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Higher manufacturing yields using DOE.

Design of experiments (DOE) is a proven technique that efficiently identifies key parameters in a circuit or system, and explains how they should be modified to improve nominal performance and reduce variability. Used in manufacturing environments for many years, DOE has recently been added to circuit simulation software. This article discusses the benefits of DOE and how it may be applied to circuit design, and provides a design example.


Design engineers know their designs must include some margin for performance variation in production. However, the design process is too often based on nominal performance, with the estimate of the required margin based on the performance of a small number of prototypes. Designers may set specifications based on prototype performance, which leads to yield problems later in volume manufacturing. These problems occur because a small number of prototypes cannot simulate the parameter variations seen over the manufacturing lifetime of a product.

A better design approach is to consider the parameter variations that are expected to occur in manufacturing in the design process. The DOE tool assists in this process. Figure 1 shows a Monte Carlo simulation displaying how DOE improved performance and reduced variability of a simulated amplifier.

A designer's main tasks are to determine a topology, set parameter values such as bias voltages and resistor values, and decide how tightly component tolerances must be set to meet performance objectives. Perhaps the longest part of this process is determining the best nominal values for the many parameters in a circuit. Often this process is not done methodically and a designer may not be sure the best solution was found. DOE provides a methodical, efficient method of finding the best nominal parameter values in a circuit and setting their tolerances.

DOE is an organized way of setting up a series of simulations and keeping track of the results to reveal which parameters and parameter interactions are important. A graphical description of an interaction is shown in Figure 2. In this hypothetical experiment, an interaction is present between the two dopants. When dopant B is at its high value, the effect of dopant A on the gain is much greater. Without a tool like DOE, it is difficult to uncover interactions and quantify them, especially if a circuit has many parameters. Designers then use this information to set parameter values and their tolerances to optimize circuit characteristics and minimize variations.

When DOE is carried out in a manufacturing environment, diverse physical parameters are varied in a particular sequence, depending on the type of DOE analysis, and a number of experiments are run. Then the experimenter has to display the results manually and interpret them. Now that DOE has been added to circuit simulation software, the process of carrying out DOE analysis on a circuit or system has been made easier. The designer enters a circuit schematic, decides which parameters to vary, and chooses a type of DOE analysis. The simulator runs the necessary sequence of simulations, and the results of the experiment are presented automatically.

DOE Benefits

Some of the benefits derived from using DOE are that it enables the design of circuits that are insensitive to parameter variations and gives estimates of performance variation in manufacturing before starting production. DOE helps manufacturing engineers troubleshoot existing designs and improve their yield. It efficiently reveals which parameters or components dominate a circuit's nominal performance and variability, and reveals how interactions between parameters or components affect a circuit. It also helps set tolerances by revealing which ones are critical and presents data in a way that makes trade-offs clear.

Run #   Resistor   Resistor    Gain    Response
           A          B

1         900        450       9 dB       G1
2        1100        450      10 dB       G2
3         900        550      9.5 dB      G3
4        1100        550      12 dB       G4


The objective of DOE analysis is to optimize a circuit's characteristics by finding the best nominal parameter values and their tolerances. A DOE analysis is carried out by varying parameter values in a circuit, and seeing how the circuit's characteristics change. Designers often perform this procedure when attempting to improve their designs. The big contributions of DOE are that it enables a designer to investigate the effects of many parameters simultaneously and efficiently, and it displays the results using tables and plots that clearly indicate what must be done to improve the circuit, including trade-offs that might be necessary.

For example, in an amplifier circuit design, the characteristics to be optimized are gain, stability factor and output match. DOE may be used to find parameter values and tolerances that minimize variations in these characteristics. In DOE terminology, these characteristics are response variables. The parameters to be varied might include bias resistor, stability resistor, inductor and matching network element values. In DOE terminology, these are factors.

In a DOE analysis, the parameters (factors) are varied and multiple simulations are run. The DOE outputs show how the circuit characteristics (response variables) depend on the parameters. In the most simple DOE analysis, the simulator uses two levels for the factors, the nominal value plus a percentage or absolute value, and the nominal value minus a percentage or absolute value.

As a simple example, if a designer were investigating how to set the nominal values of two resistors in an amplifier circuit to optimize the gain, a DOE simulation would be set up as listed in Table 1. This example is shown in Figure 3. Four simulations are run, each with a different possible combination of the factor values. The relationship between the gain and the two resistors is actually a surface, and the DOE simulation finds four points on this surface.

