This paper discusses fundamentals of varactor multipliers, with the analysis primarily concerned with the case of a charge-controlled multiplier using an abrupt junction diode. The use of idler circuits to obtain high-order multipliers and various circuit configurations is examined. Several multiplier circuits used by the author are treated from the designer's point of view, and a practical satellite application is considered in some detail. T. Thompson VARACTOB he need for small, lightweight, reliable R-F the square of the breakdown voltage, as will be T power sources in communications systems has shown later. High breakdown voltage can be led to many advances in the field of solid-state achieved with a low-impurity concentration. This devices. In recent years, advances in varactor diode leads to high resistivity, however; therefore, the technology have made solid-state UHF and micro­ semiconductor manufacturer will usually need to wave generators practical for such systems. Modern compromise between high power-handling capabil­ varactors can achieve high conversion efficiency ity and high efficiency. The simple equivalent cir­ and large power-handling capability. Single varac­ cuit used to represent the varactor is shown in tor circuits with power outputs of several watts in Fig. 1. The series resistance R s is considered as the low-gigacycle region are becoming common­ constant. Although this is not strictly true, effects place. At the current rate of improvement in varac­ owing to its variation with voltage do not appear tor technology many dramatic advances can be to contribute significantly to multiplier perform­ expected in the near future. ance. The Varactor Diode R. cry) ~--~W~'----~7~~~------~ The varactor is a p-n junction diode, with a reverse voltage applied, used as a capacitor. The Fig. I-Varactor equivalent circuit. diode depletion region expands with increasing back-bias voltage, and this causes a voltage variable In addition to the reverse characteristics, there capacitance effect. In the voltage region between are phenomena associated with the forward con­ forward conduction and reverse breakdown the ducting region that affect the operation of varac­ 1 capacitance voltage relationship can be expressed tor multipliers. , 2 The charge-storage effect is par­ as ticularly important. When the junction becomes forward-biased, minority carriers are injected into C = K(¢ - V),\ (1) both sides of the junction. If these carriers remain where C is capacitance, cp is the contact potential, in their respective injection regions long enough, V is the applied voltage, and K is a constant of they will combine with majority carriers and thus appropriate dimensions (farads per volt to the ex). cause a current flow. However, if the minority­ For applied voltages in the range near forward carrier lifetime is long with respect to the period conduction, this expression is incorrect; however, of the applied voltage, these carriers will be swept for voltages of several volts in the back direction it out during the next half-cycle and recombination is quite accurate. The exponent ex ranges roughly will not take place. This effect, therefore, repre­ from -1/ 5 to -1/ 2, depending on the impurity sents a recoverable stored charge. An analysis by 1 profile. In addition to the nonlinear capacity effect, Leeson of this effect shows that there is an in- the bulk and contact resistance of the semicon­ ductor material lead to an additional resistance 1 D. B. Leeson, " Large-Signal Analysis of Parametric Frequency Multipliers," Stanford Electronics Laboratories Technical Report component. Since this resistance effect leads to loss No. 1710-1 , Stanford, Cal., 1962. in harmonic generators, it is desirable to hold it 2 G. Luettgenau, J. Williams, and H . Miyahira, "A Practical Ap­ proach to the Design of Parametric Frequency Multipliers," Wescon to a minimum. On the other hand, the power­ Convention Record, Space Technology Laboratories, Inc., Los handling capability of a varactor is proportional to Angeles, 1961. 2 APL Technical Digest ULTI LIERS MULTIPLIERS crease in efficiency when it is used. The amount of ferred to Leeson's work.1 We will limit ourselves to the increase is not, however, the major deviation consideration of the charge-controlled circuit, using from simple varactor multiplier theory, which does only the abrupt-junction diode with a square-law not include the stored-charge effect. The most characteristic. Conduction in the reverse direction dramatic effect is the power-handling capability. in the diode is sufficiently small to be neglected in It has been shown experimentally that the power this discussion. The effect of forward conduction input at which maximum efficiency is achieved will not be included, and asa further simplification with an abrupt junction diode ranges from 2 to 6 we will omit the contact potential from