Chapter 3 Implications of transistor mismatch on analog circuit design and system performance 3.1 Introduction In a signal processing system several operations or computations are performed on a signal in di erent stages sequentially. Each of these operations have to emphasize a wanted component or property of the signal without adding too much unwanted extra components. These are due to the non-idealities of the circuit implementation compared to the speci ed operation. Circuit non-idealities can be divided in two groups: random and systematic errors. The random errors are the result of the stochastic nature of many physical processes. The stochastic behavior of charge carriers in a conductor, for instance, results in various types of noise signals and the stochastic nature of the physical phenomena that take place during the fabrication of integrated circuits, results in a random variation of the properties of the fabricated on-chip devices and mismatches between identically designed devices. The systematic errors occur because a typical circuit implementation only approximates an ideal signal processing operation to a limited extent. These errors are caused, for instance, by the non-linear operating characteristics of devices or by the in uence of parasitics in the signal path or device structure. The e ect of these non-idealities can be of di erent kinds. The noise signals limit the minimal signal that can be processed with the system. Device mismatch limits the accuracy of the circuit behavior and again limits the minimal signal or energy that is required to execute meaningful signal operation functions. For linear systems, the non-linearities of devices generate distortion components of the signals or modulate unwanted 'noise' signals into the used signal band. This typically limits the maximal signal that can be processed correctly. The circuit designer can reduce the e ect of the distortion non-idealities by using small 19 20 CHAPTER 3. IMPLICATIONS OF MISMATCH ON ANALOG VLSI modulation indices for the bias signals; by using large device sizes the impact of mismatch is lowered and by using low impedance levels, the thermal noise signals are reduced. These measures have, however, very important consequences on the power consumption and operation speed of the system. Therefore the quality of a circuit realization is evaluated from the obtained accuracy, noise level or linearity relative to the used power and the speed of operation. The designer will try to achieve for a given speed the best performance with a minimal power consumption. The fundamental impact of noise on the overall system performance has been studied extensively in literature see e.g. [Vit 90b] [Voo 93, and its references]. In this chapter we investigate the impact of transistor mismatch on the total performance of analog circuits and systems. First, we discuss the characterization and modeling of transistor mismatch and describe a new extraction method to derive the matching quality of sub-micron CMOS technologies. This quantitative model information is very important for the design of analog circuits since it allows the designer to accurately predict the accuracy performance. Furthermore, it forms the basis for the evaluation of the impact of transistor mismatch on the analog performance. The implications of transistor mismatch on the design of basic analog building blocks is then discussed in detail in sections 3.3 and 3.4. The speed, accuracy and power consumption performances of analog circuits are linked due to the e ect of mismatch on the circuit design; guidelines for the optimal design of circuits are derived. In section 3.5 we generalize these results and prove that mismatch puts a fundamental limitation on the maximal total performance of analog signal processing systems. The SpeedAccuracy2/Power ratio is xed by technological constants that express the matching quality of the technology. For circuit building blocks with high accuracy requirements thermal noise is considered as the limiting factor for performance improvement or power consumption reduction [Vit 94, Dij 94] but we show that the impact of transistor mismatch on the minimal power consumption is more important for present-day CMOS technologies than the impact of thermal noise for high speed analog circuits and massively parallel analog systems. The matching performance is technology dependent and the scaling of the circuit performance with the down-scaling of the technology size is discussed in section 3.5.4. The techniques to reduce the impact of mismatch and their e ect on the performance of systems is reviewed in section 3.6. The last section (3.8) treats the implications for the VLSI design of massively parallel analog systems and discusses the advantages of analog VLSI implementation over the digital VLSI implementation for massively parallel analog systems. 3.2 Modeling and characterization of transistor mismatch 3.2.1 What is transistor mismatch Two identical designed devices on an integrated circuit have random di erences in their behavior and show a certain level of random mismatch in the parameters which model their 3.2. MODELING AND CHARACTERIZATION OF TRANSISTOR MISMATCH 21 behavior. This mismatch is due to the stochastic nature of physical processes that are used to fabricate the device. In [Pel 89] the following de nition for mismatch is given: mismatch is the process that causes time-independent random variations in physical quantities of identically designed devices. 3.2.2 Modeling of CMOS transistor mismatch Mismatch in device parameters can be modeled by using di erent techniques. Several authors [Laks86] [Shy 84] [Miz 94] start from the physical background of the parameters to calculate and model the device mismatch dependence on technology parameters and device size. Or a black box approach can be used by supposing the statistical properties of the di erent mismatch generation processes and calculating their in uence on di erent circuit parameters [Pel 89]. The mismatch of two CMOS identical transistors is characterized by the random variation of the di erence in their threshold voltage VT0 , their body factor and their current factor (the de nitions of these parameters can be found in appendix A). For technologies with a minimal device size larger than typically 2 m, a widely accepted and experimentally veri ed model [Pel 89] [Bas 95] [Pav 94] for these random variations is a normal distribution with mean equal to zero and a variance dependent on the gate-width W and gate-length L and the mutual distance D between the devices: 2 VT0 + S 2 D2 2(VT0 ) = AWL (3.1) VT0 A2 (3.2) 2( ) = WL + S 2D2 ( ) 2 = A2 + S 2D2 (3.3) WL AVT0 , A , A , SVT0 , S and S are process-dependent constants. In table 3.1 and 3.2 the proportionality constants for several processes are summarized. Experimental data show that the correlation between the VT0 and mismatch is very low although both parameters depend on the oxide thickness[Pel 89] [Bas 95]. The last two columns of table 3.2 contain corner distances at which the distance dependent term in the parameter mismatch becomes dominant over the size dependent term. The corner distances Dm is de ned as the distance for which the mismatch due to the distance e ect on a parameter p is equal to the mismatch due to the size dependence for a minimal size device (W = T and L = T , where T is the minimal size of the technology) and is calculated as: Dm = A S (3.4) T p For devices with an area of Af times the minimal area the critical distance Dm is Af times smaller. The obtained critical distances Dm for the present-day processes are very 22 CHAPTER 3. IMPLICATIONS OF MISMATCH ON ANALOG VLSI AVT0 A (VGS ; VT )m [mVm ] [%m ] [V] 2.5m [Pel 89] nMOS 30 2.3 2.6 pMOS 35 3.2 2.2 1.2m [Bas 95] nMOS 21 1.8 2.3 pMOS 25 4.2 1.2 0.7m nMOS 13 1.9 1.4 pMOS 22 2.8 1.6 Table 3.1: The matching proportionality constants for size dependence for di erent industrial CMOS processes. The parameter (VGS ; VT )m is de ned in (3.33) in section 3.3. Technology Type Technology Type SVT0 S DVTm D m T [V /m ] [ppm/m ] [mm] [mm] 2.5m [Pel 89] nMOS 4 2 3 5 pMOS 4 2 3.5 13 1.2m [Bas 95] nMOS 0.3 3 58 5 pMOS 0.6 5 35 12 0.7m nMOS 0.4 2 46 14 pMOS 3 13 Table 3.2: The matching proportionality constants for distance dependence for di erent industrial CMOS processes. 3.2. MODELING AND CHARACTERIZATION OF TRANSISTOR MISMATCH 23 large compared to the typical size of an analog circuit. Therefore the distance dependence of the parameter mismatch will be neglected in the discussion of the impact of transistor mismatch on analog circuit and system performance. In equations (3.1), (3.2) and (3.3) the standard deviation of the di erence of the parameters of two transistors is given. For a random variable Z de ned as Z = X ; Y the variance is 2(Z ) = 2(X ) + 2(Y ). The following relations are thus obtained for the variance of the absolute parameters of a single transistor: p (VT0 ) = 2(VT0 ) p ( ) = 2( ) ( ) = p2 ( ) (3.5) (3.6) (3.7) 3.2.3 Characterization of transistor mismatch The matching behavior of transistors is very strongly dependent on the used IC technology. Therefore an in-house characterization procedure has been set-up [Bas 95]. A new direct extraction algorithm has been developed to extract the VT0 and of a transistor pair I DS from their measured relative current di erence IDS in saturation. The big advantage of measuring currents in saturation is the much lower sensitivity to parasitics in the set-up, which becomes more and more important for sub-micron and deep sub-micron technologies. Also, the model of VT0 mismatch of minimal sized devices in sub-micron technologies has been improved. In this paragraph a short overview is given; for more details the reader is referred to [Stey94b] [Bas 95], [Bas 96c], [Bas 96b], and [Bas 96a]. A) Test Circuits Test-circuits are processed to experimentally check the validity of the models and to determine the proportionality constants of the size and distance dependence of transistor mismatch. In gure 3.1 a micro-photograph of the nMOS test-chip for a 1.2 m CMOS technology is presented. The test-chip contains a matrix of transistors. On a row identical transistors are spaced at di erent mutual distances to examine the mismatch spatial dependence. The di erent rows contain transistors with di erent sizes to determine the mismatch dependence on device size. All sources are connected to a common point. Transistors in the same row have their gates connected and transistors in the same column have their drains connected. Special attention has to be paid in the layout to obtain a very low resistance in the source path to eliminate systematic errors during the measurements; very wide source metal connections are used and can be clearly distinguished in gure 3.1. Two separate test-chips for the characterization of the nMOS transistors and pMOS transistors have been designed. B) Measurement set-up The measurements are carried out on packaged test-circuits using a HP4062A Semiconductor Parametric Test System including a switch matrix and 24 CHAPTER 3. IMPLICATIONS OF MISMATCH ON ANALOG VLSI Figure 3.1: Micro-photograph of the mismatch characterization test-chip in a 1.2 m CMOS technology. 3.2. MODELING AND CHARACTERIZATION OF TRANSISTOR MISMATCH 25 two voltage source units. The drain current of the di erent transistors in a row are accessed sequentially through a switch matrix which connects the drain of the transistor under test to the current meter; the drains of the other transistor in the row are left open. The switch-matrix connects the gate of the transistor under test to the gate voltage source and connects the gates of the other rows to ground. The transistors are biased in strong inversion by using gate voltages larger than VT ; the current is measured in saturation by applying a constant drain voltage larger than the maximal (VGS ; VT ). The drain voltage is applied by using a 4 point technique where two separate sense wires, which carry no current, monitor the drain voltage and the current ow is through two separate force wires. For the C12 technology, for instance, the gate voltage is swept in 26 steps from 0.75 V to 2.0 V with a constant drain voltage of 2.0 V. Since the di erent transistors are measured sequentially the DC repeatability of the DC gate voltage source must be larger than the smallest gate-voltage mismatch we want to measure. The repeatability of the source in our set-up was better than 6 digits which is more than sucient. For the extraction of the transistor mismatch we are interested in the current di erences between the di erent transistor pairs. The current di erence can only be obtained by measuring the currents separately and then subtracting the current measurements. This procedure is very sensitive to errors, but it is the only way to obtain a measurement of the current di erence of two transistors. To have an accurate estimation of the current di erence, the individual currents have to be measured very The necessary accurately. ( I ) m DS relative accuracy on the the drain current measurement IDS is dependent on the relative current di erence II we want to measure and on the wanted relative accuracy IDS ) : for the current di erence m(IDS m(IDS ) = p1 I m(IDS ) (3.8) IDS IDS 2 I The necessary number of digits in the current measurement must be larger as the number of digits we want in the current di erence measurement plus the relative current di erence we are measuring expressed in a number of digits. In our set-up a HP3457 multi-meter with a 5 to 7 1/2 digits resolution is used. Using this procedure the current through all transistors in the array as function of the gate voltage is measured and stored. C) New Mismatch Parameter Extraction Technique The relative current di er- ence of a transistor pair is dependent on both the current factor matching, the threshold voltage VT0 matching and the gate voltage; in section 3.3.2 the following dependence is derived for transistors biased in saturation: IDS = ; 2VT0 (3.9) I (V ; V ) DS GS T Since the current measurements are performed in saturation, the current of a transistor is in rst order not dependent on the drain voltage but only on the gate-source voltage. 