmax was se-lected for the distortion analysis, where Vmax is the maximum volt-age. This range implies that in the case of a sinusoidal signal, thesinusoid has a peak amplitude and a DC-level both equal to Vmax/2. Eq. (4) was then applied to the above mentioned synthetic I–Vcurve using the (k-dependent) coefficients a, b and c as given by(9)–(11). The Fourier coefficients H1, H2 and H3 are obtained usingvalues of k = 1, 2 and 3, respectively. It is important to point outthat a numerical integration algorithm, which would produce neg-ligible error, should be used as described in Appendix C.

-30 -25 -20 -15 -10 -5 0 5 10-140-120-100

-80-60-40-20

02040

Out

put-P

ower

(dBm

)

Input-Power (dBm)

Symbols: FourierLines: FSIM

H1

H2

H3

Fig. 1. Harmonic coefficients H1, H2, and H3, as functions of input-power, as calc-ulated with the proposed FSIM (lines) and by traditional Fourier analysis (symbols)using synthetic data. Values in dBm are referred to 50 X, and correspond to sinu-soidal input signals with increasing and equal peak amplitude and DC-level valuesof Vmax/2, in a given range 0 < V < Vmax.

10

1094 R. Salazar et al. / Solid-State Electronics 52 (2008) 1092–1098

Traditional Fourier analysis was also used to calculate the samecoefficients. Fig. 1 shows the comparison between FSIM and tradi-tional Fourier analysis extracted H1, H2 and H3, as input power isincreased. The results presented in Fig. 1 confirm that the FSIMand traditional Fourier analysis produce identical results. Noticethat the coefficients follow a linear trend as input-power is varied.These straight lines corresponding to H1, H2 and H3 have slopes val-ues of 1, 2 and 3, respectively.

3. Comparison of FSIM to Fourier analysis using experimentaldata

3.1. Experimental I–V characteristics

The output ID–VDS characteristics in the triode region for a set ofn-channel MOSFETs having a channel width W = 10 lm, oxide

I D(m

A)

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4Lm = 1 μm

VD (V)0.0 0.1 0.2 0.3 0.4 0.5

I D (m

A)

0

2

4

6

8 Lm = 120 nm

0.5 V 1.0 V

1.5 V

2.5 V

2.0V

VG = 3.0 V

VG = 3.0 V

2.5 V

2.0V

1.5 V1.0 V

0.5 V

Fig. 2. Measured triode region output characteristics for two n-MOSFETs withW = 10 lm, tox = 3.3 nm and two channel lengths, at various gate voltages.

thickness tox = 3.3 nm, and eight different channel lengths rangingfrom 1 lm down to 120 nm, were measured at six different gatevoltages from 0.5 to 3.0 V, using a drain voltage sweep from 0 Vto 0.5 V with increments of 10 mV. Fig. 2 presents the results ofthe experimentally measured output characteristics for two spe-cific devices (Lm = 120 nm and Lm = 1 lm). We have chosen to studythe harmonic distortion in the triode region of the MOSFETs’ out-put characteristics as an illustrative example. However, this choicein no way restricts the method’s applicability. The FSIM may be ap-plied to transfer or output characteristics in any region of opera-tion, or to any other type of semiconductor device’s I–Vcharacteristics, or even to a more complex system, such as anamplifier circuit.

3.2. Comparison of FSIM to traditional Fourier analysis

When the FCs are calculated for a system or device with an I–Vtransfer characteristic represented for an nth-order polynomialwhere the polynomial coefficients are constants, the output-powerof the FCs varies linearly as the input-power is varied until the in-put-power is large enough to induce compressive or expansivebehavior. Such linear dependence was shown in Fig. 1 when syn-thetic data from a 3rd-order polynomial was used. However, it isnot reasonable to expect that the FCs calculated from an experi-mental I–V curve would have a fully linear variation through an en-tire input-power range (when the input-power range is largeenough). This is illustrated in Fig. 3 where the FCs for experimentalI–V curves have been calculated using both; the traditional Fourieranalysis and the FSIM.

