Selected Papers from the Midwest Symposium on Circuits and Systems
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Yan Zhu, UFat Chio, HeGong Wei, SaiWeng Sin, SengPan U, R. P. Martins, "Linearity Analysis on a SeriesSplit Capacitor Array for HighSpeed SAR ADCs", VLSI Design, vol. 2010, Article ID 706548, 8 pages, 2010. https://doi.org/10.1155/2010/706548
Linearity Analysis on a SeriesSplit Capacitor Array for HighSpeed SAR ADCs
Abstract
A novel Capacitor array structure for Successive Approximation Register (SAR) ADC is proposed. This circuit efficiently utilizes charge recycling to achieve highspeed of operation and it can be applied to highspeed and lowtomediumresolution SAR ADC. The parasitic effects and the static linearity performance, namely, the INL and DNL, of the proposed structure are theoretically analyzed and behavioral simulations are performed to demonstrate its effectiveness under those nonidealities. Simulation results show that to achieve the same conversion performance the proposed capacitor array structure can reduce the average power consumed from the reference ladder by 90% when compared to the binaryweighted splitting capacitor array structure.
1. Introduction
The SAR ADC is widely used in many communication systems, such as ultrawideband (UWB) and wireless sensor networks which require lowtomediumresolution converters with low power consumption. Traditional SAR ADCs are difficult to be applied in highspeed design; however the improvement of technologies and design methods have allowed the implementation of highspeed, lowpower SAR ADCs that become consequently more attractive for a wide variety of applications [1, 2].
The power dissipation in an SAR converter is dominated by the reference ladder of the DAC capacitor array. Recently, a capacitor splitting technique has been presented, which was proven to use 31% less power from the reference voltage and achieve better DNL than the binaryweighted capacitor (BWC) array. The total power consumption of a 5โb binaryweighted split capacitor (BWSC) array is 6โmW which does not take into account the power from the reference ladder [3]. However, as the resolution increases, the total number of input capacitance in the binaryscaled capacitive DAC will cause an exponential increase in power dissipation as well as a limitation with reduction of speed due to a large charging timeconstant. Therefore, small capacitance spread for DAC capacitor arrays is highly desirable in highspeed SAR ADCs [4].
This paper presents a novel structure of a split capacitor array for optimization of the power efficiency and the speed of SAR ADCs. Due to the series combination of the split capacitor array both small capacitor ratios and powerefficient charge recycling in the DAC capacitor array can be achieved, leading to fast DAC settling time and low power dissipation in the SAR ADC. The parasitic effects, the position of the attenuation capacitor, as well as the linearity performance (INL and DNL) of the proposed structure will be theoretically discussed and behavioral simulations will be performed. Different from the BWSC array, which only achieves better DNL (but not INL) than the BWC array, the proposed capacitor array structure can have both better INL and DNL than the series capacitor (SC) array. The design and simulations of an 8โb 180MS/s SAR ADC in 1.2V supply voltage are presented in 90โnm CMOS exhibiting a SignaltoNoiseandDistortion Ratio (SNDR) of 48โdB, with a total power consumption of 14โmW which demonstrates the feasibility of the proposed structure.
2. The Overall SAR ADC Operation
The architecture of an SAR ADC is shown in Figure 1, consisting of a series structure of a capacitive DAC, a comparator, and successive approximation (SA) control logic. The SA control logic includes shift registers and switch drivers which control the DAC operation by performing the binaryscaled feedback during the successive approximation. The DAC capacitor array is the basic structure of the SAR ADC and it serves both to sample the input signal and as a DAC for creating and subtracting the reference voltage.
