Using Design of Experiments (DOE)

Design of experiments (commonly referred to as DOE) is a data-driven technique for robust design. In the early 1900's, DOE was used by agricultural engineers to improve crop yields. Today circuit and system designers are applying the method as a means to the same end-yield improvement.

A typical DOE includes three primary steps:

  1. Plan the experiment:
    • Assess the experimental resource budget.
    • Identify the input and response variables.
    • Assign levels (values) to input variables.
  2. Perform the experiment and collect response data.
  3. Analyze the data using statistical methods.

Sequential application of this methodology can be used to improve the statistical performance of a given circuit or system. Because of an inherent compromise between statistical performance prediction accuracy and the number of input variables, a screening experiment is used to identify variables that contribute significantly to performance variation. Next a refining experiment can be used to hone in on the target statistical response.

DOE and Computer Simulation

In a general application, DOE methods are designed to accommodate errors of the type found in any experiment. But because circuit and system simulators provide identical results for any analysis having the same input values, complexity in setting up, performing, and analyzing experiments is reduced.

Since the computer is being used to perform the experiment, a more complete characterization of input/output relationships can be realized. Finally, since the computer handles the tedious tasks of bookkeeping during the experiment, there is a further reduction in the possibility of human error.

The primary purpose of DOE is to characterize an unknown process. In circuit or system simulation, the unknown process is predicting the response of the design under test (DUT). A simple technique for characterizing a DUT is to perturb each input variable ( factor ) in turn, and to record the resulting output response. However this approach breaks down if the response due to a change in one factor depends on the value of a different factor.

Minimum DOE Requirements

Prior to performing a Design of Experiments, you need:

The design components needed for DOE are located in the Optim/Stat/Yield/DOE library or palette.

Specifying Component Parameters for DOE

The procedure for specifying components for DOE is as follows:

  1. Select and place an appropriate component from one of the component palettes or component libraries. For example, place a parallel resistor-inductor-capacitor (PRLC) from the Lumped Components palette.
  2. Double-click on the component in the design window to access its associated dialog box.
  3. From the dialog box, highlight the parameter that you want to vary in the Select Parameters box (for example R for parallel resistance), then choose the Tune/Opt/Stat/DOE Setup button, which will only appear for valid DOE parameters. The Setup dialog box appears, with the Optimization tab active. Click the DOE tab.
  4. From the DOE Status drop-down list, select Enabled so that you can set specification of the appropriate fields. Enabled causes the parameter to be varied when the simulation is run. Disabled allows you to temporarily suspend any parameter variation previously assigned, and Clear removes the values you previously applied to the design.
  5. In the Type drop-down list, accept the default DOE Value Type of DOE Discrete .
  6. From the Format drop-down list, select an appropriate statistical value format:

    min/max
    +/-Delta %
    +/-Delta

    For complete descriptions of the available format, refer to the section Value Types for DOE.

  7. If you selected +/-Delta or +/-Delta % formats, specify the deviation value. For these formats, the units can also be specified in the drop-down list next to each input field.
  8. If you selected a min/max format, you can optionally enter values for nominal, minimum, and maximum in the appropriate boxes, and select an appropriate unit assignment for each from the drop-down list next to the boxes.
    Note
    Unit specification via the Setup dialog box is not possible for variables defined in the Var/Eqn component.
  9. From the Nominal Value field and the Units drop-down list, the value and units in your design for this component are displayed. You can change these if you wish.
  10. Choose OK .

Placing an Appropriate Simulation Control Component for DOE

An appropriate simulation control component must be placed in the design prior to initiating a DOE analysis.

For Analog/RF Systems simulation, all analysis types are supported, for example place one of the following components:

For details on specifying parameters for each of these control components, Using Circuit Simulators.

Setting DOE Goals

DOE goals are specified by placing a DOE Goal component and double-clicking it to display the Goals for DOE dialog box. The Goal component can be found as follows:

You can specify and place more than one Goal if needed. The goals to be used are referenced by the DOE component, as described in the later section, Setting Job Parameters for DOE. By default, all goals placed apply to all DOE components in a design.


