LockedFP Block Diagram

<< Graphics`

Equation

\(ブロックダイアグラムを連立方程式であらわし, \(ν\_4\) \(\(を求めます\)\(.\)\)\)

Simplify[
      Solve[{(Subscript[ν, 1]) == (Subscript[ν, 0])  -  ((Subscript[ν, 2])*(Subscript[P, MC])*
                      (Subscript[H, MC, PD]) + 
                    (Subscript[N, MC]) + ((Subscript[ν, 3])*(Subscript[P, IN])*(Subscript[H, IN, PD]) + (Subscript[N, IN]))*(Subscript[C, IN])*
                      (Subscript[C, FA]))*
                (Subscript[C, 0])*((Subscript[H, PZT])*(Subscript[C, PZT]) + 
                    (Subscript[H, EOM])*(Subscript[C, EOM])),  (Subscript[ν, 2]) == (Subscript[ν, 1]) - ((Subscript[ν, MC]) - (((Subscript[ν, 3])*(Subscript[P, IN])*(Subscript[H, IN, PD]) + (Subscript[N, IN]))*(Subscript[C, IN])*(Subscript[C, MC]) + 
                        (Subscript[N, MCEND]))*
                    (Subscript[H, MC, END])), (Subscript[ν, 3]) == (Subscript[ν, 4]) - (Subscript[ν, IN]), (Subscript[ν, 4]) == (Subscript[ν, 1])*
                (Subscript[P, MC]) + ((Subscript[ν, MC]) - (((Subscript[ν, 3])*(Subscript[P, IN])*(Subscript[H, IN, PD]) + 
                                (Subscript[N, IN]))*(Subscript[C, IN])*(Subscript[C, MC]) + (Subscript[N, MCEND]))*
                      (Subscript[H, MC, END]))*(1 - 
                    (Subscript[P, MC]))},  {(Subscript[ν, 4])},  {(Subscript[ν, 1]), (Subscript[ν, 2]), (Subscript[ν, 3])}]]

\(\(N\_\(MCEND, IN, MC\)\) が無いとき\)、

T1 := ((Subscript[ν, MC]) - (Subscript[C, IN]) (Subscript[C, MC]) (Subscript[N, IN]) (Subscript[H, MC, END]) + 
            (Subscript[N, MCEND]) (Subscript[H, MC, END]) + 
            (Subscript[C, IN]) (Subscript[C, MC]) (Subscript[P, IN]) (Subscript[ν, IN]) (Subscript[H, IN, PD]) (Subscript[H, MC, END]) + 
            (Subscript[P, MC]) ((Subscript[ν, 0]) - (Subscript[ν, MC]) - ((Subscript[N, MCEND]) + 
                        (Subscript[C, IN]) (Subscript[C, MC]) (-(Subscript[N, IN]) + 
                              (Subscript[P, IN]) (Subscript[ν, IN]) (Subscript[H, IN, PD]))) (Subscript[H, MC, END])) - 
            (Subscript[C, 0]) ((Subscript[C, EOM]) (Subscript[H, EOM]) + (Subscript[C, PZT]) (Subscript[H, PZT])) (Subscript[P, MC]) ((Subscript[N, MC]) + 
                  (Subscript[C, FA]) (Subscript[C, IN]) ((Subscript[N, IN]) - 
                        (Subscript[P, IN]) (Subscript[ν, IN]) (Subscript[H, IN, PD])) - ((Subscript[ν, MC]) + ((Subscript[N, MCEND]) + (Subscript[C, IN]) (Subscript[C, MC]) (-(Subscript[N, IN]) + 
                                    (Subscript[P, IN]) (Subscript[ν, IN]) (Subscript[H, IN, PD]))) (Subscript[H, MC, END])) (Subscript[H, MC, PD])))/(1 - 
            (Subscript[C, IN]) (Subscript[C, MC]) (Subscript[P, IN]) (-1 + 
                  (Subscript[P, MC])) (Subscript[H, IN, PD]) (Subscript[H, MC, END]) + 
            (Subscript[C, 0]) ((Subscript[C, EOM]) (Subscript[H, EOM]) + 
                  (Subscript[C, PZT]) (Subscript[H, PZT])) (Subscript[P, MC]) ((Subscript[C, FA]) (Subscript[C, IN]) (Subscript[P, IN]) (Subscript[H, IN, PD]) + (1 + 
                        (Subscript[C, IN]) (Subscript[C, MC]) (Subscript[P, IN]) (Subscript[H, IN, PD]) (Subscript[H, MC, END])) (Subscript[H, MC, PD])))

T2 = 
      Simplify[T1 - Coefficient[T1, (Subscript[N, MCEND])]*(Subscript[N, MCEND]) - 
          Coefficient[T1, (Subscript[N, IN])]*(Subscript[N, IN]) - 
          Coefficient[T1, (Subscript[N, MC])]*(Subscript[N, MC])]

Simplify[Coefficient[T2, (Subscript[ν, IN])]]

この式から,\(\(ν\_4\) はMCとPrimary\ Cavity\ Servoのル\)-プゲイン\((後で説明)\)が大きくなれば
\(\(ν\_IN\) に近づくことが示せます . 
\((P\_MC = 1, \ 
          1 << \(C\_IN\)\ C\_MC\ P\_IN\ H\_\(IN, PD\)\ \(H\_\(MC, 
                END\)\) \(\(とした\)\(.\)\)\ )\)\)

