Published in IET Circuits, Devices & Systems Received on 19th August 2009 Revised on 11th April 2010 doi: 10.1049/iet-cds.2010.0030 ISSN 1751-858X Design of non-balanced cross-coupled oscillators with no matching requirements A.S. Elwakil Department of Electrical and Computer Engineering, University of Sharjah, P.O. Box 27272, Sharjah, Emirates E-mail: elwakil@ieee.org Abstract: Using two-port network transmission parameters we derive the general characteristic equation of a cross-coupled circuit topology which involves two active devices and four or six impedances. The derived equations are generic and apply both to BJT or MOS transistors and even any other active device without any matching constraints. Application to realising novel non-balanced non-matched cross-coupled oscillators is demonstrated. Spice simulations of a MOS oscillator in a 0.25 m technology are given as well as experimental results from an oscillator employing discrete Bipolar transistors. 1 Introduction Cross-coupled oscillators are very important and widely used in the realisation of transceivers. A huge amount of literature pertaining to the design and analysis of these oscillators both in their sinusoidal oscillation mode and their relaxation oscillation mode can be found [1–7]. Two typical widely used cross-coupled oscillator structures are shown in Figs. 1a and b employing MOS transistors. The same structures may well be implemented using Bipolar transistors. In both structures, the design requirements usually imposed are: 1. the two transistor should be matched in order to realise an effective differential negative resistance r =−1/g m where g m is the small signal transconductance for each individual transistor; 2. the load impedances of both transistors should be similar in order to obtain a symmetrical (balanced) structure. It is not however clear whether these design requirements are essential or not in order to obtain an oscillator. In this work, we derive using two port network transmission parameters the characteristic equation of the general four-impedance cross-coupled structure shown in Fig. 1c and the six-impedance structure shown in Fig. 1d without assuming that the employed transistors have to be matched. The derived equation is valid both for BJT and MOS transistors and indeed for any other active device. We then find the oscillation start-up condition and oscillation frequency for the two classical cases shown in Fig. 1 and for other cases in which the two load impedances (Z 2 and Z 3 ) are not similar; that is the cross- coupled oscillator in this case is not balanced. Of course, oscillators whether sinusoidal or relaxation are highly non- linear circuits and their accurate modelling and analysis implies using non-linear dynamics techniques [8–10]. However, linearised small-signal models are usually the start point for the design phase of any oscillator in order to derive the oscillation start-up condition (Hopf bifurcation condition) and also estimate the expected oscillation frequency as a function of the circuit parameters. By virtue of using transmission parameters, which are particularly suited for networks in cascade, we show how the effects of parasitic impedances can be easily incorporated into the derived characteristic equation. Two novel design examples of cross-coupled oscillators where (i) the transistors are not matched and are even of non-identical types and (ii) the loads are not balanced are given. Spice simulation using a 0.25 m MOS technology file is used to verify one example while the other is verified experimentally using discrete components. 2 Background Fig. 2a shows the typical two-port network input and output variables for which the transmission matrix (a) is IET Circuits Devices Syst., 2010, Vol. 4, Iss. 5, pp. 365–373 365 doi: 10.1049/iet-cds.2010.0030 & The Institution of Engineering and Technology 2010 www.ietdl.org