Real Transformer

Generated from LOSSY_TRANSFORMER.wxm with rfMaxima 0.2.5 on Maxima 5.46.0. All results were computed by executing the notebook; none is transcribed.

/* Put rfMaxima.mac in the same directory as this notebook, or add its
   directory to file_search_maxima. */
load("rfMaxima.mac")$

(See: https://analog-electronics.ewi.tudelft.nl/webbook/SED/_build/html/network_theory/NetworkTheory-1_Modified_Nodal_Analysis.html)

4-Port Model of Real Transformer

4-Port Y-parameters of the real transformer

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.

/* Mutual inductance of two coupled coils: M = k*sqrt(L1*L2).
   0.2.4 had sqrt(L1+L2) here.  (The 4-port Y-matrix below always used
   sqrt(L1*L2) correctly, so only this displayed definition was wrong.) */
M:k*sqrt(L1*L2);
L1 L2 k\sqrt{L_{1}}\,\sqrt{L_{2}}\,k
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)]);
(1L1 (1−k2) s−1L1 (1−k2) s−kL1 L2 (1−k2) skL1 L2 (1−k2) s−1L1 (1−k2) s1L1 (1−k2) skL1 L2 (1−k2) s−kL1 L2 (1−k2) s−kL1 L2 (1−k2) skL1 L2 (1−k2) s1L2 (1−k2) s−1L2 (1−k2) skL1 L2 (1−k2) s−kL1 L2 (1−k2) s−1L2 (1−k2) s1L2 (1−k2) s)\begin{pmatrix} {{1}\over{L_{1}\,\left(1-k^2\right)\,s}}&-{{1}\over{L_{1}\,\left(1-k^2\right)\,s}}&-{{k}\over{\sqrt{L_{1}}\,\sqrt{L_{2}}\,\left(1-k^2\right)\,s}}&{{k}\over{\sqrt{L_{1}}\,\sqrt{L_{2}}\,\left(1-k^2\right)\,s}}\\ -{{1}\over{L_{1}\,\left(1-k^2\right)\,s}}&{{1}\over{L_{1}\,\left(1-k^2\right)\,s}}&{{k}\over{\sqrt{L_{1}}\,\sqrt{L_{2}}\,\left(1-k^2\right)\,s}}&-{{k}\over{\sqrt{L_{1}}\,\sqrt{L_{2}}\,\left(1-k^2\right)\,s}}\\ -{{k}\over{\sqrt{L_{1}}\,\sqrt{L_{2}}\,\left(1-k^2\right)\,s}}&{{k}\over{\sqrt{L_{1}}\,\sqrt{L_{2}}\,\left(1-k^2\right)\,s}}&{{1}\over{L_{2}\,\left(1-k^2\right)\,s}}&-{{1}\over{L_{2}\,\left(1-k^2\right)\,s}}\\ {{k}\over{\sqrt{L_{1}}\,\sqrt{L_{2}}\,\left(1-k^2\right)\,s}}&-{{k}\over{\sqrt{L_{1}}\,\sqrt{L_{2}}\,\left(1-k^2\right)\,s}}&-{{1}\over{L_{2}\,\left(1-k^2\right)\,s}}&{{1}\over{L_{2}\,\left(1-k^2\right)\,s}} \end{pmatrix}
Y_REAL_TRANSFORMER.matrix([v1],[v2],[v3],[v4]);
(k v4L1 L2 (1−k2) s−k v3L1 L2 (1−k2) s−v2L1 (1−k2) s+v1L1 (1−k2) s−k v4L1 L2 (1−k2) s+k v3L1 L2 (1−k2) s+v2L1 (1−k2) s−v1L1 (1−k2) s−v4L2 (1−k2) s+v3L2 (1−k2) s+k v2L1 L2 (1−k2) s−k v1L1 L2 (1−k2) sv4L2 (1−k2) s−v3L2 (1−k2) s−k v2L1 L2 (1−k2) s+k v1L1 L2 (1−k2) s)\begin{pmatrix} {{k\,v_{4}}\over{\sqrt{L_{1}}\,\sqrt{L_{2}}\,\left(1-k^2\right)\,s}}-{{k\,v_{3}}\over{\sqrt{L_{1}}\,\sqrt{L_{2}}\,\left(1-k^2\right)\,s}}-{{v_{2}}\over{L_{1}\,\left(1-k^2\right)\,s}}+{{v_{1}}\over{L_{1}\,\left(1-k^2\right)\,s}}\\ -{{k\,v_{4}}\over{\sqrt{L_{1}}\,\sqrt{L_{2}}\,\left(1-k^2\right)\,s}}+{{k\,v_{3}}\over{\sqrt{L_{1}}\,\sqrt{L_{2}}\,\left(1-k^2\right)\,s}}+{{v_{2}}\over{L_{1}\,\left(1-k^2\right)\,s}}-{{v_{1}}\over{L_{1}\,\left(1-k^2\right)\,s}}\\ -{{v_{4}}\over{L_{2}\,\left(1-k^2\right)\,s}}+{{v_{3}}\over{L_{2}\,\left(1-k^2\right)\,s}}+{{k\,v_{2}}\over{\sqrt{L_{1}}\,\sqrt{L_{2}}\,\left(1-k^2\right)\,s}}-{{k\,v_{1}}\over{\sqrt{L_{1}}\,\sqrt{L_{2}}\,\left(1-k^2\right)\,s}}\\ {{v_{4}}\over{L_{2}\,\left(1-k^2\right)\,s}}-{{v_{3}}\over{L_{2}\,\left(1-k^2\right)\,s}}-{{k\,v_{2}}\over{\sqrt{L_{1}}\,\sqrt{L_{2}}\,\left(1-k^2\right)\,s}}+{{k\,v_{1}}\over{\sqrt{L_{1}}\,\sqrt{L_{2}}\,\left(1-k^2\right)\,s}} \end{pmatrix}

2-Port parameters of the real transformer

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]);
[i1=L1 L2 k vout−L2 vin(L1 L2 k2−L1 L2) s,i2=−L1 L2 k vout−L2 vin(L1 L2 k2−L1 L2) s,i3=L1 L2 k vin−L1 vout(L1 L2 k2−L1 L2) s,i4=−L1 L2 k vin−L1 vout(L1 L2 k2−L1 L2) s,v1=vin,v2=0,v3=vout,v4=0,iin=L1 L2 k vout−L2 vin(L1 L2 k2−L1 L2) s,iout=L1 L2 k vin−L1 vout(L1 L2 k2−L1 L2) s]\left[ i_{1}={{\sqrt{L_{1}}\,\sqrt{L_{2}}\,k\,\mathit{v}_{out}-L_{2}\,\mathit{v}_{in}}\over{\left(L_{1}\,L_{2}\,k^2-L_{1}\,L_{2}\right)\,s}} , i_{2}=-{{\sqrt{L_{1}}\,\sqrt{L_{2}}\,k\,\mathit{v}_{out}-L_{2}\,\mathit{v}_{in}}\over{\left(L_{1}\,L_{2}\,k^2-L_{1}\,L_{2}\right)\,s}} , i_{3}={{\sqrt{L_{1}}\,\sqrt{L_{2}}\,k\,\mathit{v}_{in}-L_{1}\,\mathit{v}_{out}}\over{\left(L_{1}\,L_{2}\,k^2-L_{1}\,L_{2}\right)\,s}} , i_{4}=-{{\sqrt{L_{1}}\,\sqrt{L_{2}}\,k\,\mathit{v}_{in}-L_{1}\,\mathit{v}_{out}}\over{\left(L_{1}\,L_{2}\,k^2-L_{1}\,L_{2}\right)\,s}} , v_{1}=\mathit{v}_{in} , v_{2}=0 , v_{3}=\mathit{v}_{out} , v_{4}=0 , \mathit{i}_{in}={{\sqrt{L_{1}}\,\sqrt{L_{2}}\,k\,\mathit{v}_{out}-L_{2}\,\mathit{v}_{in}}\over{\left(L_{1}\,L_{2}\,k^2-L_{1}\,L_{2}\right)\,s}} , \mathit{i}_{out}={{\sqrt{L_{1}}\,\sqrt{L_{2}}\,k\,\mathit{v}_{in}-L_{1}\,\mathit{v}_{out}}\over{\left(L_{1}\,L_{2}\,k^2-L_{1}\,L_{2}\right)\,s}} \right]
TwoPortMatrices(Solution);
(L1 sL1 L2 k sL1 L2 k sL2 s)\begin{pmatrix} L_{1}\,s&\sqrt{L_{1}}\,\sqrt{L_{2}}\,k\,s\\ \sqrt{L_{1}}\,\sqrt{L_{2}}\,k\,s&L_{2}\,s \end{pmatrix}
ABCD(s) := (L1L2 k−(L1 L2 k2−L1 L2) sL1 L2 k1L1 L2 k sL2L1 k)\begin{pmatrix} {{\sqrt{L_{1}}}\over{\sqrt{L_{2}}\,k}}&-{{\left(L_{1}\,L_{2}\,k^2-L_{1}\,L_{2}\right)\,s}\over{\sqrt{L_{1}}\,\sqrt{L_{2}}\,k}}\\ {{1}\over{\sqrt{L_{1}}\,\sqrt{L_{2}}\,k\,s}}&{{\sqrt{L_{2}}}\over{\sqrt{L_{1}}\,k}} \end{pmatrix}
G(s) := (1L1 s−L2 kL1L2 kL1(L2−L2 k2) s)\begin{pmatrix} {{1}\over{L_{1}\,s}}&-{{\sqrt{L_{2}}\,k}\over{\sqrt{L_{1}}}}\\ {{\sqrt{L_{2}}\,k}\over{\sqrt{L_{1}}}}&\left(L_{2}-L_{2}\,k^2\right)\,s \end{pmatrix}
H(s) := ((L1−L1 k2) sL1 kL2−L1 kL21L2 s)\begin{pmatrix} \left(L_{1}-L_{1}\,k^2\right)\,s&{{\sqrt{L_{1}}\,k}\over{\sqrt{L_{2}}}}\\ -{{\sqrt{L_{1}}\,k}\over{\sqrt{L_{2}}}}&{{1}\over{L_{2}\,s}} \end{pmatrix}
InverseABCD(s) := (L2L1 k−(L1 L2 k2−L1 L2) sL1 L2 k1L1 L2 k sL1L2 k)\begin{pmatrix} {{\sqrt{L_{2}}}\over{\sqrt{L_{1}}\,k}}&-{{\left(L_{1}\,L_{2}\,k^2-L_{1}\,L_{2}\right)\,s}\over{\sqrt{L_{1}}\,\sqrt{L_{2}}\,k}}\\ {{1}\over{\sqrt{L_{1}}\,\sqrt{L_{2}}\,k\,s}}&{{\sqrt{L_{1}}}\over{\sqrt{L_{2}}\,k}} \end{pmatrix}
S(s) := ((L1 L2 k2−L1 L2) s2+(L2−L1) Z0 s+Z02(L1 L2 k2−L1 L2) s2+(−L2−L1) Z0 s−Z02−2 L1 L2 Z0 k s(L1 L2 k2−L1 L2) s2+(−L2−L1) Z0 s−Z02−2 L1 L2 Z0 k s(L1 L2 k2−L1 L2) s2+(−L2−L1) Z0 s−Z02(L1 L2 k2−L1 L2) s2+(L1−L2) Z0 s+Z02(L1 L2 k2−L1 L2) s2+(−L2−L1) Z0 s−Z02)\begin{pmatrix} {{\left(L_{1}\,L_{2}\,k^2-L_{1}\,L_{2}\right)\,s^2+\left(L_{2}-L_{1}\right)\,Z_{0}\,s+Z_{0}^2}\over{\left(L_{1}\,L_{2}\,k^2-L_{1}\,L_{2}\right)\,s^2+\left(-L_{2}-L_{1}\right)\,Z_{0}\,s-Z_{0}^2}}&-{{2\,\sqrt{L_{1}}\,\sqrt{L_{2}}\,Z_{0}\,k\,s}\over{\left(L_{1}\,L_{2}\,k^2-L_{1}\,L_{2}\right)\,s^2+\left(-L_{2}-L_{1}\right)\,Z_{0}\,s-Z_{0}^2}}\\ -{{2\,\sqrt{L_{1}}\,\sqrt{L_{2}}\,Z_{0}\,k\,s}\over{\left(L_{1}\,L_{2}\,k^2-L_{1}\,L_{2}\right)\,s^2+\left(-L_{2}-L_{1}\right)\,Z_{0}\,s-Z_{0}^2}}&{{\left(L_{1}\,L_{2}\,k^2-L_{1}\,L_{2}\right)\,s^2+\left(L_{1}-L_{2}\right)\,Z_{0}\,s+Z_{0}^2}\over{\left(L_{1}\,L_{2}\,k^2-L_{1}\,L_{2}\right)\,s^2+\left(-L_{2}-L_{1}\right)\,Z_{0}\,s-Z_{0}^2}} \end{pmatrix}
Y(s) := (−1(L1 k2−L1) sL1 L2 k(L1 L2 k2−L1 L2) sL1 L2 k(L1 L2 k2−L1 L2) s−1(L2 k2−L2) s)\begin{pmatrix} -{{1}\over{\left(L_{1}\,k^2-L_{1}\right)\,s}}&{{\sqrt{L_{1}}\,\sqrt{L_{2}}\,k}\over{\left(L_{1}\,L_{2}\,k^2-L_{1}\,L_{2}\right)\,s}}\\ {{\sqrt{L_{1}}\,\sqrt{L_{2}}\,k}\over{\left(L_{1}\,L_{2}\,k^2-L_{1}\,L_{2}\right)\,s}}&-{{1}\over{\left(L_{2}\,k^2-L_{2}\right)\,s}} \end{pmatrix}
Z(s) := (L1 sL1 L2 k sL1 L2 k sL2 s)\begin{pmatrix} L_{1}\,s&\sqrt{L_{1}}\,\sqrt{L_{2}}\,k\,s\\ \sqrt{L_{1}}\,\sqrt{L_{2}}\,k\,s&L_{2}\,s \end{pmatrix}
SParameters(S);
−ω3 L12 L2 k2+ω L2 Z02+ω3 L12 L2Z02+ω2 L12{{-\omega^3\,L_{1}^2\,L_{2}\,k^2+\omega\,L_{2}\,Z_{0}^2+\omega^3\,L_{1}^2\,L_{2}}\over{Z_{0}^2+\omega^2\,L_{1}^2}}
Abs_S11(%omega) := (ω2 L1 L2 k2−Z02−ω2 L1 L2)2+(ω L1 Z0−ω L2 Z0)2(ω2 L1 L2 k2+Z02−ω2 L1 L2)2+(ω L2 Z0+ω L1 Z0)2{{\sqrt{\left(\omega^2\,L_{1}\,L_{2}\,k^2-Z_{0}^2-\omega^2\,L_{1}\,L_{2}\right)^2+\left(\omega\,L_{1}\,Z_{0}-\omega\,L_{2}\,Z_{0}\right)^2}}\over{\sqrt{\left(\omega^2\,L_{1}\,L_{2}\,k^2+Z_{0}^2-\omega^2\,L_{1}\,L_{2}\right)^2+\left(\omega\,L_{2}\,Z_{0}+\omega\,L_{1}\,Z_{0}\right)^2}}}
