Sigma Case Study

7/28/2019 Sigma Case Study

1/16

2003 Successful Statistics LLC 1 www.OQPD.com

Six Sigma Tolerance Design Case Study:Optimizing an Analog Circuit Using Monte Carlo Analysis

Andy SleeperSuccessful Statistics LLC

[email protected]

1. Abstract

Tolerance Design is the science of predicting the variation in systemperformance caused by variations in component values or the environment. Thisarticle shows how Monte Carlo simulation can be applied to predict and improvethe quality of a system before even one prototype has been built. Using thesemethods allows new products to be developed rapidly and introduced with fewerunexpected problems. The case study in this article is a simple analog circuit.

The analytical methods and optimization process may be successfully applied toany engineering problem where a transfer function can be derived.

2. Overview of Tolerance Design

In general, any product or process is a system converting inputs to outputs. Thisis shown graphically in Figure 1.

At the center of the system is aTransfer function, which converts theinputs (X) into outputs (Y). The

transfer function is a mathematicalequation, which may be known,estimated, or unknown.

These three types of transferfunctions are common in engineeringproblems:

White box transfer functionsare derived analytically, usingprinciples of science and

engineering. Gray box transfer functionsare estimated by simulatingthe behavior of the system,using computer programs likeSPICE. The function itself may be too complicated to derive, or it mayhave no closed-form solution.

X

Y

Part CharacteristicsProcess Characteristics

Environmental Characteristics

System Characteristics

Inputs

Transfer functionY = f(X)

Outputs

Figure 1 - Generic System


2/16


Black box transfer functions are estimated by observing the behavior of aphysical system. This is done by designing an orthogonal experiment,collecting the data, and estimating the transfer function using analysis ofvariance and linear regression methods.

This paper describes an example of Tolerance Design applied to a white-boxtransfer function. Whenever possible, white-box transfer functions are preferred,because they can be derived earlier in the development process, leading tofaster introduction of new products.

Figure 2 illustrates an effective process for tolerance design, using these fivesteps. More details on these steps will be explained later, using the case studyas an example.

1. Define tolerance for Y: Based on customer requirements for the system,define the widest limits on Y which provide tolerable performance for thesystem.

2. Develop transfer function: Derive the transfer function for the initial designof the system. Set up an Excel worksheet with formulas to calculate thetransfer function.

3. Compile variation data on X: If real data is available on the Xs, computestatistics from that data, and select distributions that represent thevariation seen in the data. Usually, no data is available, and anassumption is needed. When nothing is known about X, assume that it is

X

Y

Step 1:Define tolerance for Y

Step 3:Compile variation data on X

Step 4:Predict variation of Y

Step 5:Optimize system

Step 2:Develop transfer function

Figure 2 - Tolerance Design Process


3/16


uniformly distributed between its tolerance limits. This is a conservativeassumption, because it is worse than real life, in most cases. UsingCrystal Ball software, define Assumption cells for each input X, based onthis information.

4. Predict variation of Y: Using Crystal Ball software, define Forecast cellsfor each output Y. Set run preferences and run the simulation. In a SixSigma environment, compute capability metrics for Y, such as CP, CPK andDPMLT. If these predicted quality metrics meet Six Sigma criteria, then,stop!

5. Optimize system: If the system is not acceptable, what needs to bechanged? Consider these questions:

a. Does the tolerance for Y accurately reflect customer needs?

b. Which X contributes most to variation in Y? The sensitivity chart

produced by Crystal Ball tells you this. For the biggest contributor,either get some data to replace the default assumption, or choose adifferent component with less variation. Dont waste time fiddlingwith the small contributors on the sensitivity chart.

c. If the design needs to be changed, a new transfer function must bedeveloped. The results of the simulation and the sensitivity chartprovide clues to help in your redesign effort.

3. Case Study

The schematic shown below is part of a 5V power supply designed to detectwhen the 5V voltage drops too low. When this happens, the comparatorchanges state, resetting the processor before it starts doing evil things.