3. Capacitor Array Structure
3.1. Capacitor Structure Design
The major limitation on the speed of the SA converter is often related with the RC time constants of the capacitor array, reference ladder, and switches. For a BWC array the size of capacitors rises exponentially with the resolution in number of bits, which causes large power and RC settling time, thus limiting the speed of the overall SAR ADC. To solve this problem, Figure 2(a) shows an SC array [5], which utilizes attenuation capacitors to separate the capacitive DAC into bits MSB and bits LSB arrays. Thus, smaller capacitor ratios can be achieved when compared to the BWC array. However, chargeredistribution switching method for the SC array has been proven to be inefficient when discharging the MSB capacitor and charging the MSB/2 capacitor, which consumes 5 times more power than the chargerecycling switching method. Thus, a seriessplit capacitor (SSC) array is proposed, as shown in Figure 2(b), which can both alleviate the speed limitation and implement a chargerecycling switching approach.
(a)
(b)
The solution to perform chargerecycling for SC array is different from the BWC array, which just splits the MSB capacitor into subarrays. As illustrated in Figure 2(b), the of the SC array is split into subarrays in the MSB array, where the total capacitance of the 1 subarrays is and as a result the capacitors in LSB array and should be doubled; thus the can be calculated as
where and are the sum of LSB and MSB array capacitors, respectively. The can then be seen as two split unit capacitors attached to the right side of MSB array to maintain the capacitive ratio as . Therefore, the chargerecycling methodology in each section can perform binaryscaled feedback during the successive approximation.
3.2. Charge Recycling Implementation
In the proposed implementation the seriessplit capacitor array is designed to achieve charge recycling for the () bit capacitive DAC, as shown in Figure 2(b). During the global sampling phase, the voltage is stored in the entire capacitor array. Then, the algorithmic conversion begins by switching all upper capacitor arrays to and the lower to , respectively, instead of switching only the MSB capacitor to and others to . This implies that in the conversion phase 1 (corresponding to MSB capacitor conversion) settles to (considering only differential node voltage)
and the comparator output will be
The comparator output will decide the switching logic of and . If is low, is switched to , dropping the voltage at [] to . If is high, is switched to , raising the voltage at [] to . The above process is repeated for cycles. As is switched from to (bit decision back from โ1โ to โ0โ) the switches, from to , are kept connected to and drive [] to []. The initial charge, supplied by in phase 1, is kept stored in the capacitors which will connect to at phase 1, instead of being redistributed; so the charge formed at phase 1 can be recycled in the next phases. However, the conventional switching method that discharges MSB capacitor and charges the MSB/2 capacitor will cause charge redistribution in the capacitor array and thus consuming more power.
3.3. Linearity Performance
To analyze the linearity of the SSC and SC arrays, each of the capacitors is modeled as the sum of the nominal capacitance value and the error term, as follows:
Consider the case where all the errors are in the unit capacitors whose values are independentidenticallydistributed Gaussian random variables with a variance of
and where is the standard deviation of the unit capacitor.
The accuracy of an SAR ADC is dependent on the DAC outputs which are calculated here in the case of no initial charge on the array (). For a given DAC digital input , with equals 1 or 0 representing the ADC decision for bit , the analog output of the SSC array can be calculated as
where
Subtracting the nominal value (i.e., in (6)) from (6) the INL can be calculated as
The first and second terms are quite small when compared with the third and fourth terms in the numerator, and the third term ฮC in the denominator does not depend on the bit decision , which only causes a gain error; then they will be neglected. Thus, (8) can be simplified as
and the variance can be expressed as
To simplify the analysis only the worse INL is considered that combines all the errors together (i.e., ). For (5) it can be concluded that and . Thus (10) can be simplified as
While for the SC array, the can be calculated similarly as
Then, subtracting (12) from (13), its value will become
As a result of (14), the INL of the SSC array should be lower than the SC array which is different from the BWC and BWSC arrays that were already proven to have no difference between the INLs [1].
The maximum DNL for the SSC array is expected to occur at the step below the MSB transition [1], and the two output voltages can be calculated as
subtracting (16) from (15), they will yield
with variance
For SC array the can be calculated similarly as
thus, can be expressed as
Thus, from (20) it can be concluded that the maximum DNL of the SSC is also lower than that of the SC array.
