Agilent Measurement Journal - Issue 5 - 2008 - (Page 46)

Modeling phase noise Over the years, researchers have studied many ways to characterize the phase noise in electrical oscillators. Leeson’s model, published in 1966, was the ﬁrst attempt to predict phase noise in oscillators.1 It is described with the following equation: Equation 2. Equation 3. Qout Qi Equation 4. NKpKvcoF (s) Ns + KpKvcoF (s) NGOL (s) 1 + GOL (s) Qout QPFD Equation 5. F (s) Kvco s Kvco s ℓ (f) 2FkT Ps () f osc 2Qf ( ) 1 1 + GOL (s) ) 2 F is the noise factor of the oscillator’s gain element, k is the Boltzmann constant, T is the absolute temperature, Ps is the oscillator signal power, fosc is the carrier frequency, Q is the quality factor (formed from the resonating inductance (L) and capacitance (C) portion of the oscillator), and f is the offset frequency from the carrier. These days, highly detailed models are being used to characterize oscillator performance. For example, Razavi uses a linear time-invariant (LTI) model to describe the behavior of phase noise in oscillators, while Hajimiri relies on a more accurate linear time-variant (LTV) model.2, 3 Demir derived a nonlinear stochastic differential equation for phase error, and solves this equation in the presence of random perturbations.4 To better understand PLL noise, consider the generic PLL shown in Figure 1. It includes ﬁve key elements: phase frequency detector (PFD), charge pump (CP), loop ﬁlter (LF), VCO, and frequency divider (FD). The noise transfer function of the closed-loop PLL is derived as follows: Qout QLF Equation 6. ( 1 1 + GOL (s) ) Qout QVCO ( 1 1 + GOL (s) In this derivation, Qi, Qout, QPFD, QlF, and QVCO represent the noise signal at different stages; N is the divide ratio from the PLL divider circuit; K p and Kvco are gain attributed to the PFD and VCO, respectively; GOL(s) is the open-loop gain; and s is time in seconds. Figure 2 shows the typical noise contribution from individual blocks in the example PLL. In the ﬁgure, P refers to the charge pump stage of the PLL and t is the period of one clock cycle. Note that within the PLL loop bandwidth (w c) the phase noise is typically dominated by contributions from the frequency dividers (blue) and phase detector (green). For frequencies well below the loop bandwidth, the phase-noise plot typically ﬂattens out due to the cumulative noise being dominated by the phase frequency detector; thus, the resultant in-band noise is essentially ﬂat below the loop bandwidth. Outside of the loop bandwidth the major noise contributor is the VCO. Qe = Qi − Qout N Qi QPFD Kp PFD/CP F(s) LF QLPF K vco s QVCO Qout VCO ÷N Qout N Frequency Figure 1. Block diagram of a generic PLL with noise signals at different nodes 46 Agilent Measurement Journal

Table of Contents Feed for the Digital Edition of Agilent Measurement Journal - Issue 5 - 2008

Agilent Measurement Journal Issue 5 Finding a New Path when Assumptions Break Down Contents Testing MIPI D-PHY Protocols Oscilloscope Firmware Update DNA Replication Joining the LTE/SAE Trial Initiative Testing Proteins Praise for DC Power Analyzer Scripting Features for AMDS Testing Hybrid PCBA Systems Improving the Gene Expression System Workflow Turning a “Good Enough” Test Strategy into One That’s Reliable, Repeatable and Traceable Understanding the Effects of Limited-Bandwidth Channels on Digital Data Signals Introducing the 3GPP LTE Downlink Validating the 26 Validating the Physical and Protocol Layers in DDR Memory Interfaces Understanding Total Jitter Measurements of Low Probabilities Using Behavioral-Model Simulation to Accurately Predict First-Order PLL Performance Creating Synchronous High-Frequency Sampling across Multiple Digitizers Overcoming the Challenges of Testing FlexRay Networks Understanding Operating Life and Repeatability of Electromechanical Switches and their Effect on Total Cost of Ownership Implementing Micro and Nano LC/MS Techniques for High Sensitivity Lipidomics Analysis

Agilent Measurement Journal - Issue 5 - 2008

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