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(%i1) load("/home/vcaeken/Desktop/rfMaxima.mac")$

Result

Real Transformer

(See: http://www.analog-electronics.eu/analog-electronics/modified-nodal-analysis/modified-nodal-analysis.xhtml)

Symbol-M

1 4-Port Model of Real Transformer

4-Port Y-parameters of the real transformer

(%i2) declare(k,real)$
declare(L1,real)$
declare(L2,real)$
declare(Z_0,real)$
assume(k>0)$
assume(L1>0)$
assume(L2>0)$
assume(Z_0>0)$

k is the coupling factor of the transformer.
M is the mutual inductance of the transformer.

(%i10) M:k*sqrt(L1+L2);

Result

(%i11) Y_REAL_TRANSFORMER:1/(1-k^2)*matrix(
[1/(s*L1),-1/(s*L1),-k/(s*sqrt(L1*L2)),k/(s*sqrt(L1*L2))],
[-1/(s*L1),1/(s*L1),k/(s*sqrt(L1*L2)),-k/(s*sqrt(L1*L2))],
[-k/(s*sqrt(L1*L2)),k/(s*sqrt(L1*L2)),1/(s*L2),-1/(s*L2)],
[k/(s*sqrt(L1*L2)),-k/(s*sqrt(L1*L2)),-1/(s*L2),1/(s*L2)]);

Result

(%i12) Y_REAL_TRANSFORMER.matrix([v1],[v2],[v3],[v4]);

Result

2-Port parameters of the real transformer

(%i13) Solution:linsolve([
i1=(k*v4)/((1-k^2)*s*sqrt(L1*L2))-(k*v3)/((1-k^2)*s*sqrt(L1*L2))-v2/((1-k^2)*s*L1)+v1/((1-k^2)*s*L1),
i2=-(k*v4)/((1-k^2)*s*sqrt(L1*L2))+(k*v3)/((1-k^2)*s*sqrt(L1*L2))+v2/((1-k^2)*s*L1)-v1/((1-k^2)*s*L1),
i3=(k*v2)/((1-k^2)*s*sqrt(L1*L2))-(k*v1)/((1-k^2)*s*sqrt(L1*L2))-v4/((1-k^2)*s*L2)+v3/((1-k^2)*s*L2),
i4=-(k*v2)/((1-k^2)*s*sqrt(L1*L2))+(k*v1)/((1-k^2)*s*sqrt(L1*L2))+v4/((1-k^2)*s*L2)-v3/((1-k^2)*s*L2),
v2=0,
v4=0,
i2=-i1,
i4=-i3,
v1=v_in,
v3=v_out,
i1=i_in,
i3=i_out],[i1,i2,i3,i4,v1,v2,v3,v4,i_in,i_out]);

Result

(%i14) TwoPortMatrices(Solution);

Result

(%i15) SParameters(S);

Result

2 2-Port Model of Capacitively-Tuned Real Transformer

(%i16) declare(C,real)$
assume(C>0)$

(%i18) Solution:linsolve([
i1=(k*v4)/((1-k^2)*s*sqrt(L1*L2))-(k*v3)/((1-k^2)*s*sqrt(L1*L2))-v2/((1-k^2)*s*L1)+v1/((1-k^2)*s*L1),
i2=-(k*v4)/((1-k^2)*s*sqrt(L1*L2))+(k*v3)/((1-k^2)*s*sqrt(L1*L2))+v2/((1-k^2)*s*L1)-v1/((1-k^2)*s*L1),
i3=(k*v2)/((1-k^2)*s*sqrt(L1*L2))-(k*v1)/((1-k^2)*s*sqrt(L1*L2))-v4/((1-k^2)*s*L2)+v3/((1-k^2)*s*L2),
i4=-(k*v2)/((1-k^2)*s*sqrt(L1*L2))+(k*v1)/((1-k^2)*s*sqrt(L1*L2))+v4/((1-k^2)*s*L2)-v3/((1-k^2)*s*L2),
v2=0,
v4=0,
i2=-i1,
i4=-i3,
v_in=v1,
v_out=v3,
v_in=1/(s*C)*i5,
i_in=i5+i1,
i_out=i3],[i1,i2,i3,i4,i5,v1,v2,v3,v4,i_in,i_out]);

Result

(%i19) TwoPortMatrices(Solution);

Result

(%i20) SParameters(S);

Result

3 2-Port Model of Capacitively-Tuned Real and Lossy Transformer

(%i21) declare(R1,real)$
declare(R2,real)$
assume(R1>0)$
assume(R2>0)$

(%i25) Solution:linsolve([
i1=(k*v4)/((1-k^2)*s*sqrt(L1*L2))-(k*v3)/((1-k^2)*s*sqrt(L1*L2))-v2/((1-k^2)*s*L1)+v1/((1-k^2)*s*L1),
i2=-(k*v4)/((1-k^2)*s*sqrt(L1*L2))+(k*v3)/((1-k^2)*s*sqrt(L1*L2))+v2/((1-k^2)*s*L1)-v1/((1-k^2)*s*L1),
i3=(k*v2)/((1-k^2)*s*sqrt(L1*L2))-(k*v1)/((1-k^2)*s*sqrt(L1*L2))-v4/((1-k^2)*s*L2)+v3/((1-k^2)*s*L2),
i4=-(k*v2)/((1-k^2)*s*sqrt(L1*L2))+(k*v1)/((1-k^2)*s*sqrt(L1*L2))+v4/((1-k^2)*s*L2)-v3/((1-k^2)*s*L2),
v2=0,
v4=0,
i2=-i1,
i4=-i3,
v0-v1=R1*i_in,
v_in=v0,
v_out=v5,
v5-v3=R2*i_out,
v_in=1/(s*C)*i5,
i_in=i5+i1,
i_out=i3],[i1,i2,i3,i4,i5,v0,v1,v2,v3,v4,v5,i_in,i_out]);