Eq. (1) times that indicated by the simple theory.2,3,4 Fur­ and consider the functional relationship, thennore, the maximum input power at which an abrupt junction diode can be used at slightly re­ c = dq = KV-l/2 (2) duced efficiency increases by another factor of 2 dV ' or more.4 Step-recovery effects also appear to enter where V is the magnitude of the reverse voltage. The into multiplier perfonnance. Step recovery is a incremental capacitance C defined above, will be result of charge storage that comes about when the J used throughout this discussion. minority carriers are returned to the point of origin Common measurement techniques generally em­ in a compact bunch; this sudden cessation of cur­ ployed on varactor diodes lead to values of incre­ rent results in a wavefonn rich in harmonics. 5 It is mental, rather than total, capacitance. Therefore, possible to enhance certain of these characteristics the incremental capacitance is a more natural defi­ by shaping the impurity profile. Usually such tech­ nition in discussions involving varactor diodes. The niques result in modifying the reverse characteristic charge-voltage relationship can now be determined also. Varactors are not now characterized in an from adequate fashion with respect to the forward region. q = JcdV = JKV-l/2dV = 2KVI/2 + A. (3) Consider the circuit of Fig. 2. The ideal filters in Varactor Multiplier Analysis this case allow only w current in the input port and From the above it is apparent that any general 2w current in the output port. Allowing for an analysis of varactor multipliers must consider the arbitrary phase angle cp, -the current in the varactor varied nature of the voltage-capacitance charac­ is represented by teristic and the effects of forward conduction. Also, dq the circuit configuration or parameters representing i = II sinwt + 12 sin(2wt + ¢) (4) circuit losses must be included. The complexity of dt' the more general problem will not allow its dis­ cussion in this treatment; those interested are re- 3 T. M. Hyitin, " Varactor Frequency Multiplier Design," Texas Instruments Technical Seminar, Washington, D. C., May 1964. 4 H. C. Lee, "VHF Power Sources Using Varactor Multipliers," I , sin wt /" 12 sin (2wt + cf» RCA Application Note SMA-22, RCA Semiconductor and Materials Division, Somerville, N. J., July 1963. 5 G. Schaffner, "High Power Varactor Diodes Theory and Applica­ tion," Motorola M~ nitor, 2 , 1964. Fig. 2-Charge-controlled varactor doubler. Novembel' -DecembeT 1964 3 Therefore, third and fourth harmonic voltages. The magni­ tude of the third harmonic component depends II 12 q = - - coswt - -2 cos(2wt + ¢) + B, (5) upon the product 1112 and is considered a mixing w w of these two currents in the varactor nonlinearity. where The fourth harmonic voltage obviously results from B = q = 2KEo1/ 2 + A, a doubling of the second harmonic component. We should now consider the possibility of using and Eo represents the voltage across the diode due the above circuit for a tripler by allowing only to the average charge q. Equations (3) and (5) fundamental and third harmonic currents, lead to i = II sinwt + 13 sin(3wt + ¢). (9) v = [E}/2 - 2iw coswt - 4~w cos(2wt + ¢) J2. (6) We note from the above that for this case Expanding this expression leads to V ex: [A + B coswt + C cos(3wt +¢ )]2. (10) 2 2 2 v = Vo + VI + V2 + V3 + V4, (7) The squared terms (B2 cos wt) and (C cos wt) lead to second and sixth harmonic terms (doubling 112 122 mode). And the cross product leads to second and Vo = Eo + 8K2w2 + 32K2w2' (7a) fourth harmonic terms, the sum and difference Id2 l1Eo1/2 (mixing mode). The only fundamental and third VI = 8K2w2 cos(wt + ¢) - ~ coswt, (7b) harmonic vol tage terms resul t from 12Eol/2 112 VI = AB coswt, V2 = 2Kw cos (2wt + ¢) + 8K2w2 cos2wt, (7c) and Id2 V3 = AC cos(3wt + ¢). (11) V3 = 8K2w2 cos(3wt + ¢), (7d) and These terms are in quadrature with the input and output currents, resulting in no power transforma­ tion ; therefore, we see that the abrupt junction diode used in the simple charge-controlled circuit Consider the expressions for voltage. The time­ cannot be used as a tripler. It can be shown that independent terms include the voltage related to the this diode in a simple charge-controlled circuit can average charge as well as to the two terms resulting only be used as a doubler. However, as seen in the from the currents into the nonlinear device. The doubler case, a third harmonic voltage results from sum of these terms represents the bias voltage. If the mixing of wand 2w terms. If, in addition, we we compare the w terms, Eq. (7b), to the funda­ allowed third harmonic current, with an additional mental current, Eq. (4), we note that the second arbitrary phase angle, third harmonic power could term is in quadrature with the current and there­ be generated. This is the argument that leads to the fore represents a reactance.