3.4. Parasitic Nonlinearity Effect
One potential issue with these two series capacitor array structures (SSC and SC) is the parasitic capacitances and on the nodes and , which will deteriorate the desired voltage division ratio and result in poor linearity. The parasitic effect is caused by the bottom and topplate parasitic capacitance of as well as the topplate parasitic capacitance of MSB and LSB array capacitors which can be calculated as
where and represent the percentage of bottom and topplate parasitic capacitances of each capacitor, respectively (with metalisolatormetal (MIM) capacitor option, , ). For the SSC array, the analog output with and taken in to account can be calculated as
where denotes . This equation shows that the parasitic capacitances and in the denominator are completely uncorrelated in the bit decisions, which can cause only a gain error and have no effect into the linearity performance. However, the parasitic capacitance in the numerator contributes with a codedependent error, which degrades the linearity of the SAR ADC. Subtracting the nominal value the error term will become
The parasitic capacitance is composed of the parasitic capacitance of and . By reducing the number of bits in the LSB array, the size of can be minimized; thus the nonlinearity effect can be alleviated. But, this will enlarge the capacitor spread in the MSB array; thus the distribution of bits in both MSB and LSB arrays should consider the tradeoff between linearity, tolerance, and capacitance spread limitations.
3.5. Behavioral Simulations
Four behavioral simulations of the SSC and the SC array DAC were performed to verify the previous analysis. The values of the unit and attenuation capacitors used are Gaussian random variables with standard deviation of 1% (), and the ADC is otherwise ideal. Figure 3 shows the result of 10000time Monte Carlo runs, where the standard deviation of DNL and INL is plotted versus output code at the 8bit level. As expected, the SSC array has better INL and DNL than its SC array counterpart. Figure 4 shows the result of 1000 Monte Carlo runs with 5% topplate and 10% bottomplate parasitic capacitances, where the SNDRs are plotted versus different distribution of bits in the MSB and LSB array at the 8bit level. Comparing the SNDR shown in Figure 4, and as expected, a larger number of bits in the LSB array will cause poor linearity. Although can achieve the best SNDR, since larger capacitor ratios will both reduce the conversion speed and increase the power dissipation, will be adopted for circuit implementation due to both good linearity performance and smaller capacitor ratios. Figure 5 illustrates the result of 1000time Monte Carlo runs, where the SNDRs are plotted versus the percentage of the topplate parasitic capacitance for the SSC array structure for an 8bit ADC. With increasing, the parasitic capacitance will decrease the SNDR of the conversion performance. But with approximate variance of a good linearity performance of an SAR ADC can still be achieved. Figure 6 illustrates the result of 1000 Monte Carlo runs, where the SNDRs are plotted versus the percentage of the topplate parasitic capacitance at the 6 to 12bit level with proper bits distribution of the LSB and MSB arrays. From it we can find that the parasitic nonlinearity effect is insignificant; thus the series split structure can also be utilized in highresolution applications.
(a)
(b)
3.6. Power Consumption Analysis
The power consumption of the SAR converter is dominated by the DAC capacitor array, the comparator, and the switchesโ drivers. The arrayโs power is proportional to the sum of the array total capacitance of which the bottomplate is connected to the reference voltage supply and can be calculated as
where is the fullscale input voltage, assuming that has been fully sampled on to the capacitor array and the charge is all supplied by the reference voltage [1]. In a 8bit case, the of the proposed structure is (with ), but for a binaryweighted capacitor array the is , which can consume 5 times more power than the proposed structure. The series combination allows a significant reduction of the largest capacitor ratio; in an 8bit case, the largest capacitor of the series split and binaryweighted split capacitor array structure is and , respectively, which decreases the DAC settling time and speeds up the conversion. The total input capacitance of the proposed structure is not completely dependent on the number of bits of the ADC and can be calculated as
The power consumptions of the comparator and switch drivers are also proportional to the equivalent input capacitance . Therefore, the smaller , , and it will imply an increase in efficiency of the overall conversion performance.
