Application Report SBAA173A – November 2009 – Revised November 2010 Determining Minimum Acquisition Times for SAR ADCs When a Step Function is Applied to the Input Miro Oljaca and Keith Sanborn ......................................................................... Data Acquisition Products ABSTRACT This application report analyzes a simple method for calculating minimum acquisition times for successive-approximation register analog-to-digital converters (SAR ADCs). The input structure of the ADC is examined along with the driving circuit. The voltage on the sampling capacitor is then determined for the case when a step function is applied to the input of the driving circuit. Three different test cases are subsequently evaluated using both precise and approximated equations. Contents 1 Introduction .................................................................................................................. 1 2 SAR ADC Analog Input Equivalent Circuit ............................................................................... 2 3 Mathematical Analysis of the Equivalent Circuit ........................................................................ 3 4 Minimum Acquisition Time ................................................................................................. 4 5 Test Cases ................................................................................................................... 5 6 Conclusion ................................................................................................................... 7 7 References ................................................................................................................... 7 Appendix A ........................................................................................................................ 8 Appendix B ....................................................................................................................... 10 Appendix C ....................................................................................................................... 12 List of Figures 1 1 Typical SAR ADC Input Driving Circuit................................................................................... 2 2 Simplified SAR ADC Input Driving Circuit 3 SAR ADC Input Driving Circuit Represented as a Second-Order, Low-Pass Filter ................................ 2 4 Second-Order Filter with Voltages and Currents Defined ............................................................. 3 5 Plots of Equations (8), (9), and (10) versus Time 6 Case (a) ...................................................................................................................... 6 7 Case (b) ...................................................................................................................... 6 8 Case (c) ...................................................................................................................... 6 ............................................................................... ...................................................................... 2 4 Introduction When it comes to designing the proper input driving circuit for analog-to-digital converters (ADCs), emphasis is generally placed on the calculation of the RC filter in front of the analog input and the selection of an operational amplifier (see Reference 1). The selection of the external RC components depends on the internal structure, sampling sequence, and charge injection of the successive approximation register (SAR) ADC (see Reference 2 through Reference 4). Knowledge of the internal ADC input structure, especially the value of the sampling capacitor, will assist users in optimizing the external RC components in order to obtain the maximum specified device performance (see Reference 5). All trademarks are the property of their respective owners. SBAA173A – November 2009 – Revised November 2010 Submit Documentation Feedback Determining Minimum Acquisition Times for SAR ADCs Copyright © 2009–2010, Texas Instruments Incorporated 1 SAR ADC Analog Input Equivalent Circuit www.ti.com The calculation of the external RC filter is usually carried out with the assumption that the analog input sampling switch resistance is negligible (see Reference 5). In the following analysis, the analog input sampling switch resistance will be included. 