To set appropriate goal specifications in this dialog box:

  1. If desired, enter a name in the Instance Name field that is different from the assigned default name shown.
  2. In the Select Parameter list box on the left, click on each parameter that you want to modify, then make other associated changes in the box on the right. When you select a parameter, such as Expr, all relevant items in your design will be displayed in the box. The style of this box varies depending on the parameter, as described in the table below.
Parameter Goals for Nominal Optimization
Parameter Description Use Model
Expr A valid AEL expression that operates on the simulation results, such as mag(S11), or the name of a MeasEqn. For more information on AEL expressions, refer to the AEL manual or the Measurement Expressions manual. The list box label becomes Measurement Equations. All associated expressions are displayed in the box. Select the one you want to analyze and it will appear just below in the Selection box. For expressions not related to MeasEqns, you must type them in the Selection box.
SimInstanceName Enter the instance name for the simulation control component that you placed in your design, which will generate the data used by the Expr field. The list box label becomes Analysis Components. Select the analysis component (simulation controller), such as S-parameter, that you want to analyze and it will appear just below in the Selection box.
Min Enter a number for a minimum acceptable response value. Fields for Parameter Entry Mode and Equation editor are used as in any component parameter dialog box. Type a value in the box. Note: Both Min and Max do not have to be specified, but at least one does.
Max Enter a number for a maximum acceptable response value. Same as above.
Weight Enter a weighting valued to be used in error function calculation. Default is 1. For more information on using the weighting factor to form the error function, refer to Weighting Factors. Fields for Parameter Entry Mode and Equation editor are used as in any component parameter dialog box. Type a value in the box.
RangeVar Independent variable name. Same as above, but note that this parameter is "indexable" and can be applied to more than one independent variable.
RangeMin Minimum limit of range for independent variable during optimization. Same as above.
RangeMax Maximum limit of range for independent variable during optimization. Same as above.

Setting Job Parameters for DOE

To set job parameters, you need to specify appropriate data in the DOE Simulation dialog box.
This three-tabbed dialog appears when you place a DOE Simulation component (labeled DOE ). Do the following:

  1. Place the DOE component in the appropriate Schematic window.
  2. Double-click the component to being up the dialog box. The Setup tab is active.
  3. Make specifications in each tab (Setup, Parameters, and Display) of the dialog box, as described in the next sections.

Selecting a DOE Specification

First select the Setup tab of the DOE dialog box to set up a DOE analysis.

  1. In the Experiment Selection box, select the desired Experiment Type from the drop-down list. Refer to DOE Concepts for more information on DOE theory and experiment types. The available types are as follows:
    Available DOE Experiment Types
    Experiment Type Description
    2kmp 2 raised to the power of k minus p, where k is the number of factors and p is the fractionalization element. When p = 0, a full factorial experiment is identified.
    Plackett-Burman Allows the study of k=N-1 variables in N runs, where N is a multiple of 4.
    CCD Combines a 2-level experiment with the center point and star points along the coordinate axis. Star points lie outside the 2-level experiment and their distance from the center point is a function of the number of factors, i.e., d=2 (k/4)
    Box-Behnken Consists of the zero point (nominal) and a 2-level, 2-factor factorial design for all combinations of factors, while holding other factors at their nominal value.
    3k A 3-level full factorial experiment.
  2. If you select the 2kmp method, enter a Fractionalization element in the Fractionalization Element field.
  3. In the DOE Goal box, accept the default Use All DOE Goals in Design checkbox. This is the best approach for most designs, and all DOE components placed in a design will be implicitly associated with the DOE Goal component.
    To associate a subset of all DOE Goals with a given DOE analysis controller, deselect the Use All DOE Goals in Design checkbox. Select a DOE spec from the Edit drop-down list, which will include all DOE components that are currently placed in the design. This step is similar to the same procedure for Yield, as described in the section, Setting Up a Yield Specification. Choose Add to place in the DOE Goal box, and repeat this procedure if necessary. Choose the Cut or Paste buttons, if necessary to make any changes in the DOE Goal box.
  4. Choose Apply to retain the specifications that you have made while you enter data into the Parameters tab, as described in the next section.

