# How to Model Statistical Tolerance Analysis of Complex Circuits Using LTspice

## How to Model Statistical Tolerance Analysis of Complex Circuits Using LTspice

“LTspice® can be utilized for statistical tolerance evaluation of complicated circuits. This text describes strategies for tolerance evaluation and worst-case evaluation utilizing Monte Carlo and Gaussian distributions in LTspice. To display the effectiveness of this strategy, we mannequin a voltage regulation instance circuit in LTspice to display Monte Carlo and Gaussian distribution strategies with an inner reference voltage and suggestions resistors.

“

By: Steve Knudtsen, Area Functions Engineer, Analog Gadgets

**Abstract**

LTspice

^{®}Can be utilized for statistical tolerance evaluation of complicated circuits. This text describes strategies for tolerance evaluation and worst-case evaluation utilizing Monte Carlo and Gaussian distributions in LTspice. To display the effectiveness of this strategy, we mannequin a voltage regulation instance circuit in LTspice to display Monte Carlo and Gaussian distribution strategies with an inner reference voltage and suggestions resistors. The ensuing simulation outcomes are then in comparison with the worst-case evaluation simulation outcomes. It contains 4 appendices. Appendix A gives insights on fine-tuning the reference distribution. Appendix B gives the Gaussian distribution evaluation in LTspice. Appendix C gives a graphical view of the Monte Carlo distribution outlined by LTspice. Appendix D gives directions for modifying LTspice schematics and extracting simulation knowledge.

This text describes the statistical evaluation that may be carried out utilizing LTspice. This isn't a overview of 6-sigma design rules, the central restrict theorem, or Monte Carlo sampling.

**Tolerance Evaluation**

In system design, parameter tolerance constraints have to be thought of with a view to guarantee a profitable design. A typical strategy is to make use of a worst case evaluation (WCA), through which all parameters are adjusted to the utmost tolerance limits. In a worst-case evaluation, the efficiency of the system is analyzed to find out whether or not the worst-case outcomes are inside the system design specs. The ability of worst-case evaluation has some limitations, equivalent to:

• Worst-case evaluation requires figuring out which parameters should be maximized and which should be minimized with a view to arrive at a real worst-case end result.

• Worst-case evaluation outcomes typically violate design specs, leading to costly element alternatives to acquire acceptable outcomes.

• Statistically, the outcomes of a worst-case evaluation will not be consultant of routinely noticed outcomes; it could be crucial to make use of a lot of programs underneath check to review a system that reveals the efficiency of a worst-case evaluation.

One other various to performing system tolerance evaluation is to make use of statistical instruments to carry out element tolerance evaluation. The benefit of statistical evaluation is that the ensuing distribution of information displays which parameters are usually measured in a bodily system. On this article, we use LTspice to simulate circuit efficiency utilizing Monte Carlo and Gaussian distributions to account for parameter tolerance variations, and evaluate them to worst-case evaluation simulations.

Along with a number of the points talked about about worst-case evaluation, each worst-case evaluation and statistical evaluation can present worthwhile insights associated to system design. For a tutorial on easy methods to use worst-case evaluation when utilizing LTspice, see the article “LTspice: Worst-Case Circuit Evaluation with Minimal Simulation Runs” by Gabino Alonso and Joseph Spencer.

**Monte Carlo distribution**

Determine 1 exhibits the reference voltage modeled in LTspice, utilizing a Monte Carlo distribution. The nominal voltage supply is 1.25 V with a 1.5% tolerance. The Monte Carlo distribution was inside a 1.5% tolerance, defining 251 voltage states. Determine 2 exhibits a histogram of 251 values with 50 bins. Desk 1 presents the statistical outcomes related to this distribution.

Determine 1. LTspice schematic for a voltage supply (utilizing Monte Carlo distribution)

Determine 2. Monte Carlo simulation outcomes for a 1.25 V reference voltage, offered as a histogram of fifty bars and 251 factors

Desk 1. Statistical evaluation of Monte Carlo simulation outcomes

end result | |

common worth | 1.249933 |

minimal | 1.2313 |

most worth | 1.26874 |

commonplace deviation | 0.010615 |

optimistic error | 1.014992 |

destructive error | 0.98504 |

**Gaussian distribution**

Determine 3 exhibits the reference voltage modeled in LTspice, utilizing a Gaussian distribution. The nominal voltage supply is 1.25 V with a 1.5% tolerance. The Monte Carlo distribution was inside a 1.5% tolerance, defining 251 voltage states. Determine 4 exhibits a histogram of 251 values with 50 bins. Desk 2 presents the statistical outcomes related to this distribution.

Determine 3. LTspice schematic for a voltage supply (utilizing a 3-sigma Gaussian distribution)

Desk 2. Statistical evaluation of Gaussian reference simulation outcomes

end result | |

minimal | 1.22957 |

most worth | 1.26607 |

common worth | 1.25021 |

commonplace deviation | 0.006215 |

optimistic error | 1.012856 |

destructive error | 0.983656 |

Determine 4. 3-sigma Gaussian simulation outcomes for a 1.25 V reference voltage, offered as a histogram of fifty bars and 251 factors

The Gaussian distribution is a standard distribution represented by a bell-shaped curve, and its likelihood density is proven in Determine 5.