Arg_S11(%omega) := −atan2(2 ω3 L1 L22 Z0 k2−2 ω L1 Z03−2 ω3 L1 L22 Z0,ω4 L12 L22 k4−2 ω4 L12 L22 k2−Z04+(ω2 L12−ω2 L22) Z02+ω4 L12 L22)-{\rm atan2}\left(2\,\omega^3\,L_{1}\,L_{2}^2\,Z_{0}\,k^2-2\,\omega\,L_{1}\,Z_{0}^3-2\,\omega^3\,L_{1}\,L_{2}^2\,Z_{0} , \omega^4\,L_{1}^2\,L_{2}^2\,k^4-2\,\omega^4\,L_{1}^2\,L_{2}^2\,k^2-Z_{0}^4+\left(\omega^2\,L_{1}^2-\omega^2\,L_{2}^2\right)\,Z_{0}^2+\omega^4\,L_{1}^2\,L_{2}^2\right)
Abs_S12(%omega) := 2 ∣ω∣ L1 L2 Z0 k(ω2 L1 L2 k2+Z02−ω2 L1 L2)2+(ω L2 Z0+ω L1 Z0)2{{2\,\left| \omega\right| \,\sqrt{L_{1}}\,\sqrt{L_{2}}\,Z_{0}\,k}\over{\sqrt{\left(\omega^2\,L_{1}\,L_{2}\,k^2+Z_{0}^2-\omega^2\,L_{1}\,L_{2}\right)^2+\left(\omega\,L_{2}\,Z_{0}+\omega\,L_{1}\,Z_{0}\right)^2}}}
Arg_S12(%omega) := atan2(L1 L2 (2 ω3 L1 L2 Z0 k3+(2 ω Z03−2 ω3 L1 L2 Z0) k),L1 L2 (2 ω2 L2+2 ω2 L1) Z02 k){\rm atan2}\left(\sqrt{L_{1}}\,\sqrt{L_{2}}\,\left(2\,\omega^3\,L_{1}\,L_{2}\,Z_{0}\,k^3+\left(2\,\omega\,Z_{0}^3-2\,\omega^3\,L_{1}\,L_{2}\,Z_{0}\right)\,k\right) , \sqrt{L_{1}}\,\sqrt{L_{2}}\,\left(2\,\omega^2\,L_{2}+2\,\omega^2\,L_{1}\right)\,Z_{0}^2\,k\right)
Abs_S21(%omega) := 2 ∣ω∣ L1 L2 Z0 k(ω2 L1 L2 k2+Z02−ω2 L1 L2)2+(ω L2 Z0+ω L1 Z0)2{{2\,\left| \omega\right| \,\sqrt{L_{1}}\,\sqrt{L_{2}}\,Z_{0}\,k}\over{\sqrt{\left(\omega^2\,L_{1}\,L_{2}\,k^2+Z_{0}^2-\omega^2\,L_{1}\,L_{2}\right)^2+\left(\omega\,L_{2}\,Z_{0}+\omega\,L_{1}\,Z_{0}\right)^2}}}
Arg_S21(%omega) := atan2(L1 L2 (2 ω3 L1 L2 Z0 k3+(2 ω Z03−2 ω3 L1 L2 Z0) k),L1 L2 (2 ω2 L2+2 ω2 L1) Z02 k){\rm atan2}\left(\sqrt{L_{1}}\,\sqrt{L_{2}}\,\left(2\,\omega^3\,L_{1}\,L_{2}\,Z_{0}\,k^3+\left(2\,\omega\,Z_{0}^3-2\,\omega^3\,L_{1}\,L_{2}\,Z_{0}\right)\,k\right) , \sqrt{L_{1}}\,\sqrt{L_{2}}\,\left(2\,\omega^2\,L_{2}+2\,\omega^2\,L_{1}\right)\,Z_{0}^2\,k\right)
tau_g(%omega) := −(ω2 L1 L22+ω2 L12 L2) Z0 k2+(−L2−L1) Z03+(−ω2 L1 L22−ω2 L12 L2) Z0ω4 L12 L22 k4+(2 ω2 L1 L2 Z02−2 ω4 L12 L22) k2+Z04+(ω2 L22+ω2 L12) Z02+ω4 L12 L22-{{\left(\omega^2\,L_{1}\,L_{2}^2+\omega^2\,L_{1}^2\,L_{2}\right)\,Z_{0}\,k^2+\left(-L_{2}-L_{1}\right)\,Z_{0}^3+\left(-\omega^2\,L_{1}\,L_{2}^2-\omega^2\,L_{1}^2\,L_{2}\right)\,Z_{0}}\over{\omega^4\,L_{1}^2\,L_{2}^2\,k^4+\left(2\,\omega^2\,L_{1}\,L_{2}\,Z_{0}^2-2\,\omega^4\,L_{1}^2\,L_{2}^2\right)\,k^2+Z_{0}^4+\left(\omega^2\,L_{2}^2+\omega^2\,L_{1}^2\right)\,Z_{0}^2+\omega^4\,L_{1}^2\,L_{2}^2}}
Abs_S22(%omega) := (ω2 L1 L2 k2−Z02−ω2 L1 L2)2+(ω L2 Z0−ω L1 Z0)2(ω2 L1 L2 k2+Z02−ω2 L1 L2)2+(ω L2 Z0+ω L1 Z0)2{{\sqrt{\left(\omega^2\,L_{1}\,L_{2}\,k^2-Z_{0}^2-\omega^2\,L_{1}\,L_{2}\right)^2+\left(\omega\,L_{2}\,Z_{0}-\omega\,L_{1}\,Z_{0}\right)^2}}\over{\sqrt{\left(\omega^2\,L_{1}\,L_{2}\,k^2+Z_{0}^2-\omega^2\,L_{1}\,L_{2}\right)^2+\left(\omega\,L_{2}\,Z_{0}+\omega\,L_{1}\,Z_{0}\right)^2}}}
Arg_S22(%omega) := −atan2(2 ω3 L12 L2 Z0 k2−2 ω L2 Z03−2 ω3 L12 L2 Z0,ω4 L12 L22 k4−2 ω4 L12 L22 k2−Z04+(ω2 L22−ω2 L12) Z02+ω4 L12 L22)-{\rm atan2}\left(2\,\omega^3\,L_{1}^2\,L_{2}\,Z_{0}\,k^2-2\,\omega\,L_{2}\,Z_{0}^3-2\,\omega^3\,L_{1}^2\,L_{2}\,Z_{0} , \omega^4\,L_{1}^2\,L_{2}^2\,k^4-2\,\omega^4\,L_{1}^2\,L_{2}^2\,k^2-Z_{0}^4+\left(\omega^2\,L_{2}^2-\omega^2\,L_{1}^2\right)\,Z_{0}^2+\omega^4\,L_{1}^2\,L_{2}^2\right)
Z_IN(s) := −(L1 L2 k2−L1 L2) s2−L1 Z0 sL2 s+Z0-{{\left(L_{1}\,L_{2}\,k^2-L_{1}\,L_{2}\right)\,s^2-L_{1}\,Z_{0}\,s}\over{L_{2}\,s+Z_{0}}}
R_IN(%omega) := ω2 L1 L2 Z0 k2Z02+ω2 L22{{\omega^2\,L_{1}\,L_{2}\,Z_{0}\,k^2}\over{Z_{0}^2+\omega^2\,L_{2}^2}}
X_IN(%omega) := −ω3 L1 L22 k2+ω L1 Z02+ω3 L1 L22Z02+ω2 L22{{-\omega^3\,L_{1}\,L_{2}^2\,k^2+\omega\,L_{1}\,Z_{0}^2+\omega^3\,L_{1}\,L_{2}^2}\over{Z_{0}^2+\omega^2\,L_{2}^2}}
Z_OUT(s) := −(L1 L2 k2−L1 L2) s2−L2 Z0 sL1 s+Z0-{{\left(L_{1}\,L_{2}\,k^2-L_{1}\,L_{2}\right)\,s^2-L_{2}\,Z_{0}\,s}\over{L_{1}\,s+Z_{0}}}
R_OUT(%omega) := ω2 L1 L2 Z0 k2Z02+ω2 L12{{\omega^2\,L_{1}\,L_{2}\,Z_{0}\,k^2}\over{Z_{0}^2+\omega^2\,L_{1}^2}}
X_OUT(%omega) := −ω3 L12 L2 k2+ω L2 Z02+ω3 L12 L2Z02+ω2 L12{{-\omega^3\,L_{1}^2\,L_{2}\,k^2+\omega\,L_{2}\,Z_{0}^2+\omega^3\,L_{1}^2\,L_{2}}\over{Z_{0}^2+\omega^2\,L_{1}^2}}

2-Port Model of Capacitively-Tuned Real Transformer

declare(C,real)$
assume(C>0)$
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]);
[i1=L1 L2 k vout−L2 vin(L1 L2 k2−L1 L2) s,i2=−L1 L2 k vout−L2 vin(L1 L2 k2−L1 L2) s,i3=L1 L2 k vin−L1 vout(L1 L2 k2−L1 L2) s,i4=−L1 L2 k vin−L1 vout(L1 L2 k2−L1 L2) s,i5=C s vin,v1=vin,v2=0,v3=vout,v4=0,iin=L1 L2 k vout+s2 (C L1 L2 k2 vin−C L1 L2 vin)−L2 vin(L1 L2 k2−L1 L2) s,iout=L1 L2 k vin−L1 vout(L1 L2 k2−L1 L2) s]\left[ i_{1}={{\sqrt{L_{1}}\,\sqrt{L_{2}}\,k\,\mathit{v}_{out}-L_{2}\,\mathit{v}_{in}}\over{\left(L_{1}\,L_{2}\,k^2-L_{1}\,L_{2}\right)\,s}} , i_{2}=-{{\sqrt{L_{1}}\,\sqrt{L_{2}}\,k\,\mathit{v}_{out}-L_{2}\,\mathit{v}_{in}}\over{\left(L_{1}\,L_{2}\,k^2-L_{1}\,L_{2}\right)\,s}} , i_{3}={{\sqrt{L_{1}}\,\sqrt{L_{2}}\,k\,\mathit{v}_{in}-L_{1}\,\mathit{v}_{out}}\over{\left(L_{1}\,L_{2}\,k^2-L_{1}\,L_{2}\right)\,s}} , i_{4}=-{{\sqrt{L_{1}}\,\sqrt{L_{2}}\,k\,\mathit{v}_{in}-L_{1}\,\mathit{v}_{out}}\over{\left(L_{1}\,L_{2}\,k^2-L_{1}\,L_{2}\right)\,s}} , i_{5}=C\,s\,\mathit{v}_{in} , v_{1}=\mathit{v}_{in} , v_{2}=0 , v_{3}=\mathit{v}_{out} , v_{4}=0 , \mathit{i}_{in}={{\sqrt{L_{1}}\,\sqrt{L_{2}}\,k\,\mathit{v}_{out}+s^2\,\left(C\,L_{1}\,L_{2}\,k^2\,\mathit{v}_{in}-C\,L_{1}\,L_{2}\,\mathit{v}_{in}\right)-L_{2}\,\mathit{v}_{in}}\over{\left(L_{1}\,L_{2}\,k^2-L_{1}\,L_{2}\right)\,s}} , \mathit{i}_{out}={{\sqrt{L_{1}}\,\sqrt{L_{2}}\,k\,\mathit{v}_{in}-L_{1}\,\mathit{v}_{out}}\over{\left(L_{1}\,L_{2}\,k^2-L_{1}\,L_{2}\right)\,s}} \right]
TwoPortMatrices(Solution);
(L1 sC L1 s2+1L1 L2 k sC L1 s2+1L1 L2 k sC L1 s2+1−(C L1 L2 k2−C L1 L2) s3−L2 sC L1 s2+1)\begin{pmatrix} {{L_{1}\,s}\over{C\,L_{1}\,s^2+1}}&{{\sqrt{L_{1}}\,\sqrt{L_{2}}\,k\,s}\over{C\,L_{1}\,s^2+1}}\\ {{\sqrt{L_{1}}\,\sqrt{L_{2}}\,k\,s}\over{C\,L_{1}\,s^2+1}}&-{{\left(C\,L_{1}\,L_{2}\,k^2-C\,L_{1}\,L_{2}\right)\,s^3-L_{2}\,s}\over{C\,L_{1}\,s^2+1}} \end{pmatrix}
ABCD(s) := (L1L2 k−(L1 L2 k2−L1 L2) sL1 L2 kC L1 s2+1L1 L2 k s−(C L1 L2 k2−C L1 L2) s2−L2L1 L2 k)\begin{pmatrix} {{\sqrt{L_{1}}}\over{\sqrt{L_{2}}\,k}}&-{{\left(L_{1}\,L_{2}\,k^2-L_{1}\,L_{2}\right)\,s}\over{\sqrt{L_{1}}\,\sqrt{L_{2}}\,k}}\\ {{C\,L_{1}\,s^2+1}\over{\sqrt{L_{1}}\,\sqrt{L_{2}}\,k\,s}}&-{{\left(C\,L_{1}\,L_{2}\,k^2-C\,L_{1}\,L_{2}\right)\,s^2-L_{2}}\over{\sqrt{L_{1}}\,\sqrt{L_{2}}\,k}} \end{pmatrix}
G(s) := (C L1 s2+1L1 s−L2 kL1L2 kL1(L2−L2 k2) s)\begin{pmatrix} {{C\,L_{1}\,s^2+1}\over{L_{1}\,s}}&-{{\sqrt{L_{2}}\,k}\over{\sqrt{L_{1}}}}\\ {{\sqrt{L_{2}}\,k}\over{\sqrt{L_{1}}}}&\left(L_{2}-L_{2}\,k^2\right)\,s \end{pmatrix}
H(s) := ((L1 k2−L1) s(C L1 k2−C L1) s2−1−L1 L2 k(C L1 L2 k2−C L1 L2) s2−L2L1 L2 k(C L1 L2 k2−C L1 L2) s2−L2−C L1 s2+1(C L1 L2 k2−C L1 L2) s3−L2 s)\begin{pmatrix} {{\left(L_{1}\,k^2-L_{1}\right)\,s}\over{\left(C\,L_{1}\,k^2-C\,L_{1}\right)\,s^2-1}}&-{{\sqrt{L_{1}}\,\sqrt{L_{2}}\,k}\over{\left(C\,L_{1}\,L_{2}\,k^2-C\,L_{1}\,L_{2}\right)\,s^2-L_{2}}}\\ {{\sqrt{L_{1}}\,\sqrt{L_{2}}\,k}\over{\left(C\,L_{1}\,L_{2}\,k^2-C\,L_{1}\,L_{2}\right)\,s^2-L_{2}}}&-{{C\,L_{1}\,s^2+1}\over{\left(C\,L_{1}\,L_{2}\,k^2-C\,L_{1}\,L_{2}\right)\,s^3-L_{2}\,s}} \end{pmatrix}
InverseABCD(s) := (−(C L1 L2 k2−C L1 L2) s2−L2L1 L2 k−(L1 L2 k2−L1 L2) sL1 L2 kC L1 s2+1L1 L2 k sL1L2 k)\begin{pmatrix} -{{\left(C\,L_{1}\,L_{2}\,k^2-C\,L_{1}\,L_{2}\right)\,s^2-L_{2}}\over{\sqrt{L_{1}}\,\sqrt{L_{2}}\,k}}&-{{\left(L_{1}\,L_{2}\,k^2-L_{1}\,L_{2}\right)\,s}\over{\sqrt{L_{1}}\,\sqrt{L_{2}}\,k}}\\ {{C\,L_{1}\,s^2+1}\over{\sqrt{L_{1}}\,\sqrt{L_{2}}\,k\,s}}&{{\sqrt{L_{1}}}\over{\sqrt{L_{2}}\,k}} \end{pmatrix}