26 CHAPTER 3. IMPLICATIONS OF MISMATCH ON ANALOG VLSI Parasitic resistances in the drain path, from e.g. the switches in the switch-matrix or the current mirror, do not generate errors. However, di erences in the parasitic resistors in the source path of the two devices under test generate systematic errors; therefore wide interconnects are used on the test chip in the source connections of the transistors. First the threshold voltage is extracted for one of the transistors using a standard extraction technique [Bas 95]. Then the model of (3.9) is tted to the relative current di erence measurement using a linear least-squares algorithm. The mismatch parameters are obtained directly from the measured current di erence. VT0 and Other techniques can be used to calculate the VT0 matching and matching. In [Pel 89] the transistors are measured in the linear region and for each individual transistor the threshold voltage VT0 and the current factor are extracted using classical parameter extraction algorithms. The parameter mismatch is then calculated by subtracting the parameters of the individual transistors. Also a direct extraction technique can be derived from a more complex drain current model which includes a mobility reduction parameter . The extra mismatch parameter is however highly correlated to the and the overall modeling of the current mismatch becomes more complicated without improvements in accuracy [Bas 95]. We conclude that he new direct extraction procedure has two main advantages: The current measurements are performed in saturation so that parasitic resistances in the drain path do not generate errors. In sub-micron technologies where the current factor is large due to the thinner oxide, the equivalent resistance of the transistors biased in the linear region becomes very small so that any parasitic series resistance in the current path gives rise to important errors. Thus especially for sub-micron technologies measuring in saturation region is a more robust technique. Moreover, the majority of the transistors are biased in saturation in analog design so that the mismatch parameters are obtained under realistic conditions. The parameter mismatches are extracted directly from the current di erence mea- surements with a high accuracy; when rst the absolute parameters are extracted and the mismatch parameters are calculated as a di erence of absolute parameters, a very high accuracy in the absolute parameters is necessary to obtain accurate mismatch parameters - the same calculations (3.8) as for the current measurements can be used to calculate the necessary accuracy of the absolute parameters. A higher accuracy can be obtained in the individual absolute current measurements by using highly accurate measurement equipment than in the extraction of the absolute parameters and thus the presented direct technique gives more accurate results [Bas 95]. In this way a measurement of the VT0 and is obtained for each of the transistor pairs, which all have di erent mutual distances and di erent gate areas, for every test-chip. From this experimental data the proportionality constants for the mismatch dependence on distance and on size can be extracted for the models in (3.1) and (3.3). 3.2. MODELING AND CHARACTERIZATION OF TRANSISTOR MISMATCH 27 D) Mismatch Dependence on Distance For every row - containing transistors of the same size - the rst transistor is used as a reference and the VT0 and are extracted for the consecutive pairs as a function of the distance. For every size and every distance one sample is obtained per test-chip. Then the standard deviation of the mismatch ( ) is calculated by combining the samples of all test-chips parameters (VT0 ) and and calculating the sample variance from the MAD (median of the absolute di erences) [Rey 83] to eliminate the e ect of outliers. At this point it is important to discuss the accuracy of the extracted standard deviations. The estimation of the standard deviation is an application of the estimation of a parameter of the distribution of a random variable [Pap 91]. For a normally distributed random variable, the sample variance s2 can be used as an estimation for the variance 2 and the s2=2 ratio follows a Chi-squared distribution. This allows to determine the con dence interval for the extracted value for a given con dence level (see also [Per 95]). In our extractions we have aimed at a 20% accuracy with a 99.7% con dence, which requires a sample size of over 100 samples. For the same con dence level, over 500 samples or test-chips would be required to attain an accuracy of 10%. These numbers clearly illustrate, the very high measurement e ort that has to be done to obtain good quantitative mismatch parameters. For small transistors the distance e ect is completely masked by the large variance due to the small gate area as can be noted in (3.1) and (3.3) and from table 3.2. In the 1.2 m CMOS technology, for instance, a signi cant distance dependence is only observed for a 20=20 m=m nMOS transistor as is shown in gure 3.2. A straight line is tted to the standard deviation data points and the SVT0 and S from (3.1) and (3.3) are extracted. In table 3.2 the distance dependence model parameters for several technologies are summarized. E) Mismatch Dependence on Size The rows contain transistors of di erent sizes so that the size dependence of the parameter mismatch can be investigated. For each transistor size, the standard deviation of the parameter mismatch is again estimated from the sample obtained by combining the results of the devices at minimum distance over all test-chips. The same statistical techniques are used as for the distance dependence. Threshold voltage VT0 mismatch In gure 3.3 the standard deviation of the threshold voltage mismatch is plotted versus the square root of the e ective area for the 10 transistor sizes on the C12 test-chip. The model of (3.1) predicts a linear relation between the (VT0 ) and the square root of the e ective area. For the large transistors, which do not have a minimal width nor a minimal length, the experimental results con rm the model; this also agrees with the experimental results from other authors [Pel 89, Laks86, Mic 92]. A straight line is tted and the size proportionality constant is obtained (see table 3.1). CHAPTER 3. IMPLICATIONS OF MISMATCH ON ANALOG VLSI 28 1.5 0.8 ( )= [%] (VT0 ) [mV ] 0.7 1 0.5 0.6 0.5 0.4 0.3 0.2 0.1 0 0 200 400 Distance [m] 600 0 0 800 200 (a) 400 Distance [m] 600 800 (b) Figure 3.2: Threshold voltage mismatch (a) and current factor mismatch (b) for the 20/20 nMOS transistor versus distance in the 1.2 m CMOS technology; a straight line is tted through the measurement points to extract the mismatch distance dependence. 25 (VT0 ) [mV ] 20 15 10 5 0 0 0.2 p 0.4 0.6 0.8 1 1= WL [1=m] Figure 3.3: The standard deviation of the threshold voltage VT0 mismatch versus the square root of the gate area for nMOS transistors in the 1.2 m CMOS technology; the experimental results are represented by + and the values predicted by the model in equation (3.10) are represented by o; the straight line represents the predictions by the linear model of equation (3.1). 3.2. MODELING AND CHARACTERIZATION OF TRANSISTOR MISMATCH 29 Accurate modeling of VT0 mismatch in sub-micron technologies p Narrow chan- nel transistors with a minimal width (W = 1:4m, L = 6:2m and 1= We Le = 0:37=m) show less mismatch than predicted by the linear model (3.1) as can be veri ed from the experimental results in gure 3.3; p short channel transistors with a minimal gate length p (L = 1:2m, W = 6:2m and 1= We Le = 0:42=m; or L = 1:2m, W = 50m and 1= We Le = 0:15=m) on the other hand show a signi cant higher mismatch than predicted by the linear model. In high speed analog designs, the designer prefers to use small gate-lengths so that the highest intrinsic speed fT for the transistor is obtained [Lak 94]; accurate models for minimum sized transistors are thus necessary. For the accurate modeling of the threshold mismatch in sub-micron technologies the simple linear model has to be extended for short and narrow channel e ects. The threshold voltage is dependent on the at-band voltage, the surface potential, the depletion charge and the gate capacitance [Lak 94]. It has been veri ed experimentally that the mismatch of the threshold voltage is mainly attributed on the mismatch of the bulk depletion charges in the two devices [Pel 89, Miz 94, Laks86]. Due to the random process of the ion implantation and the drive-in di usion process, the doping ions are distributed randomly and the depletion charge uctuates randomly. The depletion charge follows a Poisson distribution: the mean of the depletion charge is proportional to the gate-area and the bulk doping level; the variance is equal to the square root of the mean of the depletion charge. This leads to the linear model (3.1) where the standard deviation of the threshold voltage is inversely proportional to the gate area. In sub-micron technologies two e ects introduce errors in the model. Due to the presence of the source and the drain di usion areas and the charge sharing e ect, part of the channel depletion charge is not controlled by the gate voltage anymore. For devices with a small gate-length, this charge is a relatively large part of the depletion charge. The depletion charge controlled by the gate is smaller and as a result, the threshold voltage lowers for small gate lengths whereas the variance of the threshold voltage or the VT mismatch increases. The depletion charge is not limited to the gate area but due to the fringing eld some of the dopant atoms on the side are also depleted. For large widths, the part of the depletion region on the sides is a small percentage of the total depletion region volume. But for narrow channel devices, the side parts are a large percentage of the depletion charge. The depletion charge controlled by the gate is now larger so that the threshold voltage increases and the VT mismatch decreases for narrow channel devices. The narrow and short channel e ects explain the deviations of the VT mismatch from the linear model in the experimental data very well. When these e ects are modeled quantitatively, the following extended model for the VT0 mismatch is obtained [Stey94b] [Bas 95]: A3VT 1VT 2VT 2(VT0 ) = AWL + AWL ; (3.10) 2 W 2L In this extended model the second term models the short channel e ect for transistors with small gate-lengths; the last term results in a lower mismatch for small gate-widths 30 CHAPTER 3. IMPLICATIONS OF MISMATCH ON ANALOG VLSI parameter nMOS pMOS A1VT 20 23 mV m A2VT 19 20 mV m3=2 A3VT 18 12 mV m3=2 Table 3.3: Mismatch tting constants for the extended model of equation (3.10) for a 1.2 m CMOS technology. 2.5 ( )= [%] 2 1.5 1 0.5 0 0 0.2 0.6 0.8 p 1= WL [1=m] 0.4 1 Figure 3.4: The standard deviation of the current factor mismatch as function of the square root of the gate-ares for nMOS devices in the 1.2 m CMOS technology. and models the narrow channel e ect. This new model is able to predict the mismatch data within the con dence limits. The model parameters for the 1.2 m technology are summarized in table 3.3 [Bas 95] and in gure 3.3 the results for the tting of the new model are represented by the circle; a very good agreement with the experimental data is obtained. Current factor mismatch In gure 3.4 the standard deviation of the current factor mismatch is plotted as function of the square root of the e ective area for nMOS devices in the C12 technology. The experimental data ts well to the linear model of (3.3) and no signi cant deviation for short or narrow channel devices is observed. F) Extraction Validation The correlation factor between the VT and mismatch also has to be computed. In all characterized technologies the correlation factor remains very low and the correlation can be neglected [Bas 95]. This agrees well with the experimental results of other authors [Pel 89, Laks86]. To check the accuracy of the characterization procedure, the measured and the predicted current mismatch are compared. In saturation the predicted variance of the current mismatch is given by (3.29) and (3.31), when the correlation between the parameters is negligible. The standard deviation of the measured 3.2. MODELING AND CHARACTERIZATION OF TRANSISTOR MISMATCH 31 7 (IDS )=IDS [%] 6 5 4 3 2 1 0.2 0.4 0.6 0.8 1 1.2 (VGS ; VT ) [V ] Figure 3.5: Standard deviation of the relative current mismatch versus the gate-overdrive voltage for a 6.2/1.2 nMOS transistor in a 1.2 m CMOS technology; the crosses indicate the measurements and the solid lines are the theoretical predictions from the extracted mismatch in the parameters. current mismatch, for each bias point and for each device size, has been calculated and is compared with the predicted value. In gure 3.5 the measured and predicted standard deviation of the current mismatch is plotted for the 6:2=1:2 m=m nMOS transistor in the 1.2 m technology is plotted. The current mismatch is predicted within 20% over the complete measurement range which is within the accuracy limits of the experimental data. Summary We conclude that the characterization of transistor mismatch is a tedious pro- cess which requires a very large measurement e ort. The design, realization and validation of the measurement set-up, the acquisition of the experimental data and the statistical processing of the data have to be performed with great care to avoid errors and systematic e ects. Moreover, when migrating towards sub-micron and deep-sub-micron technologies, the standard mismatch models have to checked for their validity; if necessary the e ects of the short or narrow channel e ects have to be accounted for in model extensions. On the other hand, in the rest of this chapter, we will demonstrate the importance of a good knowledge of the matching behavior of devices. A circuit designer can only improve the accuracy of circuits by increasing the device area of the devices. Unfortunately, the capacitive load in circuits is also proportional to the area. Generating a signal excursion across a capacitor, results in loading and unloading currents so that the circuit dissipates energy proportional to the capacitive load or the accuracy. The quality of the device matching thus is the ultimate limit for the power consumption of circuits; the impact of transistor mismatch is even more important in high speed applications than thermal noise. The better the mismatch of devices is modeled and characterized, the smaller area's the designer can safely use while keeping a high circuit yield; consequently the circuits 32 CHAPTER 3. IMPLICATIONS OF MISMATCH ON ANALOG VLSI will consume less power for the speci ed accuracy and speed. When no mismatch data is available, very conservative designs using large devices have to be used and very poor overall circuit performance is achieved. 3.3 Implications of mismatch on transistor behavior In this section the consequence of parameter mismatch on the transistor behavior is calculated for the di erent possible operation regions of the transistor. First, we introduce the necessary mathematical techniques for mismatch calculations. 3.3.1 Mathematical techniques Transistor mismatch results in small deviations of the transistor parameters from their nominal value and these deviations result in small deviations of the circuit characteristics from their nominal values. In order to calculate the e ect of transistor mismatch on circuit characteristics the following relations are used. For a circuit characteristic Z , which is de ned as Z = f (x; y), the deviation Z in Z due to a deviation x in x and y in y is calculated as: @f Z = @f x+ @x @y y (3.11) For independent and normally distributed deviations in x and y the standard deviation of Z is: 2 (Z ) = @f 2 2(x ) + @f @x @y 2 2(y ) (3.12) For two characteristics Z1 and Z2 de ned as Z1 = f (x1 ; y1 ) and Z2 = f (x2 ; y2 ) the following relations are derived applying (3.11) and (3.12), with x = x1 ; x2 and y = y1 ; y2 : @f x + y Z = Z1 ; Z2 = @f @x @y 2 2 2(x) + @f 2(y ) 2(Z ) = @f @x @y p (Z ) = 2(Z ) (3.13) (3.14) (3.15) These relations allow us to calculate the e ect of parameter mismatches and transistor mismatches on the transistor or circuit behavior. 3.3. IMPLICATIONS OF MISMATCH ON TRANSISTOR BEHAVIOR IDS1 IDS2 VGS1 M1 VGS M2 (a) M1 IDS VGS2 33 M2 IDS (b) Figure 3.6: (a) Two transistors biased with equal gate-source voltage; (b) two transistors biased with equal drain-source current. 