The nominal value of resistor A is 1000 [ohms], and the nominal value of resistor B is 500 [ohms]. Each one is set to its nominal value [+ or -]10 percent during the simulation runs. All possible combinations of resistor values are simulated. There are two levels for the factors during the simulations. Experiments of this type are "two-level full factorial."

The outputs from a DOE simulation generate a Pareto chart that shows the factors and interactions that dominate the response variable(s) (gain, in this example), main effects plots and interaction plots, which show how the factor values should be set for best performance. An analysis of variances (ANOVA) table that provides additional statistical information about the experiment is also generated. This experiment is so simple that the designer is able to choose the best parameter values directly from the table. However, in more complex DOE simulations with many factors and two response variables, examining the DOE outputs is necessary to choose the best parameter values.

Response Variables

Response variables are circuit characteristics. A response variable can be anything definable by an equation, such as the minimum value of a stability factor, gain, DC power consumption, power-added efficiency, harmonic levels or worst-case return loss. DOE can be used with all analyses, such as DC, AC, S-parameter and harmonic balance. This flexibility allows many different characteristics to be analyzed. DOE outputs for other response variables may be generated easily, without resimulating the circuit. An arbitrarily large number of response variables may be displayed simultaneously in table form, with each response variable being a column. These tables are useful when making trade-offs among circuit characteristics.

DOE outputs

There are six different DOE outputs, including the design matrix and response variable table, the ANOVA table, the Pareto chart, the main effect plot, the interaction plot and the cell-mean-average plot. The design matrix and response variable table shows the values of the factors for each simulation and the corresponding response variables. The ANOVA table indicates numerically the factors and interactions that have the largest effect on the response variables, and gives other statistical information about the experiment. The Pareto chart displays the ANOVA table information in a histogram, which clearly shows the factors and interactions that dominate. The main effects plot shows the main effects of all of the factors on the response variables. The designer uses these to determine the best nominal values for each factor. The interaction plot demonstrates how interactions affect the response variables. The designer uses these to determine factor values when interactions are significant. Cell-mean-average plots, another means of viewing the results of a DOE analysis, aid the selection of the best set of parameter values.

Applying DOE

How DOE is applied depends on where a circuit is in the design process. If acceptable nominal performance has not yet been achieved, then DOE may be used as an optimizer. DOE can also be used to evaluate variability in circuit performance and to set specifications. If an acceptable solution already exists, either from performance optimization or manual design techniques, then DOE may be used to investigate the variability of the circuit's responses. This application is particularly useful when determining whether a circuit will meet its specifications or when setting specifications.

Additionally, DOE may be used to optimize circuit performance and minimize variability simultaneously, using a nested technique. Known as the Taguchi method, it allows both the mean value of a circuit response and the response variation to be considered simultaneously when setting parameter values.

For circuits that are currently difficult to manufacture, or that sometimes fail to meet specifications, DOE may be used to determine the parameters that contribute the most to variations in circuit responses. This type of analysis would lead the designer to tighten the tolerances on the critical component(s), use a nested DOE analysis, or use the other techniques to find a better set of nominal parameter values.

A Design Example

To demonstrate how DOE can design circuits, a simple amplifier example is shown. The circuit was originally designed to be a low noise 1 GHz amplifier. The original circuit, shown in Figure 4, was designed independently considering bias, stability, noise figure and output match. Appendix A lists the amplifier's circuit values.

One method of applying DOE is as follows. A topology and initial component values are chosen. A simulation is run to check the circuit's initial characteristics. Objectives are determined for the DOE simulations, and circuit parameters (factors) that will vary are chosen. A screening experiment is set up to determine the important factors. The screening experiment is run to remove insignificant factors. A fractional factorial or full factorial experiment is then run. The best set of factor values are chosen. This step may require several iterations. The variability in circuit responses due to parameter variations is evaluated.

DOE will primarily be used to improve the amplifier's gain and stability performance in the 1 to 2 GHz band. For simplicity in this design example, a topology and nominal parameter values have been assumed. Additionally, the circuit has been simulated to get a performance baseline for future comparisons. As part of this simulation, a Monte Carlo analysis is run to check for performance variability. The original amplifier's nominal minimum gain was 12.05 dB over a 1 to 2 GHz range and the nominal minimum stability factor was 0.75 from 0.1 to 4 GHz. For unconditional stability, this factor must be greater than 1.

