IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 60, NO. 11, NOVEMBER 2012 3527 Experimental Characterization of Oscillator Circuits for Reduced-Order Models Pedro Umpiérrez, Víctor Araña, Member, IEEE, and Franco Ramírez Abstract—A new technique is presented to obtain reduced-order models of microwave oscillators from experimental measure- ments. The models can then be applied to the analysis and design of single-oscillator or multioscillator configurations such as cou- pled-oscillator systems. The first-order model is given by the derivatives of the oscillator admittance function with respect to the amplitude and frequency. The experimental technique presented here is based on the extraction of these derivatives from the injection-locked response of the oscillator, synchronized to a small-signal source. The new approach to obtain the derivatives has been validated through comparison with the standard method using commercial harmonic-balance software. The experimental setup and all the key aspects related to calibration and equip- ment requirements are explained in detail. Finally, the proposed technique is applied to characterize a voltage-controlled oscillator at 4.97 GHz. Excellent agreement has been obtained between measurements and simulations. Index Terms—Auxiliary generator (AG) technique, free-run- ning voltage-controlled oscillator (VCO), synchronized VCO, VCO derivatives. I. INTRODUCTION N EW approaches for the exploitation of injection-locked and free-running oscillators have been proposed by dif- ferent authors. In these studies, oscillator-based circuits have been presented as a suitable and convenient alternative to more conventional solutions for frequency conversion [1]–[7], beam- steering, and power combining in coupled-oscillator systems [8]–[10], signal amplification [11], [12], or, more recently, in applications for frequency and phase modulation and demodu- lation [13]–[16]. However, and beside the potential of oscillator-based circuits for the aforementioned applications, only a small amount of effort has been devoted to their investigation, which is mostly due to the complexity of their analysis. Due to their autonomous nature, harmonic balance (HB) converges by default to nonoscillatory Manuscript received April 27, 2012; revised August 02, 2012; accepted Au- gust 06, 2012. Date of publication September 27, 2012; date of current ver- sion October 29, 2012. This work was supported by the Spanish Ministry of Economy and Competitiveness under Contract TEC2011-29264-C03-01 and Contract TEC2011-29264-C03-02 and by the Ramón y Cajal Programme under Contract RYC-2008-02172. P. Umpiérrez and V. Araña are with the Departamento de Señales y Comu- nicaciones, Universidad de Las Palmas de Gran Canaria, Las Palmas 35017, Spain (e-mail: pedro_umpierrez@yahoo.es; varana@dsc.ulpgc.es). F. Ramírez is with the Departamento de Ingeniería de Comunicaciones, Escuela Técnica Superior de Ingenieros Industriales y de Telecomunica- ciones (ETSIIT), Universidad de Cantabria, Santader 39005, Spain (e-mail: ramirezf@unican.es). 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/TMTT.2012.2214054 solutions and in the case of multioscillator configurations, this undesired convergence must be avoided in each of the oscillator components. To cope with some of these limitations, the auxiliary generator (AG) technique has been proposed by Suárez [17]. The AG technique has proven to be a powerful and flexible tool for the analysis, design, and optimization of oscillating circuits in free-running and synchronized regimes. It has also been successfully applied to the analysis of coupled-oscillator systems for beam-steering applications. However, as these multioscillator circuits increase in size andcomplexity, numerical problems arise, related to the size of the equations systems and matrices to be handled in their analysis. A recent example of this is the analysis of coupled-oscillator arrays, where the HB analysis, in combination with the AG technique, is limited to a small number of individual-oscillator elements [9], [10]. In [4], a reduced-order model (semianalytical formulation) was presented for the analysis of autonomous circuits and has been successfully applied to the analysis of frequency dividers [4], injection-locked oscillators, and coupled-oscillator systems containing an arbitrary number of oscillator elements [10]. In [17] and [18], the semianalytical formulation was also applied for the calculation of the phase-noise spectrum of free-running and injection-locked oscillators. This semianalytical formulation is based on a perturbation model about the steady-state solution of the free-running or synchronized solution of the oscillator circuit. In this manner, the system solution for low injection level or weak coupling can be obtained with a much lower computational effort compared to the conventional HB analysis. The reduced-order models involve the calculation of derivatives of the total admittance or impedance function, associated to the individual free-running oscillator, with respect to different state variables. In [4], these derivatives are obtained numerically from HB simulations, using finite differences. Therefore, the reliability of the derivatives is dependent on the models used for the different circuit components. Moreover, if the circuit schematic, internal structure, or models of the linear and nonlinear components are not available, the derivatives required for the reduced-order model could not be accurately estimated. This could also happen when using commercial or third-party oscillators and the provided models do not fit well with measurements. In this paper, a procedure is demonstrated for the ex- perimental extraction of the derivatives of the admit- tance/impedance function of the elementary free-running oscillator. The new methodology is based on the character- ization of the oscillator response when injection locked to a small-signal source. A robust measurement setup will be pre- sented to obtain the derivatives that constitute the reduced-order 0018-9480/$31.00 © 2012 IEEE