+

+5

R14.99k1%

R25.36k1%

VR1AD7802.5V0.2%

R3499k 1%

R410K1%

U1LM2903Voffset = 0 15 mV

Figure 3 Undervoltage Comparator, Original Design


4/16


Step 1: Define Tolerance for Y

First, what is Y? What characteristics of this circuit are we interested in? Hereare three:

VTRIP-DOWN This is the voltage of the +5V bus when the comparator

changes state, when the +5V is going down, for instance, when the powersupply is shutting off.

VTRIP-UP This is the voltage of the +5V bus when the comparator changesstate, when the +5V is going up, for instance, when the power supplystarts up.

VHYST = VTRIP-UP VTRIP-DOWN For stability, the comparator circuit requiresa certain amount of hysteresis.

For simplicity in this article, we will only analyze VTRIP-DOWN. If you wish to practiceusing these techniques, try analyzing the other two Ys as an exercise!

So what are the customer requirements for VTRIP-DOWN? This circuit is buriedinside a product, and appears to be far away from the customer. No customer isever aware of this circuit, unless it fails to work properly. This circuit is a safetydevice, intended to prevent undesired malfunction of the digital circuitry. So thecustomer requirement for VTRIP-DOWN is to shut down the processor before itssupply voltage goes out of range at 4.75V. Therefore, 4.75V is the lowertolerance limit.

The upper tolerance limit is set by the variation of the +5V output itself. If VTRIP-DOWN is above 4.85V, and the +5V voltage is low because of load conditions or itsinherent variation, the system will not work correctly.

So the tolerance limits for VTRIP-DOWN are 4.75V to 4.85V.

Step 2: Develop Transfer Function

For many problems, this step can be the most difficult. But a few simpleguidelines help make this easier:

Do not include inputs which have negligible impact Use new symbols to represent intermediate values Keep equations short. Look for opportunities to substitute symbols for

portions of the equation

For the undervoltage comparator, there are many inputs I choose to ignore. Thisis risky, and requires some engineering judgment. There is a risk of ignoring aninput that is actually significant. So when in doubt, either leave it in, or use someother method (such as circuit simulation) to determine if the input is significant ornot.


5/16


In this case, I choose to ignore the effect of the resistor in series with thereference diode. Based on the specifications of the diode, I can calculate thatthe effect of the resistor tolerance is in the nanovolt range, which is swamped outby the voltage tolerance of the diode. So I feel safe in ignoring this input.

Likewise, the input bias current of the comparator and the load impedance of thecircuit following the comparator have effects, but these are extremely small, and Iignore them.

What follows is one way to derive the transfer function. In this derivation:VTRIP-DOWN is the +5V bus voltage at the point where the comparatorchanges stateV+ is the voltage at the + input to the comparatorV- is the voltage at the input to the comparator

1VRV

pointtriptheatVVV

-

OFFSET

=

+= +

Since we are analyzing VTRIP-DOWN , the output of the comparator before itchanges state is high, so the open-collector output of the LM2903 is floating.

( )

( )

[ ] ( )( )

+

+++

+=

+++

+=+=

++=

+

+

14R3R1R2R

4R3R1RV1VRV

2R4R3R1R

4R3R1R

2RVV1VRV

2R4R3R1R

2RVV

OFFSETDOWNTRIP

DOWNTRIPOFFSET

DOWNTRIP

This last equation is the transfer function to be analyzed.

Figure 4 illustrates an Excel worksheet containing thisformula. Here are some tips to make this processeasier:

Enter a name in the cell to the left of eachcomponent. In the next step, Crystal Ball willautomatically pick up this name for each

Assumption cell.

Format each cell with a reasonable number ofdecimal places.

Split the transfer function into small pieces tominimize errors. Here, the numerator anddenominator of the fraction were calculated

Undervoltage Comparator

VR1 2.5000

Voffset 0.0000

R1 4990

R2 5360

R3 499000

R4 10000

Numerator 2539910000

Denominator 2754986400

Vtrip-down 4.8048

Figure 4 - Worklsheet


6/16


separately.

