# How to Model Statistical Tolerance Analysis of Complex Circuits Using LTspice LTspice® can be used for statistical tolerance analysis of complex circuits. This article describes methods for tolerance analysis and worst-case analysis using Monte Carlo and Gaussian distributions in LTspice. To demonstrate the effectiveness of this approach, we model a voltage regulation example circuit in LTspice, demonstrating Monte Carlo and Gaussian distribution techniques with an internal reference voltage and feedback resistors.

By: Steve Knudtsen, Field Applications Engineer, Analog Devices

Summary

LTspice®Can be used for statistical tolerance analysis of complex circuits. This article describes methods for tolerance analysis and worst-case analysis using Monte Carlo and Gaussian distributions in LTspice. To demonstrate the effectiveness of this approach, we model a voltage regulation example circuit in LTspice, demonstrating Monte Carlo and Gaussian distribution techniques with an internal reference voltage and feedback resistors. The resulting simulation results are then compared to the worst-case analysis simulation results. It includes 4 appendices. Appendix A provides insights on fine-tuning the reference distribution. Appendix B provides the Gaussian distribution analysis in LTspice. Appendix C provides a graphical view of the Monte Carlo distribution defined by LTspice. Appendix D provides instructions for editing LTspice schematics and extracting simulation data.

This article describes the statistical analysis that can be performed using LTspice. This is not a review of 6-sigma design principles, the central limit theorem, or Monte Carlo sampling.

Tolerance Analysis

In system design, parameter tolerance constraints must be considered in order to ensure a successful design. A common approach is to use a worst case analysis (WCA), in which all parameters are adjusted to the maximum tolerance limits. In a worst-case analysis, the performance of the system is analyzed to determine whether the worst-case results are within the system design specifications. The power of worst-case analysis has some limitations, such as:

• Worst-case analysis requires determining which parameters need to be maximized and which need to be minimized in order to arrive at a true worst-case result.

• Worst-case analysis results often violate design specifications, resulting in expensive component selections to obtain acceptable results.

• Statistically, the results of a worst-case analysis are not representative of what is routinely observed; studying a system that exhibits worst-case analysis performance may require the use of a large number of systems under test.

Another alternative to performing system tolerance analysis is to use statistical tools to perform component tolerance analysis. The advantage of statistical analysis is that the resulting distribution of data reflects which parameters are typically measured in a physical system. In this article, we use LTspice to simulate circuit performance using Monte Carlo and Gaussian distributions to account for parameter tolerance variations, and compare them to worst-case analysis simulations.

In addition to some of the issues mentioned about worst-case analysis, both worst-case analysis and statistical analysis can provide valuable insights related to system design. For a tutorial on how to use worst-case analysis when using LTspice, see the article “LTspice: Worst-Case Circuit Analysis with Minimal Simulation Runs” by Gabino Alonso and Joseph Spencer.

Monte Carlo distribution

Figure 1 shows the reference voltage modeled in LTspice, using a Monte Carlo distribution. The nominal voltage source is 1.25 V with a 1.5% tolerance. The Monte Carlo distribution was within a 1.5% tolerance, defining 251 voltage states. Figure 2 shows a histogram of 251 values ​​with 50 bins. Table 1 presents the statistical results associated with this distribution. Figure 1. LTspice schematic for a voltage source (using Monte Carlo distribution) Figure 2. Monte Carlo simulation results for a 1.25 V reference voltage, presented as a histogram of 50 bars and 251 points

Table 1. Statistical analysis of Monte Carlo simulation results

 result average value 1.249933 minimum 1.2313 maximum value 1.26874 standard deviation 0.010615 positive error 1.014992 negative error 0.98504

Gaussian distribution

Figure 3 shows the reference voltage modeled in LTspice, using a Gaussian distribution. The nominal voltage source is 1.25 V with a 1.5% tolerance. The Monte Carlo distribution was within a 1.5% tolerance, defining 251 voltage states. Figure 4 shows a histogram of 251 values ​​with 50 bins. Table 2 presents the statistical results associated with this distribution. Figure 3. LTspice schematic for a voltage source (using a 3-sigma Gaussian distribution)

Table 2. Statistical analysis of Gaussian reference simulation results

 result minimum 1.22957 maximum value 1.26607 average value 1.25021 standard deviation 0.006215 positive error 1.012856 negative error 0.983656 Figure 4. 3-sigma Gaussian simulation results for a 1.25 V reference voltage, presented as a histogram of 50 bars and 251 points

The Gaussian distribution is a normal distribution represented by a bell-shaped curve, and its probability density is shown in Figure 5. Figure 5.3 – sigma Gaussian normal distribution

The correlation between the ideal distribution and the Gaussian distribution simulated by LTspice is shown in Table 3.

