HARMONIC AND INTERMODULATION PERFORMANCE OF NONLINEAR ELECTRONIC CIRCUITS

A unified technique for mathematical modelling of the input output characteristics of nonlinear electronic circuits is presented. This technique can be easily implemented using simple hand computations without recourse to standard curve-fitting techniques, which invariably requires extensive computing facilities and well-developed software. The mathematical model, basically a sine-series function, can easily yield closed-form series expressions for the amplitudes of the output components resulting from a multisinusoidal input signal to nonlinear electronic circuits. The technique is very helpful in analysing nonlinear electronic circuits under different scenarios of input signals.


I. INTRODUCTION
Electronic circuits incorporating active devices are inherently nonlinear.While this inherent nonlinearity is, generally,considered to be a disadvan- tage for example in amplifiers and filters, in some applications it may be used to advantage for example in mixers and detectors.Whenever a number of signals of different frequencies pass through a nonlinear circuit, energy is transferred to frequencies that are harmonies and sums and dif- ferences of the original frequencies; that is intermodulation products.These harmonics and intermodulation products can severly degrade the performance of an electronic circuit intentionally designed to be linear; for example an amplifier, or it can yield the useful output of an electronic cir- cuit intentionally designed to be nonlinear; for example a mixer.In all cases it is essential for a designer to have a rapid indication of the likely spectrum of output signals, including harmonics and intermodulation products, from a nonlinear electronic circuit excited by a multisinusoidal input signal.This requires a tractable mathematical model for the input-output characteristic of the electronic circuit under consideration.
A literature survey indicates that a widely used procedure for evaluating the output spectra of a nonlinear electronic circuit involves derivation of a high-order polynomial approximation to the input-output characteristic (1--4).
Two approaches are available for obtening the coefficients of this polynomial approximation.The first approach implies the use of least square-error curve-fitting techniques to approximate a measured (or simu- lated) set of input-output data points.These curve-fitting techniques invar- iably demand extensive computing facilities and well developed software.It should be noted, also, that a fit will not necessarily improve by increas- ing the order of the polynomial.Increasing the order allows the curve to wiggle more in order to come closer to the data.The least square-error cri- teflon assures good fit near an original data point, but in between succes- sive data points the fitted curve may oscillate.As the order of the polynomial function increases, the mean square error is reduced, yet the oscillations increase (5).Moreover, for an nth order polynomial, the har- monics and intermodulation products that can be evaluated are restricted to nth order only.The second approach involves a Taylor series expansion of an available, but untractable, mathematical expression about an operat- ing point (6).The problem arises, however, when the mathematical expres- sion available represents the input variable as a function of the output variable, where as what we would prefer is an expression for the output variable as a function of the input variable (7).This requires a series rever- sion to get the output variable as a function of the input variable.Another procedure, which is less widely used, is based on performing a Fourier analysis of the output variable waveform.The first step for using this procedure is to develop an accurate sketch of the variation of the output varia- ble with time (8)(9)(10)(11).Then by selecting an even number of equally time-spaced samples of this time.function,it is possible to obtain the amplitudes of the different harmonic components of the output variable.The major limitation of this procedure is that many electronic circuits have inputs at two or more independent frequencies.It is not unlikely that very often these frequencies are such that the ratio of the highest to lowest fre- quency generated by the nonlinear electronic circuit may be large.And since the size of the time step must be proportional to the highest output frequency expected while the length of the simulation interval must be proportional to the lowest output frequency expected, then the analysis of such circuits require a vast number of time points.The application of this procedure is, therefore, very limited to cases with inputs consisting of a relatively small number of independent frequencies that are not widely dif- ferent.A third procedure, which is especially suitable for use in compu- ter-aided analysis using programs such as SPICE, is to perform a transient analysis of the electronic circuit with a multi-sinusoidal input followed by a Fourier decomposition of the output signal (12).The major drawback of this technique is that it is not computationally simple as it has to be repeated for different input levels.However, it is very accurate particularly when the nonlinearity of the electronic circuit is not well behaved (e.g., nonmonotonic).Alternatively, SPICE can be used to generate a DC trans- fer characteristic of the circuit (12).The results are then applied to a least squares curve-fitting subroutine that yields a polynomial approximation for the transfer characteristic.Using this polynomial approximation, expressions can be obtained for the output products resulting from a mul- tusinusoidal input.The major advantage of this technique is its computa- tional simplicity as it requires only one simple analysis to predict the output products at any input level (12).SPICE, however, cannot be a substi- tute for the thought-process provided by a hand analysis using simple models for the transistors (13).A better insight into the operation of circuits and a systematic way of getting a preliminary understanding of the circuit performance can be gained by hand analysis using simple device models.Then circuit simulation, using SPICE, can be used to obtain more accurate results and to verify the results obtained by hand calculations (1'14).Moreo- ver, circuit simulation may suffer a wide range of problems, particularly those arising from the discrete nature of the solution methods used to ana- lyze problems of a continuously nonlinear nature (15).
A fourth procedure, which is not widely used in the literature, uses a sine-series function to represent the nonlinear input-output characteristic of the electronic circuit (16).This technique is attractive since a low-order sine-series function can yield the magnitudes of harmonics and intermodu- lation products of any order.However, similar to the polynomial approxi- mation, a large number of terms is required to fit realistic data.Although a standard least-square curve-fitting procedure can yield the coefficients of the sine-series, this demands extensive computing facilities and well developed software, and yet the desired accuracy may not be attained especially for strong nonlinearities.
From the above discussion it is obvious that a unified analytical tech- nique for analysing the harmonic and intermodulation performance of nonlinear electronic circuits, which can be easily presented in a class room, is not available.The desired features of such a technique are: 1. being suitable for implementation without recourse to extensive com- puting facilities and well developed software. 2. can yield closed-form expressions for the amplitudes of the harmonic and intermodulation products resulting from a nonlinear electronic cir- cuit excited by a large number of sinusoidal inputs.This allows the analysis of a wide variety of electronic circuits under different scenar- ios of input signals.
It is the purpose of this paper to present this technique.In principal, the proposed technique is based on representing the nonlinear input-output characteristic by a sine-series function.The coefficients of this sine-series function can be obtained using simple hand calculations without recourse to standard curve-fitting techniques.