Step 3: Compile Variation Data on Each Input X

In the ideal world, engineers would have access to vast databases with actualmeasured values from samples of all these parts. From this data, we couldselect the most appropriate probability distribution and use that distribution forthe Monte Carlo simulation.

But in real life, most engineers have no data.

For the first simulation in data-poor real life, I recommend assuming that eachcomponent is uniformly distributed between its specification limits. This is aconservative assumption, because it is usually (but not always) worse than realdata will be.

A handy way to implement this assumption with Crystal Ball is to define the

tolerance limits in worksheet cells. For each X, define a uniform distribution andenter references to the cells where the tolerance limits are located. This isillustrated in Figure 5.

Figure 5 - Defining Assumptions with Calculated Parameter Values

After defining the first assumption, use the Crystal Ball Copy Data and PasteData functions to quickly define the rest of the assumption cells.

Step 4: Predict Variation of Y

Select the cell containing the calculated value for VTRIP-DOWN and define that as aCrystal Ball forecast cell, so that Crystal Ball will keep track of the randomlygenerated values. At this point, the spreadsheet looks like Figure 6.


7/16


Next, we must decide how many trials to run. We could pick a number out of theair, but Crystal Ball provides a better approach, called precision control. Usingthis feature, the simulation runs until we have enough information.

In this case, I asked Crystal Ball to run until the mean and standard deviation ofVTRIP-DOWN are known to within 1%, with 95% confidence. For this model, thisprecision was achieved after 15,500 trials, which were completed in 10 secondson my computer.

I also selected Latin Hypercube Sampling, which tends to converge faster thanthe default simple random sampling used by Crystal Ball.

For more complicated models which require more calculation time, relaxing theprecision control to 5% or more may be needed to finish the simulation in apractical time.

Figure 6 - Spreadsheet ready for simulation


8/16


Frequency Chart

Certainty is 94.63%from 4.7500 to 4.8500

.000

.006

.012

.018

.024

0

92.75

185.5

278.2

371

4.7286 4.7666 4.8046 4.8426 4.8806

15,500 Trials 15,500 Displayed

Forecast: Vtrip-down

Figure 7 displays the frequency chart for the forecast VTRIP-DOWN. The certaintygrabbers are set at the tolerance limits, 4.75 and 4.85.

Clearly, this design has a problem. Based on this simulation, only 94.63% ofthese circuits would meet their tolerance requirements.

In a Six Sigma environment, we must calculate other metrics, such as CP, CPKand DPMLT. To do this, we need the mean and standard deviation of VTRIP-DOWNwhich Crystal Ball predicts are 4.8049 and 0.02568, respectively. I plug thesevalues into another spreadsheet to make the capability calculations. (Thisworksheet, CapMet16.xls, is available on my web site, www.OQPD.com)

Figure 7 - Predicted forecast distribution


9/16


Capability metrics

Cp 0.6490991

Cc 0.0981429

Cpu 0.5853946

Cpl 0.7128036

Cpk 0.5853946

Z-bench 1.5913082Z-st 1.7561839

Z-lt 0.2561839

Quality Prediction, assuming normal distribution

Short-Term Shifted Up Shifted down Long-Term

Defects per million, upper DPMU 39528.466 398904.492 564.664

Defects per million, lower DPML 16241.654 137.197 261603.116

Defects per million DPM 55770.119 399041.689

4.6 4.65 4.7 4.75 4.8 4.85 4.9 4.95 5

Normal probability

functionSpecification limits

Target

Shifted up

Shifted down

Figure 8 - Capability of Initial Design

This report predicts a CPK of 0.58 and a long term defect rate of 399,042 DefectsPer Million Units (DPMLT). These metrics are clearly unacceptable. The shifteddistributions in the chart illustrate the effects of inevitable shifts and drifts whichhappen during the production of a product.

Step 5: Optimize System

Clearly improvement is needed. We can revisit the tolerance VTRIP-DOWN, but forthe reasons explained above, no changes to the tolerance are possible.