In Fig. 3 the FCs obtained from Fourier analysis were calculatedusing five different sets of polynomial coefficients; one set for eachof the following input ranges: 0 V < VDS < 0.1 V; 0 V < VDS < 0.2 V;

-50

-40

-30

-20

-10

0

Input-Power (dBm)-8 -6 -4 -2 0 2 4 6 8

Out

put-P

ower

(dBm

)

-60

-40

-20

0H1

H2

H3

Symbols: FourierLines:FSIM

VG = 1VLm=120nm

VG = 1.5V

Out

put-P

ower

(dBm

)

H1

H2

H3

Fig. 3. Output harmonic coefficients (Hn) as functions of Input-Power, produced bythe triode region output characteristics of an n-channel MOSFET with Lm = 120 nm,at two gate voltage values, as calculated using the FSIM (lines) and using traditionalFourier analysis (symbols). The values in dBm are referred to 50 X, and correspondto sinusoidal input signals with increasing and equal peak amplitude and DC-levelvalues of Vmax/2, in a given range 0 < V < Vmax.

Lm (nm)200 400 600 800 1000

0.00

0.05

0.10

0.15

0.20

0.25

Vdc=Vp=0.25V

THD

, HD

2an

d H

D3

0.00

0.01

0.02

0.03

0.04

THDHD2

HD3

VG=1.5V

Vdc=Vp=50mV

Fig. 5. THD, HD2 and HD3 as functions of channel length, for a fixed gate voltage, attwo different values of the input signal level Vdc = Vp. These values were calculatedapplying the FSIM to the measured triode region output characteristics of eightexperimental n-channel MOSFETs of various channel lengths (see Fig. 2).

R. Salazar et al. / Solid-State Electronics 52 (2008) 1092–1098 1095

0 V < VDS < 0.3 V; 0 V < VDS < 0.4 V; 0 V < VDS < 0.5 V. It should beclear from Fig. 3 that both the traditional Fourier analysis andthe FSIM produce very similar results. Therefore, the FSIM can beused as an accurate and computationally efficient method for cal-culation of FCs without having to perform any kind of AC analysis.It is important to notice that since the calculation of FCs has beencarried out with 3rd-order polynomials there are no terms withhigher-order powers. Therefore, the observable change of trendsin H2 and H3 cannot be explained as the effect of higher-order har-monics exerting compression or expansion in the lower-order har-monics H2 and H3. Hence, it is important to highlight theobservation that there are variations in the lower-order FCs thatmay not be caused by higher-order harmonics acting on them.Such variations are intrinsic to the natural response of the deviceor system, which cannot be represented as a unique polynomialwith constant coefficients. In this figure, we have used a polyno-mial with constant coefficients for each input-power.

Using the FSIM we proceeded to characterize the effect of thechannel length (Lm) on the behavior of traditional figures of meritsuch as total harmonic distortion (THD), second harmonic distor-tion ratio (HD2) and third harmonic distortion ratio (HD3). Fig. 4shows the trends of THD, HD2 and HD3 as Lm is varied for a sinusoi-dal signal input with Vdc = Vp = 0.25 V, at two different gate volt-ages. On the other hand, it can be seen in Fig. 4 that forrelatively large channel lengths, the THD, HD2 and HD3 do notchange much as Lm is varied. This behavior is expected when aclassic description of the MOSFET is used. In such a classic model,all the terms of the output characteristic ID(VDS) are scaled by W/L.Therefore, it is expected that when Lm is changed, all FCs wouldchange in the same proportion. Consequently, the THD, HD2 andHD3 (which are ratios of two FCs) should not change as Lm is varied.However, it should be clear from Fig. 4 that when the length of thechannel is short enough, the trends of the THD, HD2 and HD3 devi-ate from a constant value. This suggests that such classical modelof ID(VDS) is no longer valid to describe the behavior of the outputcharacteristic at those channel lengths.