Objectives and Factors

If the nominal circuit responses are acceptable, then at this point it is appropriate to consider variability and ways to reduce it. However, the stability factor is too low, so increasing its value must be the first objective of DOE analysis. It will be necessary to keep the gain as high as possible and to consider other circuit characteristics.

The circuit parameters (factors) to be varied must also be chosen. Because improving stability is the primary objective, circuit parameters that are expected to have an effect on stability should be chosen. These parameters include the Rstable and Lstable stabilizing resistor and inductor, the Rbase1 and Rbase2 bias resistors, and the RCol. Additional parameters such as CollMatchChoke output match inductor, BaseMatchL input match inductor and various capacitors might be included as DOE factors. While these parameters are not expected to have any effect on the stability, they are included in the initial analysis as a precaution.

Important Factors

A screening experiment is run to determine the important factors. DOE full factorial experiments, in which the circuit is simulated with all possible combinations of factor values, reveal the most information, but also take the most time. For example, if there are eight factors, then a full factorial experiment would require 2[8] simulations. For this reason, it is desirable to run a smaller screening experiment to eliminate factors that do not have an effect on the circuit characteristics of interest.

There are two types of screening experiments, a sensitivity analysis and a fractional factorial experiment. In a sensitivity analysis, all circuit parameters (factors) are held at their nominal values, except one, which is set to its high value for one simulation, and its low value for another. These pairs of simulations are repeated for each of the factors. This type of analysis will not reveal any parameter interactions, but it will show the main effect of each of the factors. In a fractional factorial experiment, only part (either one-half, one-quarter or less) of a full factorial experiment is run. This experiment may also be used to screen factors. Fractional factorial experiments reveal less information than full factorial ones, but they are much faster, and reveal enough information to eliminate unimportant factors.

Insignificant Factors

The results of the experiment are used to remove the insignificant factors. Several DOE presentations are generated automatically after the user specifies which response variables (circuit characteristics such as stability factor and gain) to examine.

Since the objective of the screening experiment is to remove insignificant factors from the experiment, attention should be focused on the Pareto chart. Figure 5 shows a Pareto chart that indicates which factors have the greatest effect on the min_gain and min_MU response variables. Varying factor B (resistor rbase2) has a strong effect on min_gain, but almost no effect on min_MU. Similarly, varying factors C, D, H or I has no effect on either min_gain or min_MU, which indicates that these factors could be removed.

Sensitivity analyses only reveal main effects. Factors C, D, H and I could have interactions with some of the other factors, and the high residual indicates that interactions are present. DOE analysis computes the coefficients of a polynomial that expresses the response variables in terms of each of the factors and interactions among factors. The residual measures this equation's accuracy. If the residual is high, then a more complex DOE analysis should be done, such as a fractional or full factorial, to choose a best set of factor values.

The Best Factor Values

In this next DOE run, the change in resistors Rbase1 and Rbase2 was reduced to [+ or -] 10 percent because with the change at 25 percent the device would be biased incorrectly for some of the DOE runs. With only five factors, the designer can run a full factorial experiment that requires 32 simulations. A fractional factorial experiment, requiring only 16 simulations, was chosen as a compromise between simulation time and obtained information.

The resulting Pareto chart shows that the MU stability factor is dominated by the Rbase1 and Rbase2 bias resistors and the Lstab stabilizing inductor, but that varying the Rstab stability resistor has virtually no effect on MU. The effect of the stability resistor is undoubtedly greater when the nominal values of the other parameters in the circuit are significantly different.

This Pareto chart also shows the main effect plots and the interaction plots that determine which direction to change the various circuit parameters. The main effect plots indicate that factors A and B (Rbase1 and Rbase2), and the AB interaction influence the amplifier's gain. Factors A, B and E (Lstab) influence the stability factor. Because A and B influence both the gain and the stability factor, a trade-off may need to be made. Figure 6 shows the Pareto chart of the second DOE run.

The Pareto chart indicates the factors and interactions that are important, but does not set the factor values. For this information, the main effect and interaction plots must be examined. Figure 7 shows the main effect plots for each of the factors.

The main effect plots indicate how the circuit parameter values may be adjusted to improve the circuit responses. The Rbase1 main effect plot shows the average value of the response variables, min_gain and min_MU when the Rbase1 (A) factor is set to its low value (9.0495 K [Omega]) and its high value (11.06 K [Omega]). The plot indicates that both min_gain and min_MU are higher, a good result, when Rbase1 is set to its low value. Studying the other main effect plots indicates that the Rbase2 (B) factor should be set to its high value, and the Lstab (E) and Lload (G) factors should be set to their low values. The main effect plots in this example are somewhat unusual in that no trade-offs have to be made, that is, gain does not have to be sacrificed to attain better stability or vice versa. However, the bias current increases, which is a trade-off.