Simplify[(((Subscript[C, IN]) (Subscript[P, IN]) (Subscript[H, IN, PD]) (Subscript[C, 0]) ((Subscript[C, EOM]) (Subscript[H, EOM]) + (Subscript[C, PZT]) (Subscript[H, PZT])) (Subscript[P, MC]) ((Subscript[C, FA]) + (Subscript[C, MC]) (Subscript[H, MC, END]) (Subscript[H, MC, PD])))/(1 + (Subscript[C, 0]) ((Subscript[C, EOM]) (Subscript[H, EOM]) + (Subscript[C, PZT]) (Subscript[H, PZT])) (Subscript[P, MC]) ((Subscript[C, FA]) (Subscript[C, IN]) (Subscript[P, IN]) (Subscript[H, IN, PD]) + (Subscript[C, IN]) (Subscript[C, MC]) (Subscript[P, IN]) (Subscript[H, IN, PD]) (Subscript[H, MC, END]) (Subscript[H, MC, PD]))))]

\(しかし, 当然のことながら, 分子の中で, \(ν\_IN\) を含む項よりも他のN\_IN, 
    N\_MC, \(ν\_MC\) を含む項が大きくては, \(目標安定度であるν\_IN\) を達成できません . そこで, \(ν\_IN\) そのものと他の項の大きさを比較するために, \(\(他の項を求めます\)\(.\)\(
\)\)\)

Numerator of (Subscript[ν, 4])

\(まず, \(求めたν\_4\) の分子のうち, \(ν\_IN\) の係数を求めます . それをCoeffNuINとし, それで他の項を割ることで, \(ν\_IN\) と比較できます . 
まずν\_IN \( \(の係数を求めます\)\(.\)\)\)

(Subscript[ν, 4]) := T1

CoeffNuIN = 
      Simplify[Coefficient[Numerator[(Subscript[ν, 4])], (Subscript[ν, IN])]]

\(\(次にN\_IN\) \(\(の係数を求めます\)\(.\)\)\)

Simplify[
      Coefficient[Numerator[(Subscript[ν, 4])]/CoeffNuIN, (Subscript[N, IN]) ]]

Simplify[Coefficient[(Subscript[ν, 4]), (Subscript[N, IN]) ]]

\(\(次にN\_MC\) \(\(の係数を求めます\)\(.\)\)\)

Simplify[
      Coefficient[Numerator[(Subscript[ν, 4])]/CoeffNuIN, (Subscript[N, MC]) ]]

Simplify[Coefficient[(Subscript[ν, 4]), (Subscript[N, MC])]]

\(\(次にν\_MC\) \(\(の係数を求めます\)\(.\)\)\)

Simplify[
      Coefficient[Numerator[(Subscript[ν, 4])]/CoeffNuIN, (Subscript[ν, MC]) ]]

Simplify[Coefficient[(Subscript[ν, 4]), (Subscript[ν, MC])]]

\(\(最後にN\_MCEND\) \(\(の係数を求めます\)\(.\)\)\)

Simplify[
      Coefficient[Numerator[(Subscript[ν, 4])]/CoeffNuIN, (Subscript[N, MCEND]) ]]

Simplify[Coefficient[(Subscript[ν, 4]), (Subscript[N, MCEND])]]

Denominator of (Subscript[ν, 1])

Denominator[(Subscript[ν, 4])]

\(ちなみに, 分母は, 
    G\_MC = C\_0\ *\((C\_PZT*H\_PZT + C\_EOM\ *H\_EOM)\)*P\_MC\ *
        H\_\(MC, PD\), 
    G\_IN = C\_IN\ *P\_IN\ *
        H\_\(IN, PD\)*\((C\_FA\/H\_\(MC, PD\) + \(P\_MC\) C\_MC\ *
              H\_\(MC, END\))\) とすると, 
1 + \(G\_MC\) \((\(P\_MC\) C\_IN\ P\_IN\ H\_\(IN, PD\) + 
            G\_IN)\) となります . 普通, 
    G\_IN >> \(P\_MC\) C\_IN\ P\_IN\ \(H\_\(IN, PD\)\) で, G\_MC >> 1, 
    G\_IN >> 1, ならば1 + \(G\_MC\) \((G\_IN)\) -> \ \(G\_MC\) \(G\_IN\) \(\(になる\)\(.\)\)\)

Parameter1

\(CLIOの基本的なパラメ - タ - です . 仮定を多く含みます . \)

(Subscript[L, MC]) := 
        9.5
(Subscript[L, IN]) := 
      100
(Subscript[L, Per]) := 100
  
(Subscript[ν, YAG]) := 2.998*(10^8)/1.064*(10^6); (Subscript[ν, 0]) := 
        (10^4)*(f^(-1))

  
(Subscript[P, MC]) := 
      (4000/(4000 + f*I))
(Subscript[P, IN]) := 
      (250/(250 + f*I))
(Subscript[P, Per]) := (250/(250 + f*I))