Setting Parameter Information

You set parameter information in the Parameters tab of the DOE Simulation dialog box, such as what data to save and when the data is output.

During DOE analysis, a complete set of DOE outputs (Pareto, Effects, and Interactions diagrams) are implicitly generated for each DOE Goal component. In addition to the implicitly generated outputs, an ASCII file of the experiment results is created for each DOE analysis component. This file is stored in the / data subdirectory. You can use this file to input your DOE results into third-party spreadsheet or statistical analysis programs.

To do this, follow these steps:

  1. In the Output Data field, specify which data you want to retain in your dataset following DOE analysis.
    • Analysis outputs sends all measurements (including measurement equations) to the dataset for each trial. This can create a substantial amount of data.
    • DOE Goals sends the result of each Goal's Expr field to the dataset for each trial.
    • DOE Experiment variables sends the values of all DOE experiment variables to the dataset for each trial.
  2. In the Output Data Control field, specify whether you want to:
    • Save data for all treatment combinations. Data for all treatment combinations is saved. This can create a substantial amount of data.
      Note
      For DOE experiments, enabling this feature can slow the analysis time considerably when the experiment is large. The default is off where only the first and last treatment combinations are saved to the dataset.
    • Update display after each treatment combination updates the dataset on each DOE treatment combination so you can see the results in the Data Display window as they occur instead of waiting to the end where all the traces are displayed at once.
  3. In the Levels box, enter a number for the desired annotation level in the Status level field. Levels are 0-4, with increasing information displayed in the Status window. (2 is the default.)
  4. Choose Apply to retain the specifications that you have made while you enter data into the Display tab, as described in the next section.

Displaying Analysis Data on the Schematic

Selecting the DOE parameters that will be displayed on your schematic is done the same way as in nominal optimization. Refer to Displaying Analysis Data on the Schematic for details. Below is a DOE example.


When you have finished setting up all the tabs in the DOE Simulation dialog box, click OK .

Initiating Design of Experiments

To initiate a DOE analysis:

  1. Choose Simulate or click the Simulate button on the toolbar. The analysis status, including information about the current treatment combination number as well as result computation progress, is displayed in the Status window. Upon completion of the analysis, the simulator ceases analysis and indicates success.
    Note
    If the DOE analysis process becomes exceedingly long, you can use the Stop and Release Simulator command on the Simulate menu to interrupt the process.
  2. When the simulation is complete, you are ready to view the DOE output, which is available for each specified DOE goal.

Swept DOE

DOE can be swept as any other ADS analysis. When the DOE controller is referenced by a parameter sweep controller, the DOE is performed for each value of the sweep variable and the results output as a function of the sweep variable. For more information, refer to Swept Optimization.

DOE Terminology

Following are definitions of the most frequently used design of experiments (DOE) terms :

DOE Concepts

The following figure shows two factors, A and B, and the associated response at various values ( levels ) of the factors.


Comparison of Responses of Factors A and B

Notice that the factors have two levels: one low (−1) and one high (+1). The ±1 notation indicates the factor values are in design units , and are obtained from physical values using the following equation:

where X is the minimum (maximum) physical value of the variable, and X lo , X hi , and X mid are the minimum, middle, and maximum physical values. For example, a capacitor value might be 100pF ± 10%, leading to low, mid, and high values of 90, 100, and 110pF respectively.
If we were to note the change in response due to a change in factor A (from low to high), we would be led to believe that increasing A causes an increase in the response-the same would be observed for factor B. A model from the three response points r1, r2, and r3 can be formulated as the plane surface which contains them:
y = [(r2-r1)/2]A + [(r3-r1)/2]B + BIAS
where the BIAS term is found by equating the response at a given factor level condition, for example, if A and B are -1, then

This leads to:

However, if either factor A or B is held high, and the same experiment is performed, an inverse relationship exists between factor level and response. The plane surface model from the one-factor-at-a-time experiment would significantly overestimate response r4.

Whenever the factor-response relationship changes as a function of a different factor, there is said to be an interaction between the two factors.

To account for interactions, a modification to the simple one-factor-at-a-time experiment scheme is necessary. The factorial experiment is generally accepted as one of the most efficient methods for characterizing the effects of two or more factors.

