A simplified approach to predicting negative resistance in a microwave transistor with series feedback C. R. Poole and I. Z. Darwazeh University College London, UK Abstract— This paper addresses the question of how to predict the series feedback termination required to generate negative resistance in a microwave transistor in different circuit config- urations. Although this question is usually approached through computer modelling and simulation, we present here a simple closed form equation that predicts the value of feedback reactance most likely to generate negative resistance at the input of a 2-port sub-circuit comprising a transistor plus series feedback, based solely on the 2-port S-parameters of the transistor. This simple equation provides new insight into the effect of series feedback on a transistor and leads us to propose a general rule, namely that the series feedback reactance required to generate negative resistance in a given transistor in a given configuration depends on the sign of the imaginary part of the sum of the 2-port S- parameters for the device in that configuration. I. I NTRODUCTION Negative resistance is an important circuit property that is widely employed in the design of microwave oscillators. Tra- ditionally, two-terminal negative resistance devices have been used, where application of the correct DC bias is a sufficient condition to generate negative resistance. Modern microwave circuitry is more likely to be implemented in MMIC form, where the active devices are mostly transistors, which are not inherently negative resistance devices. Negative resistance in transistors needs to be induced by a combination of device configuration and external feedback. A single series feedback element is commonly used for this purpose [1]. For example, inductive series feedback is usually employed in transistor neg- ative resistance oscillators with the transistor in the Common Base (CB) configuration [2]. In spite of the existence of a large body of work on transistor negative resistance oscillator design, there is little in the way of coherent explanation of the mechanism of negative resistance generation, and why, for example, inductive feedback is preferable in some transistor configurations and capacitive feedback in others. In this short paper we attempt to shed a little more light on this topic by deriving a simple, closed form, equation for the value of series feedback termination most likely to induce negative resistance in a given transistor. This equation requires only the 2-port S-parameters of the active device. The ’optimum’ feedback termination thus calculated may be used as the starting point for further CAD simulation and optimisation, thereby helping to shorten the design cycle, or could be used without further refinement in some simple oscillator design scenarios. It should be noted that we have referred to bipolar transistors throughout this paper, but our observations could apply equally to a FET or any other 3-terminal microwave active device. II. DESIGN OF THE ACTIVE 2- PORT SUB- CIRCUIT Transistor feedback circuit analysis and design is best carried out with the transistor represented by a 3-port S-matrix, which can be directly measured using suitable equipment [3], or calculated from the measured CE 2-port S-parameters [4]. The CE 3-port shown in figure 1 is characterised by the following 3-port s-matrix :-   b 1 b 2 b 3   =   s 11 s 12 s 13 s 21 s 22 s 23 s 31 s 32 s 33     a 1 a 2 a 3   (1) Where a i and b i represent incident and reflected power waves at the respective ports [5]. The 3-port S-matrix for the CB and CC configurations can be constructed simply be rearranging the port assignments in (1). Port 1 Port 2 Port 3 Fig. 1. Series CE 3-port definitions for a transistor Figure 2 shows series feedback 2-port sub-circuits created by adding a passive series feedback termination, Γ 3 , to the common port of a transistor in the the three possible configu- rations, with Port 2 of the sub-circuit terminated in the system characteristic impedance, Z o . Bias circuitry has been omitted for simplicity. These sub-circuits represent the basic building block of a typical transistor negative resistance oscillator. To implement an oscillator using the sub-circuits in figure 2 we need to couple a passive resonator to Port 1. The fact that the resonator is passive implies that |Γ in | must be greater than unity [6]. The input reflection coefficient of the CE 2-port sub-circuit in figure 2(a) is calculated given the 3-port s-parameters, s ij , and an arbitrary feedback termination, Γ 3 , as follows [7] : Γ in = s 11 + s 13 s 31 Γ 3 1 - s 33 Γ 3 (2)