  
(Subscript[H, MCPend]) := 
      (10^(-9))*(f^(-4))
(Subscript[H, INPend]) := 
      (10^(-9))*(f^(-6))
(Subscript[H, PerPend]) := (10^(-9))*(f^(-6))

  
(Subscript[ν, MC]) := ((Subscript[H, MCPend]) + (10^(-18)))/(Subscript[P, IN])*
          ((Subscript[L, MC])^(-1))*(Subscript[ν, YAG]); (Subscript[ν, IN]) := 
        (Subscript[H, INPend])*((Subscript[L, IN])^(-1))*(Subscript[ν, YAG])

  
(Subscript[H, MC, END]) := ((Subscript[ν, YAG])/(Subscript[L, MC]))*4*(10^(-6))*
          (f^(-2))

  
(Subscript[H, Per, NM]) := ((Subscript[ν, YAG])/(Subscript[L, Per]))*2*(10^(-7))*(1.1^2)*
          (f^(-2))

  
(Subscript[N, IN]) := 
      15*(10^(-9))
(Subscript[N, MC]) := 
      15*(10^(-9))
(Subscript[N, MCEND]) := 
      20*(10^(-9))

Feedback Circuit

\(Feedback\ \(\(Circuitの伝達関数です\)\(.\)\)\)

(Subscript[H, MC, PD]) := 
        5*(10^(-3))
(Subscript[H, IN, PD]) := 
      3*(10^(-1))
(Subscript[H, Per, PD]) := 4.5*(10^(-1))

  
(Subscript[C, 0]) := 
        0.3  
(Subscript[C, PZT]) := 
      20*(20/(20 + f*I))*(((3.3*(10^3) + f*I)/(333 + f*I)))*((10^5)/((10^5) + f*I))*(E^(-I*2*π*f*(25/299792458)))
(Subscript[H, PZT]) := 
      1.5*(10^6)
(Subscript[C, EOM]) := (((f*I)/(500 + f*I))*((10*(10^3) + f*I)/((10^3) + f*I))*((1*(10^6))/(1*(10^6) + f*I)))*(((1*(10^3))/(1*(10^3) + f*I))*((3.2*(10^3) + f*I)/(3.2*(10^3)))*(1.6/0.5)*((5*(10^5) + f*I)/(1.6*(10^6) + f*I))*(1.6/0.5)*((700*(10^3))/(700*(10^3) + f*I)))*(250*((f*I)/((10^3) + f*I))*((10*(10^3))/(10*(10^3) + f*I))*((220*(10^3) + f*I)/(220*(10^3)))*((306*(10^3) + f*I)/(306*(10^3)))*((350*(10^3))/(350*(10^3) + f*I)))*((10^5)/((10^5) + f*I))*(E^(-I*2*π*f*(25/299792458)))
(Subscript[H, EOM]) := f*I*1*(10^(-2))

  
(Subscript[C, IN]) := 3*((10^5)/((10^5) + f*I))  
  
(Subscript[C, FA]) := ((f*I)/(30 + f*I))*((30*(10^3) + f*I)/(300*(10^3) + f*I))*((3*(10^5))/(3*(10^4)))*1.1
(Subscript[C, MC]) := -1*((250 + f*I)/(7.2*(10^3) + f*I))*((7.2*(10^3))/2500)*(2*((3.3*(10^3) + f*I)/(16*(10^3) + f*I))*((16*(10^3))/(3.3*(10^3))))*((10^4)/((10^4) + f*I))*(((10^4)/((10^4)*(E^(-I (π/4))) + f*I))*((10^4)/((10^4)*(E^(I  (π/4))) + f*I)))

  
(Subscript[C, Per]) := 
        1*((150 + f*I)/((10^3) + f*I))*((10^3)/150)*(((10^2) + f*I)/((10^3) + f*I))*(((((10^4)/((10^4)*(E^(-I ((3  π)/8))) + f*I))*((10^4)/((10^4)*(E^(I  (π/8))) + f*I))*((10^4)/((10^4)*(E^(-I (π/8))) + f*I))*((10^4)/((10^4)*(E^(I (( 3  π)/8))) + f*I))))^1)*(1/1)

(Subscript[G, Per]) := 
      (Subscript[C, Per])*(Subscript[P, Per]) *(Subscript[H, Per, PD])*
        (Subscript[H, Per, NM])
(Subscript[G, MC]) := 
      (Subscript[C, 0]) *((Subscript[C, PZT])*(Subscript[H, PZT]) + (Subscript[C, EOM]) *(Subscript[H, EOM]))*(Subscript[P, MC]) *
        (Subscript[H, MC, PD])
(Subscript[G, IN]) := 
          (Subscript[C, IN]) *(Subscript[P, IN]) *
            (Subscript[H, IN, PD])*(((Subscript[G, MC])/(1 + (Subscript[G, MC])))*((Subscript[C, FA])/(Subscript[H, MC, PD])) + 
                (Subscript[C, MC]) *(Subscript[H, MC, END]))
  
  
W := ((f*I)/(2.26 + f*I))*((f*I)/(91.11 + f*I))*((f*I)/(10.44 + f*I))*(116/(116 + f*I))*27.55