In the factorial experiment, response results are collected for all combinations of factor levels. For 2-level factorial designs, 2 k data points must be collected for each response, where k is the number of factors. The factorial experiment not only accounts for interactions, but also is formulated using average response values as opposed to the raw ones used in the one-factor-at-a-time method. The following figure shows four response points r1-r4, as well as the orientation for the plane surface used to model the factor-response relationship.


(a) Response Points and Plane Surface Orientation, and (b) Modified Surface

The orientation is found by evenly allocating any estimation error due to factor interactions-this error is labeled as d in the figure.

The four points on the plane surface m1-m4 represent the average response for the corresponding edge (i.e., m1 = [r1 + r2]/2). Without the interaction term, the two-level factorial model equation is of the form:
y + S A A + S B B + BIAS
where S A and S B represent the slope of the average response for the given variable, i.e., S A = [m4-m2]/2. The BIAS term is simply the grand average of all raw response values so that

It turns out that the two lines having endpoints m4, m2, and m3, m1 respectively, intersect in the middle of the plane surface. You can prove this by equating responses in the center of the plane surface, as follows:
m1 + S A (1) = m2 + S B (1).
To account for the interaction, an additional term must be added to the prediction equation:

To find S AB . evaluate the response at one of the response points, for example, r1, and solve for S AB :

so that

The solution is:
S AB = [r1 + r4 -r2 -r3]/4
The response depicted in part b of the previous figure shows the modified surface.

Notice that along either factor coordinate axis, the response is linear. However, the slope of the linear model changes as a function of the other factor. For example, with B = −1 the response as a function of A, y(A) indicates a positive slope. But as B increases, the slope of y(A) decreases. Also note that the off-axis response contour is quadratic. The following figure shows the wire mesh plot for the new surface.

Wire Mesh Plot of the Modified Surface of the previous figure

The new prediction equation can be restated in terms related more closely to DOE:

where ME A and MEB are, in DOE parlance, the main effect of factor A and B respectively. I AB is referred to as the interaction between A and B. The main effect for a 2-level experiment is defined as the difference in average response at the two levels of a factor. Referring to the figure above (a) Response Points and Plane Surface Orientation, and (b) Modified Surface, ME A is simply m4−m2. The interaction term for the same experiment is defined as half the difference between the main effects of one factor at the two levels of a second factor.

Again using the figure (a) Response Points and Plane Surface Orientation, and (b) Modified Surface, I AB can be taken as half the difference in the main effect of factor A when B is high- ME A (B+) and the main effect of factor A when B is low-ME A (B−), i.e., I AB = [(r4−r3) − (r2−r1)]/2. (For the 2-factor case, ME A (B+) and ME A (B−) are simply differences in response values. Usually, these terms will be differences of averages.) When there are more than two factors, it is possible to define higher order interaction terms, such as I AB C-half the difference between the two factor interaction effects at the two levels of a third factor, and so on.

There are some additional DOE terms and concepts that are easy to discuss in the context of the 2-level factorial design. First, recall that factor coefficients in the prediction equation were all divided by 1/2. Appropriately there is a DOE term called the half-effect , which is simply one half of the main effect. The prediction equation then becomes:
y = HEAA + HEBB + HEABAB.

Design Matrix

There are many types of experiments that can be applied to any given situation. Differences in these designs are readily seen by examining the design matrix . The following table shows a design matrix for the 2-factor factorial experiment.

Design matrix for 2-factor factorial experiment
tc Factors Interactions Response
A B AB
1 - - + r1
a + - - r2
b - + - r3
ab + + + r4

Notice that columns are delineated into four main groups-tc or treatment combination , factors, interactions, and response. Under the tc column, a shorthand is used to indicate the unique conditions of each experiment run (trial). Lower case letters are used to indicate the factor(s) having +1 levels for the trial. (A 1 is used to indicate the run where all factors are held low.) Factor levels are designated using another shorthand where unity is implied in the symbols + and −. The levels of the interaction columns are found by taking the product of the factors involved. The response column is simply a log of the computed response.

