One such method of "real-world" analysis is Monte Carlo analysis, which, with each new analysis run, randomly varies parameters (within their user-defined limits) to give the user a useful picture of actual circuit performance.

However, as a circuit designer, I'm most often interested in worst-case performance. That is, I want to know how a circuit performs at the extremes of component values, to ensure that I've met whatever design specification I'm designing to. And although Monte Carlo analysis can tell me what the performance is at these limits, if the circuit contains many components, it can take quite a lot of runs before its random selection of parameter values happens to simultaneously correspond to the worst-case limits of all of the components (and it's quite possible that I'll never see the true worst-case limits -- after all, it's a matter of chance).

To truly evaluate performance at a circuit's worst-case limits, we can perform a "worst-case" analysis in lieu of a Monte Carlo analysis. This analysis has an added advantage, too, in that not as many runs are required to ensure that we've truly evaluated all of the components over all possible variations in their tolerances -- after all, we don't need to measure

*between*the tolerances limits (which is what Monte Carlo analysis does). For example, if one of the components is a 10K ohm resistor with a 5% tolerance, worst-case analysis will only use resistance values of 10.5K and 9.5K for this component and not any other value between these limits (whereas Monte Carlo analysis would randomly use any value between, and including, these limits).

Linear Technology's

*LTSpice*can handle both Monte Carlo analysis as well as worst-case analysis. Unfortunately, it's not obvious how to do this from their "Help Topics." For example, "Monte Carlo", when entered into LTSpice's search field, returns no results. Not much use, that!

Nevertheless, LTSpice does indeed have a pre-defined Monte Carlo function. This is the "mc" function, and a search of the Help Topics for "mc" will point to the .PARAM topic, and under this heading we find the function

*mc (x,y)*, which, when invoked, returns a "random number between x*(1+y) and x*(1-y) with uniform distribution."

To use this function, rather than define a resistor's value as, say, 10K, we define it as "{mc(10K,0.05)}", where

*10K*is its nominal value, and

*0.05*is its tolerance (5%). (An example will follow below).

OK, so the predefined

*mc*function handles Monte Carlo analysis, but there is no pre-defined "worst case" function. We need to create this ourselves, but it's not too difficult. This can be done using Spice's ".function" directive. Here's an example of a function for worst-case analysis (we'll use this later, too):

**.function wc_a(nom,tola) if (run == 1, nom, if(flat(1)>0,nom*(1+tola),nom*(1-tola)))**

In this definition:

*.function*is the Spice directive for defining a function.*wc_a*is the name I've given this instance of this worst-case function.*nom*is the "nominal" value (e.g. 10K for a 10K resistor).*tola*is the tolerance (defined elsewhere with a .param directive (e.g. 0.1 for a 10% tolerance)).*run*is a variable (defined elsewhere in the .step directive) identifying the current run count).*flat(1)*is a Spice function that returns a random number between -1 and 1.

*.function*and

*flat*can both be found in LTSpice Help).

Take as an example a 10K ohm resistor. Normally, we'd just label its value in the LTSpice schematic as "10K". However, to vary its value between its worst-case values, we instead use a more complex label for its value. Rather than entering "10K", in this case we'll enter "{wc_a(10K,tola)}" into the component's value field. (Note the use of the curlicue brackets).

So what happens when we run the analysis?

If this is the first analysis run (

*run*is defined elsewhere in the .step directive and simply is the current run being performed), then, because run = 1, the function returns the value assigned to

*nom*, in this case, 10K.

But for each new run after the first run, and for each component defined with a

*wc_a*function, the function

*flat(1)*is re-evaluated for that component. If the random result returned from

*flat(1)*is greater than 0, then the tolerance is added to the component's nominal value (e.g. for a resistor whose nominal (

*nom*) value is 10K, if

*tola*is 0.1 (10% tolerance), the resistor's value is set to 11K). Otherwise, the tolerance is subtracted from the component's nominal value (e.g. the 10K resistor is set to 9K).

Here's a demonstration of both Monte Carlo and Worst Case analysis. Consider this basic circuit:

(Click on image to enlarge)

Given these component values, a frequency sweep of the input from 1 Hz to 1KHz shows that the circuit has the following gain and phase transfer function when measured at its Vout node:

(Click on image to enlarge)

But what happens when the component values vary over their tolerance range? Let's suppose that the resistors have 10% tolerance and the capacitors have 20% tolerance. Let's perform a

**Monte Carlo**analysis on this circuit, given these tolerance values. The same circuit, but now set up with its Monte-Carlo functions and .param directives, looks like this:

(Click on image to enlarge)

(Note that within the "mc" function, I'm not setting the tolarance field to an actual number (although I could have done it this way, too). Instead, I'm using a separate directive (.param) to define the tolerances (in this case, 10% and 20%) globally.)