Table 3. Statistical distribution of the 251-point Gaussian distribution simulated by LTspice

 simulation ideal value 1-Sigma amplitude 67.73% 68.27% 2-Sigma amplitude 95.62% 95.45% 3-sigma amplitude 99.60% 99.73%

In summary, LTspice can be used to simulate Gaussian or Monte Carlo tolerance distributions of voltage sources. This voltage source can be used to model the reference voltage in a DC-DC converter. The LTspice Gaussian distribution simulation results are in good agreement with the predicted probability density distribution.

Tolerance Analysis of DC-DC Converter Simulation

Figure 6 shows a schematic diagram of an LTspice simulation of a DC-DC converter using a voltage-controlled voltage source to simulate closed-loop voltage feedback. Feedback resistors R2 and R3 are nominally 16.4 kΩ and 10 kΩ. The internal reference voltage is nominally 1.25 V. In this circuit, the nominal regulation voltage VOUTOr the setpoint voltage is 3.3 V. Figure 6. LTspice DC-DC Converter Simulation Schematic

To simulate the tolerance analysis of the voltage regulation, the tolerance of the feedback resistors R2 and R3 is defined as 1%, and the tolerance of the internal reference voltage is defined as 1.5%. This section describes three methods of tolerance analysis: statistical analysis using the Monte Carlo distribution, statistical analysis using the Gaussian distribution, and worst case analysis (WCA).

Figures 7 and 8 show the schematic and voltage regulation histograms simulated using the Monte Carlo distribution. Figure 7. Schematic of tolerance analysis using Monte Carlo distribution Figure 8. Voltage regulation histogram simulated using Monte Carlo distribution

Figures 9 and 10 show the schematic and voltage regulation histograms simulated using a Gaussian distribution. Figure 9. Schematic of tolerance analysis using Gaussian distribution Figure 10. Histogram of Tolerance Analysis Using Gaussian Distribution Simulation

Figure 11 and Figure 12 show schematics and voltage regulation histograms simulated using worst-case analysis Figure 11. Schematic of tolerance analysis using worst-case analysis simulation Figure 12. Histogram of tolerance analysis using WCA

Table 4 and Figure 13 compare the tolerance analysis results. In this example, WCA predicts the largest deviation, and a simulation based on a Gaussian distribution predicts the smallest deviation. Specifically, as shown in the box plot in Figure 13, the box represents the 1-sigma limit, and the box whiskers represent the minimum and maximum values.

Table 4. Summary of Voltage Regulation Statistics for Three Tolerance Analysis Methods

 WCA Gaussian Monte Carlo average value 3.30013 3.29944 3.29844 minimum 3.21051 3.24899 3.21955 maximum value 3.39153 3.35720 3.36922 standard deviation 0.04684 0.01931 0.03293 positive error 1.02774 1.01733 1.02098 negative error 0.97288 0.98454 0.97562 Figure 13. Boxplot Comparison of Regulation Voltage Distribution

Summarize

This article uses a simplified DC-DC converter model to analyze three variables, using two feedback resistors and an internal reference voltage to simulate voltage set point regulation. Use statistical analysis to present the resulting distribution of voltage setpoints. Display the results with graphs. And compare with the worst-case calculation results. The resulting data suggest that the worst-case limit is statistically impossible.

Thanks

Simulations were conducted in LTspice.

Simulations are all done in LTspice.

Appendix A

Appendix A presents the statistical distribution of regulated reference voltages in integrated circuits.

Before adjustment, the internal reference voltage adopts Gaussian distribution, and after adjustment, adopts Monte Carlo distribution. The tuning process usually looks like this:

• Measure the value before adjustment. At this time, a Gaussian distribution is usually used.
• Can the chip be fine-tuned? If not, discard the chip. This step basically clips the end part of the Gaussian distribution.
• Adjust the value. This keeps the reference voltage as close to the ideal as possible; the farther the value is from the ideal, the greater the adjustment. However, the fine-tuning resolution is very precise, so that the reference voltage value close to the ideal value does not shift.
• Measure the adjusted value and lock the value if the value is acceptable.