S(s) := (−(C L1 L2 Z0 k2−C L1 L2 Z0) s3+(−L1 L2 k2−C L1 Z02+L1 L2) s2+(L1−L2) Z0 s−Z02(C L1 L2 Z0 k2−C L1 L2 Z0) s3+(L1 L2 k2−C L1 Z02−L1 L2) s2+(−L2−L1) Z0 s−Z02−2 L1 L2 Z0 k s(C L1 L2 Z0 k2−C L1 L2 Z0) s3+(L1 L2 k2−C L1 Z02−L1 L2) s2+(−L2−L1) Z0 s−Z02−2 L1 L2 Z0 k s(C L1 L2 Z0 k2−C L1 L2 Z0) s3+(L1 L2 k2−C L1 Z02−L1 L2) s2+(−L2−L1) Z0 s−Z02(C L1 L2 Z0 k2−C L1 L2 Z0) s3+(L1 L2 k2+C L1 Z02−L1 L2) s2+(L1−L2) Z0 s+Z02(C L1 L2 Z0 k2−C L1 L2 Z0) s3+(L1 L2 k2−C L1 Z02−L1 L2) s2+(−L2−L1) Z0 s−Z02)\begin{pmatrix} -{{\left(C\,L_{1}\,L_{2}\,Z_{0}\,k^2-C\,L_{1}\,L_{2}\,Z_{0}\right)\,s^3+\left(-L_{1}\,L_{2}\,k^2-C\,L_{1}\,Z_{0}^2+L_{1}\,L_{2}\right)\,s^2+\left(L_{1}-L_{2}\right)\,Z_{0}\,s-Z_{0}^2}\over{\left(C\,L_{1}\,L_{2}\,Z_{0}\,k^2-C\,L_{1}\,L_{2}\,Z_{0}\right)\,s^3+\left(L_{1}\,L_{2}\,k^2-C\,L_{1}\,Z_{0}^2-L_{1}\,L_{2}\right)\,s^2+\left(-L_{2}-L_{1}\right)\,Z_{0}\,s-Z_{0}^2}}&-{{2\,\sqrt{L_{1}}\,\sqrt{L_{2}}\,Z_{0}\,k\,s}\over{\left(C\,L_{1}\,L_{2}\,Z_{0}\,k^2-C\,L_{1}\,L_{2}\,Z_{0}\right)\,s^3+\left(L_{1}\,L_{2}\,k^2-C\,L_{1}\,Z_{0}^2-L_{1}\,L_{2}\right)\,s^2+\left(-L_{2}-L_{1}\right)\,Z_{0}\,s-Z_{0}^2}}\\ -{{2\,\sqrt{L_{1}}\,\sqrt{L_{2}}\,Z_{0}\,k\,s}\over{\left(C\,L_{1}\,L_{2}\,Z_{0}\,k^2-C\,L_{1}\,L_{2}\,Z_{0}\right)\,s^3+\left(L_{1}\,L_{2}\,k^2-C\,L_{1}\,Z_{0}^2-L_{1}\,L_{2}\right)\,s^2+\left(-L_{2}-L_{1}\right)\,Z_{0}\,s-Z_{0}^2}}&{{\left(C\,L_{1}\,L_{2}\,Z_{0}\,k^2-C\,L_{1}\,L_{2}\,Z_{0}\right)\,s^3+\left(L_{1}\,L_{2}\,k^2+C\,L_{1}\,Z_{0}^2-L_{1}\,L_{2}\right)\,s^2+\left(L_{1}-L_{2}\right)\,Z_{0}\,s+Z_{0}^2}\over{\left(C\,L_{1}\,L_{2}\,Z_{0}\,k^2-C\,L_{1}\,L_{2}\,Z_{0}\right)\,s^3+\left(L_{1}\,L_{2}\,k^2-C\,L_{1}\,Z_{0}^2-L_{1}\,L_{2}\right)\,s^2+\left(-L_{2}-L_{1}\right)\,Z_{0}\,s-Z_{0}^2}} \end{pmatrix}
Y(s) := ((C L1 k2−C L1) s2−1(L1 k2−L1) sL1 L2 k(L1 L2 k2−L1 L2) sL1 L2 k(L1 L2 k2−L1 L2) s−1(L2 k2−L2) s)\begin{pmatrix} {{\left(C\,L_{1}\,k^2-C\,L_{1}\right)\,s^2-1}\over{\left(L_{1}\,k^2-L_{1}\right)\,s}}&{{\sqrt{L_{1}}\,\sqrt{L_{2}}\,k}\over{\left(L_{1}\,L_{2}\,k^2-L_{1}\,L_{2}\right)\,s}}\\ {{\sqrt{L_{1}}\,\sqrt{L_{2}}\,k}\over{\left(L_{1}\,L_{2}\,k^2-L_{1}\,L_{2}\right)\,s}}&-{{1}\over{\left(L_{2}\,k^2-L_{2}\right)\,s}} \end{pmatrix}
Z(s) := (L1 sC L1 s2+1L1 L2 k sC L1 s2+1L1 L2 k sC L1 s2+1−(C L1 L2 k2−C L1 L2) s3−L2 sC L1 s2+1)\begin{pmatrix} {{L_{1}\,s}\over{C\,L_{1}\,s^2+1}}&{{\sqrt{L_{1}}\,\sqrt{L_{2}}\,k\,s}\over{C\,L_{1}\,s^2+1}}\\ {{\sqrt{L_{1}}\,\sqrt{L_{2}}\,k\,s}\over{C\,L_{1}\,s^2+1}}&-{{\left(C\,L_{1}\,L_{2}\,k^2-C\,L_{1}\,L_{2}\right)\,s^3-L_{2}\,s}\over{C\,L_{1}\,s^2+1}} \end{pmatrix}
SParameters(S);
−ω5 C2 L12 L2 Z02 k2+ω3 C L1 L2 Z02 k2−ω3 L12 L2 k2+ω5 C2 L12 L2 Z02−2 ω3 C L1 L2 Z02+ω L2 Z02+ω3 L12 L2∣ω4 C2 L12 Z02−2 ω2 C L1 Z02+Z02+ω2 L12∣{{-\omega^5\,C^2\,L_{1}^2\,L_{2}\,Z_{0}^2\,k^2+\omega^3\,C\,L_{1}\,L_{2}\,Z_{0}^2\,k^2-\omega^3\,L_{1}^2\,L_{2}\,k^2+\omega^5\,C^2\,L_{1}^2\,L_{2}\,Z_{0}^2-2\,\omega^3\,C\,L_{1}\,L_{2}\,Z_{0}^2+\omega\,L_{2}\,Z_{0}^2+\omega^3\,L_{1}^2\,L_{2}}\over{\left| \omega^4\,C^2\,L_{1}^2\,Z_{0}^2-2\,\omega^2\,C\,L_{1}\,Z_{0}^2+Z_{0}^2+\omega^2\,L_{1}^2\right| }}
Abs_S11(%omega) := (−ω3 C L1 L2 Z0 k2+ω3 C L1 L2 Z0−ω L2 Z0+ω L1 Z0)2+(ω2 L1 L2 k2+ω2 C L1 Z02−Z02−ω2 L1 L2)2(ω3 C L1 L2 Z0 k2−ω3 C L1 L2 Z0+ω L2 Z0+ω L1 Z0)2+(ω2 L1 L2 k2−ω2 C L1 Z02+Z02−ω2 L1 L2)2{{\sqrt{\left(-\omega^3\,C\,L_{1}\,L_{2}\,Z_{0}\,k^2+\omega^3\,C\,L_{1}\,L_{2}\,Z_{0}-\omega\,L_{2}\,Z_{0}+\omega\,L_{1}\,Z_{0}\right)^2+\left(\omega^2\,L_{1}\,L_{2}\,k^2+\omega^2\,C\,L_{1}\,Z_{0}^2-Z_{0}^2-\omega^2\,L_{1}\,L_{2}\right)^2}}\over{\sqrt{\left(\omega^3\,C\,L_{1}\,L_{2}\,Z_{0}\,k^2-\omega^3\,C\,L_{1}\,L_{2}\,Z_{0}+\omega\,L_{2}\,Z_{0}+\omega\,L_{1}\,Z_{0}\right)^2+\left(\omega^2\,L_{1}\,L_{2}\,k^2-\omega^2\,C\,L_{1}\,Z_{0}^2+Z_{0}^2-\omega^2\,L_{1}\,L_{2}\right)^2}}}
Arg_S11(%omega) := −atan2(2 ω5 C L12 L22 Z0 k4−(4 ω5 C L12−2 ω3 L1) L22 Z0 k2−(2 ω L1−2 ω3 C L12) Z03−(2 ω3 L1−2 ω5 C L12) L22 Z0,(ω4 L12 L22−ω6 C2 L12 L22 Z02) k4+((2 ω6 C2 L12−2 ω4 C L1) L22 Z02−2 ω4 L12 L22) k2+(−ω4 C2 L12+2 ω2 C L1−1) Z04+((−ω6 C2 L12+2 ω4 C L1−ω2) L22+ω2 L12) Z02+ω4 L12 L22)-{\rm atan2}\left(2\,\omega^5\,C\,L_{1}^2\,L_{2}^2\,Z_{0}\,k^4-\left(4\,\omega^5\,C\,L_{1}^2-2\,\omega^3\,L_{1}\right)\,L_{2}^2\,Z_{0}\,k^2-\left(2\,\omega\,L_{1}-2\,\omega^3\,C\,L_{1}^2\right)\,Z_{0}^3-\left(2\,\omega^3\,L_{1}-2\,\omega^5\,C\,L_{1}^2\right)\,L_{2}^2\,Z_{0} , \left(\omega^4\,L_{1}^2\,L_{2}^2-\omega^6\,C^2\,L_{1}^2\,L_{2}^2\,Z_{0}^2\right)\,k^4+\left(\left(2\,\omega^6\,C^2\,L_{1}^2-2\,\omega^4\,C\,L_{1}\right)\,L_{2}^2\,Z_{0}^2-2\,\omega^4\,L_{1}^2\,L_{2}^2\right)\,k^2+\left(-\omega^4\,C^2\,L_{1}^2+2\,\omega^2\,C\,L_{1}-1\right)\,Z_{0}^4+\left(\left(-\omega^6\,C^2\,L_{1}^2+2\,\omega^4\,C\,L_{1}-\omega^2\right)\,L_{2}^2+\omega^2\,L_{1}^2\right)\,Z_{0}^2+\omega^4\,L_{1}^2\,L_{2}^2\right)
Abs_S12(%omega) := 2 ∣ω∣ L1 L2 Z0 k(ω3 C L1 L2 Z0 k2−ω3 C L1 L2 Z0+ω L2 Z0+ω L1 Z0)2+(ω2 L1 L2 k2−ω2 C L1 Z02+Z02−ω2 L1 L2)2{{2\,\left| \omega\right| \,\sqrt{L_{1}}\,\sqrt{L_{2}}\,Z_{0}\,k}\over{\sqrt{\left(\omega^3\,C\,L_{1}\,L_{2}\,Z_{0}\,k^2-\omega^3\,C\,L_{1}\,L_{2}\,Z_{0}+\omega\,L_{2}\,Z_{0}+\omega\,L_{1}\,Z_{0}\right)^2+\left(\omega^2\,L_{1}\,L_{2}\,k^2-\omega^2\,C\,L_{1}\,Z_{0}^2+Z_{0}^2-\omega^2\,L_{1}\,L_{2}\right)^2}}}
Arg_S12(%omega) := atan2(L1 L2 (2 ω3 L1 L2 Z0 k3+((2 ω−2 ω3 C L1) Z03−2 ω3 L1 L2 Z0) k),L1 L2 (2 ω4 C L1 L2 Z02 k3+((2 ω2−2 ω4 C L1) L2+2 ω2 L1) Z02 k)){\rm atan2}\left(\sqrt{L_{1}}\,\sqrt{L_{2}}\,\left(2\,\omega^3\,L_{1}\,L_{2}\,Z_{0}\,k^3+\left(\left(2\,\omega-2\,\omega^3\,C\,L_{1}\right)\,Z_{0}^3-2\,\omega^3\,L_{1}\,L_{2}\,Z_{0}\right)\,k\right) , \sqrt{L_{1}}\,\sqrt{L_{2}}\,\left(2\,\omega^4\,C\,L_{1}\,L_{2}\,Z_{0}^2\,k^3+\left(\left(2\,\omega^2-2\,\omega^4\,C\,L_{1}\right)\,L_{2}+2\,\omega^2\,L_{1}\right)\,Z_{0}^2\,k\right)\right)
Abs_S21(%omega) := 2 ∣ω∣ L1 L2 Z0 k(ω3 C L1 L2 Z0 k2−ω3 C L1 L2 Z0+ω L2 Z0+ω L1 Z0)2+(ω2 L1 L2 k2−ω2 C L1 Z02+Z02−ω2 L1 L2)2{{2\,\left| \omega\right| \,\sqrt{L_{1}}\,\sqrt{L_{2}}\,Z_{0}\,k}\over{\sqrt{\left(\omega^3\,C\,L_{1}\,L_{2}\,Z_{0}\,k^2-\omega^3\,C\,L_{1}\,L_{2}\,Z_{0}+\omega\,L_{2}\,Z_{0}+\omega\,L_{1}\,Z_{0}\right)^2+\left(\omega^2\,L_{1}\,L_{2}\,k^2-\omega^2\,C\,L_{1}\,Z_{0}^2+Z_{0}^2-\omega^2\,L_{1}\,L_{2}\right)^2}}}
Arg_S21(%omega) := atan2(L1 L2 (2 ω3 L1 L2 Z0 k3+((2 ω−2 ω3 C L1) Z03−2 ω3 L1 L2 Z0) k),L1 L2 (2 ω4 C L1 L2 Z02 k3+((2 ω2−2 ω4 C L1) L2+2 ω2 L1) Z02 k)){\rm atan2}\left(\sqrt{L_{1}}\,\sqrt{L_{2}}\,\left(2\,\omega^3\,L_{1}\,L_{2}\,Z_{0}\,k^3+\left(\left(2\,\omega-2\,\omega^3\,C\,L_{1}\right)\,Z_{0}^3-2\,\omega^3\,L_{1}\,L_{2}\,Z_{0}\right)\,k\right) , \sqrt{L_{1}}\,\sqrt{L_{2}}\,\left(2\,\omega^4\,C\,L_{1}\,L_{2}\,Z_{0}^2\,k^3+\left(\left(2\,\omega^2-2\,\omega^4\,C\,L_{1}\right)\,L_{2}+2\,\omega^2\,L_{1}\right)\,Z_{0}^2\,k\right)\right)
tau_g(%omega) := ω4 C L12 L22 Z0 k4+((3 ω2 C L1−ω4 C2 L12) L2 Z03+((−2 ω4 C L12−ω2 L1) L22−ω2 L12 L2) Z0) k2+((ω4 C2 L12−2 ω2 C L1+1) L2+ω2 C L12+L1) Z03+((ω4 C L12+ω2 L1) L22+ω2 L12 L2) Z0(ω6 C2 L12 L22 Z02+ω4 L12 L22) k4+(((2 ω4 C L1−2 ω6 C2 L12) L22+2 ω2 L1 L2) Z02−2 ω4 L12 L22) k2+(ω4 C2 L12−2 ω2 C L1+1) Z04+((ω6 C2 L12−2 ω4 C L1+ω2) L22+ω2 L12) Z02+ω4 L12 L22{{\omega^4\,C\,L_{1}^2\,L_{2}^2\,Z_{0}\,k^4+\left(\left(3\,\omega^2\,C\,L_{1}-\omega^4\,C^2\,L_{1}^2\right)\,L_{2}\,Z_{0}^3+\left(\left(-2\,\omega^4\,C\,L_{1}^2-\omega^2\,L_{1}\right)\,L_{2}^2-\omega^2\,L_{1}^2\,L_{2}\right)\,Z_{0}\right)\,k^2+\left(\left(\omega^4\,C^2\,L_{1}^2-2\,\omega^2\,C\,L_{1}+1\right)\,L_{2}+\omega^2\,C\,L_{1}^2+L_{1}\right)\,Z_{0}^3+\left(\left(\omega^4\,C\,L_{1}^2+\omega^2\,L_{1}\right)\,L_{2}^2+\omega^2\,L_{1}^2\,L_{2}\right)\,Z_{0}}\over{\left(\omega^6\,C^2\,L_{1}^2\,L_{2}^2\,Z_{0}^2+\omega^4\,L_{1}^2\,L_{2}^2\right)\,k^4+\left(\left(\left(2\,\omega^4\,C\,L_{1}-2\,\omega^6\,C^2\,L_{1}^2\right)\,L_{2}^2+2\,\omega^2\,L_{1}\,L_{2}\right)\,Z_{0}^2-2\,\omega^4\,L_{1}^2\,L_{2}^2\right)\,k^2+\left(\omega^4\,C^2\,L_{1}^2-2\,\omega^2\,C\,L_{1}+1\right)\,Z_{0}^4+\left(\left(\omega^6\,C^2\,L_{1}^2-2\,\omega^4\,C\,L_{1}+\omega^2\right)\,L_{2}^2+\omega^2\,L_{1}^2\right)\,Z_{0}^2+\omega^4\,L_{1}^2\,L_{2}^2}}
Abs_S22(%omega) := (ω3 C L1 L2 Z0 k2−ω3 C L1 L2 Z0+ω L2 Z0−ω L1 Z0)2+(ω2 L1 L2 k2+ω2 C L1 Z02−Z02−ω2 L1 L2)2(ω3 C L1 L2 Z0 k2−ω3 C L1 L2 Z0+ω L2 Z0+ω L1 Z0)2+(ω2 L1 L2 k2−ω2 C L1 Z02+Z02−ω2 L1 L2)2{{\sqrt{\left(\omega^3\,C\,L_{1}\,L_{2}\,Z_{0}\,k^2-\omega^3\,C\,L_{1}\,L_{2}\,Z_{0}+\omega\,L_{2}\,Z_{0}-\omega\,L_{1}\,Z_{0}\right)^2+\left(\omega^2\,L_{1}\,L_{2}\,k^2+\omega^2\,C\,L_{1}\,Z_{0}^2-Z_{0}^2-\omega^2\,L_{1}\,L_{2}\right)^2}}\over{\sqrt{\left(\omega^3\,C\,L_{1}\,L_{2}\,Z_{0}\,k^2-\omega^3\,C\,L_{1}\,L_{2}\,Z_{0}+\omega\,L_{2}\,Z_{0}+\omega\,L_{1}\,Z_{0}\right)^2+\left(\omega^2\,L_{1}\,L_{2}\,k^2-\omega^2\,C\,L_{1}\,Z_{0}^2+Z_{0}^2-\omega^2\,L_{1}\,L_{2}\right)^2}}}
