Constant-k Section Analysis

Generated from CONSTANT_K_SECTION.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 "Microwave Engineering, 2nd Ed., by D. Pozar, Wiley, 1998, p. 433)

Without Assumptions for Z_1 and Z_2

ABCD-, G-, H-, S-, Y- and Z-Parameters

Solution:linsolve([v_in-v_1=Z_1/2*i_in,v_out-v_1=Z_1/2*i_out,v_1=Z_2*i_1,i_in+i_out=i_1],[i_in,i_out,i_1,v_1]);
TwoPortMatrices(Solution);
[iin=(4 Z2+2 Z1) vin−4 Z2 vout4 Z1 Z2+Z12,iout=−−4 Z2 vout−2 Z1 vout+4 Z2 vin4 Z1 Z2+Z12,i1=2 vout+2 vin4 Z2+Z1,v1=2 Z2 vout+2 Z2 vin4 Z2+Z1]\left[ \mathit{i}_{in}={{\left(4\,Z_{2}+2\,Z_{1}\right)\,\mathit{v}_{in}-4\,Z_{2}\,\mathit{v}_{out}}\over{4\,Z_{1}\,Z_{2}+Z_{1}^2}} , \mathit{i}_{out}=-{{-4\,Z_{2}\,\mathit{v}_{out}-2\,Z_{1}\,\mathit{v}_{out}+4\,Z_{2}\,\mathit{v}_{in}}\over{4\,Z_{1}\,Z_{2}+Z_{1}^2}} , i_{1}={{2\,\mathit{v}_{out}+2\,\mathit{v}_{in}}\over{4\,Z_{2}+Z_{1}}} , v_{1}={{2\,Z_{2}\,\mathit{v}_{out}+2\,Z_{2}\,\mathit{v}_{in}}\over{4\,Z_{2}+Z_{1}}} \right](2 Z2+Z12Z2Z22 Z2+Z12)\begin{pmatrix} {{2\,Z_{2}+Z_{1}}\over{2}}&Z_{2}\\ Z_{2}&{{2\,Z_{2}+Z_{1}}\over{2}} \end{pmatrix}
ABCD(s) := (2 Z2+Z12 Z24 Z1 Z2+Z124 Z21Z22 Z2+Z12 Z2)\begin{pmatrix} {{2\,Z_{2}+Z_{1}}\over{2\,Z_{2}}}&{{4\,Z_{1}\,Z_{2}+Z_{1}^2}\over{4\,Z_{2}}}\\ {{1}\over{Z_{2}}}&{{2\,Z_{2}+Z_{1}}\over{2\,Z_{2}}} \end{pmatrix}
G(s) := (22 Z2+Z1−2 Z22 Z2+Z12 Z22 Z2+Z14 Z1 Z2+Z124 Z2+2 Z1)\begin{pmatrix} {{2}\over{2\,Z_{2}+Z_{1}}}&-{{2\,Z_{2}}\over{2\,Z_{2}+Z_{1}}}\\ {{2\,Z_{2}}\over{2\,Z_{2}+Z_{1}}}&{{4\,Z_{1}\,Z_{2}+Z_{1}^2}\over{4\,Z_{2}+2\,Z_{1}}} \end{pmatrix}
H(s) := (4 Z1 Z2+Z124 Z2+2 Z12 Z22 Z2+Z1−2 Z22 Z2+Z122 Z2+Z1)\begin{pmatrix} {{4\,Z_{1}\,Z_{2}+Z_{1}^2}\over{4\,Z_{2}+2\,Z_{1}}}&{{2\,Z_{2}}\over{2\,Z_{2}+Z_{1}}}\\ -{{2\,Z_{2}}\over{2\,Z_{2}+Z_{1}}}&{{2}\over{2\,Z_{2}+Z_{1}}} \end{pmatrix}
InverseABCD(s) := (2 Z2+Z12 Z24 Z1 Z2+Z124 Z21Z22 Z2+Z12 Z2)\begin{pmatrix} {{2\,Z_{2}+Z_{1}}\over{2\,Z_{2}}}&{{4\,Z_{1}\,Z_{2}+Z_{1}^2}\over{4\,Z_{2}}}\\ {{1}\over{Z_{2}}}&{{2\,Z_{2}+Z_{1}}\over{2\,Z_{2}}} \end{pmatrix}
S(s) := (4 Z1 Z2+Z12−4 Z02(4 Z1+8 Z0) Z2+Z12+4 Z0 Z1+4 Z028 Z0 Z2(4 Z1+8 Z0) Z2+Z12+4 Z0 Z1+4 Z028 Z0 Z2(4 Z1+8 Z0) Z2+Z12+4 Z0 Z1+4 Z024 Z1 Z2+Z12−4 Z02(4 Z1+8 Z0) Z2+Z12+4 Z0 Z1+4 Z02)\begin{pmatrix} {{4\,Z_{1}\,Z_{2}+Z_{1}^2-4\,Z_{0}^2}\over{\left(4\,Z_{1}+8\,Z_{0}\right)\,Z_{2}+Z_{1}^2+4\,Z_{0}\,Z_{1}+4\,Z_{0}^2}}&{{8\,Z_{0}\,Z_{2}}\over{\left(4\,Z_{1}+8\,Z_{0}\right)\,Z_{2}+Z_{1}^2+4\,Z_{0}\,Z_{1}+4\,Z_{0}^2}}\\ {{8\,Z_{0}\,Z_{2}}\over{\left(4\,Z_{1}+8\,Z_{0}\right)\,Z_{2}+Z_{1}^2+4\,Z_{0}\,Z_{1}+4\,Z_{0}^2}}&{{4\,Z_{1}\,Z_{2}+Z_{1}^2-4\,Z_{0}^2}\over{\left(4\,Z_{1}+8\,Z_{0}\right)\,Z_{2}+Z_{1}^2+4\,Z_{0}\,Z_{1}+4\,Z_{0}^2}} \end{pmatrix}
Y(s) := (4 Z2+2 Z14 Z1 Z2+Z12−4 Z24 Z1 Z2+Z12−4 Z24 Z1 Z2+Z124 Z2+2 Z14 Z1 Z2+Z12)\begin{pmatrix} {{4\,Z_{2}+2\,Z_{1}}\over{4\,Z_{1}\,Z_{2}+Z_{1}^2}}&-{{4\,Z_{2}}\over{4\,Z_{1}\,Z_{2}+Z_{1}^2}}\\ -{{4\,Z_{2}}\over{4\,Z_{1}\,Z_{2}+Z_{1}^2}}&{{4\,Z_{2}+2\,Z_{1}}\over{4\,Z_{1}\,Z_{2}+Z_{1}^2}} \end{pmatrix}
Z(s) := (2 Z2+Z12Z2Z22 Z2+Z12)\begin{pmatrix} {{2\,Z_{2}+Z_{1}}\over{2}}&Z_{2}\\ Z_{2}&{{2\,Z_{2}+Z_{1}}\over{2}} \end{pmatrix}