PROPOSED TECHNIQUE
In general, the nonlinear input-output characteristic, shown in Fig. 1, of an electronic circuit will be available in the form of a discrete number of data points Yn, n 0, 1 N measured (or simulated) at X n, n 0, 1 N or calculated from an expression of the form X n f(Yn) (or Yn f(Xn)) where X Xmin, X N Xmax and X K 0, I<K<N.Both Xmin and Xmax may be positive or negative and the magnitude of Xmax is not necessarily equal to Xmin.Also, Ymin and Ymax may be positive or negative and are not necessarily equal in magnitude.This implies that the characteristic may or may not be symmetrical.First, we make this set of data periodic with odd symmetry by removing the offset at X K (if any) and utilizing the resulting data in mirror image to generate a complete half peroid 2B as shown in Fig (l).Secondly, we interpolate the resulting nonlinear input-output characteristic between the data points by using 4 N straight line segments joined end-to-end as shown in Fig( l).The X-values of the segment joins are termed knots.The number of knots and their positions must generally be chosen so that closer knots are placed in regions where Y is changing rapidly.FIGU Input-Output chactedsdc of an electronic Circuit (Solid line) shifted and rror-imaged to generate half peod 2B X-Y: Original axes and XtY': shifted es allows the choice of a lge number of ots to represent the fine details of the nonline input-output chactedsfic.By denoting the slope of each seg- ment by , as shown in Fig( l), it is easy, following the procedure described by eyszig ( 7), to show, without peffong y integration, that the coeffi- cients, ?2m+l, of the sine-series function of eqn(1) c be expressed by eqn(2).
Y-Ym-2m+lsin (2m+ 1)g X-Xm From eqn(2) it is obvious that, in contrast with the standard least-square-eor cue-fitting techniques, cNculation of the coefficients 2m+l of any order 2m+l requires only simple mathematical operations.Also, inspection of eqn(2) suggests that as 2m+l becomes infinite, the sine-series coefficients m+ always approach zero.
Using eqn(2), any set of measured, simulated, or calculated data points can be fitted to a sine-series function to any degree of accuracy, by using as many terms as necessy.In fact, for numerical computation using mainframe or personal computers, there is no reason to avoid increasing the number of terms M until the inclusion of the next term is seen to make a negligible contribution towds a best fit criteria; for example the ni- mum root-mean-square error.However, for hand computation using pocket calculators, only small values of M can be used.