Arg_S22(%omega) := atan2(((2 ω3 C L1−2 ω5 C2 L12) L2 Z03−2 ω3 L12 L2 Z0) k2+(2 ω5 C2 L12−4 ω3 C L1+2 ω) L2 Z03+2 ω3 L12 L2 Z0,(ω6 C2 L12 L22 Z02+ω4 L12 L22) k4+((2 ω4 C L1−2 ω6 C2 L12) L22 Z02−2 ω4 L12 L22) k2+(−ω4 C2 L12+2 ω2 C L1−1) Z04+((ω6 C2 L12−2 ω4 C L1+ω2) L22−ω2 L12) Z02+ω4 L12 L22){\rm atan2}\left(\left(\left(2\,\omega^3\,C\,L_{1}-2\,\omega^5\,C^2\,L_{1}^2\right)\,L_{2}\,Z_{0}^3-2\,\omega^3\,L_{1}^2\,L_{2}\,Z_{0}\right)\,k^2+\left(2\,\omega^5\,C^2\,L_{1}^2-4\,\omega^3\,C\,L_{1}+2\,\omega\right)\,L_{2}\,Z_{0}^3+2\,\omega^3\,L_{1}^2\,L_{2}\,Z_{0} , \left(\omega^6\,C^2\,L_{1}^2\,L_{2}^2\,Z_{0}^2+\omega^4\,L_{1}^2\,L_{2}^2\right)\,k^4+\left(\left(2\,\omega^4\,C\,L_{1}-2\,\omega^6\,C^2\,L_{1}^2\right)\,L_{2}^2\,Z_{0}^2-2\,\omega^4\,L_{1}^2\,L_{2}^2\right)\,k^2+\left(-\omega^4\,C^2\,L_{1}^2+2\,\omega^2\,C\,L_{1}-1\right)\,Z_{0}^4+\left(\left(\omega^6\,C^2\,L_{1}^2-2\,\omega^4\,C\,L_{1}+\omega^2\right)\,L_{2}^2-\omega^2\,L_{1}^2\right)\,Z_{0}^2+\omega^4\,L_{1}^2\,L_{2}^2\right)
Z_IN(s) := (L1 L2 k2−L1 L2) s2−L1 Z0 s(C L1 L2 k2−C L1 L2) s3−C L1 Z0 s2−L2 s−Z0{{\left(L_{1}\,L_{2}\,k^2-L_{1}\,L_{2}\right)\,s^2-L_{1}\,Z_{0}\,s}\over{\left(C\,L_{1}\,L_{2}\,k^2-C\,L_{1}\,L_{2}\right)\,s^3-C\,L_{1}\,Z_{0}\,s^2-L_{2}\,s-Z_{0}}}
R_IN(%omega) := ω2 L1 L2 Z0 k2∣ω6 C2 L12 L22 k4−2 ω6 C2 L12 L22 k2+2 ω4 C L1 L22 k2+ω4 C2 L12 Z02−2 ω2 C L1 Z02+Z02+ω6 C2 L12 L22−2 ω4 C L1 L22+ω2 L22∣{{\omega^2\,L_{1}\,L_{2}\,Z_{0}\,k^2}\over{\left| \omega^6\,C^2\,L_{1}^2\,L_{2}^2\,k^4-2\,\omega^6\,C^2\,L_{1}^2\,L_{2}^2\,k^2+2\,\omega^4\,C\,L_{1}\,L_{2}^2\,k^2+\omega^4\,C^2\,L_{1}^2\,Z_{0}^2-2\,\omega^2\,C\,L_{1}\,Z_{0}^2+Z_{0}^2+\omega^6\,C^2\,L_{1}^2\,L_{2}^2-2\,\omega^4\,C\,L_{1}\,L_{2}^2+\omega^2\,L_{2}^2\right| }}
X_IN(%omega) := −ω5 C L12 L22 k4+2 ω5 C L12 L22 k2−ω3 L1 L22 k2−ω3 C L12 Z02+ω L1 Z02−ω5 C L12 L22+ω3 L1 L22∣ω6 C2 L12 L22 k4−2 ω6 C2 L12 L22 k2+2 ω4 C L1 L22 k2+ω4 C2 L12 Z02−2 ω2 C L1 Z02+Z02+ω6 C2 L12 L22−2 ω4 C L1 L22+ω2 L22∣{{-\omega^5\,C\,L_{1}^2\,L_{2}^2\,k^4+2\,\omega^5\,C\,L_{1}^2\,L_{2}^2\,k^2-\omega^3\,L_{1}\,L_{2}^2\,k^2-\omega^3\,C\,L_{1}^2\,Z_{0}^2+\omega\,L_{1}\,Z_{0}^2-\omega^5\,C\,L_{1}^2\,L_{2}^2+\omega^3\,L_{1}\,L_{2}^2}\over{\left| \omega^6\,C^2\,L_{1}^2\,L_{2}^2\,k^4-2\,\omega^6\,C^2\,L_{1}^2\,L_{2}^2\,k^2+2\,\omega^4\,C\,L_{1}\,L_{2}^2\,k^2+\omega^4\,C^2\,L_{1}^2\,Z_{0}^2-2\,\omega^2\,C\,L_{1}\,Z_{0}^2+Z_{0}^2+\omega^6\,C^2\,L_{1}^2\,L_{2}^2-2\,\omega^4\,C\,L_{1}\,L_{2}^2+\omega^2\,L_{2}^2\right| }}
Z_OUT(s) := −(C L1 L2 Z0 k2−C L1 L2 Z0) s3+(L1 L2 k2−L1 L2) s2−L2 Z0 sC L1 Z0 s2+L1 s+Z0-{{\left(C\,L_{1}\,L_{2}\,Z_{0}\,k^2-C\,L_{1}\,L_{2}\,Z_{0}\right)\,s^3+\left(L_{1}\,L_{2}\,k^2-L_{1}\,L_{2}\right)\,s^2-L_{2}\,Z_{0}\,s}\over{C\,L_{1}\,Z_{0}\,s^2+L_{1}\,s+Z_{0}}}
R_OUT(%omega) := ω2 L1 L2 Z0 k2∣ω4 C2 L12 Z02−2 ω2 C L1 Z02+Z02+ω2 L12∣{{\omega^2\,L_{1}\,L_{2}\,Z_{0}\,k^2}\over{\left| \omega^4\,C^2\,L_{1}^2\,Z_{0}^2-2\,\omega^2\,C\,L_{1}\,Z_{0}^2+Z_{0}^2+\omega^2\,L_{1}^2\right| }}
X_OUT(%omega) := −ω5 C2 L12 L2 Z02 k2+ω3 C L1 L2 Z02 k2−ω3 L12 L2 k2+ω5 C2 L12 L2 Z02−2 ω3 C L1 L2 Z02+ω L2 Z02+ω3 L12 L2∣ω4 C2 L12 Z02−2 ω2 C L1 Z02+Z02+ω2 L12∣{{-\omega^5\,C^2\,L_{1}^2\,L_{2}\,Z_{0}^2\,k^2+\omega^3\,C\,L_{1}\,L_{2}\,Z_{0}^2\,k^2-\omega^3\,L_{1}^2\,L_{2}\,k^2+\omega^5\,C^2\,L_{1}^2\,L_{2}\,Z_{0}^2-2\,\omega^3\,C\,L_{1}\,L_{2}\,Z_{0}^2+\omega\,L_{2}\,Z_{0}^2+\omega^3\,L_{1}^2\,L_{2}}\over{\left| \omega^4\,C^2\,L_{1}^2\,Z_{0}^2-2\,\omega^2\,C\,L_{1}\,Z_{0}^2+Z_{0}^2+\omega^2\,L_{1}^2\right| }}

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

declare(R1,real)$
declare(R2,real)$
assume(R1>0)$
assume(R2>0)$
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]);
[i1=s (L1 L2 k vout+C R1 R2 vin−L2 vin)+C L2 R1 s2 vin−R2 vin(L1 L2 k2−L1 L2) s2+(−L1 R2−L2 R1) s−R1 R2,i2=−s (L1 L2 k vout+C R1 R2 vin−L2 vin)+C L2 R1 s2 vin−R2 vin(L1 L2 k2−L1 L2) s2+(−L1 R2−L2 R1) s−R1 R2,i3=−s (L1 vout−L1 L2 k vin)+R1 vout+C L1 L2 R1 k s2 vin(L1 L2 k2−L1 L2) s2+(−L1 R2−L2 R1) s−R1 R2,i4=s (L1 vout−L1 L2 k vin)+R1 vout+C L1 L2 R1 k s2 vin(L1 L2 k2−L1 L2) s2+(−L1 R2−L2 R1) s−R1 R2,i5=C s vin,v0=vin,v1=−s (L1 L2 R1 k vout+L1 R2 vin)+s2 (L1 (L2 vin−C R1 R2 vin)−L1 L2 k2 vin)+s3 (C L1 L2 R1 k2 vin−C L1 L2 R1 vin)(L1 L2 k2−L1 L2) s2+(−L1 R2−L2 R1) s−R1 R2,v2=0,v3=s2 (L1 L2 k2 vout−L1 L2 vout+C L1 L2 R1 R2 k vin)+s (−L2 R1 vout−L1 L2 R2 k vin)(L1 L2 k2−L1 L2) s2+(−L1 R2−L2 R1) s−R1 R2,v4=0,v5=vout,iin=s (L1 L2 k vout−L2 vin)+s3 (C L1 L2 k2 vin−C L1 L2 vin)−C L1 R2 s2 vin−R2 vin(L1 L2 k2−L1 L2) s2+(−L1 R2−L2 R1) s−R1 R2,iout=−s (L1 vout−L1 L2 k vin)+R1 vout+C L1 L2 R1 k s2 vin(L1 L2 k2−L1 L2) s2+(−L1 R2−L2 R1) s−R1 R2]\left[ i_{1}={{s\,\left(\sqrt{L_{1}}\,\sqrt{L_{2}}\,k\,\mathit{v}_{out}+C\,R_{1}\,R_{2}\,\mathit{v}_{in}-L_{2}\,\mathit{v}_{in}\right)+C\,L_{2}\,R_{1}\,s^2\,\mathit{v}_{in}-R_{2}\,\mathit{v}_{in}}\over{\left(L_{1}\,L_{2}\,k^2-L_{1}\,L_{2}\right)\,s^2+\left(-L_{1}\,R_{2}-L_{2}\,R_{1}\right)\,s-R_{1}\,R_{2}}} , i_{2}=-{{s\,\left(\sqrt{L_{1}}\,\sqrt{L_{2}}\,k\,\mathit{v}_{out}+C\,R_{1}\,R_{2}\,\mathit{v}_{in}-L_{2}\,\mathit{v}_{in}\right)+C\,L_{2}\,R_{1}\,s^2\,\mathit{v}_{in}-R_{2}\,\mathit{v}_{in}}\over{\left(L_{1}\,L_{2}\,k^2-L_{1}\,L_{2}\right)\,s^2+\left(-L_{1}\,R_{2}-L_{2}\,R_{1}\right)\,s-R_{1}\,R_{2}}} , i_{3}=-{{s\,\left(L_{1}\,\mathit{v}_{out}-\sqrt{L_{1}}\,\sqrt{L_{2}}\,k\,\mathit{v}_{in}\right)+R_{1}\,\mathit{v}_{out}+C\,\sqrt{L_{1}}\,\sqrt{L_{2}}\,R_{1}\,k\,s^2\,\mathit{v}_{in}}\over{\left(L_{1}\,L_{2}\,k^2-L_{1}\,L_{2}\right)\,s^2+\left(-L_{1}\,R_{2}-L_{2}\,R_{1}\right)\,s-R_{1}\,R_{2}}} , i_{4}={{s\,\left(L_{1}\,\mathit{v}_{out}-\sqrt{L_{1}}\,\sqrt{L_{2}}\,k\,\mathit{v}_{in}\right)+R_{1}\,\mathit{v}_{out}+C\,\sqrt{L_{1}}\,\sqrt{L_{2}}\,R_{1}\,k\,s^2\,\mathit{v}_{in}}\over{\left(L_{1}\,L_{2}\,k^2-L_{1}\,L_{2}\right)\,s^2+\left(-L_{1}\,R_{2}-L_{2}\,R_{1}\right)\,s-R_{1}\,R_{2}}} , i_{5}=C\,s\,\mathit{v}_{in} , v_{0}=\mathit{v}_{in} , v_{1}=-{{s\,\left(\sqrt{L_{1}}\,\sqrt{L_{2}}\,R_{1}\,k\,\mathit{v}_{out}+L_{1}\,R_{2}\,\mathit{v}_{in}\right)+s^2\,\left(L_{1}\,\left(L_{2}\,\mathit{v}_{in}-C\,R_{1}\,R_{2}\,\mathit{v}_{in}\right)-L_{1}\,L_{2}\,k^2\,\mathit{v}_{in}\right)+s^3\,\left(C\,L_{1}\,L_{2}\,R_{1}\,k^2\,\mathit{v}_{in}-C\,L_{1}\,L_{2}\,R_{1}\,\mathit{v}_{in}\right)}\over{\left(L_{1}\,L_{2}\,k^2-L_{1}\,L_{2}\right)\,s^2+\left(-L_{1}\,R_{2}-L_{2}\,R_{1}\right)\,s-R_{1}\,R_{2}}} , v_{2}=0 , v_{3}={{s^2\,\left(L_{1}\,L_{2}\,k^2\,\mathit{v}_{out}-L_{1}\,L_{2}\,\mathit{v}_{out}+C\,\sqrt{L_{1}}\,\sqrt{L_{2}}\,R_{1}\,R_{2}\,k\,\mathit{v}_{in}\right)+s\,\left(-L_{2}\,R_{1}\,\mathit{v}_{out}-\sqrt{L_{1}}\,\sqrt{L_{2}}\,R_{2}\,k\,\mathit{v}_{in}\right)}\over{\left(L_{1}\,L_{2}\,k^2-L_{1}\,L_{2}\right)\,s^2+\left(-L_{1}\,R_{2}-L_{2}\,R_{1}\right)\,s-R_{1}\,R_{2}}} , v_{4}=0 , v_{5}=\mathit{v}_{out} , \mathit{i}_{in}={{s\,\left(\sqrt{L_{1}}\,\sqrt{L_{2}}\,k\,\mathit{v}_{out}-L_{2}\,\mathit{v}_{in}\right)+s^3\,\left(C\,L_{1}\,L_{2}\,k^2\,\mathit{v}_{in}-C\,L_{1}\,L_{2}\,\mathit{v}_{in}\right)-C\,L_{1}\,R_{2}\,s^2\,\mathit{v}_{in}-R_{2}\,\mathit{v}_{in}}\over{\left(L_{1}\,L_{2}\,k^2-L_{1}\,L_{2}\right)\,s^2+\left(-L_{1}\,R_{2}-L_{2}\,R_{1}\right)\,s-R_{1}\,R_{2}}} , \mathit{i}_{out}=-{{s\,\left(L_{1}\,\mathit{v}_{out}-\sqrt{L_{1}}\,\sqrt{L_{2}}\,k\,\mathit{v}_{in}\right)+R_{1}\,\mathit{v}_{out}+C\,\sqrt{L_{1}}\,\sqrt{L_{2}}\,R_{1}\,k\,s^2\,\mathit{v}_{in}}\over{\left(L_{1}\,L_{2}\,k^2-L_{1}\,L_{2}\right)\,s^2+\left(-L_{1}\,R_{2}-L_{2}\,R_{1}\right)\,s-R_{1}\,R_{2}}} \right]
TwoPortMatrices(Solution);
