Harmonic Distortion Analysis for Nonlinear Circuits by Using Volterra Series and Spice Software S. El-RABAIE, and TAMER S. EL-SAYED Abstract — Nonlinear circuit analysis differs from linear circuit analysis in that the transfer characteristic between input and output is a function of signal level. .different methods were established to manipulate these circuits for different frequency ranges with different excitations. These methods were classified into Time-domain and Frequency domain methods. Volterra series is one of the frequency- domain methods. A simple circuit with a diode which is a nonlinear device was studied. Second and third harmonic distortions were calculated for this circuit. The obtained results were compared with the results from SPICE software. The effect of changing the amplitude of the excitation signal was discussed. An error curve between the obtained results was drawn. Index Terms—Harmonic distortion, kernel, Nonlinear circuits, Volterra series. I. I NTRODUCTION The development of a method which can analyze nonlinear microwave circuits evokes widespread interest. Nonlinear circuit analysis differs from linear circuit analysis in that the transfer characteristic between input and output is a function of signal level. By this meant that input signals are sinusoidal and small enough so that few harmonics are produced. This does not imply that the nonlinearities in the circuit can be neglected. Mixers and frequency doublers are examples of that circuits that fundamentally depend on nonlinear effects to operate. The significant difference between linear and nonlinear circuits could be summarized as in [1]: i. All nonlinear components in circuit must have large-signal models which describe their transfer function over the desired range of operating levels. ii. The analysis must account for any new frequency components which are generated. iii. Information on the bias sources and drive level must be accounted for the analysis. iv. The solution may not be unique, stable, or even exist. Numerous attempts to find the response of the nonlinear circuits and to study harmonic distortion are found. These attempts fall into three main classes [2]: 1) time-domain analysis, 2) frequency-domain analysis using equivalent circuits incorporating nonlinear elements, and 3) nonlinear transfer functions (Volterra series) for weakly nonlinear circuit analysis. A. Time-domain methods Time-domain methods analyze the nonlinear circuit by solving the nonlinear differential equations governing the circuit in the time-domain. Direct numerical integration of the state equations, associated discrete circuit modeling, and shooting methods are examples of Time-domain methods [1]. SPICE is a general time domain tool for analysis of transient effects in a circuit. B. Frequency-domain method The Volterra series is a method that is widely used for the analysis. This method gives the solution in analytical form. Although the algorithm is theoretically elegant, it is not so easy to derive the higher order Volterra Kernels for the large scale systems containing many nonlinear elements. They can only apply to the weakly nonlinear circuits. Further more, the nonlinear characteristics should be approximated by polynomial functions, where Taylor expansions are used to approximations. C. Mixed frequency and time domain simulation The harmonic balance is a technique for the numerical solution of nonlinear analog circuits. In this method the circuit will be partitioned into two sub circuit, one was called linear and the other nonlinear. Procedure for this method could be found in [3]. In this paper, calculating second and third harmonic distortion by using Volterra series will be manipulated. A comparison with SPICE results will be established.