3.3.2 Implications on transistor behavior A circuit designer can bias a transistor in two ways: for current biasing the current through the device is imposed and the terminal voltages are the dependent variables; for voltage biasing the terminal voltages are imposed and the current is the dependent variable. For the sake of simplicity of the equations and calculations, the source and bulk are supposed connected so that VSB = 0 and no bulk-e ect occurs. However, if a bulk-e ect does occur in the circuit, the extra mismatch due to the mismatch in the bulk-e ect coecients can in rst order simply be considered as an extra degradation of the VT matching of the transistors, so that most equations can still be used. Of course, for the optimization of the biasing, the dependence of the bulk-e ect on the bias voltages should be taken into account and slightly di erent results will be obtained. For voltage biasing the terminal voltages are imposed and the current IDS is the dependent variable as is illustrated in gure 3.6(a); the current depends on the terminal voltages and on the transistor parameters: IDS = f (VGS ; VDS ; VT0 ; ) (3.16) For a pair of transistors with an identical VGS , the di erence in their currents is calculated using (3.13): @IDS V I DS = @I@ DS + @V T0 T0 (3.17) From the device equations in appendix A one can conclude for all regions of operation that: @IDS = @IDS = ;g (3.18) m @VT0 @VGS @IDS = IDS (3.19) @ 34 CHAPTER 3. IMPLICATIONS OF MISMATCH ON ANALOG VLSI so that the relative current di erence in all operating regions of the transistor is expressed by: IDS = ; gm V (3.20) T0 IDS IDS and the variance of the current di erence is: (IDS ) 2 = ( ) 2 + gm 2 2(V ) (3.21) T0 IDS IDS In a current biasing scheme as illustrated in gure 3.6(b) the current is imposed and the gate-source voltage VGS is the dependent variable. Expressions for the calculation of the VGS as a function of the current are not always available. However (3.16) can be rewritten as an implicit function for VGS : IDS ; f (VGS ; VDS ; VT0 ; ) = 0 and the partial derivatives of VGS can be expressed as: @VGS = ; @IDS @ @ = ;IDS g m @IDS @VGS (;1) (3.22) (3.23) (3.24) @VGS = ; @IDS @IDS (;1) @VT0 @VT0 @VGS =1 so that the gate-source voltage di erence and its variance become: I DS VGS = VT0 ; g m 2 2 I ( ) DS 2 2 (VGS ) = (VT0 ) + g m (3.25) (3.26) (3.27) (3.28) 3.3.2.1 Strong inversion For a transistor biased in strong inversion and in saturation, the gm =IDS is 2=(VGS ; VT ) (see (A.22)) so that (3.21) and (3.28) can be rewritten as: (IDS ) IDS 2 = ( ) 2 2 + 4 (VT0 ) 2 (VGS ; VT ) 2 2 ( V GS ; VT ) ( ) 2 2 (VGS ) = (VT0 ) + 4 (3.29) (3.30) 3.3. IMPLICATIONS OF MISMATCH ON TRANSISTOR BEHAVIOR 35 Substituting the models of (3.1) and (3.3) in (3.29) and (3.30), we obtain for closely spaced devices: ! 2 1 A2 + 4A2VT0 = WL (VGS ; VT )2 2 A2 ! ( V ; V ) 1 GS T 2(VGS ) = WL A2VT0 + 4 (IDS ) IDS (3.31) (3.32) The accuracy of the gate voltage or drain current is dependent on the bias point of the devices or (VGS ; VT ) and on their gate-area. For a technology a corner gate-drive voltage (VGS ; VT )m is de ned for which the e ect of the VT0 and mismatch on the gate voltage or drain current is of equal size: (VGS ; VT )m = 2AVT0 =A (3.33) In (3.31) and (3.32) we observe that for a circuit with a bias point with a (VGS ; VT ) smaller than (VGS ; VT )m the e ect of the VT0 mismatch is dominant, whereas for a (VGS ; VT ) larger than (VGS ; VT )m the e ect of the mismatch dominates. In table 3.1 the values of (VGS ; VT )m are listed for a few CMOS technologies. It is clear that in practical circuits the (VGS ; VT ) will be smaller than (VGS ; VT )m so that the VT0 mismatch is dominant over the mismatch for the calculation of the accuracy of the circuit behavior. In practice, equations (3.31) and (3.32) can be approximated by: 2 4A2VT0 (3.34) WL(VGS ; VT )2 2 VT0 2(VGS ) AWL (3.35) The approximation error due to neglecting the mismatch on the standard deviation for the above equations is equal to ((VGS ; VT )=(VGS ; VT )m )2=2, and is small for typical transistor bias conditions; for a nMOS transistor in the 0.7 m technology biased with a (VGS ; VT ) of 0.2 V the error on is only 1 %. (IDS ) IDS 3.3.2.2 Weak inversion When the transistors are biased in weak inversion the gm =IDS is 1=(nUT ) (see (A.4)) so that (3.21) and (3.28) can be rewritten as: (IDS ) IDS 2(V 2 = ( ) 2 2(VT0 ) + (nU )2 T 2 2 GS ) = (VT0 ) + (nUT ) ( ) (3.36) 2 (3.37) 36 CHAPTER 3. IMPLICATIONS OF MISMATCH ON ANALOG VLSI By substituting (3.1) and (3.3) in (3.36) and (3.37), the relative current variation and gate voltage variation of two closely spaced devices become: (IDS ) 2 = 1 A2 + A2VT0 (3.38) IDS WL (nUT )2 1 ;A2 + (nU )2A2 2(VGS ) = WL (3.39) T VT0 The weak-inversion slope parameter n typically has values from 1 to 2 and UT is 25.8 mV at room temperature. From table 3.1 we can conclude that the VT0 mismatch dominates the accuracy calculations so that (3.38) and (3.39) can be simpli ed to: (IDS ) 2 A2VT0 (3.40) IDS WL(nUT )2 2 VT0 (3.41) 2(VGS ) AWL The approximation error due to neglecting the mismatch on the standard deviation for the above equations is equal to (2nUT =(VGS ; VT )m )2=2; for a nMOS transistor in the 0.7 m technology the error on the is only 0.1 % at room temperature. 3.4 Implications of transistor mismatch on the behavior and design of elementary stages In the forthcoming sections the implications of transistor mismatch on the speed, power consumption and accuracy of elementary current and voltage processing stages is studied. This allows to draw guidelines for optimal design of these circuits. Furthermore it provides the background for a discussion of the implications of transistor mismatch on the performance of general analog VLSI systems [Kin 96d]. 3.4.1 Current processing circuits The current ampli er, represented in gure 3.7, is a basic current processing stage. The output transistor M2 is a parallel connection of A unit-transistors of the same size as M1 , with a gate-width W and gate-length L, so that the current ampli cation factor is A. The speed performance of the current ampli er is the highest frequency that can be processed by the current ampli er, and depends on the bandwidth. The bandwidth of this circuit is in rst-order determined by the gm of the input transistor M1 and the parallel connection of the gate capacitors of both transistors. For transistors biased in strong inversion, the bandwidth of the ampli er is: gm1 BW = 2(CGS1 + CGS2 ) (3.42) 3IB = 2(A + 1)(VGS ; VT )Cox WL 3.4. IMPLICATIONS OF MISMATCH ON ELEMENTARY STAGES 37 A IB IB iOUT iIN M1 M2 1: A Figure 3.7: Basic current ampli er. The DC power consumption of the ampli er is : P = (A + 1)IB VDD (3.43) The relative accuracy of the current processing is determined by the maximal input signal RMS value IinRMS and the 3 value of the input referred o set current IOS and is de ned as: Accrel = IinRMS (3.44) 3(IOS ) By using the 3 value of the o set current, the accuracy speci cation is met with a probability of about 99.7%. This probability that a circuit block meets its speci cations, has a direct impact on the yield of the total chip or system. In complex systems with many stages, an even higher probability can be necessary to obtain a high yield and more than the 3 has to be accounted for in (3.44). Due to the e ect of mismatches in the transistors, an error occurs in the current mirroring and for a zero input current a non-zero output current exists. The input referred o set current IOS is by de nition the current that has to be applied to the input to obtain a zero output current; it has to compensate for the variation in the current of M1 and M2 . To calculate the IOS , we rst calculate the errors in the currents of M1 and M2 . The standard deviation of the current in transistor M1 and in a unit-transistor of M2 is derived from (3.34): 2AVT0p (IUNIT ) = p1 IB (3.45) 2 (VGS ; VT ) WL p where the term 1= 2 is necessary to calculate the variance of the parameter from the variance of the di erence of parameters { see (3.15). The total current in M2 is the sum of the currents of the individual unit transistors, which are statisticallypindependent quantities, so that the standard deviation of the current of M2 is (IOUT ) = A(IUNIT ). 38 CHAPTER 3. IMPLICATIONS OF MISMATCH ON ANALOG VLSI Since the current ampli cation is A, we can express the standard deviation of the input o set current as follows: r 2 ) + 2 (I ) (IOS ) = (IAOUT UNIT 2 r p 2 A A+1 VT0 p = IB (VGS ; VT ) WL A (3.46) p For a typical bias modulation index of 1/2 the IinRMS is IB =(2 2) and the relative accuracy (Accrel) of the current processing then becomes: p r Accrel = WL12(VAGS ; VT ) A A+ 1 VT0 (3.47) With the expressions for the di erent circuit performance parameters at hand, we can now develop a relation for the total performance or quality of the circuit design. The quality of the circuit is better if its bandwidth, gain and accuracy are large and its power consumption is low. When we consider the ratio SpeedAccuracy2/Power for the current ampli er, we obtain: BWAcc2rel (VGS ; VT ) A = 2 P 96Cox AVT0 VDD (A + 1)3 (3.48) which we can rewrite for large gains A as: Gain2BWAcc2rel (VGS ; VT ) = P 96Cox A2VT0 VDD (3.49) Very important conclusions for the design of a current ampli cation stage can be drawn from this relation: The total performance of the ampli er is only dependent on technology constants and on the chosen bias point of the stage and is independent of the transistor sizes. To obtain the best total performance, a current processing stage must be designed with a large (VGS ; VT ). It is common knowledge that to improve the accuracy of a current mirror a large (VGS ; VT ) has to be used [Lak 94]. From (3.29) one can indeed conclude that the accuracy performance of a current mirror is improved by increasing (VGS ; VT ). On the other hand, increasing (VGS ; VT ) reduces the gm of the stage so that the speed performance is degraded and that a higher bias current must be used to obtain a certain speed, resulting in higher power consumption. However, (3.49) shows that increasing (VGS ; VT ) will result in the best possible trade-o between speed, accuracy, gain and power consumption for a current processing stage ! 3.4. IMPLICATIONS OF MISMATCH ON ELEMENTARY STAGES 39 For a current processing stage the (VGS ; VT ) is typically limited to VDD /2a . As a result, the optimal performance becomes: Gain2BWAcc2rel 1 = P 192Cox A2VT0 (3.50) The di erent performance speci cations of a current ampli er are linked and depend on technological and physical constants only ! Equation (3.50) shows that a designer can only trade one speci cation for the other. Due to the impact of transistor mismatch, he cannot choose the di erent performance speci cations independently ! When we rewrite (3.50) as: P = 192Cox A2VT0 Gain2 BWAcc2rel (3.51) Transistor mismatch puts a boundary on the minimal power consumption by a current ampli cation stage for a given gain, speed and accuracy speci cation. An important limitation of current processing stages is the quadratic dependence of the power consumption on the gain of the ampli er in (3.51). The fundamental reason for the appearance of this term is related to the physics of the MOS transistor. By changing the gate voltage, the conductivity of the channel is controlled and thus the current is controlled. On the physical level, a MOS transistor acts as a voltage dependent current source or a transconductor. If we try use this device as a current ampli er, we basically operate the device in an unnatural way. The only way to make current gain is by a parallel connection of several transistors to the gate of a diodeconnected transistor, that does the current to voltage conversion. The more gain we try to make, the more load we put on this gate for the same transconductance and the speed in (3.42) reduces. Moreover, the more gain, the larger the bias current in the output transistor and the larger the power consumption in (3.43). The accuracy in (3.47) on the other hand, is in rst order independent of the realized gain. The input signal swing is only limited by the modulation index we can allow. The maximal modulation index depends on the distortion as is discussed in section 3.7. The bias current of the output stage scales with the gain so that the modulation index at the output is the same as at the input. As a result the gain does not in uence the maximal signal swing or the relative accuracy. Due to the e ect of the gain on the bandwidth and the power consumption, a quadratic dependence on the gain of the quality of the circuit in (3.49) and of the the minimal power consumption of the circuit in (3.51) emerges. a for small feature sizes, the appearance of velocity saturation [Lak 94] can further reduce the maximal (VGS ; VT ) that can be used under which the above derivations remain valid. CHAPTER 3. IMPLICATIONS OF MISMATCH ON ANALOG VLSI 40 R2 vIN R1 vOUT Figure 3.8: System schematic for a basic voltage ampli cation block. Exact case When the full expression for the variance of the relative current di erence (3.31) is used, including both the VT0 and mismatch, the following expression for the quality of a current ampli er with a large gain is derived: Gain2BWAcc2rel 1 (3.52) = 2 P 24Cox VDD A2 (VGS ; VT ) + (V4GSAVT0 ;VT ) The value of (VGS ; VT ) can now be optimized to obtain a maximal total circuit performance: starting with a small (VGS ; VT ) and increasing it, the term proportional to AVT0 in the denominator of (3.52) decreases and the quality improves; for large values of (VGS ; VT ), however, the rst term proportional to A increases and the quality decreases again. An optimal value for (VGS ; VT ) exists and can easily be calculated: the optimum in circuit performance is reached for a (VGS ; VT ) equal to (VGS ; VT )m - see (3.33), and we then obtain the best possible total circuit performance: Gain2BWAcc2rel 1 = (3.53) P 96Cox VDD A AVT0 In the previous section, taking only the e ect of VT o set into account, we concluded that for current processing circuits the (VGS ; VT ) of the stage has to be maximized to optimize performance. However, by taking also the e ect of mismatch into account, we conclude that (VGS ; VT ) should not be increased beyond (VGS ; VT )m . Furthermore, if the supply voltage allows it, a current processing stage should be biased with (VGS ; VT ) = (VGS ; VT )m which yields the optimum in the trade-o between the di erent speci cations. 3.4.2 Voltage processing circuits The inverting voltage ampli er stage of gure 3.8 is a basic voltage processing stage. It consists of an operational ampli er (opamp) with a negative resistive feedback. The input voltage is converted into a current by R1 and the virtual ground that is created at the negative input terminal of the opamp. This current is converted into the output voltage by R2 so that the closed loop ampli cation of the system ACL is determined by the ratio R2 /R1 , which is well controlled over process variations. This behavior of the system is observed as long as the open-loop gain of the opamp is much larger than the closed loop gain. 3.4. IMPLICATIONS OF MISMATCH ON ELEMENTARY STAGES vIN R1 41 IB R2 M1 vOUT (a) Vin R1 V gs R2 Vout gm Vgs CGS (b) Figure 3.9: (a) One transistor voltage ampli er implementation; (b) small signal equivalent. 