(L1 s+R1C L1 s2+1L1 L2 k sC L1 s2+1−L1 L2 (C R1 k s2−k s)C L1 s2+1−(C L1 L2 k2−C L1 L2) s3−C L1 R2 s2−L2 s−R2C L1 s2+1)\begin{pmatrix} {{L_{1}\,s+R_{1}}\over{C\,L_{1}\,s^2+1}}&{{\sqrt{L_{1}}\,\sqrt{L_{2}}\,k\,s}\over{C\,L_{1}\,s^2+1}}\\ -{{\sqrt{L_{1}}\,\sqrt{L_{2}}\,\left(C\,R_{1}\,k\,s^2-k\,s\right)}\over{C\,L_{1}\,s^2+1}}&-{{\left(C\,L_{1}\,L_{2}\,k^2-C\,L_{1}\,L_{2}\right)\,s^3-C\,L_{1}\,R_{2}\,s^2-L_{2}\,s-R_{2}}\over{C\,L_{1}\,s^2+1}} \end{pmatrix}
ABCD(s) := (−L1 s+R1L1 L2 (C R1 k s2−k s)(L1 L2 k2−L1 L2) s2+(−L1 R2−L2 R1) s−R1 R2L1 L2 (C R1 k s2−k s)−C L1 s2+1L1 L2 (C R1 k s2−k s)(C L1 L2 k2−C L1 L2) s3−C L1 R2 s2−L2 s−R2L1 L2 (C R1 k s2−k s))\begin{pmatrix} -{{L_{1}\,s+R_{1}}\over{\sqrt{L_{1}}\,\sqrt{L_{2}}\,\left(C\,R_{1}\,k\,s^2-k\,s\right)}}&{{\left(L_{1}\,L_{2}\,k^2-L_{1}\,L_{2}\right)\,s^2+\left(-L_{1}\,R_{2}-L_{2}\,R_{1}\right)\,s-R_{1}\,R_{2}}\over{\sqrt{L_{1}}\,\sqrt{L_{2}}\,\left(C\,R_{1}\,k\,s^2-k\,s\right)}}\\ -{{C\,L_{1}\,s^2+1}\over{\sqrt{L_{1}}\,\sqrt{L_{2}}\,\left(C\,R_{1}\,k\,s^2-k\,s\right)}}&{{\left(C\,L_{1}\,L_{2}\,k^2-C\,L_{1}\,L_{2}\right)\,s^3-C\,L_{1}\,R_{2}\,s^2-L_{2}\,s-R_{2}}\over{\sqrt{L_{1}}\,\sqrt{L_{2}}\,\left(C\,R_{1}\,k\,s^2-k\,s\right)}} \end{pmatrix}
G(s) := (C L1 s2+1L1 s+R1−L1 L2 k sL1 s+R1−L1 L2 (C R1 k s2−k s)L1 s+R1−(L1 L2 k2−L1 L2) s2+(−L1 R2−L2 R1) s−R1 R2L1 s+R1)\begin{pmatrix} {{C\,L_{1}\,s^2+1}\over{L_{1}\,s+R_{1}}}&-{{\sqrt{L_{1}}\,\sqrt{L_{2}}\,k\,s}\over{L_{1}\,s+R_{1}}}\\ -{{\sqrt{L_{1}}\,\sqrt{L_{2}}\,\left(C\,R_{1}\,k\,s^2-k\,s\right)}\over{L_{1}\,s+R_{1}}}&-{{\left(L_{1}\,L_{2}\,k^2-L_{1}\,L_{2}\right)\,s^2+\left(-L_{1}\,R_{2}-L_{2}\,R_{1}\right)\,s-R_{1}\,R_{2}}\over{L_{1}\,s+R_{1}}} \end{pmatrix}
H(s) := ((L1 L2 k2−L1 L2) s2+(−L1 R2−L2 R1) s−R1 R2(C L1 L2 k2−C L1 L2) s3−C L1 R2 s2−L2 s−R2−L1 L2 k s(C L1 L2 k2−C L1 L2) s3−C L1 R2 s2−L2 s−R2−L1 L2 (C R1 k s2−k s)(C L1 L2 k2−C L1 L2) s3−C L1 R2 s2−L2 s−R2−C L1 s2+1(C L1 L2 k2−C L1 L2) s3−C L1 R2 s2−L2 s−R2)\begin{pmatrix} {{\left(L_{1}\,L_{2}\,k^2-L_{1}\,L_{2}\right)\,s^2+\left(-L_{1}\,R_{2}-L_{2}\,R_{1}\right)\,s-R_{1}\,R_{2}}\over{\left(C\,L_{1}\,L_{2}\,k^2-C\,L_{1}\,L_{2}\right)\,s^3-C\,L_{1}\,R_{2}\,s^2-L_{2}\,s-R_{2}}}&-{{\sqrt{L_{1}}\,\sqrt{L_{2}}\,k\,s}\over{\left(C\,L_{1}\,L_{2}\,k^2-C\,L_{1}\,L_{2}\right)\,s^3-C\,L_{1}\,R_{2}\,s^2-L_{2}\,s-R_{2}}}\\ -{{\sqrt{L_{1}}\,\sqrt{L_{2}}\,\left(C\,R_{1}\,k\,s^2-k\,s\right)}\over{\left(C\,L_{1}\,L_{2}\,k^2-C\,L_{1}\,L_{2}\right)\,s^3-C\,L_{1}\,R_{2}\,s^2-L_{2}\,s-R_{2}}}&-{{C\,L_{1}\,s^2+1}\over{\left(C\,L_{1}\,L_{2}\,k^2-C\,L_{1}\,L_{2}\right)\,s^3-C\,L_{1}\,R_{2}\,s^2-L_{2}\,s-R_{2}}} \end{pmatrix}
InverseABCD(s) := (−(C L1 L2 k2−C L1 L2) s3−C L1 R2 s2−L2 s−R2L1 L2 k s−(L1 L2 k2−L1 L2) s2+(−L1 R2−L2 R1) s−R1 R2L1 L2 k sC L1 s2+1L1 L2 k sL1 s+R1L1 L2 k s)\begin{pmatrix} -{{\left(C\,L_{1}\,L_{2}\,k^2-C\,L_{1}\,L_{2}\right)\,s^3-C\,L_{1}\,R_{2}\,s^2-L_{2}\,s-R_{2}}\over{\sqrt{L_{1}}\,\sqrt{L_{2}}\,k\,s}}&-{{\left(L_{1}\,L_{2}\,k^2-L_{1}\,L_{2}\right)\,s^2+\left(-L_{1}\,R_{2}-L_{2}\,R_{1}\right)\,s-R_{1}\,R_{2}}\over{\sqrt{L_{1}}\,\sqrt{L_{2}}\,k\,s}}\\ {{C\,L_{1}\,s^2+1}\over{\sqrt{L_{1}}\,\sqrt{L_{2}}\,k\,s}}&{{L_{1}\,s+R_{1}}\over{\sqrt{L_{1}}\,\sqrt{L_{2}}\,k\,s}} \end{pmatrix}
S(s) := (−(C L1 L2 Z0 k2−C L1 L2 Z0) s3+(−L1 L2 k2−C L1 Z02−C L1 R2 Z0+L1 L2) s2+((L1−L2) Z0+L1 R2+L2 R1) s−Z02+(R1−R2) Z0+R1 R2(C L1 L2 Z0 k2−C L1 L2 Z0) s3+(L1 L2 k2−C L1 Z02−C L1 R2 Z0−L1 L2) s2+((−L2−L1) Z0−L1 R2−L2 R1) s−Z02+(−R2−R1) Z0−R1 R2−2 L1 L2 Z0 k s(C L1 L2 Z0 k2−C L1 L2 Z0) s3+(L1 L2 k2−C L1 Z02−C L1 R2 Z0−L1 L2) s2+((−L2−L1) Z0−L1 R2−L2 R1) s−Z02+(−R2−R1) Z0−R1 R22 C L1 L2 R1 Z0 k s2−2 L1 L2 Z0 k sL1 L2 ((C L1 L2 Z0 k2−C L1 L2 Z0) s3+(L1 L2 k2−C L1 Z02−C L1 R2 Z0−L1 L2) s2+((−L2−L1) Z0−L1 R2−L2 R1) s−Z02+(−R2−R1) Z0−R1 R2)(C L1 L2 Z0 k2−C L1 L2 Z0) s3+(L1 L2 k2+C L1 Z02−C L1 R2 Z0−L1 L2) s2+((L1−L2) Z0−L1 R2−L2 R1) s+Z02+(R1−R2) Z0−R1 R2(C L1 L2 Z0 k2−C L1 L2 Z0) s3+(L1 L2 k2−C L1 Z02−C L1 R2 Z0−L1 L2) s2+((−L2−L1) Z0−L1 R2−L2 R1) s−Z02+(−R2−R1) Z0−R1 R2)\begin{pmatrix} -{{\left(C\,L_{1}\,L_{2}\,Z_{0}\,k^2-C\,L_{1}\,L_{2}\,Z_{0}\right)\,s^3+\left(-L_{1}\,L_{2}\,k^2-C\,L_{1}\,Z_{0}^2-C\,L_{1}\,R_{2}\,Z_{0}+L_{1}\,L_{2}\right)\,s^2+\left(\left(L_{1}-L_{2}\right)\,Z_{0}+L_{1}\,R_{2}+L_{2}\,R_{1}\right)\,s-Z_{0}^2+\left(R_{1}-R_{2}\right)\,Z_{0}+R_{1}\,R_{2}}\over{\left(C\,L_{1}\,L_{2}\,Z_{0}\,k^2-C\,L_{1}\,L_{2}\,Z_{0}\right)\,s^3+\left(L_{1}\,L_{2}\,k^2-C\,L_{1}\,Z_{0}^2-C\,L_{1}\,R_{2}\,Z_{0}-L_{1}\,L_{2}\right)\,s^2+\left(\left(-L_{2}-L_{1}\right)\,Z_{0}-L_{1}\,R_{2}-L_{2}\,R_{1}\right)\,s-Z_{0}^2+\left(-R_{2}-R_{1}\right)\,Z_{0}-R_{1}\,R_{2}}}&-{{2\,\sqrt{L_{1}}\,\sqrt{L_{2}}\,Z_{0}\,k\,s}\over{\left(C\,L_{1}\,L_{2}\,Z_{0}\,k^2-C\,L_{1}\,L_{2}\,Z_{0}\right)\,s^3+\left(L_{1}\,L_{2}\,k^2-C\,L_{1}\,Z_{0}^2-C\,L_{1}\,R_{2}\,Z_{0}-L_{1}\,L_{2}\right)\,s^2+\left(\left(-L_{2}-L_{1}\right)\,Z_{0}-L_{1}\,R_{2}-L_{2}\,R_{1}\right)\,s-Z_{0}^2+\left(-R_{2}-R_{1}\right)\,Z_{0}-R_{1}\,R_{2}}}\\ {{2\,C\,L_{1}\,L_{2}\,R_{1}\,Z_{0}\,k\,s^2-2\,L_{1}\,L_{2}\,Z_{0}\,k\,s}\over{\sqrt{L_{1}}\,\sqrt{L_{2}}\,\left(\left(C\,L_{1}\,L_{2}\,Z_{0}\,k^2-C\,L_{1}\,L_{2}\,Z_{0}\right)\,s^3+\left(L_{1}\,L_{2}\,k^2-C\,L_{1}\,Z_{0}^2-C\,L_{1}\,R_{2}\,Z_{0}-L_{1}\,L_{2}\right)\,s^2+\left(\left(-L_{2}-L_{1}\right)\,Z_{0}-L_{1}\,R_{2}-L_{2}\,R_{1}\right)\,s-Z_{0}^2+\left(-R_{2}-R_{1}\right)\,Z_{0}-R_{1}\,R_{2}\right)}}&{{\left(C\,L_{1}\,L_{2}\,Z_{0}\,k^2-C\,L_{1}\,L_{2}\,Z_{0}\right)\,s^3+\left(L_{1}\,L_{2}\,k^2+C\,L_{1}\,Z_{0}^2-C\,L_{1}\,R_{2}\,Z_{0}-L_{1}\,L_{2}\right)\,s^2+\left(\left(L_{1}-L_{2}\right)\,Z_{0}-L_{1}\,R_{2}-L_{2}\,R_{1}\right)\,s+Z_{0}^2+\left(R_{1}-R_{2}\right)\,Z_{0}-R_{1}\,R_{2}}\over{\left(C\,L_{1}\,L_{2}\,Z_{0}\,k^2-C\,L_{1}\,L_{2}\,Z_{0}\right)\,s^3+\left(L_{1}\,L_{2}\,k^2-C\,L_{1}\,Z_{0}^2-C\,L_{1}\,R_{2}\,Z_{0}-L_{1}\,L_{2}\right)\,s^2+\left(\left(-L_{2}-L_{1}\right)\,Z_{0}-L_{1}\,R_{2}-L_{2}\,R_{1}\right)\,s-Z_{0}^2+\left(-R_{2}-R_{1}\right)\,Z_{0}-R_{1}\,R_{2}}} \end{pmatrix}
Y(s) := ((C L1 L2 k2−C L1 L2) s3−C L1 R2 s2−L2 s−R2(L1 L2 k2−L1 L2) s2+(−L1 R2−L2 R1) s−R1 R2L1 L2 k s(L1 L2 k2−L1 L2) s2+(−L1 R2−L2 R1) s−R1 R2−L1 L2 (C R1 k s2−k s)(L1 L2 k2−L1 L2) s2+(−L1 R2−L2 R1) s−R1 R2−L1 s+R1(L1 L2 k2−L1 L2) s2+(−L1 R2−L2 R1) s−R1 R2)\begin{pmatrix} {{\left(C\,L_{1}\,L_{2}\,k^2-C\,L_{1}\,L_{2}\right)\,s^3-C\,L_{1}\,R_{2}\,s^2-L_{2}\,s-R_{2}}\over{\left(L_{1}\,L_{2}\,k^2-L_{1}\,L_{2}\right)\,s^2+\left(-L_{1}\,R_{2}-L_{2}\,R_{1}\right)\,s-R_{1}\,R_{2}}}&{{\sqrt{L_{1}}\,\sqrt{L_{2}}\,k\,s}\over{\left(L_{1}\,L_{2}\,k^2-L_{1}\,L_{2}\right)\,s^2+\left(-L_{1}\,R_{2}-L_{2}\,R_{1}\right)\,s-R_{1}\,R_{2}}}\\ -{{\sqrt{L_{1}}\,\sqrt{L_{2}}\,\left(C\,R_{1}\,k\,s^2-k\,s\right)}\over{\left(L_{1}\,L_{2}\,k^2-L_{1}\,L_{2}\right)\,s^2+\left(-L_{1}\,R_{2}-L_{2}\,R_{1}\right)\,s-R_{1}\,R_{2}}}&-{{L_{1}\,s+R_{1}}\over{\left(L_{1}\,L_{2}\,k^2-L_{1}\,L_{2}\right)\,s^2+\left(-L_{1}\,R_{2}-L_{2}\,R_{1}\right)\,s-R_{1}\,R_{2}}} \end{pmatrix}
Z(s) := (L1 s+R1C L1 s2+1L1 L2 k sC L1 s2+1−L1 L2 (C R1 k s2−k s)C L1 s2+1−(C L1 L2 k2−C L1 L2) s3−C L1 R2 s2−L2 s−R2C L1 s2+1)\begin{pmatrix} {{L_{1}\,s+R_{1}}\over{C\,L_{1}\,s^2+1}}&{{\sqrt{L_{1}}\,\sqrt{L_{2}}\,k\,s}\over{C\,L_{1}\,s^2+1}}\\ -{{\sqrt{L_{1}}\,\sqrt{L_{2}}\,\left(C\,R_{1}\,k\,s^2-k\,s\right)}\over{C\,L_{1}\,s^2+1}}&-{{\left(C\,L_{1}\,L_{2}\,k^2-C\,L_{1}\,L_{2}\right)\,s^3-C\,L_{1}\,R_{2}\,s^2-L_{2}\,s-R_{2}}\over{C\,L_{1}\,s^2+1}} \end{pmatrix}
SParameters(S);
−ω5 C2 L12 L2 Z02 k2+ω3 C L1 L2 Z02 k2+ω3 C L1 L2 R1 Z0 k2−ω3 L12 L2 k2+ω5 C2 L12 L2 Z02−2 ω3 C L1 L2 Z02+ω L2 Z02−2 ω3 C L1 L2 R1 Z0+2 ω L2 R1 Z0+ω L2 R12+ω3 L12 L2∣ω4 C2 L12 Z02−2 ω2 C L1 Z02+Z02−2 ω2 C L1 R1 Z0+2 R1 Z0+R12+ω2 L12∣{{-\omega^5\,C^2\,L_{1}^2\,L_{2}\,Z_{0}^2\,k^2+\omega^3\,C\,L_{1}\,L_{2}\,Z_{0}^2\,k^2+\omega^3\,C\,L_{1}\,L_{2}\,R_{1}\,Z_{0}\,k^2-\omega^3\,L_{1}^2\,L_{2}\,k^2+\omega^5\,C^2\,L_{1}^2\,L_{2}\,Z_{0}^2-2\,\omega^3\,C\,L_{1}\,L_{2}\,Z_{0}^2+\omega\,L_{2}\,Z_{0}^2-2\,\omega^3\,C\,L_{1}\,L_{2}\,R_{1}\,Z_{0}+2\,\omega\,L_{2}\,R_{1}\,Z_{0}+\omega\,L_{2}\,R_{1}^2+\omega^3\,L_{1}^2\,L_{2}}\over{\left| \omega^4\,C^2\,L_{1}^2\,Z_{0}^2-2\,\omega^2\,C\,L_{1}\,Z_{0}^2+Z_{0}^2-2\,\omega^2\,C\,L_{1}\,R_{1}\,Z_{0}+2\,R_{1}\,Z_{0}+R_{1}^2+\omega^2\,L_{1}^2\right| }}