Notice that each pair of factor/interaction columns is orthogonal, e.g., linearly independent. This construct allows independent analysis of each factor as well as interactions between factors. In short, for an orthogonal design , the total variation in the response can be divided into components due to each factor and interaction, thereby making it possible to rank the importance of factors. (For additional details, refer to the section DOE References).

As a final observation concerning this 2-level, 2-factor design, consider the case where little or no interaction exists between factors A and B. In this situation, the interaction term would be superfluous. But suppose there is a third factor, C, that would be desirable to study. By letting C = AB, a three factor experiment could be conducted in half as many treatment combinations as a factorial experiment. This experimental scheme is referred to as a fractional factorial experiment. (Often the factorial experiment is referred to as a full factorial .) Obviously significant savings can be afforded by the fractional factorial, but the downside of this approach is that a priori knowledge of the strength of interactions is usually unavailable.

Since the change in the response due to factor C is aliased (or confounded ) with interaction AB, fractional-factorial designs are usually used as screening experiments to identify a small subset of variables that contribute significantly to performance variation. One such screening experiment, the Plackett-Burman design, allows the study of k=N-1 variables in N runs, where N is a multiple of 4. (Any design where k=N-1 is referred to as saturated .)

A convenient way to designate all 2-level designs is with the nomenclature 2kmp- which stands for "2 raised to the power of k minus p," where k is the number of factors and p is the so-called fractionalization component . With p equal to zero, a full factorial experiment is identified. For the example above, p equals one, which denotes a (1/2) P or 1/2 fractional factorial, indicating 1/2 the number of tc's of the full factorial are required.

Multilevel Designs

Multilevel designs are characterized as those having more than two levels. These designs are useful in detecting and modeling curvature in the response as a function of the factors. After a screening experiment is performed, and the vital few factors are identified, multilevel designs such as those in the following figure can be used to more accurately predict the response.

Common Multilevel Designs in Three Factors

The Central Composite Design (CCD) combines a 2-level experiment with the center point and so-called star points along the coordinate axis. The star points lie outside of the 2-level experiment and their distance from the center point is a function of the number of factors, i.e., d = 2 (k/4) . The Box-Behnken design consists of the zero point and a 2-level, 2-factor factorial design for all combinations of factors, while holding other factors at their middle value.

DOE Outputs

Prediction equations are not the only output from DOE. In fact, prediction equation(s) are usually obtained by examining Effects plots . Pareto diagrams are used to rank factors in order of their contribution to the total variance in the response, and interaction diagrams allow detection of interactions between factors.

Effects plots are produced by simply computing the overall average response with the factor at each of its levels. For example, in the response shown in part a of the following figure, the average response for all tc's having A low is 3, and 5 when A is high. The coefficient of the prediction equation (2-level case) comes from the half effect (slope) of the main effect.

a) Effects Plot of Factor A vs. Gain, and b) Interaction Diagram for Factors A and B vs. Gain.

Interaction diagrams are iso-plots of two factors versus the response (see response shown in part b of the previous figure). Once again, average responses are computed over all tc's but this time with one additional constraint due to the companion factor. For example, the average response over all tc's having both A and B at low levels is 4, while it is 6 for the case when both A and B are high. Notice that the slopes of the lines change as a function of B. This indicates an interaction between the factors.

Pareto diagrams are bar charts that show the percentage of the total response variance attributable to each factor and interaction. See the following figure.

Pareto diagram of factors with respect to gain.

Response Values

In a typical industrial application of DOE, it is usually no problem to identify the response variables. In an injection molding application, the response variables might be the number of cracks produced in the casting, or the hardness of the product.

There are relatively few response variables. However, in computer-aided circuit or system design, where a continuous response over frequency, power, or other swept variable is approximated by discrete samples, the number of DOE response variables can number in the hundreds. The amount of data could be overwhelming to the designer.

Currently there are two schools of thought on accommodating response complexity. In the first approach, key points in a frequency/power comb are considered as individual DOE response variables to be analyzed in parallel . For example, low, mid, and high band edge samples of gain and noise figure would require six responses to be considered simultaneously.