THD

. HD

2 an

d H

D3

0.0

0.1

0.2

0.3

0.4

0.5

THDHD2

HD3

VG=1V

Lm (nm)200 400 600 800 1000

0.000.010.020.030.040.050.06 VG=3V

Fig. 4. THD, HD2 and HD3 as functions of channel length for a sinusoidal signalinput with Vdc = Vp = 0.25 V, at two gate voltages. These values were calculatedapplying the FSIM to the measured output characteristics of eight experimental n-channel MOSFETs of various channel lengths.

Notice that the specific kind of trend followed by THD, HD2 andHD3 at short channel lengths seems to be determined by the valuesof VG and of the input signal level Vdc = Vp. Fig. 5 shows that whenVG is fixed and the input signal level Vdc = Vp is varied, the trendsare similar to those observed in Fig. 4. It is important to note inFigs. 4 and 5 that changes in VG not only modify the trends of

0.05 0.10 0.15 0.20 0.250.00

0.05

0.10

0.15

0.20 Lm=1μ m

0.00

0.05

0.10

0.15

0.20

THDHD2

HD3

VG=1.5V

Lm=120nm

Vdc

=Vp (V)

THD

, HD

2 and

HD

3

Fig. 6. FSIM calculated THD, HD2, and HD3, produced by the triode region outputcharacteristics, as functions of the input signal level Vdc = Vp, at a fixed gate voltage,for two n-channel MOSFETs of different channel lengths.

0.00

0.10

0.20

0.30

0.40

Vdc

=Vp(V)

0.05 0.10 0.15 0.20 0.250.00

0.01

0.02

0.03

0.04

0.05V

G=3V

VG=1V

Lm=120nm

THD

, HD

2 and

HD

3

0.00

0.02

0.04

0.06

0.08

0.10

0.12

THDHD2

HD3

VG=2V

Fig. 7. FSIM calculated THD, HD2, and HD3, produced by the triode region outputcharacteristics, as functions of the input signal level Vdc = Vp, at three gate voltages,for an n-channel MOSFETs of 120 nm channel length.

1096 R. Salazar et al. / Solid-State Electronics 52 (2008) 1092–1098

THD, HD2 and HD3 but also change their magnitudes significantlyas Lm is varied.

In Figs. 6 and 7 we have plotted the THD, HD2 and HD3 asfunctions of the input signal level Vdc = Vp, when Lm is variedfor a fixed VG (Fig. 6) and when VG is varied for a fixed Lm

(Fig. 7). Notice in Fig. 6 that for the same VG the decrease in dis-tortion is not very significant as Lm is increased, whereas in Fig. 7the THD, HD2 and HD3 change substantially as VG is varied for afixed Lm. This implies that at least in the range of120 nm < Lm < 1 lm, the harmonic distortion is more sensitive tochanges in VG than to changes in Lm.

Figs. 6 and 7 show that THD is very similar to HD2. Thismeans that THD is mainly caused by H2, in agreement with whatis observed in Fig. 3, where it is clear that H2� H3, andtherefore HD2 must be the dominant harmonic distortion ratio.Nevertheless, notice in Fig. 7 that for VG = 3 V, HD2 hasdecreased to a point where it is comparable to the magnitudeof HD3.

Notice also that at first glance, it seems that the minimum ofHD3 in Fig. 7 for VG = 1 V might lead to the interpretation that suchminimum is produced by the effect of higher-order harmonics.However, as pointed out before in the discussion, the minimumin HD3 cannot be caused by compression of a higher-order har-

monic since the output characteristics of these devices are beingmodeled by a set of 3rd-order polynomials.