Abs_S11(%omega) := (−ω3 C L1 L2 Z0 k2+ω3 C L1 L2 Z0−ω L2 Z0+ω L1 Z0+ω L1 R2+ω L2 R1)2+(ω2 L1 L2 k2+ω2 C L1 Z02−Z02+ω2 C L1 R2 Z0−R2 Z0+R1 Z0+R1 R2−ω2 L1 L2)2(ω3 C L1 L2 Z0 k2−ω3 C L1 L2 Z0+ω L2 Z0+ω L1 Z0+ω L1 R2+ω L2 R1)2+(ω2 L1 L2 k2−ω2 C L1 Z02+Z02−ω2 C L1 R2 Z0+R2 Z0+R1 Z0+R1 R2−ω2 L1 L2)2{{\sqrt{\left(-\omega^3\,C\,L_{1}\,L_{2}\,Z_{0}\,k^2+\omega^3\,C\,L_{1}\,L_{2}\,Z_{0}-\omega\,L_{2}\,Z_{0}+\omega\,L_{1}\,Z_{0}+\omega\,L_{1}\,R_{2}+\omega\,L_{2}\,R_{1}\right)^2+\left(\omega^2\,L_{1}\,L_{2}\,k^2+\omega^2\,C\,L_{1}\,Z_{0}^2-Z_{0}^2+\omega^2\,C\,L_{1}\,R_{2}\,Z_{0}-R_{2}\,Z_{0}+R_{1}\,Z_{0}+R_{1}\,R_{2}-\omega^2\,L_{1}\,L_{2}\right)^2}}\over{\sqrt{\left(\omega^3\,C\,L_{1}\,L_{2}\,Z_{0}\,k^2-\omega^3\,C\,L_{1}\,L_{2}\,Z_{0}+\omega\,L_{2}\,Z_{0}+\omega\,L_{1}\,Z_{0}+\omega\,L_{1}\,R_{2}+\omega\,L_{2}\,R_{1}\right)^2+\left(\omega^2\,L_{1}\,L_{2}\,k^2-\omega^2\,C\,L_{1}\,Z_{0}^2+Z_{0}^2-\omega^2\,C\,L_{1}\,R_{2}\,Z_{0}+R_{2}\,Z_{0}+R_{1}\,Z_{0}+R_{1}\,R_{2}-\omega^2\,L_{1}\,L_{2}\right)^2}}}
Arg_S11(%omega) := −atan2(2 ω5 C L12 L22 Z0 k4−(((4 ω5 C L12−2 ω3 L1) L22−2 ω3 C L1 L2 R1 R2) Z0−2 ω3 C L1 L2 R1 Z02) k2−(2 ω L1−2 ω3 C L12) Z03−(4 ω L1−4 ω3 C L12) R2 Z02−((2 ω L1−2 ω3 C L12) R22+(2 ω3 L1−2 ω5 C L12) L22) Z0,(ω4 L12 L22−ω6 C2 L12 L22 Z02) k4+((2 ω6 C2 L12−2 ω4 C L1) L22 Z02+2 ω2 L1 L2 R1 Z0+2 ω2 L1 L2 R1 R2−2 ω4 L12 L22) k2+(−ω4 C2 L12+2 ω2 C L1−1) Z04+(−2 ω4 C2 L12+4 ω2 C L1−2) R2 Z03+((−ω4 C2 L12+2 ω2 C L1−1) R22+R12+(−ω6 C2 L12+2 ω4 C L1−ω2) L22+ω2 L12) Z02+(2 R12+2 ω2 L12) R2 Z0+(R12+ω2 L12) R22+ω2 L22 R12+ω4 L12 L22)-{\rm atan2}\left(2\,\omega^5\,C\,L_{1}^2\,L_{2}^2\,Z_{0}\,k^4-\left(\left(\left(4\,\omega^5\,C\,L_{1}^2-2\,\omega^3\,L_{1}\right)\,L_{2}^2-2\,\omega^3\,C\,L_{1}\,L_{2}\,R_{1}\,R_{2}\right)\,Z_{0}-2\,\omega^3\,C\,L_{1}\,L_{2}\,R_{1}\,Z_{0}^2\right)\,k^2-\left(2\,\omega\,L_{1}-2\,\omega^3\,C\,L_{1}^2\right)\,Z_{0}^3-\left(4\,\omega\,L_{1}-4\,\omega^3\,C\,L_{1}^2\right)\,R_{2}\,Z_{0}^2-\left(\left(2\,\omega\,L_{1}-2\,\omega^3\,C\,L_{1}^2\right)\,R_{2}^2+\left(2\,\omega^3\,L_{1}-2\,\omega^5\,C\,L_{1}^2\right)\,L_{2}^2\right)\,Z_{0} , \left(\omega^4\,L_{1}^2\,L_{2}^2-\omega^6\,C^2\,L_{1}^2\,L_{2}^2\,Z_{0}^2\right)\,k^4+\left(\left(2\,\omega^6\,C^2\,L_{1}^2-2\,\omega^4\,C\,L_{1}\right)\,L_{2}^2\,Z_{0}^2+2\,\omega^2\,L_{1}\,L_{2}\,R_{1}\,Z_{0}+2\,\omega^2\,L_{1}\,L_{2}\,R_{1}\,R_{2}-2\,\omega^4\,L_{1}^2\,L_{2}^2\right)\,k^2+\left(-\omega^4\,C^2\,L_{1}^2+2\,\omega^2\,C\,L_{1}-1\right)\,Z_{0}^4+\left(-2\,\omega^4\,C^2\,L_{1}^2+4\,\omega^2\,C\,L_{1}-2\right)\,R_{2}\,Z_{0}^3+\left(\left(-\omega^4\,C^2\,L_{1}^2+2\,\omega^2\,C\,L_{1}-1\right)\,R_{2}^2+R_{1}^2+\left(-\omega^6\,C^2\,L_{1}^2+2\,\omega^4\,C\,L_{1}-\omega^2\right)\,L_{2}^2+\omega^2\,L_{1}^2\right)\,Z_{0}^2+\left(2\,R_{1}^2+2\,\omega^2\,L_{1}^2\right)\,R_{2}\,Z_{0}+\left(R_{1}^2+\omega^2\,L_{1}^2\right)\,R_{2}^2+\omega^2\,L_{2}^2\,R_{1}^2+\omega^4\,L_{1}^2\,L_{2}^2\right)
Abs_S12(%omega) := 2 ∣ω∣ L1 L2 Z0 k(ω3 C L1 L2 Z0 k2−ω3 C L1 L2 Z0+ω L2 Z0+ω L1 Z0+ω L1 R2+ω L2 R1)2+(ω2 L1 L2 k2−ω2 C L1 Z02+Z02−ω2 C L1 R2 Z0+R2 Z0+R1 Z0+R1 R2−ω2 L1 L2)2{{2\,\left| \omega\right| \,\sqrt{L_{1}}\,\sqrt{L_{2}}\,Z_{0}\,k}\over{\sqrt{\left(\omega^3\,C\,L_{1}\,L_{2}\,Z_{0}\,k^2-\omega^3\,C\,L_{1}\,L_{2}\,Z_{0}+\omega\,L_{2}\,Z_{0}+\omega\,L_{1}\,Z_{0}+\omega\,L_{1}\,R_{2}+\omega\,L_{2}\,R_{1}\right)^2+\left(\omega^2\,L_{1}\,L_{2}\,k^2-\omega^2\,C\,L_{1}\,Z_{0}^2+Z_{0}^2-\omega^2\,C\,L_{1}\,R_{2}\,Z_{0}+R_{2}\,Z_{0}+R_{1}\,Z_{0}+R_{1}\,R_{2}-\omega^2\,L_{1}\,L_{2}\right)^2}}}
Arg_S12(%omega) := atan2(L1 L2 (2 ω3 L1 L2 Z0 k3+((2 ω−2 ω3 C L1) Z03+((2 ω−2 ω3 C L1) R2+2 ω R1) Z02+(2 ω R1 R2−2 ω3 L1 L2) Z0) k),L1 L2 (2 ω4 C L1 L2 Z02 k3+(((2 ω2−2 ω4 C L1) L2+2 ω2 L1) Z02+(2 ω2 L1 R2+2 ω2 L2 R1) Z0) k)){\rm atan2}\left(\sqrt{L_{1}}\,\sqrt{L_{2}}\,\left(2\,\omega^3\,L_{1}\,L_{2}\,Z_{0}\,k^3+\left(\left(2\,\omega-2\,\omega^3\,C\,L_{1}\right)\,Z_{0}^3+\left(\left(2\,\omega-2\,\omega^3\,C\,L_{1}\right)\,R_{2}+2\,\omega\,R_{1}\right)\,Z_{0}^2+\left(2\,\omega\,R_{1}\,R_{2}-2\,\omega^3\,L_{1}\,L_{2}\right)\,Z_{0}\right)\,k\right) , \sqrt{L_{1}}\,\sqrt{L_{2}}\,\left(2\,\omega^4\,C\,L_{1}\,L_{2}\,Z_{0}^2\,k^3+\left(\left(\left(2\,\omega^2-2\,\omega^4\,C\,L_{1}\right)\,L_{2}+2\,\omega^2\,L_{1}\right)\,Z_{0}^2+\left(2\,\omega^2\,L_{1}\,R_{2}+2\,\omega^2\,L_{2}\,R_{1}\right)\,Z_{0}\right)\,k\right)\right)
Abs_S21(%omega) := 4 ω4 C2 L1 L2 R12 Z02 k2+4 ω2 L1 L2 Z02 k2(ω3 C L1 L2 Z0 k2−ω3 C L1 L2 Z0+ω L2 Z0+ω L1 Z0+ω L1 R2+ω L2 R1)2+(ω2 L1 L2 k2−ω2 C L1 Z02+Z02−ω2 C L1 R2 Z0+R2 Z0+R1 Z0+R1 R2−ω2 L1 L2)2{{\sqrt{4\,\omega^4\,C^2\,L_{1}\,L_{2}\,R_{1}^2\,Z_{0}^2\,k^2+4\,\omega^2\,L_{1}\,L_{2}\,Z_{0}^2\,k^2}}\over{\sqrt{\left(\omega^3\,C\,L_{1}\,L_{2}\,Z_{0}\,k^2-\omega^3\,C\,L_{1}\,L_{2}\,Z_{0}+\omega\,L_{2}\,Z_{0}+\omega\,L_{1}\,Z_{0}+\omega\,L_{1}\,R_{2}+\omega\,L_{2}\,R_{1}\right)^2+\left(\omega^2\,L_{1}\,L_{2}\,k^2-\omega^2\,C\,L_{1}\,Z_{0}^2+Z_{0}^2-\omega^2\,C\,L_{1}\,R_{2}\,Z_{0}+R_{2}\,Z_{0}+R_{1}\,Z_{0}+R_{1}\,R_{2}-\omega^2\,L_{1}\,L_{2}\right)^2}}}
Arg_S21(%omega) := atan2(L1 L2 ((2 ω3 L1 L2 Z0−2 ω5 C2 L1 L2 R1 Z02) k3+((2 ω−2 ω3 C L1) Z03+((2 ω−2 ω3 C L1) R2+((2 ω5 C2 L1−2 ω3 C) L2−2 ω3 C L1+2 ω) R1) Z02+((2 ω−2 ω3 C L1) R1 R2−2 ω3 C L2 R12−2 ω3 L1 L2) Z0) k),L1 L2 ((2 ω4 C L1 L2 Z02+2 ω4 C L1 L2 R1 Z0) k3+((2 ω2 C−2 ω4 C2 L1) R1 Z03+((2 ω2 C−2 ω4 C2 L1) R1 R2+2 ω2 C R12+(2 ω2−2 ω4 C L1) L2+2 ω2 L1) Z02+((2 ω2 C R12+2 ω2 L1) R2+(2 ω2−2 ω4 C L1) L2 R1) Z0) k)){\rm atan2}\left(\sqrt{L_{1}}\,\sqrt{L_{2}}\,\left(\left(2\,\omega^3\,L_{1}\,L_{2}\,Z_{0}-2\,\omega^5\,C^2\,L_{1}\,L_{2}\,R_{1}\,Z_{0}^2\right)\,k^3+\left(\left(2\,\omega-2\,\omega^3\,C\,L_{1}\right)\,Z_{0}^3+\left(\left(2\,\omega-2\,\omega^3\,C\,L_{1}\right)\,R_{2}+\left(\left(2\,\omega^5\,C^2\,L_{1}-2\,\omega^3\,C\right)\,L_{2}-2\,\omega^3\,C\,L_{1}+2\,\omega\right)\,R_{1}\right)\,Z_{0}^2+\left(\left(2\,\omega-2\,\omega^3\,C\,L_{1}\right)\,R_{1}\,R_{2}-2\,\omega^3\,C\,L_{2}\,R_{1}^2-2\,\omega^3\,L_{1}\,L_{2}\right)\,Z_{0}\right)\,k\right) , \sqrt{L_{1}}\,\sqrt{L_{2}}\,\left(\left(2\,\omega^4\,C\,L_{1}\,L_{2}\,Z_{0}^2+2\,\omega^4\,C\,L_{1}\,L_{2}\,R_{1}\,Z_{0}\right)\,k^3+\left(\left(2\,\omega^2\,C-2\,\omega^4\,C^2\,L_{1}\right)\,R_{1}\,Z_{0}^3+\left(\left(2\,\omega^2\,C-2\,\omega^4\,C^2\,L_{1}\right)\,R_{1}\,R_{2}+2\,\omega^2\,C\,R_{1}^2+\left(2\,\omega^2-2\,\omega^4\,C\,L_{1}\right)\,L_{2}+2\,\omega^2\,L_{1}\right)\,Z_{0}^2+\left(\left(2\,\omega^2\,C\,R_{1}^2+2\,\omega^2\,L_{1}\right)\,R_{2}+\left(2\,\omega^2-2\,\omega^4\,C\,L_{1}\right)\,L_{2}\,R_{1}\right)\,Z_{0}\right)\,k\right)\right)
tau_g(%omega) := (ω6 C3 L12 L22 R1 Z02+(ω6 C3 L12 L22 R12+ω4 C L12 L22) Z0+ω4 C L12 L22 R1) k4+(((3 ω4 C3 L1−ω6 C4 L12) L2 R12+(3 ω2 C L1−ω4 C2 L12) L2) Z03+(((3 ω4 C3 L1−ω6 C4 L12) L2 R12+(3 ω2 C L1−ω4 C2 L12) L2) R2+3 ω4 C3 L1 L2 R13+((2 ω4 C2 L1−2 ω6 C3 L12) L22+5 ω2 C L1 L2) R1) Z02+((3 ω4 C3 L1 L2 R13+5 ω2 C L1 L2 R1) R2+((ω4 C2 L1−2 ω6 C3 L12) L22+(2 ω2 C L1−ω4 C2 L12) L2) R12+(−2 ω4 C L12−ω2 L1) L22−ω2 L12 L2) Z0+((2 ω2 C L1−ω4 C2 L12) L2 R12−ω2 L12 L2) R2−ω4 C2 L1 L22 R13+(−2 ω4 C L12−ω2 L1) L22 R1) k2+(ω4 C3 L12−2 ω2 C2 L1+C) R1 Z04+((2 ω4 C3 L12−4 ω2 C2 L1+2 C) R1 R2+((ω6 C4 L12−2 ω4 C3 L1+ω2 C2) L2+ω4 C3 L12−ω2 C2 L1+2 C) R12+(ω4 C2 L12−2 ω2 C L1+1) L2+ω2 C L12+L1) Z03+((ω4 C3 L12−2 ω2 C2 L1+C) R1 R22+(((ω6 C4 L12−2 ω4 C3 L1+ω2 C2) L2+2 ω4 C3 L12−2 ω2 C2 L1+4 C) R12+(ω4 C2 L12−2 ω2 C L1+1) L2+2 ω2 C L12+2 L1) R2+((2 ω2 C2−2 ω4 C3 L1) L2+ω2 C2 L1+C) R13+((ω6 C3 L12−2 ω4 C2 L1+ω2 C) L22+(2−2 ω2 C L1) L2+ω2 C L12+L1) R1) Z02+(((ω4 C3 L12−ω2 C2 L1+2 C) R12+ω2 C L12+L1) R22+(((2 ω2 C2−2 ω4 C3 L1) L2+2 ω2 C2 L1+2 C) R13+((2−2 ω2 C L1) L2+2 ω2 C L12+2 L1) R1) R2+ω2 C2 L2 R14+((ω6 C3 L12−ω4 C2 L1+2 ω2 C) L22+(ω4 C2 L12+1) L2) R12+(ω4 C L12+ω2 L1) L22+ω2 L12 L2) Z0+((ω2 C2 L1+C) R13+(ω2 C L12+L1) R1) R22+(ω2 C2 L2 R14+(ω4 C2 L12+1) L2 R12+ω2 L12 L2) R2+(ω4 C2 L1+ω2 C) L22 R13+(ω4 C L12+ω2 L1) L22 R1((ω8 C4 L12 L22 R12+ω6 C2 L12 L22) Z02+ω6 C2 L12 L22 R12+ω4 L12 L22) k4+((((2 