3.4.2.1 One transistor voltage ampli er The most simple implementation for the operational ampli er in gure 3.8 is obtained with a single transistor as in gure 3.9(a). From the small signal equivalent in gure 3.9(b), we derive the frequency response of the system: A(s) = VVout in (3.54) R 1 ; 1 = ( g 2 m R2 ) = ; R 1 + 1=(g R ) + s=(g =C ) 1 m 1 m GS We observe in (3.54) that in order to de ne the ampli cation accurately, the 1/gm of the transistor has to be chosen considerably smaller than the resistors R1 and R2 . The lowfrequency closed-loop ampli cation of the stage is then ACL = R2 =R1 and the closed-loop frequency response is approximately given by: 1 R 2 (3.55) A(s) ; R 1 + s=(g =C ) 1 m GS so that we obtain the following relation for the the bandwidth of the ampli er: BW gm =(2CGS ) (3.56) A) Strong inversion First we consider the performance of this voltage ampli er when the transistor is biased in strong inversion. Using the relations for the gm and CGS in strong inversion from appendix A, we can express the bandwidth of the ampli er as follows: BW = 2(V ;3IVB )WLC (3.57) GS T ox The DC power consumption of the circuit is: P = VDD IB (3.58) The relative accuracy (Accrel) of the circuit is determined by the o set voltage of the opamp (VOS ) with respect to the maximal RMS value of the input signal (VinRMS ). The 42 CHAPTER 3. IMPLICATIONS OF MISMATCH ON ANALOG VLSI maximalp voltage swing at the output is in rst order: VoutPP = VDD , so that VinRMS = VDD =(2 2Gain). The standard deviation of the o set voltage of the one transistor opamp isp determined by equation (3.35), but from (3.15) we conclude that the o set voltage is 2 times smaller since the variation on VGS is important. The expression for the relative accuracy can now be evaluated as: Accrel = VinRMS (3.59) 3(VpOS ) WL (3.60) = 6VADD Gain VT0 To obtain a better insight in the possible trade-o s between the di erent circuit parameters, we consider again the total performance of the voltage ampli er. Combining (3.57), (3.58) and (3.60), we obtain: Gain2BWAcc2rel VDD = P 24(VGS ; VT )A2VT0 Cox (3.61) From (3.61) some very important lessons can be learned for the design of a voltage ampli er: The total performance of a voltage ampli er is maximized by lowering (VGS ; VT ). As far as speed requirements allow it the stage should be biased with the lowest possible (VGS ; VT ). Again it is common knowledge that to achieve a low o set voltage a small (VGS ; VT ) has to be used as can also be derived from (3.32) [Lak 94]. However, (3.61) shows that not only a good accuracy but the best trade-o between speed, gain, accuracy and power consumption is obtained for a small (VGS ; VT ). A typical minimal value of (VGS ; VT ) = 0:2 V is derived from device physics - see Appendix A. The best attainable performance in strong inversion is: Gain2BWAcc2rel 5VDD = P 24A2VT0 Cox (3.62) Also for voltage designs the di erent performance speci cations are linked by phys- ical and technological constants only ! The designer can - once the optimal bias is chosen - not optimize the di erent speci cations independently but can only trade one speci cation for an other. When we rewrite (3.62), we obtain that the minimal required power consumption is xed for a given gain, speed and accuracy by the impact of transistor mismatch: 24 A2 C Gain2BWAcc2 P= (3.63) rel 5VDD VT0 ox In (3.63) we observe again a quadratic dependence of the power consumption on the gain of the voltage ampli er. In a voltage ampli er nor the power consumption, nor 3.4. IMPLICATIONS OF MISMATCH ON ELEMENTARY STAGES 43 the bandwidth are in rst order dependent on the realized gain. However, when the gain is increased, the maximal input signal reduces proportionally since the maximal output swing is limited by the supply voltage. This limits the maximal input signal swing; consequently, the maximal attainable accuracy for a given power supply voltage reduces proportional to the gain (3.60) and the power consumption increases quadratically with the gain (3.63). B) Weak inversion In the previous section it was shown that the total performance of a voltage ampli er improves for smaller (VGS ; VT ). Consequently, the best total per- formance is obtained when the transistor is biased in weak-inversion. The relations for the power consumption (3.58) and for the relative accuracy (3.60) remain valid. Since source and bulk are connected, the capacitance that determines the speed performance in weak inversion is the gate-bulk capacitance, which is: 1) C WL (3.64) CGB = (n ; n ox With the expression for gm in weak inversion, we derive the following relation for the bandwidth of the ampli er operating in weak-inversion: gm = IB BW = 2C (3.65) 2UT (n ; 1)Cox WL GB And we can now calculate the total performance of the ampli er in weak inversion as: Gain2BWAcc2rel VDD = (3.66) P 72(n ; 1)UT A2VT0 Cox For the 0.7 m technology e.g. the technology constants are VDD = 5 V , Cox = 2 fF =m2, AVT0 = 13 mV m, and n is approximately 1:5 and UT = 25:8 mV at room temperature; the optimal quality in weak inversion calculated with (3.66) is then 5 times better than the best quality in strong inversion calculated with (3.62). This implies that for the same gain, bandwidth and accuracy speci cations 5 times less power is required for an ampli er operated in weak inversion. 3.4.2.2 Di erential pair voltage ampli er A simple di erential implementation of the operational ampli er in gure 3.8 is a di erential pair as is shown in gure 3.10. Similarly as for the one transistor ampli er the 1/gm of the input transistors is designed larger than the value of the resistors resistors R1 and R2 to obtain a well controlled closed-loop gain. As a result the bandwidth of the ampli er in a high precision design is limited by the pole caused by the input capacitance and it is easily calculated that: BW = gm =(2CGS ). The power consumption of the circuit is P = 2VDD I . The relative accuracy (Accrel) of the circuit is determined by the o set voltage of the opamp, or the VT0 mismatch of 44 CHAPTER 3. IMPLICATIONS OF MISMATCH ON ANALOG VLSI R2 vIN IB vOUT R1 2IB Figure 3.10: Basic voltage ampli cation stage implemented with a di erential pair amplier. the input transistors, in respect to the maximal RMS of the input signal, which is in rst p order VDD =(2 2 Gain): Accrel = p VDD 6 2 (p VOS ) Gain (3.67) V DD WL = p 6 2 AVT0 Gain For transistors biased in strong inversion and saturation, the quality of the design is again expressed as: Gain2BWAcc2rel VDD = (3.68) P 96(VGS ; VT )A2VT0 Cox The total performance of this voltage processing stage is again maximized by lowering (VGS ; VT ) towards operation in weak inversion. This result shows that the di erential implementation consumes about 4 times more power for the same gain, bandwidth and accuracy speci cations as the single transistor implementation (see equation (3.61)). The power consumption is doubled since two transistors require two times the bias current and p their o set voltage is 2 as for the single transistor (see equation (3.15)); both e ects result in 4 fold increase of the power consumption. Summary To obtain a low power consumption, voltage circuits must be operated in weak inversion as much as possible. However, the maximal attainable frequency in weak inversion is limited, since the maximal current in weak inversion is also limited. Weak inversion operation is only possible for relatively low speed applications. In section 3.4.4.2 the limitations on the use of (3.61) and (3.66) to calculate the necessary power consumption are studied in detail. A second practical, but important, limitation of the use of weak inversion, is the limited availability from industrial chip foundries of good model parameters for the simulation of transistors in weak inversion. 3.4. IMPLICATIONS OF MISMATCH ON ELEMENTARY STAGES M2a M2b â‘¡ â‘¢ vIN + 45 vIN ; M1a M1b â‘ vOUT 2 IB Figure 3.11: Load compensated OTA. Up to now we have discussed the mismatch limitation on the trade-o s between the di erent performance parameters for simple signal processing stages. This makes the analytical analysis straightforward and closed-form expressions are obtained. We will now proceed with the analysis of a more complex circuit implementation of a load compensated OTA. However, increasing the circuit complexity, increases the degrees of freedom in the design and involves the design of several stages; in the calculations some reasonable assumptions and approximations are made to obtain again closed-form expressions. 3.4.2.3 Load compensated OTA voltage processing stage The operational ampli er that is used in the basic voltage ampli er of gure 3.8 can also be implemented with the operational transconductance ampli er represented in gure 3.11. It is the basic schematic for many more so sticated load compensated operational ampliers. As such, its design and the trade-o between its speci cations, is representative for the design procedure of many multi-stage operational ampli ers. The circuit operates as follows: transistors M1a ;b transform the di erential component of the input voltages into a di erential current; they act as a voltage to current converter. The signal current is converted to a single ended output voltage by the current mirror M2a ;b and the output conductance of M1b and M2b . These ampli ers are typically used in a feedback con guration like e.g. the con guration of gure 3.8. The stability of the feedback system is then rst concern of the circuit designer; an ampli er is only useful if its stability is guaranteed. Due to the high complexity of the circuit, its transfer function is of higher order. Basically, a pole is associated with each circuit node. The presence of a second and higher order poles in the open-loop transfer function implies that the open-loop frequency response of the OTA has to be adapted in order to obtain a safe phase and gain margin for all possible feedback con gurations. The maximal speed or frequency performance that can be attained is then determined by the frequency of the second pole in the open-loop transfer function. The gain-bandwidth (GBW) of the ampli er must be made Kstab times smaller than the second pole and Kstab is at least 2 for a phase margin better than 60 degrees. 46 CHAPTER 3. IMPLICATIONS OF MISMATCH ON ANALOG VLSI The second-pole (f2 ) in the open-loop transfer function is located at node â‘¡, the gates of transistors M2a and M2b , and is calculated as: f2 = 2(C gm2+ C ) GS2a GS2b (3.69) 3 I = 4C W L (BV ; V ) ox 2 2 GS T 2 so that the maximal GBW of the ampli er becomes: GBW = Kf2 stab (3.70) = 4K C W3ILB (V ; V ) stab ox 2 2 GS T 2 The accuracy of the ampli er depends on the equivalent input referred o set voltage (VOS ). The VOS is determined by the voltage matching of the transistor pair M1a ;b and the accuracy of the current mirror M2a ;b . For the calculation of VOS , the internal voltage gain of the ampli er Ain from the input to gate of M2a ;b is an important factor. It determines the attenuation of the in uence of the errors in the second stage on the input referred o set. In order to simplify the expressions we assume that all devices have equal length (L1 = L2 ), which is a reasonable assumption for high speed designs where all signal transistors are designed with a minimal length to obtain a maximal transconductance and minimal node and input capacitances. All transistors have the same bias current, so that the Ain is expressed by: r g ( V m1 1 GS ; VT )2 Ain = g = (V ; V ) = W (3.71) W2 m2 GS T 1 The standard deviation of the o set voltage is calculated as follows: ( VT02 ) 2 2 2 (3.72) (VOS ) = (VT01 ) + A in 2 2 VT0n + AVT0p (3.73) =A W1 L1 W2 L2 A2in ; = W L1 A2 A2VT0n + A2VT0p (3.74) 2 2 in Equation (3.72) shows that by increasing the internal gain Ain , the e ect of the mismatch in the current mirror on the input referred o set is lowered. However, from (3.71) we conclude that the gain can only be increased by decreasing the width W2 and consequently by decreasing the area of the current mirror transistors since the transistors have the same lengths; the increase in gain goes at the cost of a reduction of the matching; therefor the nMOS as well as the pMOS matching, expressed by respectively AVT0n and AVT0p , are equally important in the nal expression for the o set voltage (3.74). 3.4. IMPLICATIONS OF MISMATCH ON ELEMENTARY STAGES 47 Using the expression (3.74), the relative accuracy (Accrel) is calculated from (3.59). The DC power consumption of the ampli er is given by: P = 2IB VDD (3.75) The quality of the circuit implementation is evaluated from: Gain2GBWAcc2rel V A DD in = 192K C (A2 + A2 ) (V ; V ) P stab ox VT0n GS T 1 VT0p (3.76) where Gain is the gain in the closed loop con guration determined by the applied feedback. The speed performance of the total voltage processing circuit, as depicted in gure 3.8, with an OTA is determined by the maximal frequency for which the loop gain is larger than 1. Since the GBW of open-loop transfer function of the OTA is xed, we obtain the following expression for the BW of the total ampli er: BW = GBW (3.77) Gain Substituting this result in (3.76), we derive the following relation for the voltage processing system: Gain3BWAcc2rel V A DD in = 192K C (A2 + A2 ) (V ; V ) P stab ox VT0n GS T 1 VT0p (3.78) This expression leads to the following conclusions: The quality of the design improves by biasing the input transistors M1a ;b , which operate in voltage mode, with a low (VGS ; VT ). By choosing a high internal gain and consequently by biasing the transistors M2a ;b , which operate in current mode, with a high (VGS ; VT ) better performance is ob- tained. The choice of the gain in multi-stage ampli ers is further examined in section 3.5.1. The minimal power to obtain a given speed and accuracy depends on the cubic of the gain of the voltage processing block. As in the one transistor implementation a second order dependence is due to the limited output swing of the ampli er, which results in a maximal input signal and relative accuracy inversely proportional to the gain. However an extra gain dependence is due to the presence of a second pole in the ampli er; the power consumption is proportional to the GBW of the open-loop but in closed loop the bandwidth is the gain times smaller. Due to the higher complexity which brings extra constraints relating to the stability into the design, more power is consumed. CHAPTER 3. IMPLICATIONS OF MISMATCH ON ANALOG VLSI 48 Cin vIN R1 R2 (a) CL Vin Cin R1 Ve R2 fVout gm Ve Vout ro R2 CL (b) Figure 3.12: (a) OTA based voltage processing system schematic; (b) small signal equivalent where f = R1 =(R1 + R2 ). 