Determine 5.3 – sigma Gaussian regular distribution

The correlation between the perfect distribution and the Gaussian distribution simulated by LTspice is proven in Desk 3.

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

simulation | preferrred worth | |

1-Sigma amplitude | 67.73% | 68.27% |

2-Sigma amplitude | 95.62% | 95.45% |

3-sigma amplitude | 99.60% | 99.73% |

In abstract, LTspice can be utilized to simulate Gaussian or Monte Carlo tolerance distributions of voltage sources. This voltage supply can be utilized to mannequin the reference voltage in a DC-DC converter. The LTspice Gaussian distribution simulation outcomes are in good settlement with the expected likelihood density distribution.

**Tolerance Evaluation of DC-DC Converter Simulation**

Determine 6 exhibits a schematic diagram of an LTspice simulation of a DC-DC converter utilizing a voltage-controlled voltage supply to simulate closed-loop voltage suggestions. Suggestions resistors R2 and R3 are nominally 16.4 kΩ and 10 kΩ. The inner reference voltage is nominally 1.25 V. On this circuit, the nominal regulation voltage V

_{OUT}Or the setpoint voltage is 3.3 V.

Determine 6. LTspice DC-DC Converter Simulation Schematic

To simulate the tolerance evaluation of the voltage regulation, the tolerance of the suggestions resistors R2 and R3 is outlined as 1%, and the tolerance of the interior reference voltage is outlined as 1.5%. This part describes three strategies of tolerance evaluation: statistical evaluation utilizing the Monte Carlo distribution, statistical evaluation utilizing the Gaussian distribution, and worst case evaluation (WCA).

Figures 7 and eight present the schematic and voltage regulation histograms simulated utilizing the Monte Carlo distribution.

Determine 7. Schematic of tolerance evaluation utilizing Monte Carlo distribution

Determine 8. Voltage regulation histogram simulated utilizing Monte Carlo distribution

Figures 9 and 10 present the schematic and voltage regulation histograms simulated utilizing a Gaussian distribution.

Determine 9. Schematic of tolerance evaluation utilizing Gaussian distribution

Determine 10. Histogram of Tolerance Evaluation Utilizing Gaussian Distribution Simulation

Determine 11 and Determine 12 present schematics and voltage regulation histograms simulated utilizing worst-case evaluation

Determine 11. Schematic of tolerance evaluation utilizing worst-case evaluation simulation

Determine 12. Histogram of tolerance evaluation utilizing WCA

Desk 4 and Determine 13 evaluate the tolerance evaluation outcomes. On this instance, WCA predicts the most important deviation, and a simulation based mostly on a Gaussian distribution predicts the smallest deviation. Particularly, as proven within the field plot in Determine 13, the field represents the 1-sigma restrict, and the field whiskers symbolize the minimal and most values.

Desk 4. Abstract of Voltage Regulation Statistics for Three Tolerance Evaluation Strategies

WCA | Gaussian | Monte Carlo | |

common worth | 3.30013 | 3.29944 | 3.29844 |

minimal | 3.21051 | 3.24899 | 3.21955 |

most worth | 3.39153 | 3.35720 | 3.36922 |

commonplace deviation | 0.04684 | 0.01931 | 0.03293 |

optimistic error | 1.02774 | 1.01733 | 1.02098 |

destructive error | 0.97288 | 0.98454 | 0.97562 |

Determine 13. Boxplot Comparability of Regulation Voltage Distribution

**Summarize**

This text makes use of a simplified DC-DC converter mannequin to research three variables, utilizing two suggestions resistors and an inner reference voltage to simulate voltage set level regulation. Use statistical evaluation to current the ensuing distribution of voltage setpoints. Display the outcomes with graphs. And evaluate with the worst-case calculation outcomes. The ensuing knowledge recommend that the worst-case restrict is statistically unattainable.

**Thanks**

Simulations have been performed in LTspice.

Simulations are all carried out in LTspice.

**Appendix A**

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

Earlier than adjustment, the interior reference voltage adopts Gaussian distribution, and after adjustment, adopts Monte Carlo distribution. The tuning course of normally seems like this:

• Measure the worth earlier than adjustment. Presently, a Gaussian distribution is normally used.

• Can the chip be fine-tuned? If not, discard the chip. This step principally clips the top a part of the Gaussian distribution.

• Regulate the worth. This retains the reference voltage as near the perfect as potential; the farther the worth is from the perfect, the larger the adjustment. Nevertheless, the fine-tuning decision may be very exact, in order that the reference voltage worth near the perfect worth doesn't shift.

• Measure the adjusted worth and lock the worth if the worth is appropriate.

Evaluating the ensuing distribution with the unique Gaussian distribution exhibits that some values are unchanged, whereas others are as near preferrred as potential. The ensuing histogram resembles a column with a curved high, as proven in Determine 14.