2 SAR ADC Analog Input Equivalent Circuit A typical analog input driving circuit for the ADC includes an operational amplifier (op amp) as well as an input RC filter composed of RIN and CIN as shown in Figure 1. The signal is then fed through the sampling switch SW with an equivalent on-resistance RSW to the sampling capacitor CSH. The input switch is composed of a CMOS transmission gate or similar structure. The equivalent on-resistance of the transistors is not linear and depends on the input signal level (see Reference 6). For this analysis, the average on-resistance of the switch measured in the linear region of operation will be used. ADC RIN RSW VIN(t) SW CESD CIN CSH Figure 1. Typical SAR ADC Input Driving Circuit Furthermore, the op amp is assumed to have ideal characteristics. As a result, it can be modeled as an ideal voltage source. By modeling the op amp in this way, the circuit from Figure 1 can be simplified as Figure 2 shows. ADC RIN RSW CESD CIN VIN(t) SW CSH Figure 2. Simplified SAR ADC Input Driving Circuit The ESD protection circuit at the input of the ADC has an equivalent capacitance CESD. This capacitance is the parallel combination of the protection circuit from the input pin to the power-supply rail and ground. The equivalent capacitance of CESD is in the range of 4 pF to 10 pF. On the other hand, the input filter capacitance CIN is in the range of 1 nF to 10 nF. If CIN >> CESD, then CESD can be ignored. Besides treating the op amp in Figure 1 as ideal, this analysis investigates the case of the input signal to the converter changing state after the sampling switch SW has closed. This situation may occur if the input signal suddenly changes during the acquisition period for a SAR ADC with a single input channel. SAR ADCs with an integrated multiplexer may also experience this situation when changing input channels. Under these conditions, the input signal can be represented as a unit step function with voltage VIN. Furthermore, the circuit in Figure 2 can be represented as a second-order, low-pass filter. The circuit for this case with updated variables is shown in Figure 3. R1 = RIN t=0 VIN(t) = VIN ´ u(t) R2 = RSW C1 = CIN + CESD C2 = CSH Figure 3. SAR ADC Input Driving Circuit Represented as a Second-Order, Low-Pass Filter 2 Determining Minimum Acquisition Times for SAR ADCs SBAA173A – November 2009 – Revised November 2010 Submit Documentation Feedback Copyright © 2009–2010, Texas Instruments Incorporated Mathematical Analysis of the Equivalent Circuit www.ti.com The worst case occurs when the input signal switches from zero or negative full-scale (NFS) to the input voltage VIN or positive full-scale (PFS). In order to analyze the circuit in Figure 3 under worst-case conditions, the initial voltages on capacitors C1 and C2 are set to zero or NFS. Figure 4 shows the Laplace transform of the circuit in Figure 3 with the initial conditions, reference currents, and voltages that are used in the analysis. R1 I(s) VIN/s R2 V1(s) 1/sC1 V2(s) I2(s) 1/sC2 I1(s) Figure 4. Second-Order Filter with Voltages and Currents Defined The primary goal of this analysis is to determine the minimum acquisition time (tACQ) for the voltage on capacitor C2 to settle within 1/2 LSB of the input signal for an N-bit SAR ADC as a function of R1, C1, R2, and C2. In order for this analysis to be performed, an expression for the voltage V2 across capacitor C2 as a function of time must be calculated. The next section in this application report focuses on this calculation. 