So what is causing most of the variation in this system? The Crystal Ballsensitivity chart, shown in Figure 9, has the answer.


10/16


The biggestcontributor tovariation isVOFFSET, followedclosely by R1 and

R2.

So the first changeto the systemshould be toimprove VOFFSET.

Revision 1: Better

Comparator

For a modest

increase in parts cost, theLM 2903 comparator can bereplaced with a LM293,which controls offset voltageto 0 9 mV overtemperature.

The organization of the Excelworksheet used in this casestudy makes revisions very

convenient. By

changing thetolerance in cell C5to .009, theparameters of theVoffset assumptionare automaticallyupdated.

After repeating thesimulation withthese settings, CPK

is now 0.68 andDPMLT is now290,947. Its better,but not good yet.

The sensitivity chart in Figure 11 shows that R1 and R2 are now the big culprits.Further improvement to the comparator would not be cost-effective.

Target Forecast: Vtrip-down

Voffset .65

R1 .50

R2 -.50

VR1 .22

R4 -.02

R3 .00

-1 -0.5 0 0.5 1

Measured by Rank Correlation

Sensitivi ty Chart

Figure 9 - Sensitivity Chart

Figure 10 - Revision 1

Figure 11 - Sensitivity Chart - Revision 1


R2 -.60

R1 .59

Voffset .44

VR1 .24

R4 -.01

R3 .00

-1 -0.5 0 0.5 1Measured by Rank Correlation

Sensitivity Chart


11/16


Revision 2: Using 0.1% resistors for R1 and R2

It is possible (at high cost) to purchase 0.1% resistors. What if these were usedin place of R1 and R2? Its easy to find out. Change the values in cells C6 andC7 to 0.1% and repeat the simulation.

Frequency Chart

.000

.008

.016

.024

.033

0

100.7

201.5

302.2

403

4.7500 4.7750 4.8000 4.8250 4.8500



Figure 12 - Frequency chart - Revision 2

Figure 12 shows the predicted frequency chart with the tolerance limits set as thelimits of the plot. None of the trials fell outside of tolerance limits. As a result, CP= 1.44, CPK = 1.31 and DPMLT = 7,829. These numbers are better, and out of allthe simulated units, none failed.

But there are still two big problems with this design:

First, the odd-value 0.1% resistors are expensive, and using them creates costlyproblems for procurement and inventory. If these are not part of the standard

parts stocked for assembly, additional equipment and setup will be necessary.

Second, this quality level is still not good enough for Six Sigma. To meet DesignFor Six Sigma (DFSS) standards, CPK must be 2 or greater. After a product goesinto production, shifts and drifts caused by components, processes anduncontrolled environmental factors may shift the average by 1.5 standarddeviations or more, without being detected. A DFSS product must be designedso that quality is good even after the average values are shifted by 1.5 standarddeviations.

What is good enough? For a normally distributed process, if CPK = 2.00, then the

long term defect rate (DPMLT) is 3.4 Defects Per Million Units. Thats world-classquality for this type of product.

So what can we do if the system is already too costly and still does not meetquality requirements? Redesign it.


12/16


13/16


But if we have no samples and no data, we must make an assumption. Areasonable assumption is that the values of R1 and R2 are uniformlydistributed within their tolerance zones.

Figure 14 illustrates the tolerance zone of

these two resistors. Each part must bewithin 0.1% (10 ohms) of the nominalvalue, and the ratio is controlled to within0.025%. So if R1 = 10,000 ohms, thetolerance for R2 is 9,997.5 to 10,002.5ohms.

One way to express this to Crystal Ball isto use the following trick:

Specify R1 as 10,000 0.1%

In the transfer function, replace R2 by (R1+ R2A). Specify R2A as 0 2.5 ohms.

The new transfer function is shown below:

[ ]

+

+++

++

+= 14R3R1R

4R3R

5RA2R1R

1RV1VRV OFFSETDOWNTRIP

Simulating this transfer function leads to this frequency chart:

Now, CP = 1.45, CPK =

1.29 and DPMLT = 8,863,about the same as theprevious revision.