Characteristic Impedance, Z_0, and Propagation Constant, %gamma

solve(S(s)[1,1]=0,Z_0);
doallmxops:true$
solve(determinant(ABCD(s)-pow_e_gamma*ident(2))=0,pow_e_gamma);
[Z0=−4 Z1 Z2+Z122,Z0=4 Z1 Z2+Z122]\left[ Z_{0}=-{{\sqrt{4\,Z_{1}\,Z_{2}+Z_{1}^2}}\over{2}} , Z_{0}={{\sqrt{4\,Z_{1}\,Z_{2}+Z_{1}^2}}\over{2}} \right][powe,γ=−4 Z1 Z2+Z12−2 Z2−Z12 Z2,powe,γ=4 Z1 Z2+Z12+2 Z2+Z12 Z2]\left[ \mathit{pow}_{e,\gamma}=-{{\sqrt{4\,Z_{1}\,Z_{2}+Z_{1}^2}-2\,Z_{2}-Z_{1}}\over{2\,Z_{2}}} , \mathit{pow}_{e,\gamma}={{\sqrt{4\,Z_{1}\,Z_{2}+Z_{1}^2}+2\,Z_{2}+Z_{1}}\over{2\,Z_{2}}} \right]

Assuming Z_1=s*L and Z_2=1/(s*C)