Running the Monte Carlo analysis 1000 times (via the directive ".step param run 1 1000 1") gives us the following spread of gain and phase plots:

(Click on image to enlarge)

But we can't be sure that we've truly evaluated worst-case performance. So let's instead use our new "worst-case" function for a worst-case evaluation. Because there are two component tolerances for the components in the schematic (the resistors are 10% and capacitors are 20%), we'll define two worst-case functions:

*wc_a*and

*wc_b*. The function

*wc_a*will vary the value of those components whose tolerance is labeled "tola" (in this example, tola = 10%). Similarly, the function

*wc_b*will vary the value of those components whose tolerance is labeled "tolb" (in this example, tolb = 20%).

The schematic, with its new Spice directives and functions, now looks like this:

(Click on image to enlarge)

And the analysis output, after 40 runs, looks like this:

(Click on image to enlarge)

Note the discrete intervals between plots. This is because the worst-case analysis is only using component values that are at the +/- tolerance limits for each component, and not any intermediary values (except for the first of the 40 plots, which uses the nominal component values for its analysis).

**Comments:**

Further information on Worst-Case and Temperature Analysis can be found here:

http://k6jca.blogspot.com/2012/07/worst-case-and-temperature-analysis.html

Also -- there may be easier ways of doing both Monte Carlo and Worst Case analysis with LTSpice. The approaches above are the methods I've gleaned from LTSpice's Help and from Googling the Internet. If there are better ways to do these, please let me know!

**[Note: (21 September 2012) For a possible way of simplifying worst-case analysis, see the comment section below.]**

**Resources:**

LTSpice is available for free from Linear Technology. You can find it here.

## 26 comments:

Hi Jeff,

It's very helpful and just what I was looking for these days.

One question for you:

In the worst case analysis, you defined two functions for the resistor and capacitor respectively: wc_a() and wc_b(). But since their forms are the same, is one function wc_a() good enough for both resistor and capacitor?

Just a thought.

Thank you,

Sam

Hi Sam,

Thanks for the comment. I think you're right.

That is, you could define one "worst case" function, for example:

.function wc(nom,tol) if(run==1 ...(etc.))

and then for the component values use (wc(value1,tolb)}, {wc(value2,tola)},{wc(value3, tola)}, etc., where tola and tolb are defined in .param statements.

I'd want to try this first, but I can't think of any reason why it wouldn't work.

Thanks again!

- Jeff

I certainly enjoyed the way you explore your experience and knowledge about the subject.Thanks for this nice sharing keep posting more..

Hello Jeff,

Thanks for this blog post, I found it very useful. However, there are a couple of things that concern me:

1.) The built-in Monte-Carlo function "mc" uses a uniform probability distribution, rather than Gaussian. It's therefore probably better to use the following function for Monte-Carlo analysis:

.function normal(nom,tol) if(run==1, nom, nom*(1+gauss(tol/4)))

This uses a normal distribution and assumes manufacturer's quoted tolerance is based on a 4-sigma confidence (99.99% of shipped product will lie in quoted tolerance band).

2.) For the worst-case analysis, since the "wc" function randomly changes a component value, we cannot guarantee that all possible combinations of component values will be explored. You can bring the probability (that all possible combinations are explored) arbitrarily close to unity by increasing the number of simulation runs, but you can never make it actually unity! After some head-scratching I came up with the following functions:

.function binary_digit(x,powerOfTwo) floor(x/(2**powerOfTwo))-2*floor(x/(2**(powerOfTwo+1)))

.function wc(nom,tol,powerOfTwo) if(run==y, nom, if(binary_digit(run,powerOfTwo),nom*(1+tol),nom*(1-tol)))

Let's say you have 4 components that you wish to vary over their tolerance range. You require 2^4 + 1 runs (i.e., 17) to cover all the variations, plus 1 run with all values equal to their nominal. Set the number of runs as follows:

.step param run 0 16 1

Now, assign the values of the components as follows:

component 1: {wc(nominal_value,tolerance,0)}

component 2: {wc(nominal_value,tolerance,1)}

component 3: {wc(nominal_value,tolerance,2)}

component 4: {wc(nominal_value,tolerance,3)}

Finally, replace "y" in the "wc" function above with "16"

Now when you do your simulation run, you are guaranteed to have simulated all 16 different possible combinations of component value extremes exactly once each. Another benefit of this approach is that when looking at the results you can determine the actual component values used for a given result.