Comparing the resulting distribution with the original Gaussian distribution shows that some values ​​are unchanged, while others are as close to ideal as possible. The resulting histogram resembles a column with a curved top, as shown in Figure 14. Figure 14. Distribution of reference voltage values ​​after adjustment

While this looks a lot like a random distribution, it is not. If the product is trimmed after encapsulation, its profile at room temperature is shown in Figure 14. If the product is fine-tuned during wafer sorting, the distribution will spread out again when assembled into a plastic package. The result is usually a skewed Gaussian distribution.

Appendix B

Appendix B briefly reviews the Gaussian distribution commands available in LTspice. The distributions at sigma = 0.00333 and sigma = 0.002 will be reviewed, along with some numerical comparisons between the ideal distribution and the simulated Gaussian distribution. The purpose of this appendix is ​​to provide a graphical and numerical analysis of the simulation results.

Figure 15 shows a schematic diagram of the 1001 point Gaussian distribution of resistor R1. Figure 15.5 – Schematic diagram of sigma Gaussian distribution

It is worth noting the modification to the .function statement to define the tolerance of the Gaussian function as tol/5.This results in a standard deviation of 0.002, or at 1% tolerance the deviation is1⁄5. The histogram is shown in Figure 16. Figure 16. Histogram of 1001-point, 5-sigma Gaussian distribution with 50 bar intervals

Table 5 shows the statistical analysis of the 1001 point simulation. Notably, the standard deviation is 0.001948, while the prediction deviation is 0.002.

Table 5.5 – Statistical analysis of sigma distribution simulation

 result average value 1.000049 standard deviation 0.001948 minimum 0.99315 maximum value 1.00774 Median 1.00012 model 1.00024 1 point in Sigma 690 (68.9%) Figure 17. Histogram of 1001-point, 3-sigma Gaussian distribution with 50 bar intervals

Figure 17 and Table 6 give similar results with sigma = 0.00333, or when the tolerance is defined as 1%1⁄3.

Table 6.3 – Statistical analysis of Sigma Gaussian distribution simulation

 result average value 1.000080747 standard deviation 0.003247278 minimum 0.988583 maximum value 1.0129 Median 1.0002 model 1.00197 1 point in Sigma 690 (68.93%)

Appendix C

Figures 18 to 21 and Table 7 show the schematics of the 1001-point Monte Carlo simulation. Figure 18. LTspice schematic for 1001-point Monte Carlo distribution simulation

Table 7. Statistical analysis of Monte Carlo distribution simulations shown in Figures 18 to 21

 result average value 1.000014 minimum 0.990017 maximum value 1.00999 standard deviation 0.005763 Median 1.00044 model 1.00605 Figure 19. 1000 Bar Interval Histogram of 1001 Point Monte Carlo Distribution Figure 20. 500 Bar Interval Histogram of 1001 Point Monte Carlo Distribution Figure 21. 50-Bar Interval Histogram of 1001 Point Monte Carlo Distribution

Appendix D

Appendix D Review:

• how to edit LTspice schematics for tolerance analysis, and
• How to use the .measure command and SPICE error logs.

Figure 22 shows a schematic of a Monte Carlo tolerance analysis. The red arrows indicate tolerances for components defined in the .param statement. The .param statement is a SPICE directive. Figure 22. Monte Carlo tolerance analysis in LTspice

The resistor value of R1 can be edited by right-clicking on the component. As shown in Figure 23. Figure 23. Editing Resistor Values ​​in LTspice

Enter {mc(1, tol)} to define the nominal value of the resistance as 1, and the Monte Carlo distribution is defined by the parameter tol. The parameter tol is defined as a SPICE instruction.

The SPICE directives shown in Figure 22 can be entered using the SPICE Directive icon in the control bar. As shown in Figure 24. Figure 24. Entering SPICE instructions in LTspice

The .meas command provides a very useful GUI for entering relevant parameters. As shown in Figure 25. To access this GUI, enter SPICE directives as .meas commands. Right click on the .meas command and the GUI will pop up. Figure 25. GUI for entering relevant parameters

Measurement data is recorded in the SPICE error log. Figure 26 and Figure 27 show how to access the SPICE error log. Figure 26. Accessing the LTspice error log

The error log can also be accessed directly from the schematic by right-clicking on the schematic, as shown in Figure 27. Figure 27. Accessing the LTspice error log

Opening the SPICE error log displays the measured values, as shown in Figure 28. These measurements can be copied and pasted into Excel for numerical and graphical analysis. Figure 28. Graphical representation of SPICE error log with data from .meas command