Analytical phase noise study of a back-gate coupled colpitts quadrature VCO Emad Ebrahimi a, * , Sasan Naseh a , Ali Ebrahimi b , Mohammad Maymandi-Nejad c a IC Design Research Lab, Faculty of Electrical Engineering and Robotic, Shahrood University of Technology, Shahrood, Iran b Faculty of Electrical Engineering, K.N. Toosi University of Technology, Tehran, Iran c Faculty of Electrical Engineering, Ferdowsi University of Mashhad, Mashhad, Iran ARTICLE INFO Keywords: Impulse sensitivity function LC QVCO Phase noise Noise modulating function ABSTRACT This paper presents a detailed study of phase noise of an LC Quadrature Voltage-Controlled Oscillator (QVCO). This QVCO is made of coupling two identical cross-connected Colpitts oscillators as core oscillators. MOSFETs are used as varactors, the substrates of which are used for coupling the two core Colpitts oscillators. Phase noise improvement resulted from substrate coupling is not only due to elimination of noise sources (as no extra passive or active coupling devices are used) but also due to lower effective thermal noise of the substrates. To verify this, a rigorous phase noise analysis of the circuit using the Linear Time Variant (LTV) model is performed for the 1/f 2 region. It indicates that the improvement is not only due to lower thermal noise of the substrate, but also a better noise modulating factor (NMF). The proposed QVCO was designed and simulated in a 0.18 μm RF-CMOS Technology at frequency of 5.35 GHz with a power consumption of 8 mW, tuning range of 18% and phase noise of 142.8 dBc/Hz at 5 MHz frequency offset. The presented analysis indicates a phase noise of 143 dBc/ Hz at an offset of 5 MHz, which shows a good agreement with the simulation. The difference between the simulated and calculated phase noise at high frequency offsets is less than 0.2 dB. 1. Introduction The Quadrature VCO is widely used in modern portable transceivers to reduce bandwidth and image rejection limitations [1]. Besides, the demand for zero IF or direct conversion transceivers is increased rapidly due to their small size and low power consumption [2]. In fact, quad- rature oscillator is a key building block in zero IF and modern trans- ceivers and has received much attentions in the recent RFIC designs. The most important parameter in the state-of-the-art QVCOs is phase noise and has been well studied in Ref. [3–5]. Hajimiri’s phase noise theory [3] has a different approach toward the stationary noise of transistors in the oscillators. According to that theory, the transconductance of transistors and thus the channel thermal noise of transistors have variations within each period. In other words, the noise of such devices is cyclostationary. Andreani has developed Hajimiri’s theory for a quadrature oscillator [4,5] and investigates the phase noise of the conventional cross-coupled oscillators. He has shown that the coupling transistors inject noise to the oscillator, which affects the phase noise adversely. After that, based on Hajimirie’s theory, different coupling techniques and noise shaping methods have been presented in order to reduce the effect of noise of the coupling devices on the total phase noise of QVCOs [6–10]. In this paper, the analysis of phase noise in a Colpitts QVCO which uses back-gate coupling is presented and verified by simulation. The main goal of this work is to rigorously analyze the phase noise perfor- mance of back-gate coupling method by providing a closed formulae, which clearly shows how the use of substrates for coupling of two core Colpitts oscillators improves the phase noise. The paper is organized as follows: in the next section the substrate coupling method is presented. The phase noise of the proposed QVCO is analyzed in section 3. Analysis and simulation of the phase noise are compared in section 4. The conclusion of the work is presented in section 5. 2. Description of the proposed coupling method Different topologies have been presented for quadrature signals generation, among which LC QVCOs are preferred due to their good phase noise performances. In LC QVCOs, two LC cross-connected oscil- lators are coupled to each other, by mutual injection of the first har- monics [11,12] or the second harmonics [13]. The injection is done through coupling devices such as transistors, capacitors, or on-chip transformers. The coupling devices can dissipate power and inject extra * Corresponding author. E-mail address: eebrahimi@shahroodut.ac.ir (E. Ebrahimi). Contents lists available at ScienceDirect Microelectronics Journal journal homepage: www.elsevier.com/locate/mejo https://doi.org/10.1016/j.mejo.2020.104784 Received 12 September 2019; Received in revised form 28 February 2020; Accepted 7 April 2020 Available online 13 April 2020 0026-2692/© 2020 Elsevier Ltd. All rights reserved. Microelectronics Journal 100 (2020) 104784