EXAMPLES
To illustrate the effectiveness of the proposed technique, mathematical modelling of the nonlinear input-output characteristics of several elec- tronic circuits will be considered in this section.

*Vcc
Using the procedure described in the preceding section, eqns(3) and ( 4) can be represented by the sine-series function of eqn (1) with Y i0/I T and X vi/(1+N) V T for the BJT unbalanced input stage and X vi/2(1+N) V T for the BJT balanced input stage for any value of the parameter N. Sam- ples of the results obtained are shown in Table I for different values of N.

JFET Input Stages
The nonlinear input-output characteristic of the JFET unbalanced input stage shown in Fig( 4) can be expressed as (18)   2V/-vi (5) and the nonlinear input-output characteristic of the JFET balanced input stage shown in Fig( 5) can be expressed as (18)   (1+ where N= Vp Vp is the pinch-off voltage of the JFET, I T , and IDSs the drain current at pinch-off for zero gate-to-source voltage. Using the procedure described in the preceding section, eqns( 5) and (6) can be represented by the sine-series function of eqn(1), with Y=   The nonlinear input-output characteristic of the BJT emitter-follower out- put stage shown in Fig( 6) can be expressed as (6), Vr Vr N where N= R E IQ/VT, VBEQ2is the base-to-emitter voltage of Q2, and IQ is the dc collector current of Q2.
Using the procedure described in the preceding section, eqn(7) can be represented by the sine-series function of eqn(1) with Y= vo/V T and Vi-VBEQ2 for any value of the parameter N. Samples of the results

VT
obtained are shown in Table III

MOSFET differential amplifier
The nonlinear input-output characteristic of the MOSFET differential amplifier shown in Fig( 7) can be expressed as (19)   --x/ vil X/I_1/2(Vid/V/IS/K)2 for via <1 (8) is the transconductance parameter, kt is the effective carder mobility, CoxiS the capacitance per unit gate area, W is the channel width, and L is the channel length.
Using the procedure described in the preceding section, eqn( 8) can be rep- io resented by the sine-series function of eqn( 1) with Y =-sss and Vid X Samples of the results obtained are shown in Table IV.
I'ss/K  4. HARMONIC AND INTERMODULATION PRODUCTS One of the potential applications of the expression of eqn( 1) is in the cal- culation of the harmonic and intermodulation products of nonlinear elec- tronic circuits excited by a multisinusoidal signal of the form N X(t) gb + E An sintont (9) n=l where A n is the normalized amplitude of the input sinusoid of frequency to n and A b is the normalized biasing.Substituting eqn(9) into eqn(1) we get M ((2m+l) ( An sin tont_ Xax)) Y(t) Ymax E 2m+ sin 2B (10) m=0 n=l where Xmax Xmax Ab.