ABCD-, G-, H-, S-, Y- and Z-Parameters

declare(C,real)$
declare(L,real)$
assume(C>0)$
assume(L>0)$
assume(Z_0>0)$
Z_1:s*L$
Z_2:1/(s*C)$
TwoPortMatrices(Solution);
(C L s2+22 C s1C s1C sC L s2+22 C s)\begin{pmatrix} {{C\,L\,s^2+2}\over{2\,C\,s}}&{{1}\over{C\,s}}\\ {{1}\over{C\,s}}&{{C\,L\,s^2+2}\over{2\,C\,s}} \end{pmatrix}
ABCD(s) := (C L s2+22C L2 s3+4 L s4C sC L s2+22)\begin{pmatrix} {{C\,L\,s^2+2}\over{2}}&{{C\,L^2\,s^3+4\,L\,s}\over{4}}\\ C\,s&{{C\,L\,s^2+2}\over{2}} \end{pmatrix}
G(s) := (2 C sC L s2+2−2C L s2+22C L s2+2C L2 s3+4 L s2 C L s2+4)\begin{pmatrix} {{2\,C\,s}\over{C\,L\,s^2+2}}&-{{2}\over{C\,L\,s^2+2}}\\ {{2}\over{C\,L\,s^2+2}}&{{C\,L^2\,s^3+4\,L\,s}\over{2\,C\,L\,s^2+4}} \end{pmatrix}
H(s) := (C L2 s3+4 L s2 C L s2+42C L s2+2−2C L s2+22 C sC L s2+2)\begin{pmatrix} {{C\,L^2\,s^3+4\,L\,s}\over{2\,C\,L\,s^2+4}}&{{2}\over{C\,L\,s^2+2}}\\ -{{2}\over{C\,L\,s^2+2}}&{{2\,C\,s}\over{C\,L\,s^2+2}} \end{pmatrix}
InverseABCD(s) := (C L s2+22C L2 s3+4 L s4C sC L s2+22)\begin{pmatrix} {{C\,L\,s^2+2}\over{2}}&{{C\,L^2\,s^3+4\,L\,s}\over{4}}\\ C\,s&{{C\,L\,s^2+2}\over{2}} \end{pmatrix}
S(s) := (C L2 s3+(4 L−4 C Z02) sC L2 s3+4 C L Z0 s2+(4 C Z02+4 L) s+8 Z08 Z0C L2 s3+4 C L Z0 s2+(4 C Z02+4 L) s+8 Z08 Z0C L2 s3+4 C L Z0 s2+(4 C Z02+4 L) s+8 Z0C L2 s3+(4 L−4 C Z02) sC L2 s3+4 C L Z0 s2+(4 C Z02+4 L) s+8 Z0)\begin{pmatrix} {{C\,L^2\,s^3+\left(4\,L-4\,C\,Z_{0}^2\right)\,s}\over{C\,L^2\,s^3+4\,C\,L\,Z_{0}\,s^2+\left(4\,C\,Z_{0}^2+4\,L\right)\,s+8\,Z_{0}}}&{{8\,Z_{0}}\over{C\,L^2\,s^3+4\,C\,L\,Z_{0}\,s^2+\left(4\,C\,Z_{0}^2+4\,L\right)\,s+8\,Z_{0}}}\\ {{8\,Z_{0}}\over{C\,L^2\,s^3+4\,C\,L\,Z_{0}\,s^2+\left(4\,C\,Z_{0}^2+4\,L\right)\,s+8\,Z_{0}}}&{{C\,L^2\,s^3+\left(4\,L-4\,C\,Z_{0}^2\right)\,s}\over{C\,L^2\,s^3+4\,C\,L\,Z_{0}\,s^2+\left(4\,C\,Z_{0}^2+4\,L\right)\,s+8\,Z_{0}}} \end{pmatrix}
Y(s) := (2 C L s2+4C L2 s3+4 L s−4C L2 s3+4 L s−4C L2 s3+4 L s2 C L s2+4C L2 s3+4 L s)\begin{pmatrix} {{2\,C\,L\,s^2+4}\over{C\,L^2\,s^3+4\,L\,s}}&-{{4}\over{C\,L^2\,s^3+4\,L\,s}}\\ -{{4}\over{C\,L^2\,s^3+4\,L\,s}}&{{2\,C\,L\,s^2+4}\over{C\,L^2\,s^3+4\,L\,s}} \end{pmatrix}