An alternate approach involves the so-called Taguchi [4] loss-function. The overall DOE response is computed by combining each measurement's loss-function. The loss-function formulation depends on the relational operator used in defining each specification statement.

The ">" and "<" operations are interpreted as bigger is better and smaller is better , respectively. The equality constraint suggests that the response average be put "on target" with as little variance about the target as possible. The following loss-function formulations are used in the implementation of DOE:

Smaller is better (<): -10 Log 10 (Sum(x 2) / n)

Larger is better (>): -10 Log 10 (Sum(1/x 2) / n)

Target is best (=): 10 Log 10 (M 2/s 2), where s 2 = Sum(x-M) 2 / n-1

DOE Basic Example

The following example provides a basic introduction to the use of the software's DOE feature.

Set up the Schematic window as shown in the following figure, or copy it from the ADS Examples: $HPEESOF_DIR/examples/Tutorial/doe2_prj.

Note
Since this example has already been set up, the steps shown in this section can be followed to learn the general approach to using the DOE feature.

Schematic Used as Starting Point for DOE Example

Setup the DOE Goal Components

  1. Each DOE Goal will use a pre-specified measurement. In this case, VSWR1, VSWR2, and dB_s21. To setup each DOE Goal Component, double click the component. The Goal for Design of Experiment dialog box appears. The Goal for dB_s21 is shown below.
  2. In our example, the DOE goal for the S21 measurement is already setup. The steps needed to setup a DOE goal were described in Setting DOE Goals, and are similar to setting up optimization goals. In this example, note the following fields:
    • The desired measurement is Expr="db_S21" .
    • The Min and Max fields control the target DOE value. The Min field is set to -10 and the Max field is left blank, which means that the target value is ≥ -10.
    • Leave the Weight field at its default setting ( 1 ).
    • Leave the other fields blank in this example.
    • Click OK when done.
  3. The procedure described in step 2, above, is repeated for the other two DOE goals for the VSWR measurements, by setting the goals as follows:
    • VSWR1 max = 1.075
    • VSWR2 max = 1.075
Note
During DOE analysis, a complete set of DOE outputs (Pareto, Effects, and Interaction diagrams) are generated for each DOE goal component. For the case when there are several measurements (perhaps over frequency and/or power), the DOE response is computed by combining each measurement's loss-function. The loss-function, as defined by Taguchi [4] depends on the Min and/or Max fields used in defining each DOE goal. (Refer to the subsection "Response values" under DOE Outputs).

If there is only a single measurement and only one sweep point, the DOE response is computed from the actual measurement minus the number in the Value field of the DOE specification. If the actual response value is desired, leave the Value field fixed and the DOE specification blank (or zero).

The variable values for resistors is set using the VAR component as follows:

  1. Double-click the Var/Eqn component to bring up the Variables and equations dialog box, shown below.
  2. In the Name field, enter C .
  3. Click the Optimization, Statistics/DOE button.
  4. In the dialog box that appears, choose the DOE tab.
  5. In the DOE status field, select Enabled and Format to min/max .
  6. Set the Nominal Value fields as follows and as shown in the figure below.
    • Nominal Value: 30
    • Minimum Value: 20
    • Maximum Value: 40
  7. Choose OK . The following appears in the Select Parameter list box:
    C= 30 doe {20 to 40}
  8. Repeat step 7, changing C to B , then choose Add .
  9. Repeat step 7, changing B to A , then choose Add .
  10. Choose OK to dismiss the dialog box.

Setup the DOE Component

Next, we will setup the DOE Component.

  1. Double click the DOE Component. The DOE Simulation dialog box appears with the Setup tab active, as shown below.

    In DOE, the type of experiment to run is both problem-dependent and subjective. Because there are only three factors in the current example, the 2kmp is initially used (refer to the section DOE Concepts).
  2. Select 2kmp in the Experiment Type drop-down list and leave the Fractionalization Element at its default setting, 0 .
  3. Click Apply .
  4. Select the Parameters tab.
  5. The Parameters tab controls output data. Accept the defaults and click Apply .
  6. Select the Display tab.
  7. The Display tab controls which parameters are displayed on the schematic. Accept the defaults and click OK.