Enjoy!

Harry.

Thanks very much for your great comments, Harry!

Your reply raises an interesting question (for which I don't know the answer): are actual component values within a manufacturer's "Value" bins have a uniform or a Gaussian distribution? I suspect some (such as resistors) might have a uniform distribution, but I don't know. It probably depends upon the type of component and the manufacturing process.

Thanks again!

- Jeff

Jeff and Harry,

Thank you for this post and the comment. I found it very useful especially with the run-saving binary enhancement by Harry. I made one minor tweak for the binary count in "run": I started run at -1, and incremented by 1. This prevents the need to edit the wc() function every time the number of runs is changed. The nominal case is calculated when run==-1.

Jeff and Harry and all,

Thank you very much for this very interesting post.

Concerning your question Jeff, here is the answer :

On a process batch, all the resistors initially have a Gaussian distribution.

The manufacturer measures every resistance and sort them per tolerance.

This leads resistors to actually have a VERY strange distribution in the real world :

For a 100 Ohm resistor batch:

All the [99;101] Ohms resistors will be 1% resistors.

All the [95;99] and [101;105] resistors will be 5% resistors.

And so on.

This means that if you take a basket of 5% 100 Ohms resistances, you should never be able to find a 100 Ohm in it! In this basket, you'll get a gaussian distribution with a hole in the [99;101], and no "tails" in the distribution, since those resistors will belong to the 10% basket!

I found an article on that, and had a "holy sh.." moment when I understood how obvious yet bizarre that was.

I'll try to unburry it in all my stuff!

Hi,

really great post! Is x in the function: .function binary_digit(x,powerOfTwo) floor(x/(2**powerOfTwo))-2*floor(x/(2**(powerOfTwo+1))) equal to run?

Thanks

Sascha

Hi Sascha,

I believe the answer is yes.

If you look at Harry's equations, you'll see:

.function wc(nom,tol,powerOfTwo) if(run==y, nom, if(binary_digit(run,powerOfTwo),nom*(1+tol),nom*(1-tol)))

In this equation the variable run has been substituted for the x that we see used in the function definition: binary_digit().

- Jeff

Vey helpful, thanks a lot for the article and the comments.

Another point worth considering is that the tolerances given by the manufacturer apply to the product as it was shipped. Most circuits don't utilize resistors and capacitors hanging in the air, though. Therefore, the tolerances need to be widened based on expected change in value due to soldering-induced stresses. In my worst-case calculations, I double the resistor tolerance, and multiply the capacitor tolerance by 1.5 for NP0 and X7R and similar parts.

Yet another worthwhile point is to include the amplifier's GBW (really the 2nd open-loop pole) in the simulation. Depending on the sensitivity of the circuit to amplifier GBW, this may be important. GBW can vary wildly - by as much as factor of 2 or more.

Finally, if you expect variable output loading - say in an analog output exposed to the user - you should include at least the load extremes in the simulation. With good models, you should see effects of amplifier's AOL lowering under increasing load, and of the amplifier's output resistance.

Dear all,

Thank you for really great and useful posts.

However I have a simple question.

After mc or wc analysis how one can know what plot corresponds to which set of data (corner)?

Thanks,

P5K0

Hey there. Thanks for the awesome post. I tried using your example in DC but it always fails. I can't figure out why. Any suggestions?

That's an excellent question -- I usually use SPICE only for AC or Transient analysis, and so I've never tried it with DC. I'm traveling and out of town at the moment, but when I return home I'll check into it.

- Jeff

An update regarding LTSpice Monte Carlo analysis and DC, per the question above...

I created a simple voltage-divider circuit (components: one voltage source, two resistors) and gave each of these three components its own tolerance.

I then ran a 100-run Monte Carlo analysis on this circuit, looking at the voltage at the Voltage-Divider node.

Monte Carlo analysis ran fine using the Operating Point command (.op). It used all three tolerances.

Monte Carlo analysis ran with the DC Sweep command (.dc), in which the voltage source amplitude was swept, but in this case it did not use the tolerance I had assigned to the voltage source.

Monte Carlo analysis did not seem to work for the Transfer Function command (.tf).

Of course, the this last result might be due to an error on my part.

- Jeff

One more note regarding Monte Carlo analysis when running a DC Sweep...

As I mentioned in the previous comment, LTSpice ignores the "tolerance" assigned to the DC voltage source being swept.

