Electronic Reliability Technology: Worst Case Analysis Method (Part 2)

　　Figure 4: Comparison of initial tolerance, typical value, and worst-case minimum gain.

　　Other factors affecting the worst case scenario

　　Other factors that must be considered are the interface connection, mainly the input power, input signal and load of the module circuit, etc. These factors have tolerance limits on both sides of the typical value. When performing WCCA, these values ​​must be set to the limit values ​​and the positive and negative directions of the limits must be considered.

　　To evaluate the worst-case value of the minimum gain at the center frequency of the bandpass filter in Figure 1 and Eq1, the worst-case maximum and minimum values ​​of the resistors and capacitors must first be determined (as shown in Figure 3).

　　All changes are considered bias variables. Note that Vi and Vo in Figure 1 are not in Eq1 and their maximum and minimum tolerances need to be set. The sensitivity of the band directionality can be determined by performing sensitivity analysis using simulation software, as shown in Table 1.

　　According to the sensitivity analysis, substituting the worst-case maximum and minimum values ​​into Eq1, the gain obtained is Af0 = 5.76V/V, which is lower than the minimum gain requirement of Af0 = 7V/V, as shown in Figure 4. In the typical value and initial tolerance conditions calculated earlier, Af0 is greater than 7V/V. It is important to see that there are significant differences between the typical value, initial tolerance, and worst-case results.

　　It is not necessary for all resistors and capacitors to be at their worst-case values ​​to cause Af0 to be less than 7 V/V. A combination of some device parameters exceeding the initial tolerance will cause the gain to be less than 7 V/V. This method of substituting the worst-case maximum and minimum values ​​of the device into the circuit equation is called extreme value analysis (EVA).

　　Other WCCA technologies

　　Two other methods for performing WCCA are root sum square (RSS) analysis and Monte Carlo analysis. Both techniques give more optimistic results than EVA.

　　RSS is a statistical technique for combining standard deviations, which is based on the law of large numbers (central limit theory). RSS means that if multiple variables are combined, the resulting distribution is normal, regardless of the distribution form of the combined variables. Therefore, mathematical methods can be used to effectively calculate the standard deviation of circuit performance when multiple variables are combined. The standard deviation of each device is based on the sensitivity amplitude of the circuit performance for each device parameter. First, calculate the standard deviation ST of the output variable, and then multiply the result by 3 (99.7% probability) to get the worst-case value.

　　Table 1: Using simulation software, the sensitivity of each device can be obtained.

　　Monte Carlo analysis is considered to be an empirical judgment of the statistical results of multiple evaluations of circuit performance under various conditions. The parameters of each device are randomly selected under various conditions. Using Monte Carlo analysis, the average and standard deviation (δ) of the circuit can be calculated. 3δ (99.7%) is also considered to be the worst-case value. Fortunately, many simulation software can perform Monte Carlo analysis.

　　Comparison of three WCCA techniques

　　EVA is the simplest technique and is the easiest to estimate the worst-case circuit performance, but the results are the most pessimistic. EVA requires the development of a database of worst-case parameter variations for all devices in the circuit. The format that EVA requires as input is the worst-case device variation (maximum and minimum) limit (3δ), plus the circuit's sensitivity direction. The format of the circuit input result is the worst-case maximum and minimum values.

　　The results of RSS are relatively more realistic, but there may be errors inside because it assumes that the sensitivity is linear and the distribution is normal. The input format of RSS is the standard deviation of the probability distribution of device parameters (generally unavailable) and the sensitivity of the circuit to device changes. The output format is the mean and standard deviation of the probability distribution of circuit performance.

　　Monte Carlo analysis requires the understanding of the device parameter distribution (which is usually not available) before giving accurate results. It requires the help of a computer program. The input format of Monte Carlo analysis is the probability distribution of each device parameter (sensitivity analysis is not required). The output format of the result is a histogram of the probability distribution of circuit performance.

　　It can be seen that the two statistical methods, RSS and Monte Carlo, can predict the probability that the circuit performance is within the specification range, which is very important. EVA cannot give this probability result.

　　Conclusion

　　Electronic product hardware needs to work reliably within a certain lifespan, which cannot be achieved by simply designing the typical and initial tolerance values ​​of the device. The device parameters will shift after being assembled on the circuit board. If a worst-case device parameter variation database is developed, designers can easily obtain these databases, so that electronic engineers can not only do typical circuit design and analysis, but also do WCCA.