4. Conclusions

We have presented a new integral-based method, which is afurther development in our previously proposed line of integralfunction-based methods. This method has been named the ‘‘FullSuccessive Integral Method” (FSIM) because it is based on succes-sive integrals of the function being analyzed. The present methodis capable of unambiguously extract all the coefficients of any poly-nomial function of any order, within any arbitrary range. Therefore,it can also be used to calculate the corresponding Fourier coeffi-cients (Hk) for a any sinusoidal input signal of any amplitude anddc bias applied to a non-linear I–V characteristic.

The results show presented here have shown that the FSIM is asaccurate as traditional Fourier analysis methods in calculating thevalues of the harmonic distortion terms when processing both syn-thetic data (using a polynomial function) or experimentally mea-sured data.

Since the operator used by the FSIM contains exclusivelyterms with successive integrals of the I–V characteristics understudy, it is expected that this method will be a powerful toolfor distortion analysis, especially in practical situations where itwill inherently filter out the unavoidable noise normally presentin experimental data. This feature alone constitutes a clearadvantage over derivative-based methods of distortion analysis.Furthermore, since the FSIM requires only the I–V characteristicof the device or system to calculate its ensuing Fourier coeffi-cients, all the conventional figures of merit, such as IP2, IP3,THD, HDk, Dynamic Range, and P�1 dB, can be readily and accu-rately calculated without having to make use of any kind of Fou-rier or AC analysis.

Additionally, a set of n-MOSFETs with gate-oxide thickness of3.3 nm and different channel lengths ranging from 120 nm to1 lm was characterized in terms of distortion applying the FSIMto their output characteristics. The results show that increasingthe gate voltage produces significant increase of harmonic distor-tion, while changes in the channel length would not have much ef-fect on THD, HD2 and HD3.

Appendix A

Although there are available formulas [14] to evaluate the har-monic components produced by a non-linear function representedby a given polynomial, to facilitate the quick implementation of theFSIM, we have chosen to repeat in this appendix those for a 5th-or-der polynomial.

Let the non-linear function y be a 5th-order polynomial of x:

y ¼X5

j¼1

ajxj; ðA:1Þ

where aj are the polynomial coefficients and

x ¼ xdc þ xp cosðxtÞ ðA:2Þ

is a generic sinusoidal input signal with a dc component xdc, a peakamplitude xp, and a frequency x. Substituting (A.2) into (A.1) andafter some algebraic and trigonometric manipulations, we obtainfor the output signal:

y ¼X5

k¼0

Hk cosðkxtÞ; ðA:3Þ

where Hk are the harmonic distortion coefficients corresponding tothe harmonic components of the output signal with frequencies kx:

R. Salazar et al. / Solid-State Electronics 52 (2008) 1092–1098 1097

H0 ¼ ða1xdc þ a2x2dc þ a3x3

dc þ a4x4dc þ a5x5

dcÞ

þ a22 þ 3

2 a3xdc þ 3a4x2dc þ 5a5x3

dc

� �x2

p þ 38 ða4 þ 5a5xdcÞx4

p

; ðA:4Þ

H1 ¼ þða1 þ 2a2xdc þ 3a3x2dc þ 4a4x3

dc þ 5a5x4dcÞxp

þ3 a34 þ a4xdc þ 5

2 a5x2dc

� �x3

p þ5a5

8

� �x5

p

; ðA:5Þ

H2 ¼a2

2þ 3

2a3xdc þ 3a4x2

dc þ 5a5x3dc

� �x2

p þ12ða4 þ 5a5xdcÞx4

p; ðA:6Þ

H3 ¼a3

4þ a4xdc þ

52

a5x2dc

� �x3

p þ5

16a5

� �x5

p; ðA:7Þ

H4 ¼18ða4 þ 5a5xdcÞx4

p; ðA:8Þ

and

H5 ¼a5

16x5

p: ðA:9Þ

Appendix B

In what follows we will rigorously prove that the harmoniccomponents obtained through the FSIM are the same as those cal-culated in Appendix A. For simplicity’s sake, but without loss ofgenerality, we will present the derivation this time for a 3th-orderpolynomial:

y ¼X3

j¼1

ajxj; ðB:1Þ

where aj are the polynomial coefficients and the input signal is

x ¼ xdc þ xp cosðxtÞ: ðB:2Þ

First, the origin is selected at the smallest value of the input sig-nal: x = (xdc � xp) and y = y(xdc � xp) so that the translated axis arext = x � (xdc � xp) and yt = y � y(xdc � xp). Following the FSIM, weextract the coefficients by using the operator EN

