3

I have been looking around the forum but did not find anything close to what I want to accomplish. I wanted to draw this nomogram used in control systems but just do not know where to begin. Can you help me draw this:

enter image description here

It seems to be generated from a z-plane grid of constant damping factors and natural frequencies

These are some forumlas:

enter image description here

enter image description here

Matlab has the z-grid command that plots this graph.

I have just started with the barebones environment:

\documentclass{standalone}
\usepackage{pgfplots}
\usepackage{amsmath} % Required for \varPsi below
\usetikzlibrary{tikzmark,calc,arrows,shapes,decorations.pathreplacing,pgfplots.groupplots}
\pgfplotsset{compat=newest, title/.append style={align =center}}
\tikzset{every picture/.style={remember picture}}

\begin{document}
\begin{tikzpicture}



\end{tikzpicture}
\end{document} 
  • 1
    Do you know the maths behind these lines? Googling "nomograph" and "control systems" doesn't show any similar result. – Ignasi Jun 1 '17 at 12:20
  • I checked, but did not get much info. I'll keep looking around again and update my post. Thanks! – Joe Jun 1 '17 at 12:25
  • For sure this can be done. But of course you need to have the data to do so. So either you have the equations of the lines or a data table/file. If you don't have at least of of them, you could "extract" the data from the graph e.g. by using markummitchell.github.io/engauge-digitizer. – Stefan Pinnow Jun 1 '17 at 12:28
  • This I suppose is something that is closer to the Nicholas chart (in the control context). – Raaja Jun 1 '17 at 12:28
  • @Ignasi, I have updated some equations and Matlab plot function where this is generated. Hope this gives more help to a final convergence of the desired output?? Thanks for your time! – Joe Jun 1 '17 at 18:38
8

Proposed grid options, to put in axis brackets:

  height=15cm,
  unit vector ratio = 1 1,
  xmin=-1.05,
  xmax=1.1,
  ymin=0,
  ymax=1.1,
  samples=100,
  %axis lines=center,
  %ticks=none,
  minor tick num=4,
  xtick distance=.25,
  ytick distance=.1,
  major grid style={thick},
  xticklabels={,-1,,-0.5,,0,,0.5,,1},
  yticklabels={,0,,0.2,,0.4,,0.6,,0.8,,1},
  grid=both,

Yielding

enter image description here

Clean version

enter image description here

\documentclass[12pt,tikz,border=2pt]{standalone}
\usepackage{pgfplots}
\usepgfplotslibrary{polar}

\begin{document}
\begin{tikzpicture}
  \begin{axis}
    [
      height=15cm,
      unit vector ratio = 1 1,
      ymin=0,
      xmax=1.1,
      ymax=1.1,
      samples=100,
      axis lines=center,
      ticks=none,
    ]

    \pgfplotsinvokeforeach{0,...,9}
    {
      \def\zet{(.1*#1)}
      \pgfmathsetmacro{\factor}{\zet/sqrt(1-\zet^2)}
      \addplot[data cs=polar,domain=0:.35*sqrt(1-\zet^2)] (180*\x,{exp(-pi*\factor*\x})
          node[at end, sloped, anchor=south,font=\tiny, inner sep=0pt] {$\zeta{=}0.#1$};
      \addplot[data cs=polar,domain=.35*sqrt(1-\zet^2):sqrt(1-\zet^2)] (180*\x,{exp(-pi*\factor*\x});
    }

    \pgfplotsinvokeforeach{.1,.2,...,1}
    {
      \def\a{#1}
      \addplot[data cs=polar,domain=0:90] ({180*\a*cos(\x)},{exp(-pi*\a*sin(\x))});
    }
  \end{axis}
\end{tikzpicture}
\end{document}
|improve this answer|||||
  • Sincerest Thank you for all your time and help! – Joe Jun 5 '17 at 2:00
  • Shall we finish the graph by adding the \zeta=1 line, adding axis labels and the nodes for \a? If you don't want to do it yourself, do you allow me to do it and edit your answer accordingly? – Stefan Pinnow Jun 5 '17 at 10:33
  • @StefanPinnow Sure, go ahead and edit ! I didn't add nodes for the values of \a because 1. I didn't know what unknown to write. 2. using \foreach inside a pgfplots feels like esoterism to me. Furthermore, if you have a better solution than mine for placing the \zeta= nodes at 35% of the domain, I would be interested ! – marsupilam Jun 5 '17 at 10:56
  • @marsupilam, because I changed some more stuff than I first wanted to do, I decided to write this as a separate answer. And I think you have found the perfect way to add the ζ labels! – Stefan Pinnow Jun 6 '17 at 20:14
2

This is basically the same answer as marsupilam's. The (main) differences are:

  • added $\zeta=1$ line
  • added $\omega$ labels
  • added axis labels
  • provided an alternative, more automated way to add the xticklabels and yticklabels

For more details please have a look at the comments in the code.