Start the Experiment

  1. In the Schematic window, select Simulate > Simulate .

You see messages in the Status window showing the current treatment combination number. For the 2kmp with k = 3 and p = 0, there are 8 treatment combinations necessary for the experiment. Once the number of simulations is complete, a message appears, reporting the progress of DOE data computation and display.

Analyze the Experiment

Once the treatment combinations have been simulated and the DOE data computation and display task is complete, you can access the three main DOE reporting tools (refer to the section DOE Outputs for explanations of these plots):

Pareto Diagrams

To examine the Pareto diagrams:

  1. Select Window > New Data Display .
  2. Select a Rectangular Plot and place it in the middle of the window. The Plot Traces & Attributes dialog box appears.
  3. From the Datasets and Equations drop-down list, select s21_db.pareto .
  4. Choose the Add button.
  5. Choose OK .
  6. A diagram showing s21_db versus design variables is shown.
    Note that the Datasets and Equations drop-down list contains a series of plots and diagrams for each goal. Using the Plot Traces & Attributes dialog box, you can choose the diagram you are interested in, and add or delete traces in the plots.
    Let's select a Pareto diagram for a different DOE goal:
  7. Double-click the Rectangular Plot and select vswr1.pareto from the Datasets and Equations drop-down list that appears. The plot is shown in the following figure.

    Pareto Diagram for VSWR1 Measurement

Effects Plots

Next let's examine the Effects plots for the S21 DOE goal:

  1. From the opened Data Display window, Choose File > New .
  2. Place a Rectangular Plot in the window. The Plot Traces & Attributes dialog box appears.
  3. From the Datasets and Equations drop-down list, select s21_db.A.effects and click the Add button.
  4. Next, select s21_db.B.effects and click Add .
  5. Lastly, select s21_db.C.effects and click Add .
  6. Click OK . The Effect plot appears as shown in the following figure.

    Effects Plot for DOE Goal S21

Notice that factor C has the largest slope (Effect) of the three factors. Notice also that factors A and B have identical effects. Finally, note that the response has been offset by the specified goal appearing in the DOE goal components. For example, the goal for S21 is −10 dB. Because the goal is subtracted from each response, the target response on the Effects plot is zero.

Interaction Diagrams

Interaction diagrams are used to examine the effect of one factor on a different factor.

To obtain the Interaction diagram for factors AB on the S21 DOE goal:

  1. Select Window > New Data Display .
  2. Select a Rectangular Plot and place it in the middle of the window. The Plot Traces & Attributes dialog box appears.
  3. From the Datasets and Equations drop-down list, select s21_db.AB.interaction_B .
  4. Choose the Add button.
  5. Choose OK . The AB interaction diagram for DOE goal S21 appears, as shown in the following figure.

    AB Interaction Diagram for S21 Measurement

Notice that the traces are parallel, indicating that there is little or no interaction between factors A and B – the change in average response as a function of factor A does not change as a function of factor B. However, there is an offset indicating that to achieve the (adjusted) target goal of zero, both A and B should be set to the low levels.

Optimizing Using DOE Outputs

In the DOE implementation, there are two methods to accomplish design improvement. The first and easiest to apply involves examining Effects plots and making approximate changes to the design factors in an effort to put the response on target. The second method involves solution of the set of model equations. In this section, the first method is examined, with the same example used previously in this topic.

By examining the Effects plot for the DOE goal S21, it is clear that an increase in C and a decrease in the value of A and B will work toward putting the nominal response on the (adjusted) target value of zero. However, while this observation is true for the S21 goal, it may not be accurate for the VSWR goals.

Let's continue following along with our example (../examples/Tutorial/doe2_prj). To view the optimized DOE output, select the design doe2b.dsn (from the Schematic window, choose File and select doe2b from the file history list at the bottom of the menu).

To examine the Effects plot for Vswr1:

  1. Select Window > New Data Display .
  2. Select a Rectangular Plot and place it in the middle of the window. The Plot Traces & Attributes dialog box appears.
  3. From the Datasets and Equations drop-down list, select vswr1.A.effects .
  4. Choose the Add button.
  5. Using the scroll bar of the list, locate, select and Add the first three components associated with Effects of Vswr1:
    vswr1.B.effects
    vswr1.C.effects
  6. Click OK to review the Vswr1 Effects plot.