Result

(%i26) TwoPortMatrices(Solution);

Result

(%i27) SParameters(S);

Result

(%i28) L1:2e-9$
L2:2e-9$
k:0.9$
Z_0:50$

(%i32) fres:2e9$
C:1/((2*%pi*fres)^2*L1)$
float(C);
Q1:10$
Q2:10$
R1:2*%pi*fres*L1/Q1$
float(R1);
R2:2*%pi*fres*L2/Q2$
float(R2);

Result

(%i41) PlotSmithChart([
    parametric,
    realpart(S(%i*2*%pi*f)[1,1]),
    imagpart(S(%i*2*%pi*f)[1,1]),
    [f,1e6,20e9],
    [nticks,100]]);
PlotSmithChart([
    parametric,
    realpart(S(%i*2*%pi*f)[1,2]),
    imagpart(S(%i*2*%pi*f)[1,2]),
    [f,1e6,20e9],
    [nticks,100]]);
PlotSmithChart([
    parametric,
    realpart(S(%i*2*%pi*f)[2,1]),
    imagpart(S(%i*2*%pi*f)[2,1]),
    [f,1e6,20e9],
    [nticks,100]]);
PlotSmithChart([
    parametric,
    realpart(S(%i*2*%pi*f)[2,2]),
    imagpart(S(%i*2*%pi*f)[2,2]),
    [f,1e6,20e9],
    [nticks,100]]);

Result

(%i45) wxplot2d(ev(20*log10(Abs_S11(2*%pi*f*1e9))),
[f,0.1,20],
[legend,false],
[xlabel,"Frequency [GHz]"],
[ylabel,"dB20(|S11|)"],
[style,[lines,3,2]],
[gnuplot_preamble,"set grid"])$
wxplot2d(ev(20*log10(Abs_S12(2*%pi*f*1e9))),
[f,0.1,20],
[legend,false],
[xlabel,"Frequency [GHz]"],
[ylabel,"dB20(|S12|)"],
[style,[lines,3,2]],
[gnuplot_preamble,"set grid"])$
wxplot2d(ev(20*log10(Abs_S21(2*%pi*f*1e9))),
[f,0.1,20],
[legend,false],
[xlabel,"Frequency [GHz]"],
[ylabel,"dB20(|S21|)"],
[style,[lines,3,2]],
[gnuplot_preamble,"set grid"])$
wxplot2d(ev(20*log10(Abs_S22(2*%pi*f*1e9))),
[f,0.1,20],
[legend,false],
[xlabel,"Frequency [GHz]"],
[ylabel,"dB20(|S22|)"],
[style,[lines,3,2]],
[gnuplot_preamble,"set grid"])$
wxplot2d(ev(Arg_S11(2*%pi*f*1e9)*180/%pi),
[f,0.1,20],
[legend,false],
[xlabel,"Frequency [GHz]"],
[ylabel,"S11 (°)"],
[style,[lines,3,2]],
[gnuplot_preamble,"set grid"])$
wxplot2d(ev(Arg_S12(2*%pi*f*1e9)*180/%pi),
[f,0.1,20],
[legend,false],
[xlabel,"Frequency [GHz]"],
[ylabel,"S12 (°)"],
[style,[lines,3,2]],
[gnuplot_preamble,"set grid"])$
wxplot2d(ev(Arg_S21(2*%pi*f*1e9)*180/%pi),
[f,0.1,20],
[legend,false],
[xlabel,"Frequency [GHz]"],
[ylabel,"S21 (°)"],
[style,[lines,3,2]],
[gnuplot_preamble,"set grid"])$
wxplot2d(ev(Arg_S22(2*%pi*f*1e9)*180/%pi),
[f,0.1,20],
[legend,false],
[xlabel,"Frequency [GHz]"],
[ylabel,"S22 (°)"],
[style,[lines,3,2]],
[gnuplot_preamble,"set grid"])$

Result

(%i53) wxplot2d(ev((R_IN(2*%pi*f*1e9))),
[f,0.1,20],
[legend,false],
[xlabel,"Frequency [GHz]"],
[ylabel,"R_IN [Ohm]"],
[style,[lines,3,2]],
[gnuplot_preamble,"set grid"])$
wxplot2d(ev((X_IN(2*%pi*f*1e9))),
[f,0.1,20],
[legend,false],
[xlabel,"Frequency [GHz]"],
[ylabel,"X_IN [Ohm]"],
[style,[lines,3,2]],
[gnuplot_preamble,"set grid"]);
wxplot2d(ev((R_OUT(2*%pi*f*1e9))),
[f,0.1,20],
[legend,false],
[xlabel,"Frequency [GHz]"],
[ylabel,"R_OUT [Ohm]"],
[style,[lines,3,2]],
[gnuplot_preamble,"set grid"])$
wxplot2d(ev((X_OUT(2*%pi*f*1e9))),
[f,0.1,20],
[legend,false],
[xlabel,"Frequency [GHz]"],
[ylabel,"X_OUT [Ohm]"],
[style,[lines,3,2]],
[gnuplot_preamble,"set grid"])$
wxplot2d(ev((tau_g(2*%pi*f*1e9))*1e12),
[f,0.1,20],
[legend,false],
[xlabel,"Frequency [GHz]"],
[ylabel,"Group Delay [pS]"],
[style,[lines,3,2]],
[gnuplot_preamble,"set grid"])$

Result


Created with wxMaxima.