# SNDR Limits of Oscillator-Based Sensor Readout Circuits

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Phase Noise of an Autonomous Oscillator

_{carrier}is the total signal power around ${f}_{0}$, which is also equivalent to the power of the fundamental harmonic of the noiseless oscillation. Definitions (1) and (2) are approximately equivalent for low phase fluctuations, but they differ at low offset frequencies.

## 3. Input Referred Noise Model of a VCO

## 4. Prototype Measurements

^{®}emulating the behavior ot the XOR-based F2D converter shown in Figure 8.

## 5. VCO Simulation and SNDR Estimation

^{®}Virtuoso

^{®}Design Environment version IC6.1.6.500.6). The model proposed in Section 3 and validated in Section 4 is specially useful to characterize the performance of the VCO in terms of phase noise. At the end of this section we also show some simple calculations that allow to estimate the distortion of the oscillator.

^{®}Spectre

^{®}RF Option provides the Harmonic Balance (HB) analysis and the Shooting Newton method to calculate the periodic steady-state (PSS) of oscillators. The Shooting Newton method calculates the time-domain PSS and it is suitable for highly nonlinear circuits such as ring oscillators, relaxation oscillators, and frequency dividers. HB performs a frequency-domain analysis, which is more efficient for weak and midly nonlinear circuits such as LC oscillators [37]. The VCRO simulated is a strongly nonlinear circuit with sharp transitions, so the Shooting Newton method is in principle more suitable. A PSS simulation can determine in a few seconds that the oscillation frequency of this VCO is ${f}_{0}$ = 60.57 MHz. Taking advantage of the PSS sweep tool, we can perform several PSS analysis while sweeping the input voltage in order to calculate the VCO sensitivity, which in this case is ${k}_{d}$ = 2.28 V${}^{-1}$.

## 6. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 1.**Example of two oscillator-based sensor readout circuits. The sensing element can be integrated into the oscillator (

**a**) or can be connected to an analog interface that generates an intermediate signal ${v}_{in}$ that modulates the oscillator (

**b**). In both cases, the input measurand $x\left(t\right)$ modulates the frequency of the oscillation $v\left(t\right)$ (

**c**).

**Figure 2.**Effects of noise in the spectra of a VCO-ADC. (

**a**) Spectrum of $v\left(t\right)$; (

**b**) Spectrum of the output of the converter $y\left[n\right]$ assuming a sinusoidal input.

**Figure 3.**Classical approach to estimate the influence of phase noise in the performance of first-order VCO-ADCs.

**Figure 7.**(

**a**) Diagram of a real VCO with phase noise added to the phase of the oscillator; (

**b**) Equivalent block diagram of the VCO with the phase noise referred to the input.

**Figure 9.**(

**a**) Power spectrum of the oscillation ${S}_{v}\left(f\right)$; (

**b**) Phase noise and phase fluctuation power spectral density; (

**c**) IRPN and output data power spectral density.

**Figure 13.**(

**a**) ${S}_{v}\left(f\right)$ measured with an spectrum analyzer; (

**b**) Phase noise derived from ${S}_{v}\left(f\right)$; (

**c**) Comparison between the IRPN calculated from $\mathcal{L}\left(\mathsf{\Delta}f\right)$ and the DFT of the measured ADC output.

**Figure 17.**(

**a**) Performance of circuit shown in Figure 15 calculated using transient simulations and estimated from PSS and Pnoise simulations; (

**b**) SNDR difference between both methodologies.

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**MDPI and ACS Style**

Cardes, F.; Quintero, A.; Gutierrez, E.; Buffa, C.; Wiesbauer, A.; Hernandez, L. SNDR Limits of Oscillator-Based Sensor Readout Circuits. *Sensors* **2018**, *18*, 445.
https://doi.org/10.3390/s18020445

**AMA Style**

Cardes F, Quintero A, Gutierrez E, Buffa C, Wiesbauer A, Hernandez L. SNDR Limits of Oscillator-Based Sensor Readout Circuits. *Sensors*. 2018; 18(2):445.
https://doi.org/10.3390/s18020445

**Chicago/Turabian Style**

Cardes, Fernando, Andres Quintero, Eric Gutierrez, Cesare Buffa, Andreas Wiesbauer, and Luis Hernandez. 2018. "SNDR Limits of Oscillator-Based Sensor Readout Circuits" *Sensors* 18, no. 2: 445.
https://doi.org/10.3390/s18020445