318 IEEE TRANSACTIONS ON EDUCATION, VOL. 52, NO. 3, AUGUST 2009 On the SCTC-OCTC Method for the Analysis and Design of Circuits Stefano Salvatori, Member, IEEE, and Gennaro (Rino) Conte Abstract—This paper discusses guidelines that emphasize the relevance of short-circuit- and open-circuit-time constant (SCTC and OCTC, respectively) methods in the analysis and design of electronic amplifiers. It is demonstrated that it is only necessary to grasp a few concepts in order to understand that the two short- and open-circuit cases fall into a single case that can be easily addressed by low-pass and high-pass filters duality. Instead of just teaching the methods as “recipes” for frequency-performance systems analysis, SCTC and OCTC can be efficiently explained in basic analog electronic courses so as to elucidate their underlying principles. The discussion presented here will include two analysis and design examples, used to highlight the benefits gained by the approximated analysis technique, as well as a section that examines the accuracy of the technique. Index Terms—Low-pass- and high-pass-filter duality, RC net- work reduction, short-circuit and open-circuit time-constant accuracy, short-circuit- and open-circuit-time constant (SCTC and OCTC). I. INTRODUCTION A NALOG-CIRCUIT analysis and design received an im- portant contribution from the so-called open-circuit-time constant (OCTC) technique first introduced in the 1960s [1]. In the 1940s, related to the design of wideband vacuum tube ampli- fiers, Elmore introduced a method to measure the transient re- sponse of linear systems [2] that is widely used today for MOS digital-system design [3]–[6]. In analog electronics, the OCTC technique represents a powerful approximate analysis tool for the estimation of the dB high-frequency limit of a circuit. The method applies for accurate hand calculation of the fre- quency performances of an amplifier. As underlined in several papers [7]–[11], the SCTC and OCTC techniques, although ap- proximate, allow the designer to relate system bandwidth per- formance to specific circuit elements, assessing their effect on the circuit. Very good discussions and interesting application insights can be found in Thompson’s textbook [12]. However, the SCTC and OCTC calculations are usually presented almost like a “recipe” in basic analog electronics textbooks [13], [14]. They can be “mechanically” applied to the circuit under test to estimate the bandwidth of the amplifier. Generally, students are more interested when they understand the principle of the method they are applying; this works best when only a few basic Manuscript received September 12, 2007; revised May 22, 2008. First pub- lished May 05, 2009; current version published August 05, 2009. The authors are with the Engineering Electronic Department, University Rome Tre, Rome 00146, Italy (e-mail: salvator@uniroma3.it). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TE.2008.928218 Fig. 1. Frequency response of a first-order LPF (continuous line) and of a first-order high-pass filter (dotted line). The symmetry of the two responses is revealed recalculating the HPF transfer characteristic as a function of (see text). principles have to be remembered. As shown by the examples reported in the final two sections of this paper, the methods allow students to exercise a tight control over their circuit analysis or design, and to develop a real insight into the circuit functionality (especially relevant in basic analog electronics courses). II. A BRIEF INSIGHT INTO THE LOW-PASS AND HIGH-PASS FILTERS DUALITY In the Laplace domain, the transfer function of a first-order low-pass filter (LPF) is simply represented by (1) where and are the gain at zero-frequency and the time- constant of the filter, respectively. The latter, given by the cir- cuit elements product, can be also expressed as its in- verse quantity , in which is the corner frequency where the system gain decreases by 3 dB. In the same Laplace domain, a high-pass filter (HPF) simply shows the presence of a zero in the origin so that the transfer function modifies to (2) which has been written to be exactly comparable to the function. The intrinsic duality of the two transfer functions is easily revealed by observing the two equations modules as Bode plots, as reported in Fig. 1. “Reading” the HPF response in a reverse mode (i.e., from high to low frequencies), the same be- havior as that of an LPF is observed: a constant gain followed by a “decrease” of 20 dB per decade of frequency. Again, the two 0018-9359/$26.00 © 2009 IEEE