ω6 C3 L1−2 ω8 C4 L12) L22+2 ω4 C2 L1 L2) R12+(2 ω4 C L1−2 ω6 C2 L12) L22+2 ω2 L1 L2) Z02+((2 ω4 C2 L1 L2 R12+2 ω2 L1 L2) R2+(2 ω6 C3 L1 L22+2 ω4 C2 L1 L2) R13+(2 ω4 C L1 L22+2 ω2 L1 L2) R1) Z0+(2 ω4 C2 L1 L2 R13+2 ω2 L1 L2 R1) R2−2 ω6 C2 L12 L22 R12−2 ω4 L12 L22) k2+((ω6 C4 L12−2 ω4 C3 L1+ω2 C2) R12+ω4 C2 L12−2 ω2 C L1+1) Z04+(((2 ω6 C4 L12−4 ω4 C3 L1+2 ω2 C2) R12+2 ω4 C2 L12−4 ω2 C L1+2) R2+(2 ω2 C2−2 ω4 C3 L1) R13+(2−2 ω2 C L1) R1) Z03+(((ω6 C4 L12−2 ω4 C3 L1+ω2 C2) R12+ω4 C2 L12−2 ω2 C L1+1) R22+((4 ω2 C2−4 ω4 C3 L1) R13+(4−4 ω2 C L1) R1) R2+ω2 C2 R14+((ω8 C4 L12−2 ω6 C3 L1+ω4 C2) L22+ω4 C2 L12+1) R12+(ω6 C2 L12−2 ω4 C L1+ω2) L22+ω2 L12) Z02+(((2 ω2 C2−2 ω4 C3 L1) R13+(2−2 ω2 C L1) R1) R22+(2 ω2 C2 R14+(2 ω4 C2 L12+2) R12+2 ω2 L12) R2+(2 ω4 C2−2 ω6 C3 L1) L22 R13+(2 ω2−2 ω4 C L1) L22 R1) Z0+(ω2 C2 R14+(ω4 C2 L12+1) R12+ω2 L12) R22+ω4 C2 L22 R14+(ω6 C2 L12+ω2) L22 R12+ω4 L12 L22{{\left(\omega^6\,C^3\,L_{1}^2\,L_{2}^2\,R_{1}\,Z_{0}^2+\left(\omega^6\,C^3\,L_{1}^2\,L_{2}^2\,R_{1}^2+\omega^4\,C\,L_{1}^2\,L_{2}^2\right)\,Z_{0}+\omega^4\,C\,L_{1}^2\,L_{2}^2\,R_{1}\right)\,k^4+\left(\left(\left(3\,\omega^4\,C^3\,L_{1}-\omega^6\,C^4\,L_{1}^2\right)\,L_{2}\,R_{1}^2+\left(3\,\omega^2\,C\,L_{1}-\omega^4\,C^2\,L_{1}^2\right)\,L_{2}\right)\,Z_{0}^3+\left(\left(\left(3\,\omega^4\,C^3\,L_{1}-\omega^6\,C^4\,L_{1}^2\right)\,L_{2}\,R_{1}^2+\left(3\,\omega^2\,C\,L_{1}-\omega^4\,C^2\,L_{1}^2\right)\,L_{2}\right)\,R_{2}+3\,\omega^4\,C^3\,L_{1}\,L_{2}\,R_{1}^3+\left(\left(2\,\omega^4\,C^2\,L_{1}-2\,\omega^6\,C^3\,L_{1}^2\right)\,L_{2}^2+5\,\omega^2\,C\,L_{1}\,L_{2}\right)\,R_{1}\right)\,Z_{0}^2+\left(\left(3\,\omega^4\,C^3\,L_{1}\,L_{2}\,R_{1}^3+5\,\omega^2\,C\,L_{1}\,L_{2}\,R_{1}\right)\,R_{2}+\left(\left(\omega^4\,C^2\,L_{1}-2\,\omega^6\,C^3\,L_{1}^2\right)\,L_{2}^2+\left(2\,\omega^2\,C\,L_{1}-\omega^4\,C^2\,L_{1}^2\right)\,L_{2}\right)\,R_{1}^2+\left(-2\,\omega^4\,C\,L_{1}^2-\omega^2\,L_{1}\right)\,L_{2}^2-\omega^2\,L_{1}^2\,L_{2}\right)\,Z_{0}+\left(\left(2\,\omega^2\,C\,L_{1}-\omega^4\,C^2\,L_{1}^2\right)\,L_{2}\,R_{1}^2-\omega^2\,L_{1}^2\,L_{2}\right)\,R_{2}-\omega^4\,C^2\,L_{1}\,L_{2}^2\,R_{1}^3+\left(-2\,\omega^4\,C\,L_{1}^2-\omega^2\,L_{1}\right)\,L_{2}^2\,R_{1}\right)\,k^2+\left(\omega^4\,C^3\,L_{1}^2-2\,\omega^2\,C^2\,L_{1}+C\right)\,R_{1}\,Z_{0}^4+\left(\left(2\,\omega^4\,C^3\,L_{1}^2-4\,\omega^2\,C^2\,L_{1}+2\,C\right)\,R_{1}\,R_{2}+\left(\left(\omega^6\,C^4\,L_{1}^2-2\,\omega^4\,C^3\,L_{1}+\omega^2\,C^2\right)\,L_{2}+\omega^4\,C^3\,L_{1}^2-\omega^2\,C^2\,L_{1}+2\,C\right)\,R_{1}^2+\left(\omega^4\,C^2\,L_{1}^2-2\,\omega^2\,C\,L_{1}+1\right)\,L_{2}+\omega^2\,C\,L_{1}^2+L_{1}\right)\,Z_{0}^3+\left(\left(\omega^4\,C^3\,L_{1}^2-2\,\omega^2\,C^2\,L_{1}+C\right)\,R_{1}\,R_{2}^2+\left(\left(\left(\omega^6\,C^4\,L_{1}^2-2\,\omega^4\,C^3\,L_{1}+\omega^2\,C^2\right)\,L_{2}+2\,\omega^4\,C^3\,L_{1}^2-2\,\omega^2\,C^2\,L_{1}+4\,C\right)\,R_{1}^2+\left(\omega^4\,C^2\,L_{1}^2-2\,\omega^2\,C\,L_{1}+1\right)\,L_{2}+2\,\omega^2\,C\,L_{1}^2+2\,L_{1}\right)\,R_{2}+\left(\left(2\,\omega^2\,C^2-2\,\omega^4\,C^3\,L_{1}\right)\,L_{2}+\omega^2\,C^2\,L_{1}+C\right)\,R_{1}^3+\left(\left(\omega^6\,C^3\,L_{1}^2-2\,\omega^4\,C^2\,L_{1}+\omega^2\,C\right)\,L_{2}^2+\left(2-2\,\omega^2\,C\,L_{1}\right)\,L_{2}+\omega^2\,C\,L_{1}^2+L_{1}\right)\,R_{1}\right)\,Z_{0}^2+\left(\left(\left(\omega^4\,C^3\,L_{1}^2-\omega^2\,C^2\,L_{1}+2\,C\right)\,R_{1}^2+\omega^2\,C\,L_{1}^2+L_{1}\right)\,R_{2}^2+\left(\left(\left(2\,\omega^2\,C^2-2\,\omega^4\,C^3\,L_{1}\right)\,L_{2}+2\,\omega^2\,C^2\,L_{1}+2\,C\right)\,R_{1}^3+\left(\left(2-2\,\omega^2\,C\,L_{1}\right)\,L_{2}+2\,\omega^2\,C\,L_{1}^2+2\,L_{1}\right)\,R_{1}\right)\,R_{2}+\omega^2\,C^2\,L_{2}\,R_{1}^4+\left(\left(\omega^6\,C^3\,L_{1}^2-\omega^4\,C^2\,L_{1}+2\,\omega^2\,C\right)\,L_{2}^2+\left(\omega^4\,C^2\,L_{1}^2+1\right)\,L_{2}\right)\,R_{1}^2+\left(\omega^4\,C\,L_{1}^2+\omega^2\,L_{1}\right)\,L_{2}^2+\omega^2\,L_{1}^2\,L_{2}\right)\,Z_{0}+\left(\left(\omega^2\,C^2\,L_{1}+C\right)\,R_{1}^3+\left(\omega^2\,C\,L_{1}^2+L_{1}\right)\,R_{1}\right)\,R_{2}^2+\left(\omega^2\,C^2\,L_{2}\,R_{1}^4+\left(\omega^4\,C^2\,L_{1}^2+1\right)\,L_{2}\,R_{1}^2+\omega^2\,L_{1}^2\,L_{2}\right)\,R_{2}+\left(\omega^4\,C^2\,L_{1}+\omega^2\,C\right)\,L_{2}^2\,R_{1}^3+\left(\omega^4\,C\,L_{1}^2+\omega^2\,L_{1}\right)\,L_{2}^2\,R_{1}}\over{\left(\left(\omega^8\,C^4\,L_{1}^2\,L_{2}^2\,R_{1}^2+\omega^6\,C^2\,L_{1}^2\,L_{2}^2\right)\,Z_{0}^2+\omega^6\,C^2\,L_{1}^2\,L_{2}^2\,R_{1}^2+\omega^4\,L_{1}^2\,L_{2}^2\right)\,k^4+\left(\left(\left(\left(2\,\omega^6\,C^3\,L_{1}-2\,\omega^8\,C^4\,L_{1}^2\right)\,L_{2}^2+2\,\omega^4\,C^2\,L_{1}\,L_{2}\right)\,R_{1}^2+\left(2\,\omega^4\,C\,L_{1}-2\,\omega^6\,C^2\,L_{1}^2\right)\,L_{2}^2+2\,\omega^2\,L_{1}\,L_{2}\right)\,Z_{0}^2+\left(\left(2\,\omega^4\,C^2\,L_{1}\,L_{2}\,R_{1}^2+2\,\omega^2\,L_{1}\,L_{2}\right)\,R_{2}+\left(2\,\omega^6\,C^3\,L_{1}\,L_{2}^2+2\,\omega^4\,C^2\,L_{1}\,L_{2}\right)\,R_{1}^3+\left(2\,\omega^4\,C\,L_{1}\,L_{2}^2+2\,\omega^2\,L_{1}\,L_{2}\right)\,R_{1}\right)\,Z_{0}+\left(2\,\omega^4\,C^2\,L_{1}\,L_{2}\,R_{1}^3+2\,\omega^2\,L_{1}\,L_{2}\,R_{1}\right)\,R_{2}-2\,\omega^6\,C^2\,L_{1}^2\,L_{2}^2\,R_{1}^2-2\,\omega^4\,L_{1}^2\,L_{2}^2\right)\,k^2+\left(\left(\omega^6\,C^4\,L_{1}^2-2\,\omega^4\,C^3\,L_{1}+\omega^2\,C^2\right)\,R_{1}^2+\omega^4\,C^2\,L_{1}^2-2\,\omega^2\,C\,L_{1}+1\right)\,Z_{0}^4+\left(\left(\left(2\,\omega^6\,C^4\,L_{1}^2-4\,\omega^4\,C^3\,L_{1}+2\,\omega^2\,C^2\right)\,R_{1}^2+2\,\omega^4\,C^2\,L_{1}^2-4\,\omega^2\,C\,L_{1}+2\right)\,R_{2}+\left(2\,\omega^2\,C^2-2\,\omega^4\,C^3\,L_{1}\right)\,R_{1}^3+\left(2-2\,\omega^2\,C\,L_{1}\right)\,R_{1}\right)\,Z_{0}^3+\left(\left(\left(\omega^6\,C^4\,L_{1}^2-2\,\omega^4\,C^3\,L_{1}+\omega^2\,C^2\right)\,R_{1}^2+\omega^4\,C^2\,L_{1}^2-2\,\omega^2\,C\,L_{1}+1\right)\,R_{2}^2+\left(\left(4\,\omega^2\,C^2-4\,\omega^4\,C^3\,L_{1}\right)\,R_{1}^3+\left(4-4\,\omega^2\,C\,L_{1}\right)\,R_{1}\right)\,R_{2}+\omega^2\,C^2\,R_{1}^4+\left(\left(\omega^8\,C^4\,L_{1}^2-2\,\omega^6\,C^3\,L_{1}+\omega^4\,C^2\right)\,L_{2}^2+\omega^4\,C^2\,L_{1}^2+1\right)\,R_{1}^2+\left(\omega^6\,C^2\,L_{1}^2-2\,\omega^4\,C\,L_{1}+\omega^2\right)\,L_{2}^2+\omega^2\,L_{1}^2\right)\,Z_{0}^2+\left(\left(\left(2\,\omega^2\,C^2-2\,\omega^4\,C^3\,L_{1}\right)\,R_{1}^3+\left(2-2\,\omega^2\,C\,L_{1}\right)\,R_{1}\right)\,R_{2}^2+\left(2\,\omega^2\,C^2\,R_{1}^4+\left(2\,\omega^4\,C^2\,L_{1}^2+2\right)\,R_{1}^2+2\,\omega^2\,L_{1}^2\right)\,R_{2}+\left(2\,\omega^4\,C^2-2\,\omega^6\,C^3\,L_{1}\right)\,L_{2}^2\,R_{1}^3+\left(2\,\omega^2-2\,\omega^4\,C\,L_{1}\right)\,L_{2}^2\,R_{1}\right)\,Z_{0}+\left(\omega^2\,C^2\,R_{1}^4+\left(\omega^4\,C^2\,L_{1}^2+1\right)\,R_{1}^2+\omega^2\,L_{1}^2\right)\,R_{2}^2+\omega^4\,C^2\,L_{2}^2\,R_{1}^4+\left(\omega^6\,C^2\,L_{1}^2+\omega^2\right)\,L_{2}^2\,R_{1}^2+\omega^4\,L_{1}^2\,L_{2}^2}}
Abs_S22(%omega) := (ω3 C L1 L2 Z0 k2−ω3 C L1 L2 Z0+ω L2 Z0−ω L1 Z0+ω L1 R2+ω L2 R1)2+(ω2 L1 L2 k2+ω2 C L1 Z02−Z02−ω2 C L1 R2 Z0+R2 Z0−R1 Z0+R1 R2−ω2 L1 L2)2(ω3 C L1 L2 Z0 k2−ω3 C L1 L2 Z0+ω L2 Z0+ω L1 Z0+ω L1 R2+ω L2 R1)2+(ω2 L1 L2 k2−ω2 C L1 Z02+Z02−ω2 C L1 R2 Z0+R2 Z0+R1 Z0+R1 R2−ω2 L1 L2)2{{\sqrt{\left(\omega^3\,C\,L_{1}\,L_{2}\,Z_{0}\,k^2-\omega^3\,C\,L_{1}\,L_{2}\,Z_{0}+\omega\,L_{2}\,Z_{0}-\omega\,L_{1}\,Z_{0}+\omega\,L_{1}\,R_{2}+\omega\,L_{2}\,R_{1}\right)^2+\left(\omega^2\,L_{1}\,L_{2}\,k^2+\omega^2\,C\,L_{1}\,Z_{0}^2-Z_{0}^2-\omega^2\,C\,L_{1}\,R_{2}\,Z_{0}+R_{2}\,Z_{0}-R_{1}\,Z_{0}+R_{1}\,R_{2}-\omega^2\,L_{1}\,L_{2}\right)^2}}\over{\sqrt{\left(\omega^3\,C\,L_{1}\,L_{2}\,Z_{0}\,k^2-\omega^3\,C\,L_{1}\,L_{2}\,Z_{0}+\omega\,L_{2}\,Z_{0}+\omega\,L_{1}\,Z_{0}+\omega\,L_{1}\,R_{2}+\omega\,L_{2}\,R_{1}\right)^2+\left(\omega^2\,L_{1}\,L_{2}\,k^2-\omega^2\,C\,L_{1}\,Z_{0}^2+Z_{0}^2-\omega^2\,C\,L_{1}\,R_{2}\,Z_{0}+R_{2}\,Z_{0}+R_{1}\,Z_{0}+R_{1}\,R_{2}-\omega^2\,L_{1}\,L_{2}\right)^2}}}
