Ring-Oscillator with Multiple Transconductors for Linear Analog-to-Digital Conversion

: This paper proposes a new circuit-based approach to mitigate nonlinearity in open-loop ring-oscillator-based analog-to-digital converters (ADCs). The approach consists of driving a current-controlled oscillator (CCO) with several transconductors connected in parallel with different bias conditions. The current injected into the oscillator can then be properly sized to linearize the oscillator, performing the inverse current-to-frequency function. To evaluate the approach, a circuit example has been designed in a 65-nm CMOS process, leading to a more than 3-ENOB enhancement in simulation for a high-swing differential input voltage signal of 800-mV pp , with considerable less complex design and lower power and expected area in comparison to state-of-the-art circuit based solutions. The architecture has also been checked against PVT and mismatch variations, proving to be highly robust, requiring only very simple calibration techniques. The solution is especially suitable for high-bandwidth (tens of MHz) medium-resolution applications (10–12 ENOBs), such as 5G or Internet-of-Things (IoT) devices.


Introduction
The scaling down of CMOS processes has posed new challenges for analog-to-digital conversion, especially from the analog design perspective. Digital logic consumes less power, occupies less area, and works faster as design processes get smaller. Nevertheless, analog designs have become highly complex due to the low voltage supply and limited devices' gain, higher noise impact, mismatch, and parasitic effects [1]. Thus, the current trend is towards mostly digital implementations.
Energy-efficient wide-band ADCs are essential for applications such as portable battery-powered devices or radio-receivers. Flash ADCs are usually implemented for high-speed analog-to-digital conversion [2]. However, the power consumption increases exponentially with the number of bits, making them less power-efficient for more than 10 ENOBs [3]. Additionally, they are extremely sensitive to mismatch phenomena when implemented with minimum-size devices and not scalable. To minimize this impact, devices' dimensions are often increased at the cost of larger occupied area and higher power consumption [4,5].
SAR (Successive Approximation Register) ADCs are low-power, simple, and friendlydigital architectures. They are more energy-efficient than Flash-based architectures due to their mostly-digital implementation suitable with modern processes nodes [6,7]. Nevertheless, their sampling rate is limited by the need of a high-speed clock for the control front-end circuit. The main disadvantage is that the input signal attenuation directly entails a lower dynamic range.
non-linear curve non-linear curve . f = k v; where k is a gain term. The possibility of going towards a fully-synthesizable architecture might be considered in the future, but it is currently out of the scope of the present manuscript, mainly due to the analog nature of the transconductors. All the digital parts can be implemented by using digital synthesis [3,6,7,[40][41][42][43][44][45]. Tools for automated analog design are on-the-spot now and may become of application for the current architecture [46][47][48][49].
Our proposal is evaluated by transient simulation with a low power (LP) TSMC 65-nm CMOS process, showing excellent performance in power and area, especially compared to digital calibration techniques [32], significantly reducing the total harmonic distortion (THD) power. In addition, PVT variation and Monte Carlo simulations show that the solution is robust against those variations, requiring simple calibration to achieve substantial distortion enhancement and proper resolution improvement for high-swing input signals. Finally, the proposed circuit is completely scalable as the calibration is also performed digitally, making it suitable for PVT/mismatch compensation in very deep submicron CMOS nodes such as 16-nm or 7-nm.