3 Mathematical Analysis of the Equivalent Circuit The Laplace transform of voltage V2 in Figure 4 is: space V2(s) = A(s) ´ VIN (1) where: 1 1 A(s) = wn2 ´ ´ 2 s s + 2zwns + wn2 (2) The calculations for Equation 1 and Equation 2 are shown in Appendix A. The inverse Laplace transform of Equation 2 is: A(t) = wn2 ´ 1 - z2 -z 1 -zw t -zw t ´ e n ´ sin(wn 1 - z2 t) ´ e n ´ cos(wn 1 - z2 t) + 2 + wn2 wn2(z2 - 1) wn 1 - z2 (3) After simplifying and applying Euler's formula, Equation 3 can be re-written as follows (see Appendix B for further details): 1 2 2 2 2 ´ z + z - 1 ´ e-w (z - z - 1)t - z - z - 1 ´ e-w (z + z - 1)t A(t) = 1 2 2 z -1 n ( ( ( ( n (4) Equation 4, in turn, can be expressed as: - t - t 1 ´ (z + z2 - 1) ´ e t1 - (z - z2 - 1) ´ e t2 A(t) = 1 2 2 z -1 (5) where time constants t1 and t2 are defined as Equation 6 and Equation 7, respectively: t1 = 1 wn(z - z2 - 1) (6) t2 = 1 wn(z + z2 - 1) (7) SBAA173A – November 2009 – Revised November 2010 Submit Documentation Feedback Determining Minimum Acquisition Times for SAR ADCs Copyright © 2009–2010, Texas Instruments Incorporated 3 Minimum Acquisition Time www.ti.com In order to observe the effects of these two time constants, Equation 5 can be rewritten as: A(t) = 1 - [k1(t) - k2(t)] (8) where: k1(t) = z + z2 - 1 2 z2 - 1 - t t1 ´e (9) and k2(t) = z - z2 - 1 2 z2 - 1 - t t2 ´e (10) The plots of Equation 8, Equation 9, and Equation 10 as a function of time are shown in Figure 5. The following values were used in Figure 5: R1 = 100 Ω, R2 = 800 Ω, C1 = 1000 pF, and C2 = 40 pF. These component values set a = 100 ns, b = 4 ns, and c = 32 ns. These values, in turn, establish wn = 17.678 Mrad/s and z = 1.202. Furthermore, the time constants are calculated to be t1 = 105.721 ns and t2 = 30.267 ns. 1.50 A(t) = 1 - k1(t) + k2(t) k1(t) k2(t) Amplitude (Normalized) 1.25 1.00 0.75 0.50 0.25 0 0 200 400 600 800 1000 1200 1400 Time (ns) Figure 5. Plots of Equations (8), (9), and (10) versus Time As shown in Figure 5, k2(t) is going to decay faster than k1(t) when t2 << t1. In fact, Equation 6 and Equation 7 show that t1 will always be greater than t2 . Under these conditions, Equation 8 can be approximated as a function with only time constant t1, or: A(t) » 1 - z + z2 - 1 2 z2 - 1 - t t1 ´e (11) 4 Minimum Acquisition Time In order for the voltage on capacitor C2 in Figure 3 to settle within 1/2 LSB of the input signal for an N-bit SAR ADC: 1 A(t) ³ 1 - N+1 2 (12) If k1(t) >> k2(t) at the minimum acquisition time, then A(t) in Equation 12 may be approximated by Equation 11. When this approximation is done, the minimum acquisition time tACQ for an N-bit ADC is (see Appendix C for calculations): 4 Determining Minimum Acquisition Times for SAR ADCs SBAA173A – November 2009 – Revised November 2010 Submit Documentation Feedback Copyright © 2009–2010, Texas Instruments Incorporated Test Cases www.ti.com 1 wn(z - z2 - 1) ´ N ´ ln(2) + ln ( z + z2 - 1 z2 - 1 ( tACQ ³ (13) 5 Test Cases In order to evaluate if the approximation derived in Equation 11 is valid, the following test cases were analyzed for a 16-bit ADC (N = 16): (a) R1C1 = R2C2 × 100 (b) R1C1 = R2C2 (c) R1C1 = R2C2 / 100 The results of these cases are displayed in Table 1. Table 1. Results of Three Test Cases Case Parameter (b) (c) Units R1 100 100 10 Ω C1 1000 1000 1000 pF R2 20 2000 2000 Ω C2 50 50 50 pF 1.59 1.59 159 MHz MHz f1 = 1 2pR1C1 f2 = 1 2pR2C2 159 1.59 1.59 f2/f1 100 1 0.01 (1) a 100 100 1 ns b (1) 5 5 0.5 ns (1) 1 100 100 ns Mrad/s c (1) 100 10 100 z (1) 5.300 1.025 5.075 t1 105.048 125.000 100.505 ns t2 0.952 80.000 0.995 ns tACQ 1.239 1.601 1.185 ms wn (1) (a) Refer to Appendix A for equations. By using the acquisition times from Table 1, the final voltage on the sampling capacitor of the ADC from Figure 1 was calculated for each test case by using Equation 11 and Equation 8. The difference in the final voltage calculated with Equation 11 and Equation 8 for each test case is negligible. This investigation clearly shows that using the simplified Equation 11 to calculate the final voltage on the sampling capacitor does not