3.4.3 Feedback systems and general OTA design In the previous sections we showed that the SpeedAccuracy2/Power product of the basic voltage and current circuit blocks is xed by technological constants. This result is obtained because in the simple blocks the same transistor is responsible for the accuracy, speed and power speci cation of the circuit. The bandwidth limiting capacitor, for instance, is inversely proportional to the o set so that the SpeedAccuracy2 product becomes independent of the transistor sizing. In this section we will further generalize these ndings to general analog systems; we study in detail how the di erent performance speci cations are related in a general feedback voltage processing system, implemented with an OTA. Similar results can be obtained for other types of systems. In gure 3.12(a) a schematic for a voltage processing system is shown. The output voltage is fed back in series to the input so that a high input impedance is obtained, which is desirable in voltage processing systems [Gra 84, Lak 94]. The power consumption of this system is determined by the current consumption of the OTA; the bias current of the OTA is proportional to the required gmIN of the input stage, which is on its turn dependent on the wanted gain-bandwidth product of the system. General feedback theory and general opamp/OTA theory [Lak 94, Gra 84] show that in a system as in gure 3.12(a), the gain in closed loop ACL times the bandwidth in closed loop BWCL is a constant called the gain-bandwidth product GBW. An expression for the GBW of many types of OTA's is [Lak 94, Gra 84]: mIN GBW = ACLBWCL = 2gC (3.79) d where Cd is the capacitor that is associated with the dominant pole of the open-loop transfer function of the OTA and gmIN is the transconductance of the input stage. If we assume an input stage operating in strong inversion and we derive the following expression for the power consumption by using (3.58) and (3.79): P = ACLBWCLCd VDD (VGS ; VT ) (3.80) The relative accuracy of the system is expressed by (3.59). The o set voltage of the OTA is in rst order determined by the area of the input devices and consequently is 3.4. IMPLICATIONS OF MISMATCH ON ELEMENTARY STAGES 49 related to the input capacitance as is shown in (3.101) derived in section 3.5.1.pThe output swing is limited to VDD so that the maximal input signal is VinRMS = VDD =(2 2ACL) and the relative accuracy is then given by: 2 Acc2rel = 24AV2DDC CAin2 (3.81) CL ox VT0 and we can rewrite the power consumption as: ( V C GS ; VT ) d 2 3 2 P = 24 (3.82) VDD Cox AVT0 ACLBWCLAccrel Cin We obtain a relationship between the speci cations of a general voltage processing system implemented with an OTA very similar to the relationship obtained for the basic voltage processing stage in (3.61). If a relation between the the input capacitance Cin and the capacitance of the dominant pole Cd exists, the trade-o between the di erent circuit speci cations is, also for the general system, determined by technological and physical constants only. When the stability of the system in gure 3.12(a) is studied it becomes apparent that in any OTA design the fraction Cd =Cin will be larger than or equal to 1. From the small signal equivalent for an implementation with an OTA in gure 3.12(b), the loop transfer function is derived: R1 T (s) = (1 + sR Cgm)(1 (3.83) + sCin R1 ) 2 L using the following approximations: R 2 (3.84) ACL = R + 1 > 1 or R2 > R1 1 and ro R2 (3.85) Due to the presence of two capacitors in the system, we obtain two polesb: fa = 2R1 C 1 in 1 fb = 2R C 2 L (3.86) (3.87) To evaluate the position of these poles we need an estimation of the relative magnitude of the Cin versus CL. The load capacitance of the voltage processing stage is determined b the voltage input source resistance is assumed to be zero: Rs = 0; a non-zero Rs is in series with R1 for the calculation of the pole fa . In (3.86) we have thus assumed that R1 Rs and the location of the pole can be chosen by the designer. In the case that Rs R1 , the location of fa is xed by Rs and the input capacitance Cin , so that no design freedom exists for the position of fa . That case is covered in section 3.5.1 and equation (3.104). 50 CHAPTER 3. IMPLICATIONS OF MISMATCH ON ANALOG VLSI by the input capacitance of the next stage. As is explained in section 3.5.1, the input capacitance of the second stage CL will be smaller as the Cin , since the o set requirements for the second stage will be less severe. In order to meet a certain accuracy speci cation in a multi-stage system, we have to divide the allowed total o set over the di erent stages as can be concluded from (3.97); (3.101) indicates that the ratio of CL =Cin will be proportional p to 2(Vos1 )=2(Vos2 ). A good design compromise is to make (Vos1 )=(Vos2 ) = 1= ACL, since then the o set of the total system is dominated by the o set of the rst stage only. When we approximate the exact expression for ACL in (3.84) by R2 =R1 , we derive the following relative position of the two poles: fb = pfa (3.88) ACL so that fb is the dominant pole and that CL plays the role of Cd in (3.82). To assure a stable feed-back system the second pole fnd = fa must be at least larger or equal than the unity-gain frequency of the loop-transfer function T(s) to have a phase-margin of at least 45 degrees so that we obtain the following condition: fnd 1 GBW 1 2CL 1 (3.89) 2R1 Cin gm CL g R = T ! 1 Cin m 1 @DC To guarantee stability the ratio CL /Cin must be made larger as gm R1 which is the loop gain at low frequencies as can be concluded from (3.83) and which is always at least larger than 1. So to attain stability, we have to increase the load capacitance CL since we cannot make Cin smaller due to the accuracy speci cation. Due to stability requirements, the ratio of capacitors in (3.82) will always be larger than 1. Also for other design choices of (Vos1 )=(Vos2 ), stability requirements impose that Cd =Cin 1. We can conclude that the power consumption of a voltage feedback system is thus larger than: (V ; V ) P 24 GS T Cox A2VT0 A3CL BWCL Acc2rel (3.90) VDD Equation (3.90) leads to the following conclusions: the minimal power consumption is limited by the e ect of transistor mismatch and the quality of the technology is expressed by Cox A2VT0 ; and also the maximal attainable SpeedAccuracy2/Power product is limited by the input transistor; to optimize the total combined performance of a voltage processing system the input stages have to biased with a low (VGS ; VT ) or in weak inversion; 3.4. IMPLICATIONS OF MISMATCH ON ELEMENTARY STAGES 51 in open-loop stagesc the minimal power consumption is proportional to the square of the Gain in (3.51), (3.63) and (3.66); when feedback is used the minimal power is dependent on the cubic of the closed loop gain (Gain) in (3.78) and ACL in (3.90) due to the e ect of the limited gain-bandwidth of an opamp or OTA and the stability requirements. 3.4.4 Circuit design guidelines We have studied several basic circuit stages for current or voltage signal processing. In this section we summarize the conclusions of the di erent designs and we highlight the limitations of the applicability of the di erent trade-o relationships. 3.4.4.1 Current processing stages From the design of the current ampli er it can be concluded that in order to obtain optimal total performance i.e. high speed, high accuracy and low power consumption, a current processing stage must be biased with a high (VGS ; VT ) as long as VT mismatches are dominant. However the (VGS ; VT ) of the stage should not be increased above the (VGS ; VT )m of the technology. The overall best performance is obtained when the current processing stage is biased at (VGS ; VT )m . In practical situations this will be impossible due to other speci cations like, for instance, signal swing and limited power supply voltage. 3.4.4.2 Voltage processing stages A voltage processing stage must be biased with a low gate-overdrive voltage (VGS ; VT ) in order to obtain an optimal total performance; the lowest power consumption is obtained in weak-inversion. However the maximal attainable intrinsic speed in a transistor decreases for small (VGS ; VT ); so the minimal (VGS ; VT ) can be imposed by the required speed performance. In the next paragraph we illustrate this for the one-transistor voltage amplier of section 3.4.2.1. Limitations on the choice of (VGS ; VT ): Lowering the (VGS ; VT ) of a transistor in strong inversion, lowers its maximal cut-o frequency fT which is de ned as [Lak 94]: gm = 3 (VGS ; VT ) (3.91) fT = 2C 4 L2 GS Since the bandwidth and thus the maximal operating frequency of a voltage processing stage is proportional to the fT of the transistors, see (3.56), lowering the (VGS ; VT ) limits the maximal operating frequency of the circuit. The minimal gate-overdrive voltage (VGS ; VT ), which guarantees a strong inversion behavior of the transistor is typically c although the simple processing stage in section 3.4.2.1 has feedback, its performance trade-o s are those of an open-loop stage since no second pole is taken into account in the calculations. 52 CHAPTER 3. IMPLICATIONS OF MISMATCH ON ANALOG VLSI 0.2 Volts (appendix A). We de ne the maximal frequency that can be processed for a (VGS ; VT ) of 0.2 Volts, as the corner frequency fcoII : (3.92) fcoII = fT j(VGS ;VT )=0:2= 203 L2 As long as the operating frequency is lower than the corner frequency fcoII and a minimal gate length is used, equation (3.63) for the calculation of the power consumption is valid. For frequencies higher than fcoII , however, the (VGS ; VT ) must be made proportional to the required bandwidth. When we substitute this result into (3.61) the minimal power consumption for the one transistor ampli er satis es: 2 2 P = 8 Cox A2VT0 L Acc2relBW2Gain2 (3.93) 3 VDD The power consumption is now proportional to the square of the operating frequency ! Equation (3.91) suggests that the absolute maximal frequency response that can be achieved in a MOS technology is limited by the minimal length L and the maximal (VGS ; VT ) that can be used. But the maximal speed of electrons in silicon is limited to vsat ( 105m=s) so that at high (VGS ; VT ) biases, the transistor goes in velocity saturation [Lak 94]. The voltage current relation is not quadratic in (VGS ; VT ) anymore; the transconductance gm of a transistor with xed dimensions does not increase with the current anymore so that the maximal cut-o frequency fcomax becomes a constant, independent of biasing: fcomax = 21 vLsat (3.94) This is the absolute maximal frequency that can be achieved in a technology. In section 3.4.2.1 the lowest power consumption is obtained for a transistor biased in the weak-inversion or sub-threshold regime. The maximal bias current IMwi that still biases the devices in weak-inversion is limited to [Lak 94]: 2 U (3.95) IMwi = 0:2nCox W T L The maximal cut-o frequency in weak inversion is then derived by substituting (3.95) into (3.65) and is given by: T fcoI = 10(nU (3.96) n ; 1)L2 so that for frequency requirements below fcoI , the transistor can be biased in weak-inversion and the power consumption is limited by (3.66) for a given accuracy and gain. Using the technology parameters from appendix A, we calculate the values of the different cut-o frequencies for the 0.7 m CMOS technology; the maximal frequency fcoI 3.4. IMPLICATIONS OF MISMATCH ON ELEMENTARY STAGES Power 53 Accuracy up fcoI fcoII fcomax BW Figure 3.13: Evolution of the minimal power consumption as a function of the required bandwidth for the circuit, for di erent accuracies. in weak-inversion is 240 MHz; the fcoII lies at 4.6 GHz, and the absolute maximal frequency fcomax is 22 GHz. These numbers are very optimistic however, since in the rst order analysis we have only taken into account the loading by the gate-source capacitances CGS . In practical circuits, extra loads due to the e.g. the parasitic drain-bulk capacitors and interconnect or biasing sources parasitic capacitors reduce the frequency response of the circuits. Moreover, the derived expressions are for an open-loop ampli er, whereas in closed loop systems extra margins of safety have to be considered to assure system stability so that the maximal achievable bandwidth further reduces. Since the dominant parasitic load capacitors are typically also proportional to the transistor sizing (see discussion of DB in appendix A), the qualitative evolution of the power consumption discussed in this section remains valid for more complex circuits. When the minimal power consumption for a one transistor voltage ampli er is plotted as a function of the required bandwidth BW, four operating regions can be distinguished, as is illustrated in gure 3.13: for BW fcoI , the transistor can be biased in weak inversion and the power consumption is proportional to BW; for fcoI < BW fcoII , the transistor can be biased in strong inversion with the minimal (VGS ; VT ) of 0.2 Volts and the power consumption is proportional to BW but is the proportionality constant is about 5 times higher than in weak-inversion as derived in section 3.4.2.1; for fcoII < BW fcomax , the transistor must be operated in strong inversion with a (VGS ; VT ) proportional to the BW so that the power consumption scales with the square of BW; 54 CHAPTER 3. IMPLICATIONS OF MISMATCH ON ANALOG VLSI a BW > fcomax is not achievable in the technology due to velocity saturation e ects in silicon. For all regions the power consumption scales with the square of the accuracy and the square of the gain. The discrete jump in the power consumption at fcoI is due to the limited accuracy of the transistor models we have used in the derivation of the minimal power consumption. We have considered a xed limit between strong inversion and weakinversion operation; however a smooth transition region called moderate inversion [Tsi 88] exists between the two regions so that also a smooth transition is observed in practical circuits for the power consumption; unfortunately simple and accurate hand-calculation models for the transistor behavior in this region are not available. Limiting the value of (VGS ; VT ) to obtain a minimal power consumption is thus limited by the required speed performance of the circuit block. The more demanding the speed requirements are and the more they approach the technology limits, the more excessive the power consumption becomes. This conclusion can be generalized for all voltage processing stages. In section 3.4.3, for instance, the bandwidth for a general OTA design in a feedback con guration is given by (3.79) and determined by the transconductance of the input stage and the dominant capacitor. As is shown, the dominant capacitor is larger than or equal to the input capacitance for stability reasons. The speed performance of the stage is thus limited by the intrinsic speed of the input device. The optimal power consumption is achieved for minimal (VGS ; VT ) in the input stage, but also in that design reducing (VGS ; VT ) can only be tolerated as long as the bandwidth speci cation allows it. The power consumption as a function of the required speed performance will have a similar characteristic as in gure 3.13. 3.5 Implications of mismatch on analog system performance In the previous sections we have developed expressions for the trade-o s between the di erent circuit performance speci cations of simple building blocks by studying their detailed design equations. From (3.51), (3.63) and (3.78) it is clear that a minimal power consumption is imposed for a given operation frequency (or bandwidth) and a given precision by the impact of device mismatch. This minimal power consumption is proportional to the matching quality of the technology which is expressed by Cox A2VT0 . In systems containing many building blocks the relations between the di erent speci cations are more complex since more degrees of freedom exist. There is extra room for circuit and topology optimization but the transistor mismatch has basically the same impact. In this section we will derive relations for the power consumption as a function of the other speci cations from basic general design equations. We show that a general relation can be derived for the minimal power consumption of a signal processing system due to the e ect of the mismatch in the components. 