Contains a report on a research project.United States Air Force (Contract AF19(604)-5200)United States NavyUnited States ArmyLincoln Laboratory (Purchase Order DDL B-00283

Huibonhoa, R.

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XVII. CIRCUIT THEORYProf. S. J. Mason Prof. R. D. Thornton B. L. DiamondProf. H. J. Zimmermann J. T. Andreika R. HuibonhoaProf. C. L. Searle W. C. SchwabA. HIGH-ORDER FREQUENCY MULTIPLICATION WITH VARACTORSFrequency multiplication with 100 per cent efficiency has been predicted by theManley-Rowe power relation for a nonlinear reactance (1). Indeed, an efficiency ofapproximately -1 db for a frequency doubler using the varactor (nonlinear reactance)in the pure capacitive mode has been reported (2, 3). However, for high-order multi-pliers, the efficiency drops off rapidly and eventually falls well below the 1/N 2 limitfor an ideal rectifier (4). On the other hand, with the varactor used as a rectifier,efficiencies close to the 1/N 2 limit have been attained (3). Besides the filter losses,the other source of loss can be attributed to the inherent series resistance of thevaractor. It was felt that the loss caused by this series resistance is so serious inthe capacitive mode that the rectifier mode would be more desirable in the high-ordercase.Theoretical investigations have been carried out to determine the effect of the seriesresistance on the efficiency of the varactor multiplier. In particular, the relativemerits of the pure capacitive mode, the rectifier mode, and the capacitive-rectifiermode have been investigated.An equivalent-circuit model for the varactor is shown in Fig. XVII-1. From energyconservation it can be shown that the efficiency is of the formn =  2 + (H -H )- P +N2N N2 W (1)N I c r sN/ N2where N is the harmonic number that is of interest; W is the input power at the funda-mental frequency fl = (wl)/(Z1); P s l and PsN are the power dissipated by the seriesthresistance at the fundamental and at the N harmonic, respectively; and H and Hc rare two quantities associated with the nonlinear capacitance and the nonlinear resistance,respectively. These quantities have the form:H dl qc(Y)c 2 vc[q(Y)] 3 dy (2)0 dyThis work was supported in part by Purchase Order DDL B-00283 with LincolnLaboratory, a center for research operated by Massachusetts Institute of Technologywith the joint support of the U.S. Army, Navy, and Air Force under Air Force Con-tract AF 19(604)-5200.175(XVII. CIRCUIT THEORY)1 dv (i) dir(Y)r 2T di y L dy (3)rwhere y = wIt; and vc(qc ) and vr (ir) are the nonlinear characteristics for the capacitorand the resistor. For positive capacitance and positive resistance, it can be shown thatboth H and H are positive.c rWith reference to Eq. 1, it is seen that in order for r > 1/N Z, the conditionH - H > P + N2Pc r sl sN (4)must be satisfied.These conclusions can be deduced from this condition. First, the presence of theterm H can only render the inequality difficult to meet. Thus, the capacitive mode ofoperation is the more desirable. Second,the rectifier mode is made more effectiveby the presence of the nonlinear capacitance(H >0). Third, the N2 term on the right-C(v) R(v) hand side of Eq. 4 reveals the rapid deteri-oration of the high-order multiplier in theRs  presence of the varactor series resistance.However, since the quantities in Eq. 4Fig. XVII-1. Equivalent-circuit model are all functions of the actual circuit param-for a varactor. eters, the series resistance, the inputpower, and the harmonic number, realiza-bility of the condition of optimal operation can best be investigated by an analysis of theactual circuit. This investigation is now proceeding.R. HuibonhoaReferences1. J. M. Manley and H. E. Rowe, Some general properties of non-linear elements.Part I. General energy relations, Proc. IRE 44, 903-913 (1956).2. D. B. Leeson and S. Weinreb, Frequency multiplication with nonlinearcapacitors - A circuit analysis, Proc. IRE 47, 2076 (1959).3. Interim Progress Report on Contract No. NAsw-33, Research Laboratory ofElectronics, M.I.T., Dec. 1, 1959; see Sections 2 and 3.4. C. H. Page, Harmonic generation, Proc. IRE 46, 1738 (1958).176

Circuit Theory

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Circuit Theory

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