% used PGFPlots v1.14
\documentclass[12pt,border=2pt]{standalone}
\usepackage{pgfplots}
    % load the `polar' library so we can use `data cs=polar'
    \usepgfplotslibrary{polar}
    % use this `compat' level or higher to use the advanced axis label positioning
    \pgfplotsset{compat=1.3}
\begin{document}
\begin{tikzpicture}[
    % create a style for the common options of the labels
    Label/.style={
        font=\tiny,
        inner sep=1pt,
    },
]
    \begin{axis}[
        height=15cm,
        axis equal image=true,
        xmin=-1.05,
        xmax=1.05,
        ymin=0,
        ymax=1.05,
        xlabel=Real,
        ylabel=Imaginary,
        samples=61,             % <-- reduced number of samples and added `smooth'
        smooth,
        xtick distance=0.25,
        ytick distance=0.1,
        minor tick num=4,
        major grid style={thick},
        grid=both,
        % ---------------------------------------------------------------------
        % giving every second ticklabel manually ...
%        xticklabels={,-1,,-0.5,,0,,0.5,,1},
%        yticklabels={,0,,0.2,,0.4,,0.6,,0.8,,1},
        % ... and here an automatic way
        xticklabel={%
            \pgfmathsetmacro{\TickNum}{ifthenelse(mod(\ticknum,2)==0,1,0)}
            \ifdim\TickNum pt=0pt % a TeX \if -- see TeX Book
                $\pgfmathprintnumber{\tick}$%
            \else
            \fi
        },
        yticklabel={%
            \pgfmathsetmacro{\TickNum}{ifthenelse(mod(\ticknum,2)==0,1,0)}
            \ifdim\TickNum pt=0pt % a TeX \if -- see TeX Book
                $\pgfmathprintnumber{\tick}$%
            \else
            \fi
        },
        % ---------------------------------------------------------------------
        data cs=polar,          % <-- moved common `addplot' options here
        clip=false,             % <-- added so the labels aren't clipped
    ]

        % constant $\zeta$ contours
        % (we cannot also calculate the $\zeta = 1$ line directly, because
        %  this will lead to a "division by zero" error)
        % the lines will be plotted in two parts to place the labels at
        % a "good" position
        % (because we want to add them `sloped' it is not an option to add
        %  the nodes separately at the calculated positions)
        \pgfplotsinvokeforeach{0,0.1,...,0.9} {
                % calculate a factor in advance
                \pgfmathsetmacro{\factor}{#1/sqrt(1-#1^2)}
            % plot the first part of the $\zeta$ contour lines ...
            \addplot [
                domain=0:0.35*sqrt(1-#1^2),
            ] (180*x,{exp(-pi*\factor*x)})
                % ... and add the labels
                node [
                    Label,
                    at end,
                    sloped,
                    anchor=south,
                ] {$\zeta =
                    % because of some math inaccuracies we need to format the
                    % numbers when we use the `\pgfmathprintnumber'
                    \pgfmathprintnumber[
                        fixed,
                        fixed zerofill,
                        precision=1,
                    ]{#1}$
                }
            ;
            % plot the second part of the $\zeta$ contour lines
            \addplot [
                domain=.35*sqrt(1-#1^2):sqrt(1-#1^2),
            ] (180*x,{exp(-pi*\factor*x)});
        }

        % now add the $\zeta = 1$ line
        \addplot [
            domain=exp(-pi):1,
            samples=2,
            data cs=cart,
        ] (x,0)
            node [
                Label,
                pos=0.31,       % <-- found due to testing
                anchor=south,
            ] {$\zeta = 1$}
        ;

        % constant $\omega$ contours
        \pgfplotsinvokeforeach{0.05,0.1,0.2,...,1.0} {
            \addplot [
                domain=0:90,
            ] ({180*#1*cos(x)},{exp(-pi*#1*sin(x))})
                % add the nodes again
                node [
                    Label,
                    at start,
                    anchor=180*(#1-1),
                ] {%
                      % we don't want to plot the "1" so we need a special
                      % handler
                      % (unfortunately `\pgfmathprintnumber' seems to *need*
                      %  to have a number and thus we cannot do something
                      %  like
                      %     \pgfmathparse{ifthenelse(abs(#1-1)<0.01,,#1)}%
                      %     $\frac{\pgfmathprintnumber[fixed]{#1}\,\pi}{T}$
                      % )
                      \ifdim#1 pt>0.99pt
                          $\frac{\pi}{T}$
                      \else
                          $\frac{\pgfmathprintnumber[fixed]{#1}\,\pi}{T}$
                      \fi
                  }
            ;
        }
    \end{axis}
\end{tikzpicture}
\end{document}

image showing the result of above code

|improve this answer|||||
  • Ok, you're not the lazy type, are you ? ;) Glad I "let" you do the work ! Thanks, I will look into it. – marsupilam Jun 6 '17 at 20:30
  • 1
    @marsupilam, not really. But honestly: You did the great start and -- as I already mentioned in the comment to your answer -- you found the great solution for positioning the ζ labels, which I would have never come up with. I just gave your solution some "finishing". – Stefan Pinnow Jun 6 '17 at 20:31

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