BUT, if there are other voltage sources (with their own tolerances) in the circuit, the Monte Carlo analysis WILL vary these with each successive run, according to the tolerance values they've been assigned. (Note that this assignment is made in a component's VALUE field, e.g. the "mc(10K,tola)}" assigned as R1's value).

Hi,

is there any chance to read out any component values of a specific plot? Similar like it is possible with the .step param list function plots.

Cheers,

Chris

Hi Chris,

Unfortunately, I personally don't know of any way to link a specific plot to its component values. But that is not to say that there isn't a way...if there is one, however, I don't know it.

- Jeff

Dear Jeff, dear all,

I do not agree on the following sentence:

"we don't need to measure between the tolerances limits (which is what Monte Carlo analysis does)".

Suppose there is a LC circuit somewhere. Using your worst-case function wc(), you might find that L and C never resonate (for any combination (L,C): (L_min,C_min), (L_min,C_nominal), (L_min,C_max), etc.)... But there might be some (L,C) neither nominal nor minimum or maximum that actually resonate! And that condition has to be simulated in a worst case analysis, don't you think?

I am just pointing out that your worst-case function does not consider potential local extrema.

Regards,

Thomas.

Thanks for your comment, Thomas. And I agree -- if you want to check for unanticipated resonances (or other forms of "local extrema") in a circuit, then Monte Carlo analysis would be a more appropriate method of analysis.

Best Regards,

- Jeff

Thank you very much for sharing this....

Hello Harry, Jeff,

Thanks for the great post. As I was trying to understand Harry's binary function:

function binary_digit(x,powerOfTwo) floor(x/(2**powerOfTwo))-2*floor(x/(2**(powerOfTwo+1)))I put it in excel and it was giving 0000 and 1111 alternatively for 4 components. I was expecting it to give 0000,0001,0010,..1111. Am I missing anything?

Thanks,

R

Hi R,

Thanks for the comment.

I just tested the binary_digit function in Excel, and it seems to work. Here is what I did:

Define cell A2 to contain the powerOfTwo value.

Define cells A4 to A19 to contain an integer count, increasing by 1 with each row, from 0 in cell A4 to 15 in cell A19.

In cell C4, define its contents to be:

=FLOOR(A4/(2^$A$2),1)-2*FLOOR(A4/(2^($A$2+1)),1)

And then copy this formula into cells C5 through C19. Note that the copy function should change the A4 in the formula to A5 when copied from cell C4 to cell C5, etc.

If you set powerOfTwo to be 0, you will see the values in column C alternating 0,1,0,1,0...etc.

If you set powerOfTwo to be 1, you will see the values in column C alternating 0,0, 1,1,0,0,1,1...etc.

If you set powerOfTwo to be 2, you will see the value in column C alternating 0,0,0,0,1,1,1,1,0,0,0,0,1,1,1,1.

and if you set powerOfTwo to be 3, you will see the values in column C alternating 0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1.

An important note to keep in mind: in Harry's comment above, each component's WC function has a *unique* powerOfTwo defined for it (it is the last value in the WC function's field). So, each component will be toggling between its max and min values at a different rate, according to the the appropriate power-of-two sequence (per the sequences I just listed, above) selected for it.

Hope this helps,

- Jeff

Excellent article, thanks for sharing. I have one point regarding tolerance distributions, although a single component may have a peculiar distribution as discussed in the comments, if you are simulating a circuit where several components vary, the central limit theorem means the end result should be close to Gaussian regardless so a uniform assumption per component is probably as good as any.

The problem I have is that the end result of a run is a display with a band results composed of many lines but the density makes no difference. Is there any way to turn this into a distribution to extract a mean and standard deviation for a parameter such as the gain or 3dB point in your example filter. Alternatively how about a histogram? I'm thinking maybe some way of exporting data points and processing in a spreadsheet would probably be needed.

Best regards,

George

Hi George,

That's a great question!

I personally don't know of a simple way to do it (usually I'm only interested in Min and Max values to ensure that a design meets specs).

From a quick Google search, it looks like one could download individual traces to individual text files and then import these files into, say, Excel for analysis. But I can see this becoming unwieldy for any significant number of traces.

Best regards,

- Jeff

Thanks for the response Jeff,

I use min/max to guarantee results but the mean +/- spread is used for "typical" figures so I'd like both.

I've been running typically 1000 to 3000 traces to get a reasonable density so I doubt that would be practical. Exporting one point per run calculated from the curve (e.g. for the 3dB point) would be possible but I haven't found how to export a value per run automatically.

Merry Christmas,

George

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