After examining the main effect plots, it is necessary to examine the interaction plots. The Pareto chart indicates that only the AB interaction, shown in Figure 8, is significant. This interaction plot indicates that the Rbase2 (B) factor should be set to its high value and the Rbase1 (A) factor should be set to its low value to achieve the highest gain and stability factor.

Often, the interaction plots indicate that the factors should be set to conflict with what the main effect plots show. If this occurs, the designer should refer to the design matrix and responses table to choose the best factor values.

Appendix B lists the values of the factors during each DOE run, as well as the corresponding circuit responses. The listing columns for gain_delta (maximum gain variation), output_match and ICC have been added for a better choice of the factor values. Each row represents a single simulation, and shows the values of each factor, and the corresponding response variable values, min_gain and min MU. Presenting the results in this format enables trade-offs to be made quickly when choosing a best set of parameter values.

The best circuit performance may not be with all factors set to the high or low values used in the DOE experiment. It may be possible to achieve better performance by setting some of the factor values outside the range of the experiment or somewhere in between the high and low values. These possibilities can be investigated by carrying out additional DOE runs.

After several more iterations, a satisfactory set of parameter values was found, giving a minimum gain of 12.7 dB, minimum stability factor MU of 1.056, output match of -10.4 dB, and gain_delta gain variation of 5.2 dB. The last DOE simulation that was run indicated that the Lstab inductor and the Rstab resistor could be adjusted to improve the gain or stability, but not both simultaneously.

Evaluate Variability

The next step is another DOE simulation to see whether the circuit's performance is sufficiently intolerant to variations in parameters. The value of each parameter is assumed to have a Gaussian distribution, and the parameter's high and low values will be set to the nominal value [+ or -] 1 standard deviation. The results of the DOE simulation runs will vary, and from this variation, a mean value and standard deviation for each performance characteristic of the circuit may be computed. From the mean value and standard deviation, specifications can be set for each performance characteristic, or, if the specifications already exist, it can be determined whether or not the circuit will meet them. If a specification cannot be met, the DOE results can be analyzed, particularly the Pareto chart, to see the parameters that dominate the responses. The tolerances of these parameters may be tightened until the circuit meets its specifications, and conversely, the parameter tolerances that do not matter may be loosened.

Several transistor model parameters have been added as factors. Adding correlations among known parameters would increase the accuracy of this type of simulation. These transistor model scaling coefficients all have a nominal value of 1. For example, Q1.bfscale scales the forward beta of the device. Other transistor model parameters vary as well, and a screening experiment can be run to see if any important ones have been neglected.

The Pareto chart, shown in Figure 9, indicates that variations in transistor model parameters contribute the most to variations in the gain and stability factor. Factors J and K, which model variations in the transistor model's [C.sub.jc0] base-collector zero-bias depletion capacitance and the [T.sub.F] ideal forward transit time, contribute the most to variations in gain and stability factor. The computed mean value of the minimum gain was 12.7 dB, and the standard deviation was 0.152 dB. These values indicate that the minimum gain specification should not be set higher than 12.24 dB, which is the mean value of the response variable minus three times the standard deviation. In some cases, the more conservative standard of six times the standard deviation may be used when computing the specification limit. The mean value of the MU minimum stability factor was 1.055, and the standard deviation was 0.00332, which indicate that the amplifier should remain stable. The mean value of the output match was -10.3 dB, and the standard deviation was 0.498 dB, implying that the output match specification could be set near -8.8 dB.


                                Gain          Factor
                                (min)          (min)

Mean                          11.8 dB           0.68

Deviation                    0.674 dB           0.27

Resulting                                    potentially
Specification                  9.8 dB        unstable


                                Gain           Factor
                                (min)           (min)

Mean                          12.7 dB            1.06

Deviation                    0.157 dB        3.4 x [10.sup.-3]

Specification                 12.2 dB          stable

A comparison of the 250 Monte Carlo simulation results of both the original circuit and the circuit after applying DOE analysis shows that even though the mean value of the minimum gain has only improved by 0.9 dB, the standard deviation has been reduced from 0.674 dB to 0.15 dB. Clearly, the DOE-optimized amplifier's gain specification may be set much higher (12.26 dB vs. 9.8 dB, using the mean minus three standard deviations as the specification). The difference is even larger if the more conservative mean minus six standard deviations is used to set the specification. The results of the DOE analysis agree closely with the Monte Carlo simulations. This correlation will not be true for all circuits, as an estimate of performance variability from the Monte Carlo analysis is more accurate. The amplifier's characteristics before and after applying high yield design are listed in Tables 2 and 3, respectively.