Determine 14. Distribution of reference voltage values after adjustment

Whereas this seems rather a lot like a random distribution, it's not. If the product is trimmed after encapsulation, its profile at room temperature is proven in Determine 14. If the product is fine-tuned throughout wafer sorting, the distribution will unfold out once more when assembled right into a plastic bundle. The result's normally a skewed Gaussian distribution.

**Appendix B**

Appendix B briefly evaluations the Gaussian distribution instructions obtainable in LTspice. The distributions at sigma = 0.00333 and sigma = 0.002 will likely be reviewed, together with some numerical comparisons between the perfect distribution and the simulated Gaussian distribution. The aim of this appendix is to offer a graphical and numerical evaluation of the simulation outcomes.

Determine 15 exhibits a schematic diagram of the 1001 level Gaussian distribution of Resistor R1.

Determine 15.5 – Schematic diagram of sigma Gaussian distribution

It's price noting the modification to the .perform assertion to outline the tolerance of the Gaussian perform as tol/5.This ends in a typical deviation of 0.002, or at 1% tolerance the deviation is^{1}⁄5. The histogram is proven in Determine 16.

Determine 16. Histogram of 1001-point, 5-sigma Gaussian distribution with 50 bar intervals

Desk 5 exhibits the statistical evaluation of the 1001 level simulation. Notably, the usual deviation is 0.001948, whereas the prediction deviation is 0.002.

Desk 5.5 – Statistical evaluation of sigma distribution simulation

end result | |

common worth | 1.000049 |

commonplace deviation | 0.001948 |

minimal | 0.99315 |

most worth | 1.00774 |

Median | 1.00012 |

mannequin | 1.00024 |

1 level in Sigma | 690 (68.9%) |

Determine 17. Histogram of 1001-point, 3-sigma Gaussian distribution with 50 bar intervals

Determine 17 and Desk 6 give related outcomes with sigma = 0.00333, or when the tolerance is outlined as 1%^{1}⁄3.

Desk 6.3 – Statistical evaluation of Sigma Gaussian distribution simulation

end result | |

common worth | 1.000080747 |

commonplace deviation | 0.003247278 |

minimal | 0.988583 |

most worth | 1.0129 |

Median | 1.0002 |

mannequin | 1.00197 |

1 level in Sigma | 690 (68.93%) |

**Appendix C**

Figures 18 to 21 and Desk 7 present the schematics of the 1001-point Monte Carlo simulation.

Determine 18. LTspice schematic for 1001-point Monte Carlo distribution simulation

Desk 7. Statistical evaluation of Monte Carlo distribution simulations proven in Figures 18 to 21

end result | |

common worth | 1.000014 |

minimal | 0.990017 |

most worth | 1.00999 |

commonplace deviation | 0.005763 |

Median | 1.00044 |

mannequin | 1.00605 |

Determine 19. 1000 Bar Interval Histogram of 1001 Level Monte Carlo Distribution

Determine 20. 500 Bar Interval Histogram of 1001 Level Monte Carlo Distribution

Determine 21. 50-Bar Interval Histogram of 1001 Level Monte Carlo Distribution

**Appendix D**

Appendix D Overview:

• easy methods to edit LTspice schematics for tolerance evaluation, and

• Methods to use the .measure command and SPICE error logs.

Determine 22 exhibits a schematic of a Monte Carlo tolerance evaluation. The crimson arrows point out tolerances for components outlined within the .param assertion. The .param assertion is a SPICE directive.

Determine 22. Monte Carlo tolerance evaluation in LTspice

The resistor worth of R1 will be edited by right-clicking on the element. As proven in Determine 23.

Determine 23. Modifying Resistor Values in LTspice

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

The SPICE directives proven in Determine 22 will be entered utilizing the SPICE Directive icon within the management bar. As proven in Determine 24.

Determine 24. Getting into SPICE directions in LTspice

The .meas command gives a really helpful GUI for getting into related parameters. As proven in Determine 25. To entry this GUI, enter SPICE directives as .meas instructions. Proper click on on the .meas command and the GUI will pop up.

Determine 25. GUI for getting into related parameters

Measurement knowledge is recorded within the SPICE error log. Determine 26 and Determine 27 present easy methods to entry the SPICE error log.

Determine 26. Accessing the LTspice error log

The error log will also be accessed straight from the schematic by right-clicking on the schematic, as proven in Determine 27.

Determine 27. Accessing the LTspice error log

Opening the SPICE error log shows the measured values, as proven in Determine 28. These measurements will be copied and pasted into Excel for numerical and graphical evaluation.

Determine 28. Graphical illustration of SPICE error log with knowledge from .meas command

**Concerning the Creator**

Steve Knudtsen is a Senior Area Functions Engineer at Analog Gadgets in Colorado, USA. He's a graduate of Colorado State College with a BS in Electrical Engineering and has been with Linear Expertise and Analog Gadgets since 2000. Contact info:[email protected]

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