Nonlinearity in Ring-Oscillators
The conventional structure for a pseudo-differential open-loop VCO-based ADC is depicted in Figure 2a, built with one ring-oscillator per branch followed by some digital logic that samples and computes the first difference of the output phases (Figure 2b [35]). The VCO is composed of a transconductor-based front-end stage (g m ) and an N-phase CCO. The voltage-to-current conversion could be performed with a single NMOS transistor. Here, the instantaneous oscillation frequency ( f osc (t)) of the ring-oscillator follows the expression: where f o is the rest oscillation frequency, K VCO is the ring-oscillator gain, g m is the transconductance of the front-end circuit (the single NMOS device in Figure 2b), and x(t) is the input voltage centered at zero. Level shifters are required to saturate amplitude-modulated oscillator output signals before being sampled and operated in discrete time. As open-loop configuration corresponds to a highly technology-scalable architecture; this ring-oscillator is implemented with conventional CMOS logic gates and the circuitry afterward is composed of registers and XOR gates. However, the voltage-to-frequency response of the VCO is not linear due to the nonlinear time-delay dependence of the logic gates of the oscillator with respect to the flowing current [50], and the nonlinear voltage-to-current relation in the transconductance g m .
To have some intuition about the ring-oscillator nonlinear characteristic and its effect on the output data, a behavioral model of Figure 2 was designed in a 65-nm CMOS technology. The model included a 45-phase ring-oscillator with the rest of the blocks modeled with Verilog-A. The sampling frequency ( f s ) was 2 GHz with an analog bandwidth (ABW) of 50 MHz. The oscillation parameters were f o = 450 MHz and K VCO · g m = 1 GHz/V, for a 3 MHz sinusoidal differential input signal of 800 mV pp . The nominal voltage supply was 1.2 V. Figure 3 depicts the spectrum of the output data y[n] resulting from a transient simulation. The harmonic distortion is clearly visible, with the third and fifth harmonic distortion terms (HD3 and HD5) equal to −44 dBc and −68 dBc, respectively (the second harmonic distortion term, HD2, term was −39 dBc in a single-ended configuration). The signal-to-noise ratio (SNR) is 63 dB and the signal-to-noise-distortion ratio (SNDR) is 44 dB, which means a degradation of approximately three ENOBs due to distortion. As seen, a peak SNDR higher than 50 dB becomes impossible, making this architecture unsuitable for next generation WLAN standard or 5G, where a peak SNDR higher than 65 dB over wide bandwidth is required [51].  Figure 4 displays the voltage-to-frequency response of the oscillator simulated above. The ideal linear response has also been plotted. Assuming the ring-oscillator configuration of Figure 2b, we may distinguish between three different regions for the NMOS transistor: the saturated region, where the ring-oscillator shows a behavior approximately linear; and the sub-threshold and ohmic regions, where the nonlinearity is clearly visible. When having the NMOS working at such regions, the harmonic distortion will increase due to the joint action of both the nonlinear voltage-to-current conversion and the nonlinear time delay dependence of the ring-oscillator. This restricts most of the linear operating region to a small input voltage range, which might be suitable for low-swing input voltage applications [37], but not for high-swing ones. To increase the linear voltage range, we can inject more current into the oscillator in the ohmic region to move the curve up (see the arrows in Figure 4), and drain current in the sub-threshold region to move the curve down.  With that purpose in mind, we propose a circuit-level solution to extend the linear response of the ring-oscillator. The solution is based on injecting current into the ringoscillator with several transconductors, instead of a single one, shifting the saturation regions of them throughout the whole desired input voltage range. On the one hand, we assume that we have at least one device working in saturation at any point of the input voltage range, getting an approximately linear voltage-to-frequency response. On the other hand, we have more flexibility to control the injected current into the ring-oscillator and perform a voltage-to-current conversion that mitigates the distortion due to the nonlinear time dependence of the logic gates. Looking at Figure 4, two current-based operations are needed: injection and draining, requiring respectively both NMOS and PMOS-based devices in the front-end circuit. Thus, the current flowing through the ring-oscillator I RO (t) can be expressed as follows:

Proposed Multiple-Transconductor Ring-Oscillator
where I N,i (t) is the current provided by the i-th NMOS device, and I P,j (t) is the current drained by the j-th PMOS device.
Using both PMOS and NMOS devices may suppose issues arising from matching and more complexity in making the calibration of the circuit. This is why, for a proof of concept, it was decided to correct and extend only the ohmic region by means of NMOS devices. Several NMOS devices are then connected in parallel. Each of these devices has its own offset component to control the point when they get into the saturation region. Figure 5 depicts a diagram of the proposed solution with a ring-oscillator whose input current is provided by M NMOS devices, where x off,i represents the offset voltage for each of the transconductors. These offset values must be allocated throughout the desired input voltage range to feed the proper current that approximates the inverse current-frequency relation of the ring-oscillator. Linearity is kept mainly because there is always at least one transconductor which is providing sufficient transconductance. Although the rest of transconductors are still providing some current gain, these values are negligible and do not affect the approximation made.
... Apart from the mitigation of the nonlinearity, the proposed structure could also be used to enhance the oscillator gain K VCO . Note that, for the conventional approach (Figure 4), the oscillator gain is limited because the transconductor g m drops into the ohmic region and is not able to provide sufficient current to keep increasing the oscillation fre-quency with the same slope as in the saturation region. With the proposed solution, at any point of the selected input range, there is at least one transconductor providing enough current to keep a linear relation, so that the highest achievable oscillation frequency increases.
Finally, jitter requirements are alleviated due to the open-loop structure, in comparison to closed-loop architectures [21].