\(LogLinearPlot[{Arg[E\^\(\(-I\)*2*π*f*25\/299792458\)]*180\/π}, \ {f, \ 10\^2, 10\^7}, \ Frame -> \ True, \ 
      PlotRange -> {{10\^2, 10\^7}, {\(-180\), \ 180}}, \ 
      PlotPoints\  -> \ 1000, \ 
      PlotStyle\  -> \ {RGBColor[1, 0, 0]}, 
      GridLines -> {{{400000, {Dashing[{ .01,  .01}]}}}, {{0, {Dashing[{ .01,  .01}]}}}}, 
FrameLabel -> {"Frequency [Hz]", 
          Phase[Deg]}]\)

MC Servo Loop Gain

Normal gain

(Subscript[C, 0]) := 1.5;

\(\(\(LogLogPlot[{Abs[\ 
          C\_0\ *\((C\_PZT*H\_PZT + C\_EOM\ *H\_EOM)\)*P\_MC\ *
            H\_\(MC, PD\)], 
        Abs[\ C\_0\ *\((C\_EOM\ *H\_EOM)\)*P\_MC\ *H\_\(MC, PD\)], 
        Abs[\ C\_0\ *\((C\_PZT*H\_PZT)\)*P\_MC\ *H\_\(MC, PD\)]}, \ {f, \ 
        10\^2, 10\^7}, \ Frame -> \ True, \ 
      PlotRange -> {{10\^2, 10\^7}, {10\^\(-2\), \ 10\^6}}, \ 
      PlotPoints\  -> \ 3000, \ 
      PlotStyle\  -> {RGBColor[0, 0, 0], RGBColor[1, 0, 0], 
          RGBColor[0, 0, 1]}, 
      GridLines -> {{{400000, {Dashing[{ .01,  .01}]}}}, {{1, {Dashing[{ .01,  .01}]}}}}, 
FrameLabel -> {"Frequency [Hz]", 
          Gain}]\)\(
\)
    \)\)
\(LogLinearPlot[
      Arg[\ C\_0\ *\((C\_PZT*H\_PZT + C\_EOM\ *H\_EOM)\)*P\_MC\ *
            H\_\(MC, PD\)]*180\/π, \ {f, \ 10\^2, 10\^7}, \ 
      Frame -> \ True, \ 
      PlotRange -> {{10\^2, 10\^7}, {\(-180\), \ 180}}, \ 
      PlotPoints\  -> \ 3000, \ 
      PlotStyle\  -> \ {RGBColor[1, 0, 0]}, 
      GridLines -> {{{400000, {Dashing[{ .01,  .01}]}}}, {{0, {Dashing[{ .01,  .01}]}}}}, 
FrameLabel -> {"Frequency [Hz]", 
          Phase[Deg]}]\)

Gain up

(Subscript[C, 0]) := 
      1.5*((1*(10^5) + f*I)/(f*I));

\(\(\(LogLogPlot[{Abs[\ 
          C\_0\ *\((C\_PZT*H\_PZT + C\_EOM\ *H\_EOM)\)*P\_MC\ *
            H\_\(MC, PD\)], 
        Abs[\ C\_0\ *\((C\_EOM\ *H\_EOM)\)*P\_MC\ *H\_\(MC, PD\)], 
        Abs[\ C\_0\ *\((C\_PZT*H\_PZT)\)*P\_MC\ *H\_\(MC, PD\)]}, \ {f, \ 
        10\^2, 10\^7}, \ Frame -> \ True, \ 
      PlotRange -> {{10\^2, 10\^7}, {10\^\(-2\), \ 10\^6}}, \ 
      PlotPoints\  -> \ 3000, \ 
      PlotStyle\  -> {RGBColor[0, 0, 0], RGBColor[1, 0, 0], 
          RGBColor[0, 0, 1]}, 
      GridLines -> {{{400000, {Dashing[{ .01,  .01}]}}}, {{1, {Dashing[{ .01,  .01}]}}}}, 
FrameLabel -> {"Frequency [Hz]", 
          Gain}]\)\(
\)
    \)\)
\(LogLinearPlot[
      Arg[\ C\_0\ *\((C\_PZT*H\_PZT + C\_EOM\ *H\_EOM)\)*P\_MC\ *
            H\_\(MC, PD\)]*180\/π, \ {f, \ 10\^2, 10\^7}, \ 
      Frame -> \ True, \ 
      PlotRange -> {{10\^2, 10\^7}, {\(-180\), \ 180}}, \ 
      PlotPoints\  -> \ 3000, \ 
      PlotStyle\  -> \ {RGBColor[1, 0, 0]}, 
      GridLines -> {{{400000, {Dashing[{ .01,  .01}]}}}, {{0, {Dashing[{ .01,  .01}]}}}}, 
FrameLabel -> {"Frequency [Hz]", 
          Phase[Deg]}]\)

Primary Cavity Servo Loop Gain

Normal gain

(Subscript[C, 0]) := 1.5;
(Subscript[C, IN]) := 2*((10^5)/((10^5) + f*I));