SPECIAL CASE
In this section, the special case of an input signal formed of two equal-amplitude sinusoids of the form X(t) Ao + A(sin o)lt + sino)2t) (13) will be considered.Using eqn(12a), the amplitude of an output product of frequency o) (or o) 2 will be given by M YI,O-- )t2m+l cos ( (2m+l)n3(!2B maxjJ1 ((2m+a)A2B/Jo\(2m+a)A)2B ( and the amplitude of an output third-order intermodulation product of fre- quency 2o)1+ o)2 (or 2o) 2 + o)1) will be given by Y1,2 Y2,1 m--0 Similarly, using eqn(12b) the amplitude of an output second-order inter- modulation product of frequency 1 2 will be given by In a simil way, the amplitudes of honics and intermodulation prod- ucts of any order c be cflculated using eqn (12).
Combining eqns(l16), the relative second-order inteodulation dis- to.ion can be exressed as Y1,1 IM2 20 log 10 y 1,0 and the relative third-order inteodulation distoion can be defined as Y2,1 (18) IM3 20 log 10 Y1,0 Using eqns(l18) and Tables (I-IV) the inteodulation peffoance of BJT input stages, JT input stages, BJT etter-follower output stage, and the MOSFET differential amplifier were calculated d the results e shown in Figs( 8)-(ll).From Figs( 8)-(ll), it is obvious that for the bal- anced BJT and JFET input stages and e MOST differential amplifier, the second-order inteodulation products e ve smfll as expected.Moreover, it is obvious that as N increases, the third-order intermodulation distortion decreases.This means that increasing N improves the linearity of the balanced BJT and JFET input stages.This is expected as the increase in N implies an increase in RE, which means an increase in the amount of negative feedback, thus reducing the nonlinear distortion effects.Also, it is obvious that the unbalanced BJT and JFET input stages and the BJT emitter-follower output stage suffer from relatively large amounts of second-order intermodulation products.These results are in good agreement with previously published results (18).However, care is necessary if these results are used to compare the large signal performance of circuits considered; a direct comparison is valid only for same sinusoi- dal input amplitudes.It is worth mentioning also that the nonlinear input-output characteristics of the balanced BJT input stage, balanced JFET input stage, and the MOSFET differential amplifier are symmetric.Thus, no even-order harmonic and intermodulation products are expected.However, the results obtained using the present analysis show nonzero values for the even-order products.These nonzero values arise because of the imprecision inherent in digital computer calculations and should be ignored.It is, therefore, important to read the results carefully.However, in practical circuits, even-order products may result due to the mismatch between the transistors.

SIMULATION RESULTS
Using the PSPICE student version, the harmonic and intermodulation per- formance of the circuits of Figs.2-4 can be studied.However, in contrast with the technique presented in this paper, this requires a new run for every change in the input levels.In PSPICE, Probe can be used to perform a Fourier-transform on the data obtained from a tansient analysis.Fig. 12 shows a typical plot for the output current (spectral domain) obtained from the circuit of Fig. 2 with V b 0, V V2 8.6mV, R E 0, I T 1 mA, Vcc 5V, VEE 0.75V, fl 1KHz, f2 0.8KHz, RL 1K, C 100ktE This corresponds to Ab= 0, X/xmax 0.5.From Fig. 12, the second-order inter- modulation IM2 can be calculated, and it is equal to 18.0 dB.This com- pares very well with the -18.5 dB obtained from Fig. 8(a).However, because the third order intermodulation IM3 is very small, it is hard to measure it from Fig. 12.Similar results were obtained for the other circuits and the difference between the simulation and calculation is always within + ldB.FIGURE 12 Aplot for the output Current (Spectral domain) obtained from the Circuit of Fig. 2 with V b 0; V1=V2=8.6 mV, fl 1KHz, f2 0.8KHz.IT= lmA; RE= 0; RE= 1K, Vcc= 5V, VEE=-0.75V,C= 100ktF 7. CONCLUSION In this paper, a simple, yet powerful, technique has been presented for obtaining a mathematical model for the nonlinear input-output characteris- tic of electronic circuits.The model can be easily implemented using sim- ple hand calculations without recourse to standard curve-fitting techniques, that invariably demand extensive computing facilities and well-developed software.The parameters of the model can be extracted from measured, simulated data, or closed-form analytical expressions.The model can easily yield the amplitudes of the output products resulting from multisinusoidal excitation of nonlinear electronic circuits.In general, calculations of these amplitudes require the computation of ordinary Bessel functions either using subroutines available in most mainframe com- puters or using simple approximate expressions that are more appropriate for hand calculations using scientific programmable calculators.

FIGURE 8
FIGURE 8 Intcrmodulation performance of BJT input stages IM IM 2

FIGURE 9
FIGURE 9 Intermodulation performance of JFET input stages " IM :IM 2 The knots are not necessarily equally spaced; this

TABLE II
3.3.BJT emitter-follower output stage for different values of N.

TABLE III
Parameters of BJT Emitter Follower Output Stage.