3.5. IMPLICATIONS OF MISMATCH ON ANALOG SYSTEM PERFORMANCE 55 Rs vIN A1 C1 VOS1 A2 C2 VOS2 A3 C3 VOS Figure 3.14: Schematic representation for a multi-stage voltage processing system. 3.5.1 General Multi-stage voltage designs A general multi-stage voltage processing system is represented in gure 3.14. For a current signal processing system or a system architecture with currents and voltages intermixed similar schematics can be drawn and similar relationships can be derived. In 3.14 the o set voltage, input capacitance and gain of each stage is indicated. The relative accuracy - see (3.59) - of the total system is determined by the input signal RMS value and the equivalent input referred o set voltage which is calculated as follows: s ) )2 + : : : (3.97) (VOSeq ) = 2(VOS1 ) + ( (VAOS2 ) )2 + ( A(VOS3 1 1 A2 Mismatch e ects put a lower boundary on the smallest signal that can be processed in a system. As a result, their in uence is most important at the input stage where the signal levels are small. Therefore a designer will try to make the largest possible gain in the rst stage A1 so that the in uence of the following stages on the accuracy speci cation is negligibled. The expression for (VOSeq ) is dominated by the term of the o set of the rst stage so that the relative accuracy is approximately: (3.98) Accrel = V(VinRMS ) 3V(inRMS VOS1 ) OSeq The o set voltage of a stage is inversely proportional to the area of the input devices; the input capacitance is proportional to the area of the same input devices; thus a general relation between the input capacitance and the o set voltage exists. If we assume an input stage with a single transistor biased in strong inversion and since VT mismatch is dominant, we obtain the following expressions: Cin = 2=3Cox WL (3.99) VT0 2(VOS ) = A (3.100) 2WL 2 Cin = 3Cox2(AVVT0) (3.101) OS d Since the power consumption of a block increases with the gain (see e.g. (3.63)), whereas the common required supply voltage for all blocks is determined by the wanted input swing and total gain, it is most power ecient to use as few stages as possible with the largest gain possible. 56 CHAPTER 3. IMPLICATIONS OF MISMATCH ON ANALOG VLSI p For a stage with a di erential input, the o set voltage is 2 times larger but the input capacitance is 2 times smaller so that the relation (3.101) between input capacitance and o set voltage remains valid. By applying (3.98) and (3.101), the input capacitance of the multi-stage system becomes: 2 C1 3C2ox(VAVT0 ) (3.102) OSeq The maximal speed of the system is of course determined by the bandwidth of the ampli cation stages; however, the pole generated by the input capacitor C1 and the source resistance Rs is the upper boundary for the system speed: flim = 2R1 C s 1 (3.103) 2 flim = 3=2 2R VCos1 A2 s ox VT0 From (3.98) it can be concluded that: 2 flim Acc2rel 6RVinRMS (3.104) s Cox A2VT0 For the design of a multi-stage system the following conclusions are important: In high accuracy systems the best matching speci cations have to be achieved in the rst or input stages where the signal levels are the smallest; and the largest possible signal ampli cation has to be done as soon as possible. In a general design, (3.104) proves that the speed and the accuracy of the system are interdependent for a given input source impedance and their combination cannot be chosen freely but is dependent on the matching quality of the technology. In this section we assumed that the input signal source drives the input capacitance of the signal processing system directly; then the power required for the signal processing is delivered by the input signal source. Many practical input voltage signal sources have relatively high output impedances, so that from (3.104) it is clear that an unbu ered input is not possible for high speed systems. The system schematic has to be changed so that the input capacitance is bu ered for the input signal source and so that a high input impedance is obtained over a wide frequency range. Due to this extra bu ering the power required to drive the input capacitance, is now delivered by the rst block of the signal processing system itself, which we will study in the next section. 3.5.2 Limit imposed by mismatch on minimal power consumption Mismatch is a random process and and its e ect on the circuit behavior can be translated into time-invariant random DC error signals called o set voltages or currents. The random 3.5. IMPLICATIONS OF MISMATCH ON ANALOG SYSTEM PERFORMANCE 57 VDD iDD vIN Active Part C iSS VSS Figure 3.15: General schematic for an active circuit driving a capacitive load. variations or the variance of the o set signals will be smaller if large devices are used since an averaging and smoothing occurs of the spatial errors sources responsible for the device mismatch. A certain accuracy will thus require a certain device area. However, this will lead to unavoidable parasitic capacitors at the input of the circuit proportional to the device area and thus inversely proportional to the o set signals as is demonstrated in (3.101). When we want to bu er a certain voltage signal vS = Vs sin(2ft) and drive it across a capacitor C, a current has to be delivered by the active part proportional to the voltage swing and the conductance of the capacitor at the signal frequency f: iC = Vs 2fCcos(!t). In gure 3.15 a general schematic is represented for an active part driving a capacitor. Although no power is dissipated by the capacitor since its current and voltage have a 90 degrees phase-shift, for all active parts operating from a practical power voltage supply, power will be dissipated in the active part. The power consumption of the active part will depend on its eciency which is a function of the mode of operation. For a class A operation a DC bias is added to all signals equal to their amplitude. The minimum required supply voltage for the active part is VDD ; VSS = 2Vs , whereas the minimum required DC bias current is IB = 2Vs !C so that the following relation for the power consumption in class A is obtained: P = 16fCVsRMS 2 (3.105) When an active part operating in class B is used, no DC o sets are used but the active devices draw the signal current from the VDD and deliver it into the capacitive load and they return the current from the load to the VSS . The current signals iDD from the VDD supply and iSS to the VSS supply are depicted in gure 3.16. The average DC current owing from the positive to the negative supply is IDC = (1=)(2fCVs ) so that the power consumption in class B is given by: P = 8fCVsRMS 2 (3.106) Other classes of operating modes exist for the implementation of active systems, like class C, E, F, S; they have much higher eciencies but can only be used for very speci c types of signals and applications whereas class A and B are almost generally applicable. From (3.105) and (3.106) we can conclude that due to the presence of capacitors in the system a CHAPTER 3. IMPLICATIONS OF MISMATCH ON ANALOG VLSI iDD & iSS 58 Time Figure 3.16: The supply currents for a class B active circuit driven with a sinusoidal input signal: ({) iDD and ({ {) iSS . certain amount of power will be dissipated in the system to perform the signal processing at a certain speed or frequency f. At this point we have all the relations available to estimate the power consumption of a system for a certain signal processing operation. From the de nition of the dynamic range of a building block as the largest signal that it can process over the smallest signal, we have DR = VsRMS =(3(VOS )) and using (3.101) and (3.106) we obtain the following expression for the power consumption: P = 24Cox A2VT0 fDR2 (3.107) This relation leads to the following conclusion: an analog signal processing system will at least consume the power expressed in (3.107) due to the e ect of mismatch to perform a signal processing operation at a given frequency f and with a certain accuracy or DR. Equation (3.107) is the most general expression of the fact that the SpeedAccuracy2/Power ratio is xed by the technology mismatch quality. 3.5.3 Noise vs mismatch limits on minimal power consumption Thermal noise and mismatch are both random processes and put a limit on the smallest signal that can be processed in a circuit; both physical phenomena will impose a minimal power consumption to achieve a certain DR speci cation and speed. In this section the minimal power consumption due to noise is derived and the limits imposed by mismatch and imposed by thermal noise are compared. In [Vit 90b] the e ects of thermal noise on the power consumption of a circuit is studied, which we repeat here for completeness. At a node with a capacitance C and driven by an impedance R, the total integrated thermal noise is: VnRMS 2 = kT (3.108) C with k the Boltzmann constant and T the absolute temperature. The dynamic range of the building block is given by DR = VsRMS =VnRMS , so that the minimal power consumption to 3.5. IMPLICATIONS OF MISMATCH ON ANALOG SYSTEM PERFORMANCE 59 Technology Type Mismatch Noise 24Cox A2VT0 8kT [fJ] [fJ] 2.5m [Pel 89] nMOS 4.3e-2 3.3e-5 pMOS 3.3e-5 1.2m [Bas 95] nMOS 2.12e-2 3.3e-5 pMOS 3.3e-5 0.7m nMOS 6.3e-3 3.3e-5 pMOS 3.3e-5 Table 3.4: The minimal energy per cycle imposed by mismatch and by noise for a dynamic range of 1. achieve a certain DR imposed by thermal noise is given by: P = 8kTfDR2 (3.109) To be able to compare these fundamental limits to the performance of realized circuits which all have di erent operating frequencies, not the power consumption but the energy per cycle P =f is evaluated as a function of the DR. In gure 3.17 the noise limit and mismatch limit for the minimal energy consumption are plotted as a function of the DR. The mismatch limit is technology dependent, since the product Cox A2VT0 is a technology parameter and not a physical constant like kT . For present-day sub-micron CMOS technologies the limit on power consumption imposed by mismatch is about two orders of magnitude more important than the limit imposed by thermal noise, which can also be concluded from the values in table 3.4. Analog Filters In a high order analog lter the minimal power consumption per pole of the lter by the impact of noise is given by (3.109). In gure 3.17 the high dynamic range analog lter circuits referenced in [Vit 90b] are also represented. The dynamic range of lters is in rst order not sensitive to matching or o set voltages. Mismatch will mainly in uence the accuracy of the lter coecients, the distortion performance and the power supply noise or common mode rejection characteristics of the lter which do not directly in uence the dynamic range. So lter realizations only optimized to low power and high dynamic range or distortion, can consume less than predicted (3.107). High speed A/D converters, are a better benchmark because o sets or mismatch limits the bit accuracy directly. So the minimal power consumption of A/D converters is clearly limited by mismatch. In gure 3.17 the energy per cycle of several A/D converters is represented. The high accuracy architectures have typically lower speeds and include digital or analog error correction circuitry, so they are only about 2 orders of magnitude from optimal performance. The lower bit converters use very high speed architectures and CHAPTER 3. IMPLICATIONS OF MISMATCH ON ANALOG VLSI 60 Power/Frequency [J] 1E-6 1E-10 1E-14 1E-18 20 40 60 80 Dynamic Range [dB] 100 Figure 3.17: Comparison of the impact of thermal noise and mismatching on the power consumption of analog systems: ({) noise limit; mismatch limits: ({ {) 1.2 m and ({o{) 0.7 m CMOS; Realized Filters ( ) [Vit 90b] and A/D converters (). 3.5. IMPLICATIONS OF MISMATCH ON ANALOG SYSTEM PERFORMANCE 61 cannot rely on error correction so their performance is 2 to 4 orders from the optimal. The limit performance takes only the consumption of the input stage into account but in practical circuits the subsequent stages also consume considerable amounts of power. The large discrepancies are further also due to the extra power taken up by parasitics. The derived performance limit caused by mismatch is of course only valid for converter architectures for which the accuracy relies on component matching. Many converter architectures, like e.g. ; or algorithmic converters, exchange conversion speed for accuracy and are made insensitive to component matching at the cost of lower conversion speeds. Their performance is typically limited by noise [Dij 94]. Also in [Pel 94], the impact of transistor mismatch on the power consumption of high-speed A/D converters is discussed. The analysis is based on a di erent calculation path but the predicted minimal power consumption is of the same order of magnitude as our results. Summary From the results in this section we conclude that in high speed, high accu- racy analog systems, the e ect of transistor mismatch imposes a higher minimal power consumption than the e ect of thermal noise on the circuit speci cations. It is important, however, to clearly understand the assumptions that lead to this conclusion. Thermal noise is a fundamental physical limit to the minimal signal energy that can be used in electrical signal processing systems that arises from statistical thermodynamics [Mei 95]. Combining this limit with the best possible eciency of a circuit, leads to the limit of (3.109). This limit cannot be broken by any circuit unless a more ecient architecture becomes available to transfer energy to a capacitive load from a DC power supply or other types of power supplies become available. The origins of transistor mismatch on the other hand, are linked to the device structure and device physics and to the fabrication technology of integrated circuits; device mismatch originates from the stochastic nature of physical processes used for the fabrication like ion-implantations, di usions or etching; the device structure using a channel in a doped material and its operation by modulating the channel resistance result in random uctuations of the device properties and operation. For integrated circuit technologies as we use and fabricate them today, these physical limitations are very fundamental and device mismatch is unavoidable. In this perspective, the limits imposed by device mismatch, are restricted to signal processing systems realized using integrated circuits.e As such they are of course very important in the quest for minimal power consumption in integrated circuits but are of no importance for the fundamental physical limits of information processing. As was demonstrated in this section, the mismatch limit is even more important than the noise limit in high speed analog systems. For high speed signal processing systems realized in present-day CMOS technologies the power consumption will be rather limited by the implications of device mismatch on the design and sizing of circuits to achieve a e In other integrated circuit technologies then CMOS like bipolar Si or GaAs, the same or similar fabrication principles are used. Fluctuations in the device operation or device mismatches also occurs and are governed by similar relations as those described in this chapter so that we can generalize this conclusion to all integrated circuit technologies. 