\(LogLogPlot[{Abs[\ 
          C\_IN\ *\((G\_MC/\((1 + G\_MC)\)*C\_FA/H\_\(MC, PD\) + 
                P\_MC*C\_MC\ *H\_\(MC, END\))\)*P\_IN\ *H\_\(IN, PD\)], 
        Abs[C\_IN\ *G\_MC/\((1 + G\_MC)\)*\ C\_FA/H\_\(MC, PD\)*P\_IN\ *
            H\_\(IN, PD\)], 
        Abs[C\_IN\ *C\_MC\ *H\_\(MC, END\)*P\_IN\ *H\_\(IN, PD\)]}, \ {f, \ 
        10\^1, 10\^7}, \ Frame -> \ True, \ 
      PlotRange -> {{10\^1, 10\^7}, {10\^\(-2\), \ 10\^8}}, \ 
      PlotPoints\  -> \ 3000, \ 
      PlotStyle\  -> \ {RGBColor[0, 0, 0], RGBColor[1, 0, 0], 
          RGBColor[0, 0, 1]\ }, 
      GridLines -> {{{125000, {Dashing[{ .01,  .01}]}}}, {{1, {Dashing[{ .01,  .01}]}}}}, 
FrameLabel -> {"Frequency [Hz]", 
          Gain}]
\)
\(LogLinearPlot[
      Arg[\ C\_IN\ *\((G\_MC/\((1 + G\_MC)\)*C\_FA/H\_\(MC, PD\) + 
                P\_MC*C\_MC\ *H\_\(MC, END\))\)*P\_IN\ *
            H\_\(IN, PD\)]*180\/π, \ {f, \ 10\^1, 10\^7}, \ 
      Frame -> \ True, \ 
      PlotRange -> {{10\^1, 10\^7}, {\(-180\), \ 180}}, \ 
      PlotPoints\  -> \ 3000, \ 
      PlotStyle\  -> \ {RGBColor[1, 0, 0]}, 
      GridLines -> {{{125000, {Dashing[{ .01,  .01}]}}}, {{0, {Dashing[{ .01,  .01}]}}}}, 
FrameLabel -> {"Frequency [Hz]", 
          Phase[Deg]}]\)

Gain up

(Subscript[C, 0]) := 1.5*((1*(10^5) + f*I)/(f*I));  
(Subscript[C, IN]) := 
      2*((10^5)/((10^5) + f*I))*((1*(10^5) + f*I)/(f*I))*((3*(10^4) + f*I)/(f*I))*((3*(10^4) + f*I)/(f*I)); (* Gain up *)

\(LogLogPlot[{Abs[\ 
          C\_IN\ *\((G\_MC/\((1 + G\_MC)\)*C\_FA/H\_\(MC, PD\) + 
                P\_MC*C\_MC\ *H\_\(MC, END\))\)*P\_IN\ *H\_\(IN, PD\)], 
        Abs[C\_IN\ *G\_MC/\((1 + G\_MC)\)*\ C\_FA/H\_\(MC, PD\)*P\_IN\ *
            H\_\(IN, PD\)], 
        Abs[C\_IN\ *C\_MC\ *H\_\(MC, END\)*P\_IN\ *H\_\(IN, PD\)]}, \ {f, \ 
        10\^1, 10\^7}, \ Frame -> \ True, \ 
      PlotRange -> {{10\^1, 10\^7}, {10\^\(-2\), \ 10\^8}}, \ 
      PlotPoints\  -> \ 3000, \ 
      PlotStyle\  -> \ {RGBColor[0, 0, 0], RGBColor[1, 0, 0], 
          RGBColor[0, 0, 1]\ }, 
      GridLines -> {{{125000, {Dashing[{ .01,  .01}]}}}, {{1, {Dashing[{ .01,  .01}]}}}}, 
FrameLabel -> {"Frequency [Hz]", 
          Gain}]
\)
\(LogLinearPlot[
      Arg[\ C\_IN\ *\((G\_MC/\((1 + G\_MC)\)*C\_FA/H\_\(MC, PD\) + 
                P\_MC*C\_MC\ *H\_\(MC, END\))\)*P\_IN\ *
            H\_\(IN, PD\)]*180\/π, \ {f, \ 10\^1, 10\^7}, \ 
      Frame -> \ True, \ 
      PlotRange -> {{10\^1, 10\^7}, {\(-180\), \ 180}}, \ 
      PlotPoints\  -> \ 3000, \ 
      PlotStyle\  -> \ {RGBColor[1, 0, 0]}, 
      GridLines -> {{{125000, {Dashing[{ .01,  .01}]}}}, {{0, {Dashing[{ .01,  .01}]}}}}, 
FrameLabel -> {"Frequency [Hz]", 
          Phase[Deg]}]\)

Secondary Cavity Servo Loop Gain

\(LogLogPlot[{Abs[\ G\_Per]}, \ {f, \ 10\^1, 10\^4}, \ 
        Frame -> \ True, \ 
        PlotRange -> {{10\^1, 10\^4}, {10\^\(-2\), \ 10\^4}}, \ 
        PlotPoints\  -> \ 300, \ 
        PlotStyle\  -> \ RGBColor[0, 0, 0], 
        GridLines -> {{{400, {Dashing[{ .01,  .01}]}}}, {{1, {Dashing[{ .01,  .01}]}}}}, 
FrameLabel -> {"Frequency [Hz]", 
            Gain}]\)