Circuit Validation
To evaluate the proposed architecture, a version of the circuit of Figure 5 was designed in a 65-nm CMOS process with a pseudo-differential configuration ( Figure 2a) and 45 phases in the ring-oscillator. The front-end circuit of the ring-oscillator, together with the circuit used to trim the offset component of each transconductor, is shown in Figure 6. The offset component of the input signal is one of the offset references needed, and the other voltage references are generated by means of fixed bias currents and diode-connected transistors M 1 and M 2 . Due to the limited headroom voltage, M 2 operates in the sub-threshold region. In our case, three offset references were needed, with the values displayed in Figure 6. The size distribution of the transconductors was chosen according to a static behavior, performing a sweep of several transient simulations with different DC input values to get the voltage-to-frequency relation and quantify the nonlinear coefficients. Low-power opamps connected in a buffer configuration were used to avoid kick-back noise. The opamp does not require high DC gain, hence its design does not lead to a technology scaling issue. The gain-bandwidth product value was 200 MHz and the margin phase was 55 • , ensuring the proper operation of the system. Figure 6b  and partially for x op,N,3 , which gets distorted for voltage values lower than the threshold voltage (around 0.25 V). This is not relevant because, for gate voltage values lower than the threshold voltage, N 3 operates in a sub-threshold, and the current provided is negligible and does not significantly modify the approximation to the inverse oscillator's current-tofrequency function. Opamp's circuit is depicted in Figure 7, which consists of a two-stage Miller-compensated opamp with low offset (3 mV simulated, which is sufficiently low for proper performance). Transient simulations were used to check the nonlinearity mitigation. Digital demodulation and sampling logic were modeled in VerilogA language. 10-fF capacitors were placed at the outputs of the ring-oscillator to simulate the input capacitance of the digital logic. Oscillation parameters were kept similar to Figure 3, except K VCO · g m , which was equal to 1.25 GHz/V, slightly higher than before due to linearity compensation. Figure 8 depicts the result of the nominal transient simulation. The HD3 and HD5 were equal to −67 and −75 dBc, respectively, leading to an SNDR of 63 dB (the HD2 value observed in a single-ended configuration was −63 dBc). The distortion is substantially mitigated in comparison to Figure 3. The same simulation was performed with output capacitors of 30 and 50 fF, achieving SNDR values of 62.8 dB and 61.7 dB, respectively, due to the slight decrease in the oscillator gain. The proposed solution achieved a THD value of −66.4 dBc for a differential input of 800 mV pp , [32] reported a THD chip measurement of −63.6 dBc for a differential input of 566 mV pp , and [52] reported a THD of −65.3 dBc for a differential input of 400 mV pp in simulation.
Integral nonlinearity (INL) performance, as a static characterization, was also analyzed. The results of a linear ramp test, using the best-fit straight-line approximation, are shown in Figure 9. The single-ended peak-to-peak INL error was [−0.44 0.26], within 1 LSB. In [40], a fully-synthesizable VCO-ADC is presented, where the INL achieved after digital correction was in the range (−1.

Circuit-Level Impairments
To achieve a correct performance from the proposed solution, proper offset references for each of the transconductors need to be selected. The ring-oscillator might be affected by mismatch effects and PVT variations. This will result in variations in the oscillator current-to-frequency relation and in the required voltage-to-current function. The trimming circuit will be affected by these effects as well, modifying the transconductors' bias points. To reduce mismatch effects between devices, the diode-connected transistors (M 1 and M 2 ) were designed to be large. To analyze the variation of x off,N,1 , x off,N,2 and x off,N,3 ; and the oscillation frequency due to mismatching a set of 300 runs of Monte Carlo simulations was performed. The SNDR was normally distributed with a mean value of 63 dB and a standard deviation of 1.8 dB, with a worst-case SNDR value, was equal to 59.5 dB.
Apart from the mismatch verification, the impact of PVT variations in the VCO linearity was also checked. We noticed that the function of Figure 4 is mainly shifted throughout the horizontal axis, but its shape did not vary significantly for different PVT conditions. Consequently, centering the input offset in the linear region is the easiest requirement to keep the oscillator working in a linear manner. Figure 10 shows the SNDR values achieved for different PVT cases with the offset of the input signal (x off,N,2 ) correctly tuned. SNDR degradation can be mainly observed for high temperatures. This degradation occurs when all the transconductors drop into the ohmic region due to the limited V ctrl ( Figure 6), which is of special relevance for the FF case where the working linear region is dramatically reduced. To improve the linearity of the proposed solution in these cases, the voltage supply could be increased at the expense of increasing the power consumption or resizing the ring-oscillator at the expense of reducing the gain. All of the simulation results depicted in Figure 10 were obtained with a nominal voltage supply of 1.2 V. The same simulations were repeated including a variation of ±50 mV in the voltage supply. The resulting SNDR values did not differ from the ones shown in Figure 8, which means strong robustness against voltage supply variations. Additionally, periodic steady-state and phase-noise analyses were performed to evaluate the limitation that the oscillator imposes to the system in terms of phase noise [53]. The estimated value of SNDR regarding the oscillator phase noise was 71 dB, much lower than the quantization noise based SNDR limit observed in Figure 8.