So cost has improved,but quality has not.

To plan the next step,once again look at thesensitivity chart, shownin Figure 16

9990

10000

10010

9990

10000

10010

Figure 14 - Tolerance zone of R1,R2


Frequency Chart

.000

.008

.016

.025

.033

0

97

194

291

388

4.7500 4.7750 4.8000 4.8250 4.8500




14/16



Voffset .88

VR1 .45

R5 -.07

R2 -.03

R1 -.01

R3 .01

R4 .01

-1 -0.5 0 0.5 1

Measured by Rank Correlation

Sensitivity Chart

Figure 16 - Sensitivity chart - Revision 3

Once again, VOFFSET is the biggest culprit, while all the resistors now have a trivialimpact.

Revision 4: Define Assumption Based on Real Data

It is time to question the default assumption that each component is uniformlydistributed between its tolerance limits. After all, the comparator comes from acompany who publicly champions its Six Sigma program. It should be of highquality.

So, a sample of 50 LM293 parts are drawn from stock, including samples fromdifferent date codes. The offset voltage is measured on all these parts. Thefigure below shows a histogram of this data.

Histogram of Sample 1

-0.01 -0.005 0 0.005 0.01

Figure 17 - Histogram of Voffset data


15/16


The Crystal Ball Batch Fit tool may be used to select a distribution model whichbest fits this data. In this case, we decide to use a normal distribution, with

parameters set based on the statistics of this sample: = 3 x 10-6 and =.00207

In the spreadsheet model of the transfer function, we change the assumption forVOFFSET to a normal distribution with the parameters listed above, and repeat thesimulation.

The results are shown below:

Frequency Chart

.000

.013

.026

.039

.052

0

183

366

549

732

4.7500 4.7750 4.8000 4.8250 4.8500




Quality Prediction, assuming normal distributionShort-Term Shifted Up

Shifteddown Long-Term

Defects per million, upper DPMU 0.000 0.342 0.000

Defects per million, lower DPML 0.000 0.000 0.000

Defects per million DPM 0.000 0.342

Figure 19 - Six Sigma Capability Plot for Revision 4


16/16


Now, CP = 2.43, CPK = 2.15 and the predicted long-term defect rate is 0.3 DPM.Figure 19 illustrates that even with a 1.5-sigma shift added to this process,quality levels are extremely good.

4. Summary

In this article, an analog circuit design is used to illustrate the power of ToleranceDesign techniques and Monte Carlo simulation. The initial design proved to beunsatisfactory, and through a series of revisions, we generated a new design ofextremely high quality at reasonable cost. Here are the steps we followed:

1. We analyzed the initial design using Crystal Ball Monte Carlo simulation,assuming that each component is uniformly distributed between itstolerance limits. The results showed unacceptably high variation.

2. The sensitivity chart identified the biggest cause of variation, so wereplaced it with a tighter tolerance part. This reduced variation, but not

enough.3. We tried 0.1% resistors, which further improved quality, but atunacceptable parts cost.

4. We recognized that the transfer function depends heavily on the ratioR1/R2. Instead of discrete 0.1% resistors, we used a resistor network withcontrolled ratio. This reduced parts cost to acceptable levels, but variationwas still too high.

5. Again, the sensitivity chart identified the biggest cause of variation. Wegathered a sample of parts and measured them, using actual data insteadof the default assumption. This change brought the predicted quality to anacceptable level.

New product design is always iterative. To introduce products more quickly,these iterations must be done rapidly, in the analysis phase. Later, in theprototype phase, revisions are slow and costly. This case study illustrates how adesign may be fully optimized before building a single prototype.

Tolerance Design and Monte Carlo simulation are the keys to a safe, robust andsuccessful new product.

About the Author

Andy Sleeper is a DFSS Master Black Belt and General Manager of SuccessfulStatistics LLC. Andy provides training and consulting services to engineers innew product development. Andy holds a BS degree in Electrical Engineeringand a MS degree in Statistics. For more information, please [email protected] or call 970-420-0243.

Sigma Case Study

Documents