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6 Example 8-22 OGATA 4ed(Phase and gain margins, Bode plot)



    Let's calculate the phase and gain margins. Checks and Bode plot by Scilab

    \begin{displaymath}G(s)=\frac{20\cdot(s+1)}{s\cdot (s+5)\cdot (s^{2}+2\cdot s+10...
... (\frac{s}{5}+1)\cdot (\frac{s^{2}}{10}+\frac{2\cdot s}{10}+1)}\end{displaymath}

    Solution:
    Gain Table (dB)

    \begin{displaymath}20\cdot log(\frac{20}{5\cdot 10})=-7.95\end{displaymath}


    w   1   $\sqrt(10)$   5  
    $\frac{1}{jw}$ (-20) 0 (-20)   (-20)   (-20)
    $(jw+1)$ (0) 0 (20)   (20)   (20)
    $\frac{1}{(\frac{s^{2}}{10}+\frac{2\cdot s}{10}+1)}$ (0) 0 (0) 0 (-40)   (-40)
    $\frac{1}{\frac{jw}{5}+1}$ (0) 0 (0) 0 (0) 0 (-20)
    $G(jw)$ (-20) -7.95 (0) -7.95 (-40) -15.91 (-60)



    The phase's equation:

    \begin{displaymath}\lfloor{G(j\cdot w)}=arctg(w)-90-arctg(\frac{w}{5})-arctg(\frac{2\cdot w}{10-w^{2}})\end{displaymath}


    Different phase values:

    w phase
    0.1 -98
    1 -68.83
    $\sqrt(10)$ -139.85
    5 -202.61
    10 -236.61



    Gain crossover frequency is in $w<1$.

    \begin{displaymath}-7.95-20\cdot log(wcg)=0\end{displaymath}


    \begin{displaymath}wcg=10^{\frac{-7.95}{20}}=0.4\end{displaymath}



    The gain crossover frequency is $w_{cg}=0.4$. The phase's frequency is:

    \begin{displaymath}\lfloor{G(j\cdot 0.4)}=arctg(0.4)-90-arctg(\frac{0.4}{5})-arctg(\frac{2\cdot 0.4}{10-0.4^{2}})=21.8-90-4.57-4.64=-68.12\end{displaymath}


    The phase margin is: $180-68.12=111.87$



    The phase crossover frequency is obtained from this equation:

    \begin{displaymath}-180=arctg(w_{cf})-90-arctg(\frac{w_{cf}}{5})-arctg(\frac{2\cdot w_{cf}}{10-w_{cf}^{2}})\end{displaymath}


    \begin{displaymath}-90=arctg(w_{cf})-arctg(\frac{w_{cf}}{5})-arctg(\frac{2\cdot w_{cf}}{10-w_{cf}^{2}})\end{displaymath}


    \begin{displaymath}-90-arctg(w_{cf})=-arctg(\frac{w_{cf}}{5})-arctg(\frac{2\cdot w_{cf}}{10-w_{cf}^{2}})\end{displaymath}


    \begin{displaymath}tg(90+arctg(w_{cf}))=tg(arctg(\frac{w_{cf}}{5})+arctg(\frac{2\cdot w_{cf}}{10-w_{cf}^{2}}))\end{displaymath}



    \begin{displaymath}\frac{tg(90)+w_{cf}}{1-tg(90)\cdot w_{cf} }=\frac{\frac{w_{cf...
...}}{1-\frac{w_{cf}}{5}\cdot \frac{2\cdot w_{cf}}{10-w_{cf}^{2}}}\end{displaymath}



    \begin{displaymath}\frac{1+\frac{w_{cf}}{tg(90)}}{\frac{1}{tg(90)}-w_{cf} }=\fra...
...}}{1-\frac{w_{cf}}{5}\cdot \frac{2\cdot w_{cf}}{10-w_{cf}^{2}}}\end{displaymath}



    \begin{displaymath}\frac{1+0}{0-w_{cf} }=\frac{w_{cf}\cdot (10-w_{cf}^{2})+2\cdo...
...{cf}\cdot 5}{(10-w_{cf}^{2})\cdot 5 -w_{cf}\cdot 2\cdot w_{cf}}\end{displaymath}



    \begin{displaymath}-1=\frac{w_{cf}^{2}\cdot (10-w_{cf}^{2})+2\cdot w_{cf}^{2}\cdot 5}{(10-w_{cf}^{2})\cdot 5 -w_{cf}\cdot 2\cdot w_{cf}}\end{displaymath}



    \begin{displaymath}-(10-w_{cf}^{2})\cdot 5 + 2\cdot w_{cf}^{2}=10\cdot w_{cf}^{2}-w_{cf}^{4}+10\cdot w_{cf}^{2}\end{displaymath}


    \begin{displaymath}-50+5\cdot w_{cf}^{2} + 2\cdot w_{cf}^{2}=10\cdot w_{cf}^{2}-w_{cf}^{4}+10\cdot w_{cf}^{2}\end{displaymath}


    \begin{displaymath}w_{cf}^{4}-13\cdot w_{cf}^{2}-50=0\end{displaymath}


    We obtain $w_{cf}=4.01$.
    We replace the frequency in the following equation:

    \begin{displaymath}-7.95-40\cdot log(\frac{w_{cf}}{\sqrt{10}})=-12.08\end{displaymath}


    We obtain a gain margin of 12.08dB



    Because the gain and phase margins are positive, the system is stable.






    Scilab program
    
    s=%s/(2*%pi);
    g=20*(s+1)/((s+0.00000000000001)*(s+5)*(s^2+2*s+10));
    gs=syslin('c',g)
    [gg,wcp]=g_margin(gs)
    [pg,wcg]=p_margin(gs)
    clf;
    bode(gs);
    show_margins(gs);
    
    Results:
    -->[gg,wcp]=g_margin(gs)
    wcp  =
     
        4.0130645  
     gg  =
     
        9.9292942  
     
    -->[pg,wcg]=p_margin(gs)
     wcg  =
     
        0.4426366  
     pg  =
     
        103.65727
    
    Image Ejemplo8-22