Calibration Circuit
A calibration circuit is required to correctly select the offset of the input signal x off,N,2 , approximate the inverse current-to-frequency ring oscillator's function, and keep proper linearity. With that purpose in mind, we propose to use a digital foreground calibration circuit enabled periodically. Therefore, static power is not increased. The circuit is depicted in Figure 11. The idea is measuring the rest oscillation frequency f o , identifying the PVT operating point of the prototype and correctly selecting the best input signal offset, similar to what was done for Figure 10 elaboration. The rest oscillation frequency is measured by means of a digital delay chain (NAND gates) that measures the semiperiod of the oscillating signal coming from one of the phases of the ring-oscillator. The outputs of the NAND gates are stored with flip-flops, whose outputs are thermometically encoded and represent an estimation of the semiperiod of f o . This digital estimation gets into a Look-Up- Table (LUT) with previously defined values that select the best offset of the input signal. This offset value is finally controlled by means of an opamp whose output offset component can be digitally tuned.

Discussion
The proposed architecture does not add extra relevant power consumption. The main contributor to the power breakdown is the oscillator (1.5 mW per oscillator-in line with other previous publications). The power consumption of each opamp is of 115 µW, and the branch needed to generate the offset references consumes 3.6 µW. As a consequence, the power consumption associated with the extra circuitry for linearization is less than 19% of the total required power, while, in [32], it is 60%. Although this power estimation will no doubt increase in an experimental prototype, it is expected that the power ratio between these elements remains. In relation to the area, it is not expected that our solution adds extra area growth in comparison to other digital calibration techniques.
The architecture in [32] introduces a VCO-based nonuniform sampling (NUS) ADC, which involves on-chip nonlinearity estimation and off-chip nonlinearity correction embedded within a non-uniform digital signal processing (DSP). The area of the set of structures dedicated to the calibration occupies almost the same of the ADC itself: 0.13 mm 2 and 0.14 mm 2 , respectively. In [36], a reconfigurable CTSDM for analog-to-digital conversion is presented with an on-chip digital background calibration and self-canceling dither techniques. The calibration unit occupies 64% of the area of the whole chip and the voltageto-current converter and the ring-oscillator consume less than a fifth part of the total power dissipation [54]. Our solution for nonlinearity mitigation does not involve background continuous intensive digital operations and is performed in the analog domain, reducing the required power consumption and also the occupied area.
The VCO-based open-loop configuration ADC in [37] presents a resistive network to tune the ring-oscillator voltage-to-frequency function and reduce harmonic distortion. The principle of the resistive divider scheme was originally introduced in [39], and compromises ring-oscillator's gain by attenuating the input amplitude. Our solution does not restrict oscillator's gain.
The structure explored in [19] contains a VCO as a quantizer within a ∆Σ loop. Nonlinearity is mitigated through a high-gain loop filter. The VCO-based quantizer only occupies 3.7% of the active area and consumes 13% of the total power, but it is hardly extended to low voltage supply environments. While enough voltage supply is granted to allow digital switching in the ring-oscillator, our solution can be implemented for lower voltage supply applications. Table 1 summarizes the performance of the proposed design and provides a comparison to prior works. Different VCO-based ADCs structures (a fully-synthesizable design [40] is also included), hybrid SAR-VCOs, and traditional Flash and SAR architectures are characterized as competitive alternatives. To achieve a fair comparison between the reference solutions and the proposed design, area, power, and Figure-of-Merit (FoM) values only include the ring-oscillator and the associated linearization blocks for the cases where data are available ( * ).

Conclusions
A circuit-level solution to linearize ring-oscillators-based ADCs is proposed. The solution is based on making use of several transconductors connected in parallel with different bias conditions to implement a voltage-to-current function that approximates the inverse nonlinear current-to-frequency function of the ring-oscillator. To evaluate the approach, a ring-oscillator-based ADC with the proposed circuit was designed and simulated in 65-nm. Nonlinearity was strongly reduced resulting in an ENOB enhancement of more than three bits for high-swing inputs. Additionally, mismatch effects, PVT variations, and noise impact were assessed. The proposal exhibited great robustness without resorting to a complex circuit design and just requiring simple foreground digital calibration. The new VCO-based ADC structure benefits from important power savings in comparison to state-of-the-art digital calibration circuits conventionally used to mitigate distortion. It is also expected to achieve area savings, but this needs to be confirmed through experimental prototypes. The proposal is particularly intended for high-bandwidth and medium-resolution applications, such as 5G or IoT (Internet-of-Things) modules.

Data Availability Statement:
No new data were created or analyzed in this study. Data sharing is not applicable to this article.

Conflicts of Interest:
The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.

Abbreviations
The following abbreviations are used in this manuscript: ADC