# Getting the worst-case circuit analysis with a minimal number of LTspice simulation runs

September 14, 2017

## For worst-case analysis we?d prefer not to use a distribution approach. Rather, the maximum deviation from the nominal value of each component is used in the calculations.

`When designing a circuit in LTspice, you might want to assess the impact of component tolerances, such as the gain error introduced by non-ideal resistors in an op amp circuit. Here, we’ll illustrate a method that reduces the number of simulations needed, and, therefore, speeds time to results.`

LTSpice provides several ways to vary a parameter’s value:

- .step param: a parameter sweep of a user-defined variable
- gauss(x): a random number from Gaussian distribution with a sigma of x
- flat(x): a random number between -x and x with uniform distribution
- mc(x,y): a random number between x*(1+y) and x*(1-y) with uniform distribution.

These functions are very useful, especially if you want to look at results in terms of a distribution. But if you’re just looking for worst-case conditions, they may not be the quickest way to get a result.

Using gauss(x), flat(x) and mc(x,y), for example, will require a simulation to run for a statistically significant number of times. From there, a distribution can be looked at and worst-case values calculated in terms of standard deviations. However, for worst-case analysis we’d prefer not to use a distribution approach. Rather, the maximum deviation from the nominal value of each component is used in the calculations.

# Running minimal simulations

Let’s say that we want to look at the worst-case impact of a *R1 = 22.5 kΩ* resistor with a *1 percent* tolerance. In this case, we really only want to run the simulation with *R1 = 22.5 kΩ * (1 - 0.01)* and *22.5 kΩ * (1 + 0.01)*. A third run with an ideal *22.5-kΩ* resistor would also be handy.

If we were just varying one resistor value, the ‘.step param’ method would work very well. But what if we have more? The classic difference amplifier has four resistors (Figure 1).

**Figure 1.**Difference amp configurationIf a discrete difference amplifier were to be designed, each of these would have some tolerance (e.g., 1 percent or 5 percent). For an example, take the application shown in the LT1997-3 datasheet and implement it in LTspice with a discrete LT6015 op-amp and some non-ideal resistors (Figures 2, 3).

**Figure 2.**LT1997-3 application example

**Figure 3.**LT6015 difference amp with non-ideal resistorsNotice that the values of resistors R1 through R4 are replaced by a function call `wc(nominalvalue, tolerance, index)`

, which is defined in the simulation by a `.func`

statement:

`.func wc(nom,tol,index) if(run==numruns,nom,if(binary(run,index),nom*(1+tol),nom*(1-tol)))`

This function, in conjunction with the next `binary(run, index)`

function shown, vary the parameter for each component between its maximum and minimum values and, for the last run, the nominal value.

`.func binary(run,index) floor(run/(2**index))-2*floor(run/(2**(index+1)))`

The `binary`

function toggles each `index`

*'*ed component in the simulation so that all possible combinations of `nom*(1+tol) `

and `nom*(1-tol)`

are simulated. Note that the index of components should start with `0`

*.* The following table highlights the operation of the `binary()`

function with results to each `index`

and `run`

, where 1 represents `nom*(1+tol) `

and 0 represents `nom*(1-tol)`

.

.step param run 0 16 1The number of runs is determined by *2N +1*, where *N* equals the number of indexed components needed to cover all of the device’s max and min combinations and the nominal. In our case, we need *17* simulations run and we can define this using the `.step`

* *command and the `.param`

statments:

`.param numruns=16`

Lastly, we need to define the `tola`

and `tolb`

for the simulation via `.param`

statments:

`.param tola=.01`

.param tolb=.05

# Plotting the .step'ed results

If the transient analysis simulation is run, see WorstCase_LT6015.asc file, where we can observe our results. For a 250-mA test current, the Vout net should settle to 250 mV. But now with our `wc()`

function, we get a spread from 235 mV to 265 mV (Figure 4).

**Figure 4.**Difference amplifier transient analysis# Plotting the .step'ed .meas statement

At this point, we could zoom in and look at the peak-to-peak spread. But let’s take a lesson from another LTspice blog, "Plotting a Parameter Against Something Other Than Time (e.g. Resistance)." This blog covers how to run a simulation several times and plot a parameter against something other than time. In this case, we want to plot the `V(out)`

versus simulation run index.

In this simulation we have add a `.meas`

statement to calculate the average voltage of the output:

`.meas VoutAvg avg V(out)`

To plot the `V(out) `

versus run parameter, we can view the *SPICE Error Log* (Ctrl-L), right click, and select *Plot .step’ed .meas data* (Figure 5).

**Figure 5.**Spice error log interfaceThe plot in Figure 6 shows the results of our *.step*'ed .*meas* data.

**Figure ****6. **Output voltage versus simulation run

The trace shows us that the results vary from a maximum-worse case of 265 (run 9) to a minimum-worse case of 235 (run 6) or roughly a ±6 percent error. This makes sense since we had both 1 percent and 5 percent resistors in this example. The last run (16) shows the ideal result (250 mV), which uses ideal resistors. Another faster approach to this particular circuit is to use the .*op* simulation (instead of the .trans) to perform a DC operating point solution which will plot the results of our stepped .*meas* data directly.

# The Value of Matched Resistors

When designing a difference amplifier, not only is the appropriate op-amp needed, but matching the resistors is equally as important. The following references do a good job of explaining this topic (and associated math) in detail:

However, you can neither achieve good Common Mode Rejection Ratio (CMRR) or Gain Error without appropriately matched resistors.

Linear Technology has a number of precision amplifier products that also include matched resistors, such as the __LT1997-3__, with 91 dB minimum DC CMRR and 0.006 percent maximum gain error.

## References

Monte Carlo and Worst-Case Circuit Analysis using LTSpice

`Joseph Spencer is a Field Applications Engineer and Gabino Alonso is a Strategic Marketing Engineer, both for Linear Technology Corp.`

Linear Technology, part of Analog Devices

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`This article originally appeared on the Linear Technology website.`