Simulation                Factor A         Factor B       Responses

Tolerance-1a             low-sdevA        low-SdevB           G1a
Tolerance-1b             low+sdevA        low-sdevB           G1b
Tolerance-1c             low-sdevA        low+sdevB           G1c
Tolerance-1d             low+sdevA        low+sdevB           G1d

Improving Performance and Minimizing Variability Simultaneously

The amount of variation in a circuit's characteristics often depends on both the tolerances and the nominal values of the components. Nested DOE analysis, the Taguchi method, allows the nominal values of many parameters to be varied and a solution to be found that gives not only circuit characteristics with good mean values but also low variation in the characteristics.

In a nested DOE analysis, design parameters are chosen that have nominal, selectable values. Process and component tolerances are also specified and typically the high and low values are selected to be one standard deviation above and below the nominal values. A main DOE simulation is run where the design parameters are varied. For each set of factor values in the main DOE simulation, a tolerance DOE simulation will be run to examine variation of the circuit's performance characteristics. One of the four tolerance simulations is listed in Table 4. From this nested DOE analysis, it is possible to evaluate both the mean values of the circuit performance characteristics and the variability of the performance characteristics simultaneously.

The tolerance DOE simulation is used to compute the mean value and the standard deviation of each circuit characteristic for each set of main DOE simulation factor values. In the Taguchi analysis design matrices, for each set of factor values in the main design matrix, another DOE is run to see the variability in the circuit responses. In this case the variability is quantified as the standard deviation of the four responses, G1a - G1d. The sdevA and sdevB are standard deviations of factors A and B due to manufacturing variations.


This article describes the benefit of the design of experiments analysis technique, explains how DOE works and shows an application of DOE to a design. In the amplifier design example, DOE reveals the parameters that should be varied to improve the circuit.

DOE is a powerful analysis tool that efficiently shows a designer whether a circuit will perform as required, including the variability seen in manufacturing. If improvement TABULAR DATA OMITTED is desired, DOE reveals the changes that should be made, as well as any trade-offs that might be necessary.


Design of experiments is available in the HP 85149A/AN High Yield Software Package. This software must be used with HP MDS/RFDS release 6.0 or later. DOE is also available in HP EEsof Series IV, Version 5.0.


1. Robert L. Mason, Richard F. Gunst and James L. Hess, Statistical Design and Analysis of Experiments with Applications to Engineering and Science, New York, NY, John Wiley & Sons, 1989.

2. Douglas C. Montgomery, Design and Analysis of Experiments, 2nd ed., New York, NY, John Wiley & Sons, 1984.

3. Thomas B. Barker, Quality by Experimental Design, New York, NY, Marcel Decker, 1985.

4. Thomas B. Barker, Engineering Quality by Design, New York, NY, Marcel Decker, 1985.

5. Stephen R. Schmidt and Robert G. Launsby, Understanding Industrial Designed Experiments, 3rd ed., Colorado Springs, CO, Air Academy Press, 1992.

6. Keki R. Bhote, World Class Quality (Understanding Design of Experiments to Make it Happen), New York, NY, AMACOM.

7. Madhav S. Phadke, Quality Engineering Using Robust Design, Englewood Cliffs, New Jersey, Prentice Hall, 1989.

8. HP Product Note 85150-5, "Using the High Yield Software Package to Create Robust Designs," literature number 5962-9271E.

Appendix A: Circuit Values for the 1 GHz Low Noise Amplifier

RBase1 = 10.055 K RBase2 = 9.6 RCol = 200 Linput = 22 nH Lstab = 22 nH Rstab = 33 Lload = 15 nH Cblockin = 12 pF Cblockout = 3.3 pF Cstab = 1 nF Q1bfscale = 1 Q1cjescale = 1 Q1cjscale = 1 Q1tfscale = 1 Q1lisscale = 1
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Title Annotation:design of experiments
Author:Howard, Andy
Publication:Microwave Journal
Date:Jul 1, 1994
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