62 CHAPTER 3. IMPLICATIONS OF MISMATCH ON ANALOG VLSI certain speed and accuracy speci cation than by the impact of thermal noise. However, when the system architecture or the application domain allows it, device mismatch can be circumvented by using o set compensating schematics as is discussed in section 3.6. As such the mismatch imposed limit is not important for all integrated circuits but unfortunately for a large part of high performance circuits. 3.5.4 Scaling of mismatch with technology feature size 3.5.4.1 Scaling of mismatch power limit The minimal energy per cycle imposed by mismatch is proportional to Cox A2VT0 , as can be concluded from (3.107). The oxide capacitance per unit area is determined by the permittivity of silicondioxide (mathitox) and the oxide thickness (tox ): (3.110) Cox = tox ox For smaller feature size technologies proportionally thinner gate-oxides are used so that Cox is technology dependent. Variations in the xed oxide charge, silicon-silicondioxide surface-states charge and the depletion charge are the main causes for threshold voltage mismatch [Pel 89, Miz 94, Laks86, Shy 84]. These variations in charge are transformed in threshold voltage variations by the gate capacitance. Therefore the threshold voltage proportionality constant AVT0 is expected to be proportional to the gate-oxide thickness tox . Experiments where the oxide thickness is varied and all other technology parameters are kept constant, are reported in [Miz 94]; the VT0 mismatch is indeed proportional to the gate oxide thickness and for a zero gate oxide thickness a zero mismatch is extrapolated. In gure 3.18 the AVT0 of several processes with di erent feature sizes is plotted versus the gate-oxide thickness tox f; a linear relation is indeed observed between AVT0 and tox for technologies with a feature size down to 0.5 m. A linear least-squares t yields a slope of 0:5 mV m=nm and a non-zero intercept of 3:4 mV m. The physical origin of the non-zero intercept is not yet clear. However the gate-oxide thickness is not the only technological parameter that changes with the feature size. For smaller line-widths and thinner gate-oxides larger substrate doping levels (Na ) have to be used to avoid punchthrough; larger doping levels increase the variance in the depletion charge and result in 1 = 4 higher mismatching. AVT0 then becomes proportional to tox Na [Miz 94]. For a constant eld scaling rule, the gate oxide thickness scales with 1/K and the substrate doping level scales with K so that the AVT0 scales with 1=K 3=4. If the depletion charge mismatch is the dominant cause for threshold mismatch, and a constant eld scaling can be assumed for the evolution of CMOS technologies, AVT0 is then expected to scale proportional to 3 = 4 tox when di erent processes are compared as in gure 3.18 and not linearly. This could be a possible explanation for the non-zero intercept when a linear tting is erroneously f I would like to thank Dr. M. Pelgrom of Philips Research in Eindhoven (NL.) for making available his measurement results reported in [Pel 89], which are also included in gure 3.18. 3.5. IMPLICATIONS OF MISMATCH ON ANALOG SYSTEM PERFORMANCE 63 60 AVT0 [mV m] 50 40 30 20 10 0 0 20 40 60 80 100 tox [nm ] Figure 3.18: The experimental value of the threshold voltage mismatch proportionality constant AVT0 versus the gate-oxide thickness for di erent processes. assumed. However other explanations, justifying a non-zero intercept have also been proposed [Pel 89]. It is clear however that in the presented gure many parameters change from technology to technology and many di erent scaling laws are in use which could all have an impact on the AVT0 . From the previous paragraph we can conclude that to evaluate the scaling of the technology mismatch Cox A2VT0 we can assume an almost linear dependence of AVT0 on tox ; the technology mismatch Cox A2VT0 thus scales with tox and the matching behavior of transistors improves for smaller feature sizes. This is also con rmed by the experimental data in table 3.1. For the analog systems, a migration towards smaller line-widths should improve the circuit performance. However, the maximal supply voltage also reduces for smaller line-widths [Mea 94, Hu 93], so that smaller signal levels have to be used. Especially in voltage processing systems, this results in a deterioration of the performance (see also [Pel 96]). If we examine equations (3.62) and (3.66), we conclude that the quality of the circuit will decrease if the supply voltage is reduced. Consequently the scaling advantage for the quality of voltage designs with smaller technology line-widths is reduced. In gure 3.19 the quality of submicron technologies, normalized to the quality of the 0.7 m technology, is plotted versus the technology line-width using the process data from [Hu 93] and appendix A; the AVT0 is approximated as 0:5 mV m=nm tox . The ;o; line uses 1=(Cox A2VT0 ) as quality measure whereas the ;x; line uses VDD =(Cox A2VT0 ). The reduction of the supply voltage clearly reduces the scaling advantage of voltage processing stages. For a current processing stage, the maximal signal swing is much less dependent on the available supply voltage. The voltage swing at the input depends only on the chosen bias modulation index. However, since the threshold voltage cannot be scaled proportionally to the supply voltage in order to limit the transistor cut-o leakage currents, the maximal (VGS ; VT ) that can be used in a current processing stage reduces for lower supply voltages and the circuit performance will also degrade. However, in rst order current processing 64 CHAPTER 3. IMPLICATIONS OF MISMATCH ON ANALOG VLSI 5 4 3 2 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Feature size [m] Figure 3.19: Scaling of the technology threshold matching quality as a function of the feature size; ;o; does not take into account the decrease of the supply voltage for downscaled technologies, whereas ;x; accounts for the reduced quality of voltage processing stages due to the reduced available voltage swing in down-scaled technologies; the matching qualities are normalized to the value for the 0.7 m technology. stages are less sensitive to power supply voltage scaling. Short channel e ects also have a negative impact on the matching behavior as was demonstrated for the VT0 matching in section 3.2.3.E. Moreover, the increasing substrate doping levels in deeper sub-micron technologies make the parasitic drain to bulk and source to bulk capacitors relatively more and more important compared to the gate-oxide capacitance. This results in extra capacitive loading of the signal nodes and requires extra power to attain high speed operation. We conclude that although the intrinsic matching quality of the technology improves for sub-micron and deep-sub-micron technologies, practical limitations make the theoretical boundary harder to achieve. 3.5.4.2 Relative importance of current factor and threshold voltage mismatches At the start of the mismatch analysis we compared the relative importance of threshold voltage VT0 and current factor mismatches on the behavior of transistors; for present-day processes the impact of the VT0 mismatch was clearly dominant. In the previous section the linear dependence of the AVT0 on the gate-oxide thickness was introduced so that the VT0 mismatch decreases for deeper sub-micron processes. The proportionality constant A for the current factor has no clear relation to process parameters. The variation in can be due to edge roughness, variations in the oxide thickness and mobility variations. The clear linear relation of the mismatch with the gate-area excludes edge roughness as a dominant cause [Pel 89, Bas 95]; the low correlation 3.5. IMPLICATIONS OF MISMATCH ON ANALOG SYSTEM PERFORMANCE 65 4 3.5 A [%m] 3 2.5 2 1.5 1 0.5 0 0 20 40 60 80 tox [nm ] Figure 3.20: The experimental value of the current factor mismatch proportionality constant A versus the gate-oxide thickness for di erent processes. with the VT0 mismatch excludes oxide variations as the dominant cause so that local mobility variations are the most probable dominant factor for mismatches. In gure 3.20 the proportionality constants A are plotted for di erent processesg and we can conclude that the A is almost constant from technology to technology and has an approximate constant value of 2%m. When the scaling trends for AVT0 and A are compared, it is evident that the mismatch gains in importance for deeper sub-micron technologies. This trend is con rmed by the values of the corner gate-overdrive voltage (VGS ; VT )m in table 3.1 for di erent feature size technologies. For a slope in AVT0 with tox of 0:5 mV m=nm and a constant value of A of 2%m a value of 200 mV is reached for (VGS ; VT )m for a gate-oxide thickness of 4 nm ; technologies with a feature size of about 0.25 m are expected to use this tox [Mea 94, Hu 93]. For this technology the mismatch will be at least as important and even more important as the VT0 mismatch for the calculation of the accuracy of circuits in the whole strong inversion region. The circuit design guidelines derived in section 3.4.4 are no more valid at that point. When mismatch is dominant, current mirrors have to biased with the smallest possible (VGS ; VT ) to obtain optimal total performance. But current mirrors can probably be biased at the optimal (VGS ; VT ) of (VGS ; VT )m - see section 3.4.1 - thanks to the low value of (VGS ; VT )m . At that bias point the minimal power consumption of current processing stages for a given speed and accuracy is proportional to Cox AVT0 A from (3.52); this indicates that a further scaling of the technology would not further improve the performance but would result in a constant performance as far as the assumptions of a constant A and an AVT0 proportional to tox remain valid. The total performance of voltage processing stages as in (3.61) becomes proportional to the cubic of (VGS ; VT ) when mismatch g I would like to thank Dr. M. Pelgrom of Philips Research in Eindhoven (NL.) for making available his measurement results reported in [Pel 89], which are also included in gure 3.20. 66 CHAPTER 3. IMPLICATIONS OF MISMATCH ON ANALOG VLSI is dominant over VT0 mismatch so that also the smallest possible (VGS ; VT ) should be used. The minimal power consumption of voltage circuits then becomes proportional to Cox A2 ; this indicates that a further reduction of the oxide thickness would result in a worse performance for voltage processing circuits. 3.5.4.3 Summary Thanks to the reduction of the VT0 mismatch with the oxide thickness, the move towards deeper sub-micron technologies will improve the intrinsic matching quality of the technology and should make lower power consumption possible. However, up to now no clear scaling of the mismatch has occurred, so that for deep-sub-micron technologies of 0.25 m and below, the current factor mismatch will become dominant; at that point further scaling of the technology is thought not to improve anymore the total performance of analog systems. 3.6 Techniques to reduce impact of mismatch Many techniques have been developed to reduce the impact of mismatches on the performance of analog building blocks. The time-invariant nature of the o set signals caused by mismatches, allows the sampling of the error signals and subtracting them from the output. However, due to the limited availability of only accurate very short term analog memories, this compensation has to be repeated at regular time intervals during the system operation. Auto-zero In comparators using an auto-zero compensation technique, the o set voltage of the comparator is rst sampled in a dead period and stored dynamically on a capacitor; this voltage is then subtracted from the input voltage before the comparison. A simple implementation of this scheme and the timing of the control signals is represented in gure 3.21 [All 87, Lak 94]. Similar compensation techniques have been used in A/D converters and switched capacitor circuits to improve the accuracy. However several limitations exist for the applicability of this technique. First of all, dead periods in the system operation have to be available so that the analog building block can be disconnected from the signal processing chain and its o set can be sampled. Not all system architectures have dead periods available. After the sampling of the o set, it has to be stored on an analog memory. For integrated circuits, storing an analog signal as a charge quantity on a capacitor is the most practical technique. Every analog memory only has a limited retention time due to unavoidable leakage currents. Especially in high accuracy applications, where a highly accurate storage is required, the retention time becomes short so that the memory has to be refreshed with short intervals; the continuous operation period of the system becomes small for high accuracy signal processing. 3.6. TECHNIQUES TO REDUCE IMPACT OF MISMATCH 1 2 Vin 1 CAZ 2 1 (a) 1 67 Auto-zero 2 (b) t Figure 3.21: Auto-zero comparator schematic (a) to eliminate the e ect of DC o set and (b) the timing for the switches. The switching of the capacitor in 3.21, which is necessary to recon gure the system back into an operational mode, introduces additional errors due to clock-feed-through and charge injection [VPg 88, Vit 90c]. Via the overlap capacitors between the hold capacitor and clock lines, an error charge is put on the holding capacitor when the clock signals change. These errors can be compensated by using complementary switches and complementary clock signals but they increase the system complexity even more. When the switch-transistors are turned o , the mobile carriers that made up the conductive channel ow away to the source and drain so that a charge is injected onto the hold-capacitor; this introduces an error in the sampled o set voltage. Extra compensation switches can be used to reduce the charge-injection but again increase the complexity of the system [VPg 86]. The error reduction is limited by the matching between the switches and the compensation switches. Again transistor mismatch puts a boundary on the possible accuracy, but now it impact will be at least on order of magnitude less important. Chopping Chopper stabilization is another common technique for the reduction of the e ect of o set voltages and 1/f noise on the accuracy of the signal processing [All 87]. The band-limited input signal of the ampli er is chopped by a multiplier driven with a 1/-1 signal, so that in the frequency domain it is modulated around the odd harmonics of the chopping frequency. This modulated signal is then ampli ed. The output signal is then again chopped so that is demodulated. The o set errors at the input of the ampli er are una ected by the chopping and cause o sets in the output signal. The output chopping modulates these o set signals to high frequencies so that their e ect in the passband is very small. This techniques strongly reduces the e ect of mismatch and 1/f noise. It requires however a high frequency operation of the ampli ers and the multipliers whereas the signal passband is only limited. Therefore, it is practically only useful for the design of high precision low frequency applications. In practical implementations a di erential implementation of this e ect is preferable in order to reduce the e ects of power supply noise, 68 CHAPTER 3. IMPLICATIONS OF MISMATCH ON ANALOG VLSI charge injection and clock feed-through. Trimming Trimming techniques are typically applied after the fabrication of the inte- grated system. The accuracy of the system is evaluated and is corrected by changing certain component parameters. Resistor values can be changed with laser-trimming. Discrete trimming can be achieved by switching extra resistors or capacitors in parallel to trim the accuracy. This can be achieved with digitally controlled switches. When the calibration is done once, long term digital memory cells have to be available which require for instance the availability of EEPROM devices in the technology, which is far from standard. When a self-calibration scheme is applied, the component will calibrate itself at every power-up. However this requires on-chip references so that the system accuracy can be measured; the accuracy of the references must be larger than the required system accuracy so that their design can be very complex. On the other hand analog fuses can be used for a one-time calibration but they are also not standard components and increase the technology cost. The accuracy improvement will be dependent on the resolution of the switch-able elements which can lead to important extra area consumption. The most important disadvantage of o -line trimming techniques is the extra testing or calibrating time required after fabrication which increases the cost of the device substantially. The implication of the o set compensation on the power consumption of the building blocks can be very important. Basically if ideal compensation is available, mismatch has not to be taken into account in the design and the mismatch limitation on the minimal power consumption is not important. In practice however, the accuracy speci cations are relaxed and only the last bits or percents are achieved with compensation or trimming. This implies that these techniques reduce the limit imposed by mismatch typically by an order of magnitude. When the area and complexity increase of the system can be tolerated or the fabrication technology has extra trimming facilities, o set compensation techniques should be applied and will enable important power saving. 