LogLogPlot[{Abs[ (1 - (Subscript[G, Per]))/((Subscript[G, Per]))]},  {f,  
          (10^1), (10^4)},  Frame ->  True,  
        PlotRange ->  {{(10^1), (10^4)}, {(10^(-1)),  (10^3)}}, 
        PlotPoints  ->  300,  
        PlotStyle  ->  RGBColor[0, 0, 0], 
        GridLines ->  
          Automatic, 
FrameLabel -> {"Frequency [Hz]\
", Gain}]

  
GPPDper =  
        LogLogPlot[{Abs[ 
              W*(2*(10^(-7)))*(Subscript[C, Per])/(1 + (Subscript[G, Per]))]},  {f,  (10^1), 
            (10^4)},  Frame ->  True,  
          PlotRange ->  {{(10^1), (10^4)}, {(10^(-8)), (10^(-2))}}, 
          PlotPoints  ->  300,  
          PlotStyle  ->  RGBColor[1, 0, 0], 
          GridLines ->  
            Automatic, 
FrameLabel -> {"Frequency \
[Hz]", Gain}]

  
GPCoilNoise =  
        LogLogPlot[{Abs[ 
              W*(2*(10^(-7)))*(Subscript[G, Per])/(1 + (Subscript[G, Per]))]},  {f,  (10^1), 
            (10^4)},  Frame ->  True,  
          PlotRange ->  {{(10^1), (10^4)}, {(10^(-8)), (10^(-2))}}, 
          PlotPoints  ->  300,  
          PlotStyle  ->  RGBColor[0, 0, 1], 
          GridLines ->  
            Automatic, 
FrameLabel -> {"Frequency \
[Hz]", Gain}]

  
\(LogLogPlot[{\@\(Abs[\ W*\((2*10\^\(-7\))\)*G\_Per/\((1 + G\_Per)\)]\^2 + Abs[\ W*\((2*10\^\(-7\))\)*C\_Per/\((1 + G\_Per)\)]\^2\)}, \ {f, \ 10\^1, 10\^4}, \ Frame -> \ True, \ 
      PlotRange -> \ {{10\^1, 10\^4}, {10\^\(-8\), 10\^\(-2\)}}, 
      PlotPoints\  -> \ 300, \ PlotStyle\  -> \ RGBColor[0, 0, 1], 
      GridLines -> \ 
        Automatic, 
FrameLabel -> {"Frequency [Hz]", Gain}]\)

Noise Contributions

Noise Contributions(Lock Acquire Gain)

(Subscript[N, IN]),(Subscript[N, MC]),(Subscript[ν, MC]),\(と目標感度ν\_IN\)との比較.\(黄色 : ν\_IN\),\(黒色 : N\_IN\),\(赤色 : N\_MC\),ピンク色:(Subscript[ν, MC]).式から明らかなように,\(\(N\_MC\) \(やν\_MC\) は\),\(\(G\_IN\) を大きくとってやれば\),\(簡単に下げられることがわかります . 

しかし\),\(\(N\_IN\) は\),\(\(H\_\(IN, PD\)\) をあげるか\),\(\(N\_IN\) 自身を下げるしかありません . 
N\_IN  を入力換算雑音とすると\),\(\(N\_IN\) はPrimary\ Cavity\ Servo回路の初段のゲイン \((C\_IN)\) を上げられる設計にすれば\),\(普通数nV/
        rHzにはすることができますが\),そのためには,式
(Subscript[G, IN]) = 
      (Subscript[C, IN]) *(Subscript[P, IN]) *
        (Subscript[H, IN, PD])*(((Subscript[G, MC])/(1 + (Subscript[G, MC])))*((Subscript[C, FA])/(Subscript[H, MC, PD])) + 
            (Subscript[P, MC])*(Subscript[C, MC]) *
              (Subscript[H, MC, END]))
\(からわかるように, \(H\_\(MC, 
          PD\)\) を大きくしてやらないといけません . つまり, 
    MC用のPDに光をたくさん入れるか, 変調を大きくするかです  .2006  年5月18日までは, 
    MC用の変調が小さかったので, それを改善しました . もっと大きくしてもかまわないかもしれません . これで, \(回路の入力換算雑音としてのN\_IN\) はこれで落とせます . 
しかしPD自身の, 入力光がないときにすでにある雑音はこれでは減らせませんので, 現状だと, 数十nV/
      rHzの雑音が残ります . こうなると, もう, \(H\_\(IN, 
          PD\)\) を大きくすることでしか下げられません . しかし, \(G\_IN\) を一定に保つ制限があると, \(H\_\(IN, 
          PD\)\) を大きくすることと, 
    Primary\ Cavity\ Servo回路の初段のゲイン \((C\_IN)\) を上げることは, トレ - ドオフの関係になるので, 入力換算雑音を大きくしない程度に, \(C\_IN\) を設定せねばなりません . 結局それを解消するには, \(G\_IN\) のUGFをあげてル - プゲインをあげなければなりません . そうなると, \(結局G\_MC\) のUGFもあげていかねばなりません . こうなると, \(\(かなりしんどい領域に入ります\)\(.\)\(
\)\)\)

(Subscript[C, 0]) := 0.3;

    (Subscript[C, IN]) := 3*((10^5)/((10^5) + f*I)) ;