Arg_S22(%omega) := atan2(((2 ω3 C L1−2 ω5 C2 L12) L2 Z03+2 ω3 C L1 L2 R1 Z02−2 ω3 L12 L2 Z0) k2+(2 ω5 C2 L12−4 ω3 C L1+2 ω) L2 Z03+(4 ω−4 ω3 C L1) L2 R1 Z02+(2 ω L2 R12+2 ω3 L12 L2) Z0,(ω6 C2 L12 L22 Z02+ω4 L12 L22) k4+((2 ω4 C L1−2 ω6 C2 L12) L22 Z02+(2 ω2 L1 L2 R2+2 ω4 C L1 L22 R1) Z0+2 ω2 L1 L2 R1 R2−2 ω4 L12 L22) k2+(−ω4 C2 L12+2 ω2 C L1−1) Z04+(2 ω2 C L1−2) R1 Z03+((ω4 C2 L12−2 ω2 C L1+1) R22−R12+(ω6 C2 L12−2 ω4 C L1+ω2) L22−ω2 L12) Z02+((2−2 ω2 C L1) R1 R22+(2 ω2−2 ω4 C L1) L22 R1) Z0+(R12+ω2 L12) R22+ω2 L22 R12+ω4 L12 L22){\rm atan2}\left(\left(\left(2\,\omega^3\,C\,L_{1}-2\,\omega^5\,C^2\,L_{1}^2\right)\,L_{2}\,Z_{0}^3+2\,\omega^3\,C\,L_{1}\,L_{2}\,R_{1}\,Z_{0}^2-2\,\omega^3\,L_{1}^2\,L_{2}\,Z_{0}\right)\,k^2+\left(2\,\omega^5\,C^2\,L_{1}^2-4\,\omega^3\,C\,L_{1}+2\,\omega\right)\,L_{2}\,Z_{0}^3+\left(4\,\omega-4\,\omega^3\,C\,L_{1}\right)\,L_{2}\,R_{1}\,Z_{0}^2+\left(2\,\omega\,L_{2}\,R_{1}^2+2\,\omega^3\,L_{1}^2\,L_{2}\right)\,Z_{0} , \left(\omega^6\,C^2\,L_{1}^2\,L_{2}^2\,Z_{0}^2+\omega^4\,L_{1}^2\,L_{2}^2\right)\,k^4+\left(\left(2\,\omega^4\,C\,L_{1}-2\,\omega^6\,C^2\,L_{1}^2\right)\,L_{2}^2\,Z_{0}^2+\left(2\,\omega^2\,L_{1}\,L_{2}\,R_{2}+2\,\omega^4\,C\,L_{1}\,L_{2}^2\,R_{1}\right)\,Z_{0}+2\,\omega^2\,L_{1}\,L_{2}\,R_{1}\,R_{2}-2\,\omega^4\,L_{1}^2\,L_{2}^2\right)\,k^2+\left(-\omega^4\,C^2\,L_{1}^2+2\,\omega^2\,C\,L_{1}-1\right)\,Z_{0}^4+\left(2\,\omega^2\,C\,L_{1}-2\right)\,R_{1}\,Z_{0}^3+\left(\left(\omega^4\,C^2\,L_{1}^2-2\,\omega^2\,C\,L_{1}+1\right)\,R_{2}^2-R_{1}^2+\left(\omega^6\,C^2\,L_{1}^2-2\,\omega^4\,C\,L_{1}+\omega^2\right)\,L_{2}^2-\omega^2\,L_{1}^2\right)\,Z_{0}^2+\left(\left(2-2\,\omega^2\,C\,L_{1}\right)\,R_{1}\,R_{2}^2+\left(2\,\omega^2-2\,\omega^4\,C\,L_{1}\right)\,L_{2}^2\,R_{1}\right)\,Z_{0}+\left(R_{1}^2+\omega^2\,L_{1}^2\right)\,R_{2}^2+\omega^2\,L_{2}^2\,R_{1}^2+\omega^4\,L_{1}^2\,L_{2}^2\right)
Z_IN(s) := (L1 L2 k2−L1 L2) s2+(−L1 Z0−L1 R2−L2 R1) s−R1 Z0−R1 R2(C L1 L2 k2−C L1 L2) s3+(−C L1 Z0−C L1 R2) s2−L2 s−Z0−R2{{\left(L_{1}\,L_{2}\,k^2-L_{1}\,L_{2}\right)\,s^2+\left(-L_{1}\,Z_{0}-L_{1}\,R_{2}-L_{2}\,R_{1}\right)\,s-R_{1}\,Z_{0}-R_{1}\,R_{2}}\over{\left(C\,L_{1}\,L_{2}\,k^2-C\,L_{1}\,L_{2}\right)\,s^3+\left(-C\,L_{1}\,Z_{0}-C\,L_{1}\,R_{2}\right)\,s^2-L_{2}\,s-Z_{0}-R_{2}}}
R_IN(%omega) := ω2 L1 L2 Z0 k2+ω2 L1 L2 R2 k2+ω4 C L1 L22 R1 k2−ω2 C L1 R1 Z02+R1 Z02−2 ω2 C L1 R1 R2 Z0+2 R1 R2 Z0−ω2 C L1 R1 R22+R1 R22−ω4 C L1 L22 R1+ω2 L22 R1∣ω6 C2 L12 L22 k4−2 ω6 C2 L12 L22 k2+2 ω4 C L1 L22 k2+ω4 C2 L12 Z02−2 ω2 C L1 Z02+Z02+2 ω4 C2 L12 R2 Z0−4 ω2 C L1 R2 Z0+2 R2 Z0+ω4 C2 L12 R22−2 ω2 C L1 R22+R22+ω6 C2 L12 L22−2 ω4 C L1 L22+ω2 L22∣{{\omega^2\,L_{1}\,L_{2}\,Z_{0}\,k^2+\omega^2\,L_{1}\,L_{2}\,R_{2}\,k^2+\omega^4\,C\,L_{1}\,L_{2}^2\,R_{1}\,k^2-\omega^2\,C\,L_{1}\,R_{1}\,Z_{0}^2+R_{1}\,Z_{0}^2-2\,\omega^2\,C\,L_{1}\,R_{1}\,R_{2}\,Z_{0}+2\,R_{1}\,R_{2}\,Z_{0}-\omega^2\,C\,L_{1}\,R_{1}\,R_{2}^2+R_{1}\,R_{2}^2-\omega^4\,C\,L_{1}\,L_{2}^2\,R_{1}+\omega^2\,L_{2}^2\,R_{1}}\over{\left| \omega^6\,C^2\,L_{1}^2\,L_{2}^2\,k^4-2\,\omega^6\,C^2\,L_{1}^2\,L_{2}^2\,k^2+2\,\omega^4\,C\,L_{1}\,L_{2}^2\,k^2+\omega^4\,C^2\,L_{1}^2\,Z_{0}^2-2\,\omega^2\,C\,L_{1}\,Z_{0}^2+Z_{0}^2+2\,\omega^4\,C^2\,L_{1}^2\,R_{2}\,Z_{0}-4\,\omega^2\,C\,L_{1}\,R_{2}\,Z_{0}+2\,R_{2}\,Z_{0}+\omega^4\,C^2\,L_{1}^2\,R_{2}^2-2\,\omega^2\,C\,L_{1}\,R_{2}^2+R_{2}^2+\omega^6\,C^2\,L_{1}^2\,L_{2}^2-2\,\omega^4\,C\,L_{1}\,L_{2}^2+\omega^2\,L_{2}^2\right| }}
X_IN(%omega) := −ω5 C L12 L22 k4−ω3 C L1 L2 R1 Z0 k2−ω3 C L1 L2 R1 R2 k2+2 ω5 C L12 L22 k2−ω3 L1 L22 k2−ω3 C L12 Z02+ω L1 Z02−2 ω3 C L12 R2 Z0+2 ω L1 R2 Z0−ω3 C L12 R22+ω L1 R22−ω5 C L12 L22+ω3 L1 L22∣ω6 C2 L12 L22 k4−2 ω6 C2 L12 L22 k2+2 ω4 C L1 L22 k2+ω4 C2 L12 Z02−2 ω2 C L1 Z02+Z02+2 ω4 C2 L12 R2 Z0−4 ω2 C L1 R2 Z0+2 R2 Z0+ω4 C2 L12 R22−2 ω2 C L1 R22+R22+ω6 C2 L12 L22−2 ω4 C L1 L22+ω2 L22∣{{-\omega^5\,C\,L_{1}^2\,L_{2}^2\,k^4-\omega^3\,C\,L_{1}\,L_{2}\,R_{1}\,Z_{0}\,k^2-\omega^3\,C\,L_{1}\,L_{2}\,R_{1}\,R_{2}\,k^2+2\,\omega^5\,C\,L_{1}^2\,L_{2}^2\,k^2-\omega^3\,L_{1}\,L_{2}^2\,k^2-\omega^3\,C\,L_{1}^2\,Z_{0}^2+\omega\,L_{1}\,Z_{0}^2-2\,\omega^3\,C\,L_{1}^2\,R_{2}\,Z_{0}+2\,\omega\,L_{1}\,R_{2}\,Z_{0}-\omega^3\,C\,L_{1}^2\,R_{2}^2+\omega\,L_{1}\,R_{2}^2-\omega^5\,C\,L_{1}^2\,L_{2}^2+\omega^3\,L_{1}\,L_{2}^2}\over{\left| \omega^6\,C^2\,L_{1}^2\,L_{2}^2\,k^4-2\,\omega^6\,C^2\,L_{1}^2\,L_{2}^2\,k^2+2\,\omega^4\,C\,L_{1}\,L_{2}^2\,k^2+\omega^4\,C^2\,L_{1}^2\,Z_{0}^2-2\,\omega^2\,C\,L_{1}\,Z_{0}^2+Z_{0}^2+2\,\omega^4\,C^2\,L_{1}^2\,R_{2}\,Z_{0}-4\,\omega^2\,C\,L_{1}\,R_{2}\,Z_{0}+2\,R_{2}\,Z_{0}+\omega^4\,C^2\,L_{1}^2\,R_{2}^2-2\,\omega^2\,C\,L_{1}\,R_{2}^2+R_{2}^2+\omega^6\,C^2\,L_{1}^2\,L_{2}^2-2\,\omega^4\,C\,L_{1}\,L_{2}^2+\omega^2\,L_{2}^2\right| }}
Z_OUT(s) := −(C L1 L2 Z0 k2−C L1 L2 Z0) s3+(L1 L2 k2−C L1 R2 Z0−L1 L2) s2+(−L2 Z0−L1 R2−L2 R1) s−R2 Z0−R1 R2C L1 Z0 s2+L1 s+Z0+R1-{{\left(C\,L_{1}\,L_{2}\,Z_{0}\,k^2-C\,L_{1}\,L_{2}\,Z_{0}\right)\,s^3+\left(L_{1}\,L_{2}\,k^2-C\,L_{1}\,R_{2}\,Z_{0}-L_{1}\,L_{2}\right)\,s^2+\left(-L_{2}\,Z_{0}-L_{1}\,R_{2}-L_{2}\,R_{1}\right)\,s-R_{2}\,Z_{0}-R_{1}\,R_{2}}\over{C\,L_{1}\,Z_{0}\,s^2+L_{1}\,s+Z_{0}+R_{1}}}
R_OUT(%omega) := ω2 L1 L2 Z0 k2+ω2 L1 L2 R1 k2+ω4 C2 L12 R2 Z02−2 ω2 C L1 R2 Z02+R2 Z02−2 ω2 C L1 R1 R2 Z0+2 R1 R2 Z0+R12 R2+ω2 L12 R2∣ω4 C2 L12 Z02−2 ω2 C L1 Z02+Z02−2 ω2 C L1 R1 Z0+2 R1 Z0+R12+ω2 L12∣{{\omega^2\,L_{1}\,L_{2}\,Z_{0}\,k^2+\omega^2\,L_{1}\,L_{2}\,R_{1}\,k^2+\omega^4\,C^2\,L_{1}^2\,R_{2}\,Z_{0}^2-2\,\omega^2\,C\,L_{1}\,R_{2}\,Z_{0}^2+R_{2}\,Z_{0}^2-2\,\omega^2\,C\,L_{1}\,R_{1}\,R_{2}\,Z_{0}+2\,R_{1}\,R_{2}\,Z_{0}+R_{1}^2\,R_{2}+\omega^2\,L_{1}^2\,R_{2}}\over{\left| \omega^4\,C^2\,L_{1}^2\,Z_{0}^2-2\,\omega^2\,C\,L_{1}\,Z_{0}^2+Z_{0}^2-2\,\omega^2\,C\,L_{1}\,R_{1}\,Z_{0}+2\,R_{1}\,Z_{0}+R_{1}^2+\omega^2\,L_{1}^2\right| }}
X_OUT(%omega) := −ω5 C2 L12 L2 Z02 k2+ω3 C L1 L2 Z02 k2+ω3 C L1 L2 R1 Z0 k2−ω3 L12 L2 k2+ω5 C2 L12 L2 Z02−2 ω3 C L1 L2 Z02+ω L2 Z02−2 ω3 C L1 L2 R1 Z0+2 ω L2 R1 Z0+ω L2 R12+ω3 L12 L2∣ω4 C2 L12 Z02−2 ω2 C L1 Z02+Z02−2 ω2 C L1 R1 Z0+2 R1 Z0+R12+ω2 L12∣{{-\omega^5\,C^2\,L_{1}^2\,L_{2}\,Z_{0}^2\,k^2+\omega^3\,C\,L_{1}\,L_{2}\,Z_{0}^2\,k^2+\omega^3\,C\,L_{1}\,L_{2}\,R_{1}\,Z_{0}\,k^2-\omega^3\,L_{1}^2\,L_{2}\,k^2+\omega^5\,C^2\,L_{1}^2\,L_{2}\,Z_{0}^2-2\,\omega^3\,C\,L_{1}\,L_{2}\,Z_{0}^2+\omega\,L_{2}\,Z_{0}^2-2\,\omega^3\,C\,L_{1}\,L_{2}\,R_{1}\,Z_{0}+2\,\omega\,L_{2}\,R_{1}\,Z_{0}+\omega\,L_{2}\,R_{1}^2+\omega^3\,L_{1}^2\,L_{2}}\over{\left| \omega^4\,C^2\,L_{1}^2\,Z_{0}^2-2\,\omega^2\,C\,L_{1}\,Z_{0}^2+Z_{0}^2-2\,\omega^2\,C\,L_{1}\,R_{1}\,Z_{0}+2\,R_{1}\,Z_{0}+R_{1}^2+\omega^2\,L_{1}^2\right| }}
L1:2e-9$
L2:2e-9$
k:0.9$
Z_0:50$
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);
3.166286988823055×10−123.166286988823055 \times 10^{-12}2.5132741228718352.5132741228718352.5132741228718352.513274122871835
PlotSmithChart([
    parametric,
    realpart(S(%i*2*%pi*f)[1,1]),
    imagpart(S(%i*2*%pi*f)[1,1]),
    [f,1e6,20e9],
    [nticks,100]]);
/* S12 is a TRANSMISSION coefficient; a Smith chart maps REFLECTION
   coefficient.  Its magnitude is plotted against frequency further down. */
/* S21 is a TRANSMISSION coefficient; a Smith chart maps REFLECTION
   coefficient.  Its magnitude is plotted against frequency further down. */
PlotSmithChart([
    parametric,
    realpart(S(%i*2*%pi*f)[2,2]),
    imagpart(S(%i*2*%pi*f)[2,2]),
    [f,1e6,20e9],
    [nticks,100]]);
Plot generated by LOSSY_TRANSFORMER.wxm
Plot generated by LOSSY_TRANSFORMER.wxm
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"])$
Plot generated by LOSSY_TRANSFORMER.wxm
Plot generated by LOSSY_TRANSFORMER.wxm
Plot generated by LOSSY_TRANSFORMER.wxm
Plot generated by LOSSY_TRANSFORMER.wxm
Plot generated by LOSSY_TRANSFORMER.wxm
Plot generated by LOSSY_TRANSFORMER.wxm
Plot generated by LOSSY_TRANSFORMER.wxm
Plot generated by LOSSY_TRANSFORMER.wxm
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"])$
Plot generated by LOSSY_TRANSFORMER.wxm
Plot generated by LOSSY_TRANSFORMER.wxm
Plot generated by LOSSY_TRANSFORMER.wxm
Plot generated by LOSSY_TRANSFORMER.wxm
Plot generated by LOSSY_TRANSFORMER.wxm