3.7 Link with Harmonic distortion Throughout this chapter we have focused on four main speci cations of circuits and systems: accuracy, bandwidth or speed, gain and power consumption. However in many analog systems the linearity of the circuit or its distortion performance is of prime importance. In lter circuits, high linearity is necessary to avoid inter-modulation of unwanted signals into the passband or modulation of signal components. In all building blocks of telecommunication systems the distortion speci cations play an important role in the design and performance trade-o s. To satisfy the linearity speci cation, the largest signal level in the circuit has to be reduced. Distortion has a direct impact on the maximal signals and thus in uences the relative accuracy and dynamic range. Larger bias over signal ratio's have to be used so that more power is dissipated. In fact, when very high linearity 3.7. LINK WITH HARMONIC DISTORTION 69 speci cations are to be met, the power consumption will be mainly originate from the impact of non-linearities [Wam 96b, Wam 96a]. In all calculations and derivations of this chapter we have made implicit assumptions about the allowed distortion. In the basic current ampli er (section 3.4.1) we assumed a bias current modulation index of 1/2 which has direct implications on the linearity of the ampli er. In the voltage processing circuits (section 3.4.2), the limitation on the maximal input signal was only limited by the maximal output swing; when linearity is important other constraints will exist and only a lower maximal input signal will be acceptable. Since the distortion speci cation limits the maximal allowed signal, it has an impact on the relative accuracy of a circuit and consequently in uences the quality of a circuit design. In voltage processing circuits, for instance, the relative accuracy is determined by the maximal input signal VinRMS over the input referred o set signal VOS (see also (3.59)): (3.111) Accrel = 3VinRMS (VOS ) The maximal input signal depends on the allowed output signal swing and the gain (Gain) of the block. The output distortion reduces the allowed swing at the output to a fraction of the maximal swing, which is typically half the power supply VDD =2; similarly the input distortion can further reduce the allowed maximal input signal swing. The linearity speci cations and the distortion thus reduce the VinRMS to: VinRMS = disto pVDD (3.112) 2 2 Gain The parameter disto depends on the linearity speci cation and the distortion performance of the circuit or building block and is smaller than or equal to 1. After substitution into (3.111) and (3.61), the quality of the one transistor voltage ampli er becomes: Gain2BWAcc2rel 2 VDD = (3.113) disto P 24(VGS ; VT )A2VT0 Cox The more relaxed the linearity speci cations are the closer disto approaches unity and the better the total performance of the design becomes. For high linearity constraints the disto factor becomes small and as a result the power consumption for a given gain, speed and accuracy performance strongly increases due to the linearity requirements. For current processing stages a similar analysis can be done and the same conclusions can be drawn. For the performance analysis of building blocks in this chapter, optimistic assumptions have been used in view of distortion performance and linearity speci cations for the maximal attainable signal levels. In circuits that must meet high linearity speci cations, the impact of distortion will strongly reduce the maximal signal levels and results in a lower dynamic range and a lower accuracy for the same current consumption and supply voltage. Consequently the linearity speci cation further increases the power consumption of the system to attain a speci ed speed and accuracy. In this perspective, the theoretical limits derived in this chapter, become even more dicult to approach in practical signal processing systems. 70 CHAPTER 3. IMPLICATIONS OF MISMATCH ON ANALOG VLSI 3.8 Implications for analog parallel signal processing systems Accuracy speci cations The results and analysis presented in this chapter are very important for the design of analog parallel signal processing systems. In their VLSI realization we want to achieve a high density combined with a low power consumption. Both objectives are in uenced by the implication of transistor mismatch. The area of the circuits and thus their density is determined by the accuracy speci cation; but also the power consumption for a given speed is determined by the accuracy speci cations since the performance ratio SpeedAccuracy2/Power is xed by technology constants. A good accuracy speci cation of analog signal processing is thus very important; an over-speci cation results in poor speed and power performance and too loose accuracy speci cations will result in faulty system operation. Therefore in the next chapter the generation of good accuracy speci cations for the building blocks is treated in detail and the necessary theoretical evaluation methods are derived. Since the area of a circuit is proportional to the Accuracy2 A2VT0 , we can rewrite the ratio SpeedAccuracy2/Power as: Speed / 1 (3.114) DensityPower Cox so that we clearly see that the total performance of an analog parallel signal processing system is also limited by a technology constant of the used VLSI technology. Weak Inversion Operation The study of the optimal design of basic building blocks shows that current processing blocks should be biased with large gate-overdrive voltages (VGS ; VT ) and voltage processing blocks should be biased with small gate-overdrive voltages and if possible in weakinversion. This analysis thus shows that weak inversion operation of transistors is interesting for the VLSI implementation of parallel systems and neural systems but only for the voltage processing and the voltage mode blocks. Any current processing block must be biased in strong inversion to obtain good performance. An OTA circuit for instance should be designed with its input transistors in weak inversion but the biasing and current mirror stages should use strong inversion as much as possible. A more correct statement is thus that for the implementation of analog parallel systems an optimal combination of weak and strong inversion biasing is to be used. Analog or digital implementation In order to decide whether analog or digital circuits are best suited for the implementation of massively parallel signal processing systems we have to compare their speed, power and accuracy performance or equivalently their power/speed ratio as a function of the dynamic range. In gure 3.22 the fundamental limits on the energy per cycle for analog circuits imposed by transistor mismatch and thermal noise are plotted; also the energy per cycle for 3.9. CONCLUSIONS 71 Power/Frequency [J] −8 10 −10 10 −12 10 −14 10 −16 10 −18 10 20 40 60 80 100 Dynamic Range [dB] Figure 3.22: The energy consumption for a digital implementation ({) of a pole as a function of the required dynamic range; the minimal energy consumption of analog implementations due to the impact of mismatch ({o{) or noise ({x{). a typical digital implementation of a single pole transfer function is plotted after [Vit 90b]. The power consumption curve of analog circuits has a steeper slope but starts from a much lower intercept; the curve for the digital circuits is only logarithmically dependent on the dynamic range, since increasing the dynamic range basically only requires the addition of an extra bit, but it starts from a much higher intercept. Analog circuits are thus clearly advantageous for the implementation of systems with a low dynamic range requirement or low accuracy speci cations. This type of speci cations typically are required by massively parallel systems which perform perception tasks since they obtain their overall performance from the parallellism rather than from the high quality of the building blocks. Digital circuits are however clearly preferable when high dynamic ranges are required as in typical signal restitution tasks [Vit 90b, Vit 94]. However, transistor mismatch considerably lowers the dynamic range limit below which analog systems perform better compared to the limit imposed by thermal noise. For the implementation of massively parallel signal processing systems analog circuits o er the best performance. However, the implications of transistor mismatch must be carefully taken into account; therefore good accuracy speci cations are extremely important to obtain a high quality chip implementation. 3.9 Conclusions This chapter discusses the implications of transistor mismatch on the design and on the performance of analog circuits and systems. First a characterization method is developed to extract the parameters for quantitative mismatch models. Since transistor mismatch is caused by statistical phenomena, a large number of sensitive measurements have to 72 CHAPTER 3. IMPLICATIONS OF MISMATCH ON ANALOG VLSI be performed using dedicated test circuits and a dedicated automatic measurement setup. They show an inversely linear dependence of the parameter mismatch on the gate area of the transistors. When migrating to sub-micron and deep-sub-micron technologies, the in uence of the short and narrow channel e ects have to be accounted for in model extensions. This rm quantitative understanding of transistor mismatch forms the basis for the analysis of its impact on circuit and system performance. We prove that the maximal total performance or the performance ratio SpeedAccuracy2/Power of elementary voltage and current building blocks is determined by the chosen biasing point { the gate-overdrive voltage (VGS ; VT ){ and the technology matching quality. A current processing stage must be biased with large (VGS ; VT ) to obtain the best total performance whereas voltage stages must be biased with low (VGS ; VT )'s as long as the bandwidth requirements allow it. Under optimal biasing conditions, the total performance or SpeedAccuracy2/Power of the stage is xed and inversely proportional to the technology mismatching expressed by Cox A2VT0 ; the lower the Cox A2VT0 of a technology the better the circuit performance. This result thus explicitly states that a circuit designer can only trade one speci cation for another; but the best total attainable performance i.e. a combination of a high speed and a high accuracy at the same time as a low power consumption, is limited by the implication of transistor mismatch. These results are then extended for more complex circuits, including opamps and feedback systems. Moreover, for a general analog signal processing system, we show that transistor mismatch again puts a fundamental limitation on the minimal power consumption for a given frequency and accuracy performance. This technological limitation is even several orders of magnitude more important as the physical limitation imposed by the e ect of thermal noise. For high speed and massively parallel analog systems and any other analog system where no o set compensation or calibration can be done, mismatch is the performance limiting e ect, and the system design must carefully take mismatch into consideration. The analysis of the scaling of the mismatch behavior with the technology feature size, shows that the evolution towards deep-sub-micron technologies improves the matching quality of the technology. However, the smaller power supply voltages in scaled-down technologies limit the available voltage swing and reduce the scaling advantage. The more pronounced short channel e ects and velocity saturation e ects also deteriorate the performance of analog circuits and reduce the scaling bene ts. Furthermore, for very deep-submicron technologies (< 0:25 m), the current factor mismatch prevent a further quality improvement so that further down-scaling will have no positive e ect as far as the presently available data indicates. At the end of the chapter, the implications of these results for the design of massively parallel analog signal processing systems are discussed. Due to the xed ratio of performances, the importance of good accuracy speci cations is demonstrated. The generation of accuracy speci cations requires a sound understanding of the impact of random errors on the system level, a subject which is treated in the next chapter. Moreover, we show that analog circuits are indeed more power ecient for low accuracy levels than digital 3.9. CONCLUSIONS 73 implementations, but mismatch lowers the boundary where digital implementations take over. We also indicate that the sub-threshold operation of transistors, which is generally believed to be the best regime for massively parallel analog systems, is not optimal for all circuit functions. It is important to stress that the methods and analysis presented in this chapter are generally valid for the design of many types of analog systems. Transistor mismatch is an important factor in the design of analog circuits and therefore a good quantitative transistor mismatch model for the used technology should always be available to the analog designer. A good knowledge of mismatch allows a better optimization of the circuit design and avoids that very large safety margins have to be taken in the design which result in poor power and speed performance.