P , defined in Eq.(4), performing the integrations with a lower limit of xt = 0 andupper limit of 2xp, together with the values of a, b and c, given inEqs. (9)–(11). Then, the resulting the harmonic distortion coeffi-cients are

H0 ¼ ða1xdc þ a2x2dc þ a3x3

dcÞ þa2

2þ 3

2a3xdc

� �x2

p; ðB:3Þ

H1 ¼ þða1 þ 2a2xdc þ 3a3x2dcÞxp þ

3a3

4

� �x3

p; ðB:4Þ

H2 ¼a2

2þ 3

2a3xdc

� �x2

p; ðB:5Þ

and

H3 ¼a3

4

� x3

p; ðB:6Þ

which are identical to those obtained in Appendix A, for the partic-ular case of a4 = a5 = 0, demonstrating that the FSIM yields the exactexpected harmonic distortion coefficients.

Appendix C

Although there are available algorithms to perform numericalintegration of the I–V characteristics, we have chosen to offer herethe integration procedure used in this work, to facilitate the quickimplementation of the FSIM and to ensure the required precision ofthe calculations.

The present implementation of the FSIM method uses up tothree successive integrations. On the other hand, in the exampleillustrated here the I–V characteristic to be integrated is assumedto be sufficiently well represented by a third order polynomial.

Thus, after integrating three times we will end up with a sixth or-der polynomial. Therefore we need to use a numerical integrationalgorithm which would produce negligible error under these cir-cumstances. This can be accomplished for example by using aclosed Newton-Cotes formula with seven points [15]:Z x6

x0

ydx ¼ Dx140ð41y0 þ 216y1 þ 27y2 þ 272y3 þ 27y4

þ 216y5 þ 41y6Þ; ðC:1Þ

where (x0, x1, x2, x3, x4, x5, x6) have a uniform spacing Dx, and thecorresponding values in the y axis are (y0, y1, y2, y3, y4, y5, y6). Thisformula can be used for all points above the 7th, but we also needformulas for the first six points. These formulas can be obtained bynoticing [16] that Eq. (C.1) is inspired in Simpson’s rule idea: a sixthorder polynomial that approximates y(x) and passes by all thepoints. This approximate polynomial is obtained by using Lagrangepolynomials and then, analytically performing the integrations. Thefinal formulas areZ x1

x0

ydx ¼ Dx60480

ð19;087y0 þ 65;112y1 � 46;461y2

þ 37;504y3 � 20;211y4 þ 6312y5 � 863y6Þ; ðC:2ÞZ x2

x0

ydx ¼ Dx3780

ð1139y0 þ 5640y1 þ 33y2 þ 1328y3

� 807y4 þ 264y5 � 37y6Þ; ðC:3ÞZ x3

x0

ydx ¼ Dx2240

ð685y0 þ 3240y1 þ 1161y2 þ 2176y3

� 729y4 þ 216y5 � 29y6Þ; ðC:4ÞZ x4

x0

ydx ¼ 2Dx945ð143y0 þ 696y1 þ 192y2 þ 752y3

þ 87y4 þ 24y5 � 4y6Þ; ðC:5ÞZ x5

x0

ydx ¼ 5Dx12; 096

ð743y0 þ 3480y1 þ 1275y2 þ 3200y3

þ 2325y4 þ 1128y5 � 55y6Þ: ðC:6Þ

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