\(LogLogPlot[{L\_IN\/ν\_YAG*
          Abs[N\_IN*\((\(-\(\(C\_IN\ \((\(-C\_MC\)\ \((\(-1\) + 
                                  P\_MC)\)\ H\_\(MC, END\) + 
                            C\_0\ \((C\_EOM\ H\_EOM + 
                                  C\_PZT\ H\_PZT)\)\ P\_MC\ \((C\_FA + 
                                  C\_MC\ H\_\(MC, END\)\ H\_\(MC, PD\))\))\)\)\/\(1 - C\_IN\ C\_MC\ P\_IN\ \((\(-1\) + 
                              P\_MC)\)\ H\_\(IN, PD\)\ H\_\(MC, END\) + 
                        C\_0\ \((C\_EOM\ H\_EOM + 
                              C\_PZT\ H\_PZT)\)\ P\_MC\ \((C\_FA\ C\_IN\ P\_IN\ H\_\(IN, PD\) + \((1 + 
                                    C\_IN\ C\_MC\ P\_IN\ H\_\(IN, PD\)\ H\_\(MC, END\))\)\ H\_\(MC, PD\))\)\)\)\))\)], 
        L\_IN\/ν\_YAG*\ 
          Abs[N\_MC*\((\(-\(\(C\_0\ \((C\_EOM\ H\_EOM + 
                            C\_PZT\ H\_PZT)\)\ P\_MC\)\/\(1 - 
                        C\_IN\ C\_MC\ P\_IN\ \((\(-1\) + 
                              P\_MC)\)\ H\_\(IN, PD\)\ H\_\(MC, END\) + 
                        C\_0\ \((C\_EOM\ H\_EOM + 
                              C\_PZT\ H\_PZT)\)\ P\_MC\ \((C\_FA\ C\_IN\ P\_IN\ H\_\(IN, PD\) + \((1 + 
                                    C\_IN\ C\_MC\ P\_IN\ H\_\(IN, PD\)\ H\_\(MC, END\))\)\ H\_\(MC, PD\))\)\)\)\))\)], \ 
        L\_IN\/ν\_YAG*
          Abs[ν\_MC*\((\(1 + P\_MC\ \((\(-1\) + C\_0\ \((C\_EOM\ H\_EOM + C\_PZT\ H\_PZT)\)\ H\_\(MC, PD\))\)\)\/\(1 - C\_IN\ C\_MC\ P\_IN\ \((\(-1\) + P\_MC)\)\ H\_\(IN, PD\)\ H\_\(MC, END\) + C\_0\ \((C\_EOM\ H\_EOM + C\_PZT\ H\_PZT)\)\ P\_MC\ \((C\_FA\ C\_IN\ P\_IN\ H\_\(IN, PD\) + \((1 + C\_IN\ C\_MC\ P\_IN\ H\_\(IN, PD\)\ H\_\(MC, END\))\)\ H\_\(MC, PD\))\)\))\)], 
        L\_IN\/ν\_YAG*
          Abs[N\_MCEND*\((\(H\_\(MC, END\)\ \((1 + P\_MC\ \((\(-1\) + C\_0\ \((C\_EOM\ H\_EOM + C\_PZT\ H\_PZT)\)\ H\_\(MC, PD\))\))\)\)\/\(1 - C\_IN\ C\_MC\ P\_IN\ \((\(-1\) + P\_MC)\)\ H\_\(IN, PD\)\ H\_\(MC, END\) + C\_0\ \((C\_EOM\ H\_EOM + C\_PZT\ H\_PZT)\)\ P\_MC\ \((C\_FA\ C\_IN\ P\_IN\ H\_\(IN, PD\) + \((1 + C\_IN\ C\_MC\ P\_IN\ H\_\(IN, PD\)\ H\_\(MC, END\))\)\ H\_\(MC, PD\))\)\))\)], Abs[\((H\_INPend + 10\^\(-19\))\)/P\_IN]}, \ {f, \ 
        10\^1, 10\^5}, \ Frame -> \ True, \ 
      PlotRange -> {{10\^1, 10\^5}, {10\^\(-20\), \ 10\^\(-16\)}}, \ 
      PlotPoints\  -> \ 3000, \ 
      PlotStyle\  -> \ {RGBColor[0, 0, 0], \ RGBColor[1, 0, 0], \ 
          RGBColor[1, 0, 1], \ RGBColor[0, 1, 0.8], RGBColor[1, 1, 0]}, 
      GridLines -> {{{1000, {Dashing[{ .01,  .01}]}}}, {{1*10\^\(-18\), {Dashing[{ .01,  .01}]}}}}, 
FrameLabel -> {"Frequency [Hz]", Displacement\ [m\/rHz]}]\)

Noise Contributions(After GainUP)

(Subscript[C, 0]) := 0.3*((1*(10^5) + f*I)/(f*I));  
(Subscript[C, IN]) := 
        3*((10^5)/((10^5) + f*I))*((3*(10^4) + f*I)/(f*I))*((3*(10^4) + f*I)/(f*I))*((3*(10^4) + f*I)/(f*I)); (* 
      Gain @ 30  kHz and 30  kHz and 30  
        kHz *) 
 (*(Subscript[G, MC]) = 
        (Subscript[C, 0]) *((Subscript[C, PZT])*(Subscript[H, PZT]) + (Subscript[C, EOM]) *(Subscript[H, EOM]))*(Subscript[P, MC]) *
          (Subscript[H, MC, PD]); 
(Subscript[G, IN]) = 
        (Subscript[C, IN]) *(Subscript[P, IN]) *
          (Subscript[H, IN, PD])*(((Subscript[C, FA])/(Subscript[H, MC, PD])) + 
              (Subscript[P, MC])*(Subscript[C, MC]) *
                (Subscript[H, MC, END]));*) 

