• Chart Type Overview
  
• Ternary Chart (Triplot)
  
• 3D-Charts
  
• Set Axis Interval
  

SimplexNumerica supports a number of basic chart and axis types. The properties for drawing the axes, labels, and data for all these graphs are contained in different dialog boxes.

To select a graph type, call the appropriate one from the Pulldown-menu <Edit> or from the icons on the Toolswindow.

Each graph type may support a limited subset of the available axis types. The default axis type is the Physical Chart. This is also called a Cartesian coordinate system.

Some graph types require data supplied in a specific format. Furthermore, the graph type determines what effect the various style objects contained in the display component, the data lists, and the individual data items have on the graphic elements.

The following topics describe the supported graph types and present some implementation details. It is convenient to organize the graph types according to the axis types supported. The thumbnails images show how data are represented in the basic chart types. Of course, these small bitmaps do not accurately represent the visual quality of normal-sized graphs produced by SimplexNumerica.

Physical Charts

The physical coordinate system is a Cartesian system. It shows you Graphs with classic axes. It is divided into the following axes types:

·        x-linear/y-linear

·        x-logarithmically/y-logarithmically

·        x-linear/y-logarithmically

·        x-logarithmically/y-linear

In addition the following axes can be displayed:

·        2D Bottom x-Axis {Abscissa}

·        2D Left y-Axis {Ordinate}

·        2D Bottom x- && Left y-Axis

·        2D Top x-Axis {Abscissa}

·        2D Right y-Axis {Ordinate}

·        2D Apart Bottom x-Axis {Abscissa}

·        2D Apart Left y-Axis {Ordinate}

·        2D Apart Right y-Axis {Ordinate}

The Classic axis types are used to plot data along the scaled vertical and horizontal axis. Sometimes they are called ScatterPlot. The ScatterPlot ordered pairs of data values against appropriately scaled x and y-axes. The data from one or more records are plotted as a series. In SimplexNumerica there is a special <Business Diagram> whereby the data are only spaced uniformly across the x-axis by their Legend text indices.

A Contour Plot or a Contour Map like the next figure used the <2D Apart Right y-Axis {Ordinate}> with colour shading because the y-Axis coordinates are from the 3D Surface plot.

It is possible to displays a numeric y-axis also to the right, on top or on new places of the chart. Set Format Dialog to Date and Time when the data values are dates or times.

The data axes can be defined as Decimal Scale or Log Scale for logarithmic scaling. The axis can be automatically scaled to fit the data values on the coordinate system (AutoScale). The axis limits can be set to specific values by calling the appropriate dialogbox in Min Range x (xmin, ymin) and Max Range y (xmax, ymax) or make a double click on the special axis.

The appearance of the axes can be modified using above dialogs. Indeed, e.g. the y-axis labels can be completely hidden by setting this in the above described dialogs. A logarithmic data axis can be selected by the icons on the Toolswindow.

The following graph types use classic axes:

• Line and SampleData and CurveData graphs
• Step graphs
• Area graphs
• etc.

The Line graph draws a line connecting the data points of a SampleData record. The data points can be optionally covered by marker or point objects.

Also Vertical Bar Graphs belonging to the physical charts. Their axes types are used to plot data values as bars which extend vertically. The bars for the data objects in a record are spaced evenly across the horizontal axis. It displays a numeric y-axis to the left or right of the graph.

You can show also an isographic pseudo three-dimensional look to the bars in the 2D-Window. For more realistic and better 3D-Graphic please use the 3D-Graphic from the 3D-Window.

The methods described above for changing the scaling and appearance of classic axes also apply here.

Mathematical Charts

A Mathematical Chart is also based on a Cartesian coordinate system. The different to the Physical one is that the axes cross goes always through the zero point.

Some graphical and interactive functions are not supported in this chart. For manipulation of a chart goto the Physical and afterwards back to this one.

Polar coordinates system

The polar system is a special form of the mathematical coordinate system. SimplexNumerica activates automatically the mathematical coordinate system, if the polar system is selected.

The input can be made in the ArrayEditor, directly in polar coordinates. However, before the measuring data are represented, you must ensure that they are converted into Cartesian coordinates; because SimplexNumerica works internally only with Cartesian coordinates. Converting polar coordinates into Cartesian (or vice versa) can be done in the Pulldown-menu <Functions>, menu point <Polar in Cartesian Coordinates>.

The ternary system is e.g. used if one has a mixture from three chemical components and likes to see the coherencies in only two dimensions.

A Ternary Chart can be called Triplot; it is a triangular coordinate system, in which three different axes are available. These axes are however not completely independent from each other. The following relation applies to it (with format of numbers in percent):

1.    If 3 axes are given, then the sum of the 3 axes corresponds to 100%

2.    If only 2 axes are given, then results the third from x_C  = 100 - x _A  - x _B

In the accompanying dialogbox you register the number of axes and the interval

or the Number Format in percent or normalized to 1.