If you do not have a display for the S21 and Vswr2 effects, follow the above instructions to obtain any missing plot. The next three figures show the Effects plots for Vswr1, Vswr2, and S21, respectively.

Once Effects plots for S21, Vswr1, and Vswr2 are available, arrange them so that they can be viewed simultaneously. Notice that the traces of the Vswr1 and Vswr2 plots are the same. The difference is that factor A in one plot is replaced by factor B in the other. This makes sense due to the symmetry in the network – the series resistors (factors A and B) take on the same values.


Effects Plot for Vswr1


Effects Plot for Vswr2


Effects Plot for S21

How Goals Are Affected

The motivation behind viewing Vswr1, Vswr2, and S21 effects concurrently is to ensure that any factor modifications are commensurate with the overall performance goals. As mentioned previously, it appears that an increase in C and a decrease in the value of A and B will work toward satisfying the S21 goal. We must also consider how this affects the Vswr goals. Noting the significant positive slope of the AC (Vswr1) and BC (Vswr2) interaction effects, it appears that an increase in C and a decrease in A and B would be favorable to our overall goals.

The only question that remains now is how much to change the factor level nominal values. Let's try an increase in C by 1/2 unit and a decrease in A and B by the same amount.

Noting that a 1/2 unit change in design units equates to a 5-ohm change in resistance values, the nominal values for series resistors drop from 30 to 25 ohms, and the shunt resistor should be changed from 30 to 35 ohms.

To modify the nominal, minimum, and maximum values for the factors:

  1. Double-click the Var/Eqn component to bring up the Variables and equations dialog box, shown below.
  2. In the Name field, enter C .
  3. Click the Optimization, Statistics/DOE button.
  4. In the dialog box that appears, choose the DOE tab.
  5. In the DOE status field, select Enabled and Format to min/max .
  6. Set the Nominal Value fields as follows and as shown in the figure below.
    • Nominal Value: 35
    • Minimum Value: 25
    • Maximum Value: 45
  7. Choose OK . The following appears in the Select Parameter list box:
    C= 35 doe {25 to 45}
  8. Repeat step 7, changing C to B , then choose Add .
  9. Repeat step 7, changing B to A , then choose Add .
  10. Click OK . The changes in the Var/Eqn component are reflected in the schematic. The new schematic should look similar to the one shown in the following figure.

    Schematic with Modified Var/Eqn Component

Performing the DOE Confirmation Experiment

To start the experiment using the new factor nominal values:

  1. In the Schematic window, select Simulate > Simulate .
    You see messages in the Status window showing the current treatment combination number. For the 2kmp with k = 3 and p = 0, there are 8 treatment combinations necessary for the experiment. Once the number of simulations is complete, a message appears, reporting the progress of DOE data computation and display.

Analyzing the DOE Confirmation Experiment

Once the treatment combinations have been simulated and the DOE data computation and display task is complete, you can re-examine the three Effects plots created earlier for Vswr and S21 DOE goals.

To examine the Effects plot for Vswr1:

  1. Repeat the steps described in the section Effects Plots, except now we are using the results from the doe2b dataset.
  2. Repeat steps 1-5 to obtain similar plots for the first three effects of S21 and Vswr2. The Effects plots for Vswr1, Vswr2, and S21 are shown in each of the following three figures respectively.


Effects Plot for Vswr1


Effects Plot for Vswr2


Main Effects Plot for S21

Once Effects plots for S21, Vswr1, and Vswr2 are available, arrange them so that they can be viewed simultaneously. Notice that our S21 target is very nearly satisfied while the VSWR goals are off target by about 0.17. The movement in the average response for VSWR is small-from about 0.16 to 0.17. If this performance were deemed unacceptable, another iteration of the procedure could be applied.

DOE References

  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 Dekker, 1985.
  4. Thomas B. Barker, Engineering Quality by Design - Interpreting the Taguchi Approach , New York, NY; Marcel Dekker, 1990.
  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.
 

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