\(LogLogPlot[{L\_IN\/ν\_YAG*
          Abs[N\_IN*\((\(-\(\(C\_IN\ \((\(-C\_MC\)\ \((\(-1\) + 
                                  P\_MC)\)\ H\_\(MC, END\) + 
                            C\_0\ \((C\_EOM\ H\_EOM + 
                                  C\_PZT\ H\_PZT)\)\ P\_MC\ \((C\_FA + 
                                  C\_MC\ H\_\(MC, END\)\ H\_\(MC, PD\))\))\)\)\/\(1 - C\_IN\ C\_MC\ P\_IN\ \((\(-1\) + 
                              P\_MC)\)\ H\_\(IN, PD\)\ H\_\(MC, END\) + 
                        C\_0\ \((C\_EOM\ H\_EOM + 
                              C\_PZT\ H\_PZT)\)\ P\_MC\ \((C\_FA\ C\_IN\ P\_IN\ H\_\(IN, PD\) + \((1 + 
                                    C\_IN\ C\_MC\ P\_IN\ H\_\(IN, PD\)\ H\_\(MC, END\))\)\ H\_\(MC, PD\))\)\)\)\))\)], 
        L\_IN\/ν\_YAG*\ 
          Abs[N\_MC*\((\(-\(\(C\_0\ \((C\_EOM\ H\_EOM + 
                            C\_PZT\ H\_PZT)\)\ P\_MC\)\/\(1 - 
                        C\_IN\ C\_MC\ P\_IN\ \((\(-1\) + 
                              P\_MC)\)\ H\_\(IN, PD\)\ H\_\(MC, END\) + 
                        C\_0\ \((C\_EOM\ H\_EOM + 
                              C\_PZT\ H\_PZT)\)\ P\_MC\ \((C\_FA\ C\_IN\ P\_IN\ H\_\(IN, PD\) + \((1 + 
                                    C\_IN\ C\_MC\ P\_IN\ H\_\(IN, PD\)\ H\_\(MC, END\))\)\ H\_\(MC, PD\))\)\)\)\))\)], \ 
        L\_IN\/ν\_YAG*
          Abs[ν\_MC*\((\(1 + P\_MC\ \((\(-1\) + C\_0\ \((C\_EOM\ H\_EOM + C\_PZT\ H\_PZT)\)\ H\_\(MC, PD\))\)\)\/\(1 - C\_IN\ C\_MC\ P\_IN\ \((\(-1\) + P\_MC)\)\ H\_\(IN, PD\)\ H\_\(MC, END\) + C\_0\ \((C\_EOM\ H\_EOM + C\_PZT\ H\_PZT)\)\ P\_MC\ \((C\_FA\ C\_IN\ P\_IN\ H\_\(IN, PD\) + \((1 + C\_IN\ C\_MC\ P\_IN\ H\_\(IN, PD\)\ H\_\(MC, END\))\)\ H\_\(MC, PD\))\)\))\)], 
        L\_IN\/ν\_YAG*
          Abs[N\_MCEND*\((\(H\_\(MC, END\)\ \((1 + P\_MC\ \((\(-1\) + C\_0\ \((C\_EOM\ H\_EOM + C\_PZT\ H\_PZT)\)\ H\_\(MC, PD\))\))\)\)\/\(1 - C\_IN\ C\_MC\ P\_IN\ \((\(-1\) + P\_MC)\)\ H\_\(IN, PD\)\ H\_\(MC, END\) + C\_0\ \((C\_EOM\ H\_EOM + C\_PZT\ H\_PZT)\)\ P\_MC\ \((C\_FA\ C\_IN\ P\_IN\ H\_\(IN, PD\) + \((1 + C\_IN\ C\_MC\ P\_IN\ H\_\(IN, PD\)\ H\_\(MC, END\))\)\ H\_\(MC, PD\))\)\))\)], Abs[\((H\_INPend + 10\^\(-19\))\)/P\_IN]}, \ {f, \ 
        10\^1, 10\^5}, \ Frame -> \ True, \ 
      PlotRange -> {{10\^1, 10\^5}, {10\^\(-20\), \ 10\^\(-16\)}}, \ 
      PlotPoints\  -> \ 3000, \ 
      PlotStyle\  -> \ {RGBColor[0, 0, 0], \ RGBColor[1, 0, 0], \ 
          RGBColor[1, 0, 1], \ RGBColor[0, 1, 0.8], RGBColor[1, 1, 0]}, 
      GridLines -> {{{1000, {Dashing[{ .01,  .01}]}}}, {{1*10\^\(-18\), {Dashing[{ .01,  .01}]}}}}, 
FrameLabel -> {"Frequency [Hz]", Displacement\ [m\/rHz]}]\)