IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: REGULAR PAPERS, VOL. 62, NO. 3, MARCH 2015 625 Comparative Analysis of Simulation-Based Methods for Deriving the Phase- and Gain-Margins of Feedback Circuits With Op-Amps Marius Neag, Member, IEEE, Raul Oneţ, István Kovács, and Paul Mărtari Abstract—Ten methods for finding through simulations the small-signal phase and gain margins of feedback circuits based on op-amps are described and analyzed in this mostly tutorial paper. The testbenches employed by these methods are presented and the corresponding analytical expressions of the return ratio are derived and compared against their “ideal” counterpart, obtained with standard circuit analysis; the requirement that the return ratio should not depend on the point it was measured at is also verified. These analyses are performed on a fairly general case: a generic reciprocal two-port network that closes a feedback loop around an op-amp acting as the forward amplifier. The four main types of op-amps were considered. The limitations of some of the tested methods are then highlighted by simulations. Besides the detailed analysis of previously reported methods, the paper proposes a novel method for deriving the return ratio of feedback circuits, that employs only current stimuli; it is demonstrated analytically that this method can be used for bilateral circuits, not only for op-amp-based (unilateral) ones. Also, a recent method for deriving directly the phase margin of a circuit is extended to estimating the gain margin, too. Conclusions on the accuracy and suitability of the analyzed methods for practical circuit cases are drawn. These results are then extended to other circuit topologies. Index Terms—Phase and gain margins, return ratio, stability. I. INTRODUCTION N EGATIVE feedback is a key concept in circuit design, widely used to obtain reliable transfer functions by min- imizing the effects of parameter variations and external pertur- bations, to reduce nonlinearities and to improve the input/output impedances [1]–[4]. Feedback systems can be unstable, so en- suring their stability is a major design concern. The Rosenstark theorem shows that the small-signal closed- loop gain of a feedback system depends on its return ratio [2]: (1) where is the return ratio of the circuit, is the closed loop small-signal gain, is called the direct transmission term, and is the ideal closed- loop gain [3]. yields a mathematical singularity that indicates the instability of a physical circuit. As the return ratio is a fre- Manuscript received June 10, 2014; revised October 22, 2014; accepted Oc- tober 24, 2014. Date of publication December 05, 2014; date of current version February 23, 2015. This paper was recommended by Associate Editor A. M. A. Ali. The authors are with the Faculty of Electronics, Telecommunication and In- formation Technology, Basis of Electronics Department, Technical University of Cluj-Napoca, CJ RO-400027 Romania (e-mail: Marius.Neag@bel.utcluj.ro; Raul.Onet@bel.utcluj.ro; Istvan.Kovacs@bel.utcluj.ro; Paul.Martari@bel. utcluj.ro). Digital Object Identifier 10.1109/TCSI.2014.2370151 quency-dependent function, two scalar conditions must be met simultaneously for this condition to occur: (2) Thus, the stability of a feedback circuit operating at a given DC operating point can be assessed by calculating its phase- and gain (module)-margins; these metrics indicate how close the circuit is from meeting the conditions above [3]. The classical feedback theory is based on two-port analysis of the forward amplifier—with the unilateral gain —and its feed- back network—with the unilateral reverse transmission factor . The closed-loop gain of the circuit is given by: (3) Thus, the critical condition for instability is , where the product of and is called the loop gain [3], [4]. This approach is relatively easy to follow in analytical analyses and provides the expressions of the input and output impedances. However, it does not cover all feedback topologies and can yield different loop gain values for a given circuit, when the type and location of signal sources applied to the circuit changes, even if the source-free circuit does not change [5]. Moreover, it is difficult to use for finding the loop gain through simulations performed on real-life circuits, which are usually far from the unilateral model described above [6]. The return ratio associated with a dependent source [1] and (1) are better suited for analyzing feedback circuits: they are in- dependent on the feedback topology and do not depend on the type and location of the input sources. In fact, the forward am- plifier does not need to be identified, and it is not assumed to be unilateral. Most importantly, the return ratio can be measured accurately experimentally and by simulations [7]–[9]. Difficul- ties can arise when this concept is employed for circuits com- prising more than one controlled source [10] but this is not the case for the circuits discussed in this paper. In general, the return-ratio differs from the loop gain [5] but both can be used to check the stability of feedback circuits based on the phase- and gain (module)-margins [1], [3]: it was demon- strated analytically that, although the return ratio and the loop gain of an op-amp-based circuit similar to the ones analyzed here have different expressions, they reach the value of 1, crit- ical for stability, for exactly the same conditions [10]. This paper analyzes several methods for finding the return ratio, , of feedback circuits based on voltage- and current- mode op-amps through the small signal SPICE-type simula- tions, AC, and S-parameter. The loop gain is no further dis- cussed in this paper so there should be no confusion. 1549-8328 © 2014 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.