HARMONIC GENERATION BY NONDEGENERATE P-N JUNCTION VARACTOR DIODES

This paper deals with the large-signal analysis of nondegenerate p-n junction varactor diodes. Expressions are obtained for the harmonics of the varactor diode current when driven by a sinusoidal voltage signal. The special case of relatively small input amplitudes is considered and the results are compared with previously published results.


INTRODUCTION
Nondegenerate p-n junction diodes are widely used for microwave harmonic generation.Of particular interest here is the variable capacitor diode, or varactor, which is the most commonly used nondegenerate p-n junction diode [1].
It is well known that the dynamic junction capacitance of a reverse-biased p-n junction can be expressed by [1] C C (Vo + (1) where Vo is the magnitude of the contact potential barrier, VR is the reverse-bias voltage, Cb S//qNae/2, S is the cross-sectional area of the diode, q is the electronic charge, N d is the donor density in the n-type semiconductor, and e is the permitivity of the depletion layer in the junction of semiconductor.If a reverse-bias voltage of the form vR= VRo + Vs sin t%t (2) where VRo is the dc reverse-bias voltage and Vs is the amplitude of a sinusoidal voltage with frequency t%, is applied across the capacitance, then combining (1) and (2) we get Vo+V o where Co Cb/(Vo q-VRo) 1/2 is the junction capacitance without sinusoidal input voltage.From (3), it is essential to keep <_ 1 so that the capacitance C Vo+ will be real [1].Equation ( 6) can be rewritten in the form x cos t%t where y %Co(Vo + VRo) is the normalized capacitance current and x Vs is the normalized input voltage.It is obvious that (7) is nonlinear.Vo+ Therefore, the capacitor current will contain harmonics of the input frequency.
However, (7) in its present form cannot be used for predicting the amplitudes of these harmonics.Therefore, Ishii [1] specialized his interest in the small signal conditions, under which the nonlinear terms of (7) can be expanded in Taylor series.By truncating these series after the second terms and assuming that the input voltage amplitude is relatively small, Ishii [1] obtained expressions for the third-harmonic current component.The analysis of Ishii [1] cannot, therefore, predict the harmonic performance under large signal conditions.
It is the major intention of this paper to present simple approximations for the nonlinear terms of (7).These approximations, which are valid over the full useful range of input voltage, are intended to provide simple analytical expressions for the harmonic components of the capacitor current under large signal conditions.Such analytical expressions are important for evaluating the large signal performance of the vartactor diodes when used for harmonic generation.

PROPOSED APPROXIMATIONS
The development of the proposed approximations has proceeded along empirical lines by comparing the nonlinear terms shown in Fig. l(a) for Izl 0.9, and shown in Fig. l(b) for Izl -< 0.9, with the truncated Fourier-series (8) O(z) Vo + E Tk cos k=l and --T-z + Xlk sin --Z ( 10) The function O(z).(11) respectively.
The parameters /o, "/k, 1]k, o, k, [k, and T are fitting parameters selected to provide the best fit between the nonlinear terms of (8) and ( 9) and ( 10) and (11), respectively.In general, these parameters can be obtained using standard curve- fitting techniques.Alternatively, by removing the offset, at z 0 in the curves of Fig. 1, and then mirror imaging, the resulting curves can be made periodic as shown in Fig. 2. Now if we choose a number of data points, join them end to end using straight line segments, and denoting the slope of each segment by am and 13m, respectively, as shown in Fig. 2(a), (b), it is easy, following the procedure described by Kreyszig [2], to show that the coefficients /o, "k, rig, o, k, and [k can be expressed by [3] ' /k 2(kr)2 51 M-1 + E (m+l m) COS T Zm+l )) where T is the period of the periodic functions of Fig. 2, and O m and m, rn 2, 3,..., M are the values of the functions O(z) and (z) at Zm, rn 2, 3,..., M. From (13), ( 14), (16), and (17) one can see that calculation of the parameters /k, Xlk, k, and [k requires only simple mathematical operations.Also, inspection of (13), ( 14), (16), and (17) suggests that as k becomes infinite, the parameters rlk, /k, [k, and k always approach zero.For numerical computation using mainframe or personal computers, there is no reason to avoid increasing the number of terms in (10) and (11) until the inclusion of the next term is seen to make a negligible contribution towards a best fit criterion; for example, the minimum relative-root- mean square (RRMS) error.Tables I and II show the first 72 terms for approximat- ing the nonlinear terms of (8) and (9).
Using the parameters of Tables I and II and (10) and (11), calculations were made and are shown in Fig. 1 from which it is obvious that the proposed Fourier-series approximations accurately represents the nonlinear terms of (8) and (9).
M. T ABUELMA' ATTI k TABLE I First 36 terms of, /, and q,, of equation (10) for fitting (8)."o 1.18709, T 3.6 and RRMS error 0.00096.k --4.21618 0.0 x "o + Z "o -x + x Z ,,J k=l k=l the normalized second-harmonic current component will be given by y2 the normalized odd-order harmonic current component of frequency nt%, n 3, 5, 7,... will be given by yn(t) and the normalized even-order harmonic current component of frequency nt%, n 4, 6, 8 will be given by Yn(t) (X , "qk Jn-1 T x d-Jn+l T x -k=l + Jn+2 T x sinn%t (23) Using ( 20)-( 23), the amplitudes of the harmonic components of the normalized- current of any odd-or even-order can be calculated in terms of the ordinary Bessel functions available in most mainframe computers.However, for users of program- mable pocket calculators, the approximations of [4]-[6] may be useful.Moreover, for sufficiently small values of x the Bessel functions can be approximated by 'IT g Using (24) and (26), and ignoring terms containing orders of x higher than 3, the fundamental current component of frequency to s and the third-harmonic current component of frequency 3t% can be expressed as r 1 In order to compare the results obtained here with previously published results, here we recall eqn.(10.5.5) of Reference [1], from which the fundamental current component of frequency % and the third-harmonic current component of frequency 3t% can be expressed as  Inspection of (30)-(33) shows that the small signal results obtained by Ishii [1] can be obtained, with excellent accuracy, as a special case from the general large-signal analysis presented here.

CONCLUSION
In this paper, approximations using the Fourier-series, have been presented for the nonlinear terms of the current-voltage relationship of the varactor diode.The Fourier-series coefficients can be evaluated using simple calculations without recourse to numerical integration.The analytical expressions obtained for the amplitudes of the harmonic components are in terms of the ordinary Bessel functions, with arguments proportional to the amplitude of the input sinusoid, and can be easily evaluated using programmable hand calculators.The special case of relatively small-amplitude input signal was considered in detail and the results obtained in this paper are in excellent agreement with previously published results.

2 FIGURE 2 2 FIGURE 2
FIGURE 2 (a) The function O(z) of Fig. l(a) after removing the offset at Z 0, approximation by straight line segments and mirror imaging to form or complete period.

TABLE II
2 cos tOst