introduce any significant error compared to using the exact formula (Equation 8). This result is further supported by the plots in Figure 6 through Figure 8. SBAA173A – November 2009 – Revised November 2010 Submit Documentation Feedback Determining Minimum Acquisition Times for SAR ADCs Copyright © 2009–2010, Texas Instruments Incorporated 5 Test Cases www.ti.com 1.50 Amplitude (Normalized) 1.25 1.00 0.75 0.50 A(t) = 1 - k1(t) + k2(t) k1(t) k2(t) 1 - k1(t) 0.25 0 0 200 400 600 800 1000 1200 1400 Time (ns) Figure 6. Case (a) 1.50 Amplitude (Normalized) 1.25 1.00 0.75 A(t) = 1 - k1(t) + k2(t) k1(t) k2(t) 1 -1k1(t) 0.50 0.25 0 0 200 400 600 800 1000 1200 1400 Time (ns) Figure 7. Case (b) 1.50 Amplitude (Normalized) 1.25 1.00 0.75 A(t) = 1 - k1(t) + k2(t) k1(t) k2(t) 1 -1k1(t) 0.50 0.25 0 0 200 400 600 800 1000 1200 1400 Time (ns) Figure 8. Case (c) 6 Determining Minimum Acquisition Times for SAR ADCs SBAA173A – November 2009 – Revised November 2010 Submit Documentation Feedback Copyright © 2009–2010, Texas Instruments Incorporated Conclusion www.ti.com 6 Conclusion This application report provides a simple analytical method for calculating minimum acquisition times for SAR ADCs. The input structure of the ADC is analyzed together with the driving circuit. The voltage on the sampling capacitor is then determined for the case when a step function occurs on the input of the driving circuit. Three different test cases were calculated using exact equations as well as simplified ones. The difference in the final acquired voltage calculated with these two equations was negligible. 7 References The following documents are available for download through the indicated web sites. 1. Oljaca, M. and B. Baker. (2008). Start with the right op amp when driving SAR ADCs. EDN. October 16, 2008. Pp. 43-58. Download at: http://www.edn.com/article/CA6602451.html 2. Downs, R. and M. Oljaca. (2005). Designing SAR ADC drive circuitry, Part I: A detailed look at SAR ADC operation. Analog Zone. Download at: http://www.analogzone.com/acqt0221.pdf 3. Downs, R. and M. Oljaca. (2005). Designing SAR ADC drive circuitry, Part II: Input behavior of SAR ADCs. Analog Zone. Download at: http://www.analogzone.com/acqt1003.pdf 4. Downs, R. and M. Oljaca. (2005). Designing SAR ADC drive circuitry, Part III: Designing the optimal input drive circuit for SAR ADCs. Analog Zone. Download at: http://www.analogzone.com/acqt0312.pdf 5. Baker, B. and M. Oljaca. (2007). External components improve SAR-ADC accuracy. EDN. June 7, 2007. Pp. 67-75. Download at: http://www.edn.com/article/CA6447231.html 6. Oljaca, M. (2004). Understand the limits of your ADC input circuit before starting conversions. Analog Zone. Download at: http://www.analogzone.com/acqt1101.pdf SBAA173A – November 2009 – Revised November 2010 Submit Documentation Feedback Determining Minimum Acquisition Times for SAR ADCs Copyright © 2009–2010, Texas Instruments Incorporated 7 www.ti.com Appendix A The voltage and currents in the circuit of Figure 4 can be described with the following equations: I (s) V1(s) = 1 sC1 (14) V2(s) = I2(s) sC2 (15) V1(s) - V2(s) = R2I2(s) (16) VIN - V1(s) = R1I(s) s (17) I(s) = I1(s) + I2(s) (18) Equation 14, Equation 15, and Equation 17 can be rewritten as: I1(s) = sC1V1(s) (19) I2(s) = sC2V2(s) (20) I(s) = VIN V1(s) sR1 R1 (21) Substituting Equation 19 through Equation 21 into Equation 18 yields: VIN = (s2R1C1 + s)V1(s) + s2R1C2V2(s) (22) Using Equation 20 in Equation 16 produces: V1(s) = (sR2C2 + 1)V2(s) (23) Substituting Equation 23 into Equation 22 produces: VIN = (s2R1C1 + s)(sR2C2 + 1) + s2R1C2 ´ V2(s) (24) By using these constants: a = R1C1 b = R1C2 c = R2C2 Equation 24 can be simplified to: VIN = s (sa + 1)(sc + 1) + sb ´ V2(s) (25) The voltage V2(s) can be described as a function of the input step signal VIN by rearranging Equation 25 to yield: 1 1´ 1´ ´ VIN V2(s) = ac s a+b+c 1 2 + s +s ac ac (26) 8 Determining Minimum Acquisition Times for SAR ADCs SBAA173A – November 2009 – Revised