To Pt. 1:

SimplexNumerica calculates the sum and divides these afterwards by 100 from the 3 columns:

Value% = {Column1(i) + Column2(i) + Column3(i)} / 100

Then each column is set to:

Column 1 % (i) = Column 1 (i)/ Value%

Column 2 % (i) = Column 2 (i)/ Value%

Column 3 % (i) = Column 3 (i)/ Value%

To Pt. 2:

SimplexNumerica calculates the third column and from 2 columns:

Column 3 (i) = 1 - Column 2 (i) - Column 1 (i)

To Pt. 1 and 2:

• Column 1 is equal to x-Axis.
• Column 2 is equal to y-Axis.
• Column 3 is equal to z-Axis.

Afterwards the new values are shown in the chart.

Smith Diagram

In electro-technology, particularly in the high frequency and microwave engineering, the Smith diagram is very often used.

The Smith diagram is the result of the mapping of the right Gauss' number level into a circle area.

The input of the data is automatically standardized on the reference resistance (e.g. 50 ohms). Therefore, for the point (1,0) an input of x = 50 and y = 0 are necessary. The points are marked by the markers in the Smith diagram. The point (50,0) corresponds to the origin of the coordinate system in the w-level. They can represent all points of the right z-half plane in the Smith diagram.

Theory

The Smith chart is one of the most useful graphical tools for high frequency circuit applications. The chart provides a clever way to visualize complex functions and it continues to endure popularity decades after its original conception.

From a mathematical point of view, the Smith chart is simply a representation of all possible complex impedances with respect to coordinates defined by the reflection coefficient.

The domain of definition of the reflection coefficient is a circle of radius 1 in the complex plane. This is also the domain of the Smith chart.

The goal of the Smith chart is to identify all possible impedances on the domain of existence of the reflection coefficient. To do so, we start from the general definition of line impedance (which is equally applicable to the load impedance)

This provides the complex function

we want to graph.  It is obvious that the result would be applicable only to lines with exactly characteristic impedance Z₀.

In order to obtain universal curves, we introduce the concept of normalized impedance

The normalized impedance is represented on the Smith chart by using families of curves that identify the normalized resistance r (real part) and the normalized reactance x (imaginary part)

Let’s represent the reflection coefficient in terms of its coordinates

Now we can write

The real part gives

The imaginary part gives

The result for the real part indicates that on the complex plane with coordinates (Re(Γ), Im(Γ)) all the possible impedances with a given normalized resistance r  are found on a circle with

As the normalized resistance r varies from 0 to ∞, we obtain a family of circles completely contained inside the domain of the reflection coefficient | Γ | = 1.

The result for the imaginary part indicates that on the complex plane with coordinates (Re(Γ), Im(Γ)) all the possible impedances with a given normalized reactance x  are found on a circle with

As the normalized reactance x varies from -8 to 8, we obtain a family of arcs contained inside the domain of the reflection coefficient | Γ | = 1 .

The Smith chart can be used for line admittances, by shifting the space reference to the admittance location. After that, one can move on the chart just reading the numerical values as representing admittances. Let’s review the impedance-admittance terminology:

Impedance = Resistance + j Reactance

Admittance = Conductance + j Susceptance

On the impedance chart, the correct reflection coefficient is always represented by the vector corresponding to the normalized impedance. Charts specifically prepared for admittances are modified to give the correct reflection coefficient in correspondence of admittance.

Since related impedance and admittance are on opposite sides of the same Smith chart, the imaginary parts always have different sign. Therefore, a positive (inductive) reactance corresponds to a negative (inductive) susceptance, while a negative (capacitive) reactance corresponds to a positive (capacitive) susceptance. Numerically, we have

This menu option called the 3D-Grafik-Window. The description is in a separate chapter in this manual.

Set Axis

A further menu option to the physical or logarithmic coordinate system. Hereby one can set additionally axes, which can be scaled completely differently than the original coordinate system.

There are the following axes types available:

• Right ordinate:       = rights y-Axis
• Upper abscissa:      = upper one x-Axis
• New abscissa:        = free definable x-Axis
• New ordinate:        = free definable y-Axis

The new abscissa and ordinate are placed automatically in the 2D-Window. As soon as you select e.g. a new ordinate, you can set the axis scaling. The position of the axis can be changed by pressing the control key and left mouse button; press-held on the new axis and move it. The interval can be changed with the left mouse button double-click.

In the following dialogbox one can set the limits for the axis interval without an AutoScale done by the program.

Remove Axis

Axes are removed immediately from screen. In fact the axis will be hidden because the interval dimensions remain internally.