4. Circuit Implementation Details
A highspeed SAR converter imposes a stringent requirement in the clock generation; for example, an 8โb 180โMS/s SAR ADC requires an internal master clock of over 1.62โGHz. To generate such a highfrequency clock pulse the generator will consume even more power than the ADC itself. Due to the power limitations of the clock generator in a synchronous SA design an asynchronous SAR processing technique [4] will be adopted here, where only a master clock of 180โMHz is required.
The dynamic comparator [6] used in this ADC is shown in Figure 7 and it is composed of a preamplifier and a regenerative latch. The preamplifier can provide sufficient gain to suppress the relatively high input referred offset voltage of the latch. Also, the kickback noise of the latch can be isolated by the current mirror between the two stages. The dynamic operation of this circuit is divided into a reset phase and a regeneration phase. During the reset phase the two outputs ( and ) are pulled up to . After the input stage has settled, the voltage difference is then amplified to a full swing during the regeneration phase. The differential output can generate a data ready signal to indicate the completion of the comparison, which will be used to trigger a sequence of shift registers and the switch drivers to perform asynchronous conversion [4]. Dynamic logic circuits are also utilized instead of traditional static logic to release the limitation of digital feedback propagation delay in the SA loop.
5. Simulation of an 8Bit 180โMS/s SAR ADC
To verify the proposed capacitor structure of the capacitive DAC, a 1.2โV, 8โb, 180MS/s SAR ADC was designed using a 90โnm CMOS process with metalisolatormetal (MIM) capacitor option. The SAR ADC was implemented in a fully differential architecture, with a full scale differential input range of 1.2โ. Considering the parasitic capacitance of the attenuation capacitors that will reduce the linearity of the ADC, 5% topplate and 10% bottomplate, they were included in the simulations according to the data from the foundry datasheet.
Figure 8 shows a spectrum plot of the SAR ADC after a Monte Carlo simulation with an input signal of 76โMHz leading to an SNDR of 48โdB, which clearly demonstrates the tolerance to the parasitic effect caused by the . Figure 9 also shows the corresponding 30times Monte Carlo mismatch simulations where the ADC achieves a mean SNDR of 49โdB with an input signal of 76โMHz. The DNL and INL are both within as shown in Figure 10. Figure 11 shows the SNDR versus power consumption from the reference ladder in the proposed architecture, as well as in the BWSC array structure, clearly demonstrating that the BWSC results are poor in terms of SNDR mainly due to the large RC settling time. To reach the same conversion performance the BWSC array consumes 10 times more power than the proposed structure. Table 1 summarizes the overall performance of the SAR ADC with the total power consumption of 14โmW only and an FoM of 0.37โpJ/conversionstep, distinctly proving the low power dissipation feature of the proposed technique.

(a)
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6. Conclusions
A novel seriessplit capacitive DAC technique has been proposed which can both implement an efficient charge recycling SAR operation and achieve a small input capacitance. The reduction of the maximum ratio and sum of the total capacitance can lead to area savings and power efficiency, which allow the SAR converter to work at high speed while meeting a low power consumption requirement. Theoretical analysis and behavioral simulations of the linearity performance demonstrate that the proposed SSC structure can have a better INL and DNL than the traditional SC array structure. Simulation results of a 1.2โV, 8โb, 180MS/s SAR ADC were presented exhibiting an SNDR of 48โdB at a 76โMHz input with the total power consumption of 14โmW that certifies the power efficiency of the novel circuit structure.
Acknowledgments
This work was financially supported by research grants from the University of Macau and FDCT with Ref nos. RGUL/0708S/Y1/MR01/FST and FDCT/009/2007/A1.
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Copyright
Copyright © 2010 Yan Zhu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.