November 2010 Submit Documentation Feedback Copyright © 2009–2010, Texas Instruments Incorporated Appendix A www.ti.com The coefficients in Equation 26 can be represented as: a+b+c = 2zwn ac (27) and 1 = w n2 ac (28) Substituting Equation 27 and Equation 28 into Equation 26 produces: V2(s) = A(s) ´ VIN (29) where: A(s) = wn2 ´ 1 1 ´ s s2 + 2zwns + w 2 n (30) SBAA173A – November 2009 – Revised November 2010 Submit Documentation Feedback Determining Minimum Acquisition Times for SAR ADCs Copyright © 2009–2010, Texas Instruments Incorporated 9 www.ti.com Appendix B The equation: 2 A(t) = wn2 ´ 1-z -z 1 -zw t -zw t ´ e n ´ cos(wn 1 - z2 t) + 2 ´ e n ´ sin(wn 1 - z2 t) + 2 wn2 wn2(z2 - 1) wn 1 - z (31) Can be reduced to: A(t) = 1 - e -zwnt ´ cos(wn 1 - z2 t) + z 1 - z2 ´ sin(wn 1 - z2 t) (32) The arguments of the cosine and sine terms in Equation 32 can be defined as: x = wn 1 - z 2 t (33) Since: 1 - z2 = i z2 - 1 (34) Equation 33 can be re-arranged to be: x = iy (35) where: y = wn z 2 - 1 t (36) Euler's formula can be used to represent the cosine and sine terms in Equation 32 as: e-y + ey cos(iy) = 2 (37) and sin(iy) = e-y - ey 2i (38) 10 Determining Minimum Acquisition Times for SAR ADCs SBAA173A – November 2009 – Revised November 2010 Submit Documentation Feedback Copyright © 2009–2010, Texas Instruments Incorporated Appendix B www.ti.com Substituting Equation 35 and Equation 36 into Equation 37 and Equation 38 yields: -wn z2 - 1t e cos(wn 1 - z2 t) = +e 2 wn z2 - 1t (39) and sin(wn 1 - z2 t) = -wn z2 - 1t e -e 2i wn z2 - 1t (40) Using Equation 39 and Equation 40 in Equation 32 produces: A(t) = 1 - e ´ ( -wn z2 - 1t e +e 2 wn z2 - 1t -wn z2 - 1t e z + 1 - z2 -e 2i ´ wn z2 - 1t ( -zwnt (41) Substituting Equation 34 for the square-root portion in the denominator of right-hand term in Equation 41 yields: A(t) = 1 - e ( -wn z2 - 1t e ´ +e 2 wn z2 - 1t + -wn z2 - 1t e z i z2 - 1 ´ -e 2i wn z2 - 1t ( -zwnt (42) By re-arranging the terms, Equation 42 can be simplified to: 1 2 2 2 ´ z + z - 1 ´ e-w (z - z - 1)t - z - z - 1 ´ e-w (z + A(t) = 1 2 2 z -1 n ( ( ( ( n z2 - 1)t (43) SBAA173A – November 2009 – Revised November 2010 Submit Documentation Feedback Determining Minimum Acquisition Times for SAR ADCs Copyright © 2009–2010, Texas Instruments Incorporated 11 www.ti.com Appendix C For k1(t) >> k2(t), Equation 5 reduces to: A(t) » 1 - z + z2 - 1 2 z2 - 1 - t t1 ´e (44) In order for Equation 44 to satisfy the criteria in Equation 12 for the minimum acquisition time tACQ: 1 2N + 1 z + z2 - 1 ³ 2 z2 - 1 - tACQ ´ e t1 (45) Re-arranging the terms in Equation 45 and solving for tACQ yields: z + z2 - 1 z2 - 1 ( ( tACQ ³ t1 ´ N ´ ln(2) + ln (46) Using Equation 6 to replace t1 in Equation 46 produces the inequality: 1 2 wn(z - z - 1) ( ´ N ´ ln(2) + ln z + z2 - 1 z2 - 1 ( tACQ ³ (47) Revision History Changes from Original (November, 2009) to A Revision ............................................................................................... Page • • Corrected equations for test cases 1 and 3 ........................................................................................... 5 Corrected typos in Table 1; changed units for f1 and f2 to MHz from kHz ......................................................... 5 NOTE: Page numbers for previous revisions may differ from page numbers in the current version. 12 Determining Minimum Acquisition Times for SAR ADCs SBAA173A – November 2009 – Revised November 2010 Submit Documentation Feedback Copyright © 2009–2010, Texas Instruments Incorporated IMPORTANT NOTICE Texas Instruments Incorporated and its subsidiaries (TI) reserve the right to make corrections, modifications, enhancements, improvements, and other changes to its products and services at any time and to discontinue any product or service without notice. Customers should obtain the latest relevant information before placing orders and should verify that such information is current and complete. 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