Z(s) := (C L s2+22 C s1C s1C sC L s2+22 C s)\begin{pmatrix} {{C\,L\,s^2+2}\over{2\,C\,s}}&{{1}\over{C\,s}}\\ {{1}\over{C\,s}}&{{C\,L\,s^2+2}\over{2\,C\,s}} \end{pmatrix}
SParameters(S);
8 ω3 C2 L Z02−16 ω C Z02+2 ω5 C2 L3−12 ω3 C L2+16 ω L∣16 ω2 C2 Z02+4 ω4 C2 L2−16 ω2 C L+16∣{{8\,\omega^3\,C^2\,L\,Z_{0}^2-16\,\omega\,C\,Z_{0}^2+2\,\omega^5\,C^2\,L^3-12\,\omega^3\,C\,L^2+16\,\omega\,L}\over{\left| 16\,\omega^2\,C^2\,Z_{0}^2+4\,\omega^4\,C^2\,L^2-16\,\omega^2\,C\,L+16\right| }}
Abs_S11(%omega) := ∣4 ω C Z02+ω3 C L2−4 ω L∣(4 ω C Z02−ω3 C L2+4 ω L)2+(8 Z0−4 ω2 C L Z0)2{{\left| 4\,\omega\,C\,Z_{0}^2+\omega^3\,C\,L^2-4\,\omega\,L\right| }\over{\sqrt{\left(4\,\omega\,C\,Z_{0}^2-\omega^3\,C\,L^2+4\,\omega\,L\right)^2+\left(8\,Z_{0}-4\,\omega^2\,C\,L\,Z_{0}\right)^2}}}
Arg_S11(%omega) := atan2((16 ω3 C2 L−32 ω C) Z03+(4 ω5 C2 L3−24 ω3 C L2+32 ω L) Z0,−16 ω2 C2 Z04+ω6 C2 L4−8 ω4 C L3+16 ω2 L2){\rm atan2}\left(\left(16\,\omega^3\,C^2\,L-32\,\omega\,C\right)\,Z_{0}^3+\left(4\,\omega^5\,C^2\,L^3-24\,\omega^3\,C\,L^2+32\,\omega\,L\right)\,Z_{0} , -16\,\omega^2\,C^2\,Z_{0}^4+\omega^6\,C^2\,L^4-8\,\omega^4\,C\,L^3+16\,\omega^2\,L^2\right)
Abs_S12(%omega) := 8 Z0(4 ω C Z02−ω3 C L2+4 ω L)2+(8 Z0−4 ω2 C L Z0)2{{8\,Z_{0}}\over{\sqrt{\left(4\,\omega\,C\,Z_{0}^2-\omega^3\,C\,L^2+4\,\omega\,L\right)^2+\left(8\,Z_{0}-4\,\omega^2\,C\,L\,Z_{0}\right)^2}}}
Arg_S12(%omega) := −atan2(32 ω C Z03−(8 ω3 C L2−32 ω L) Z0,(64−32 ω2 C L) Z02)-{\rm atan2}\left(32\,\omega\,C\,Z_{0}^3-\left(8\,\omega^3\,C\,L^2-32\,\omega\,L\right)\,Z_{0} , \left(64-32\,\omega^2\,C\,L\right)\,Z_{0}^2\right)
Abs_S21(%omega) := 8 Z0(4 ω C Z02−ω3 C L2+4 ω L)2+(8 Z0−4 ω2 C L Z0)2{{8\,Z_{0}}\over{\sqrt{\left(4\,\omega\,C\,Z_{0}^2-\omega^3\,C\,L^2+4\,\omega\,L\right)^2+\left(8\,Z_{0}-4\,\omega^2\,C\,L\,Z_{0}\right)^2}}}
Arg_S21(%omega) := −atan2(32 ω C Z03−(8 ω3 C L2−32 ω L) Z0,(64−32 ω2 C L) Z02)-{\rm atan2}\left(32\,\omega\,C\,Z_{0}^3-\left(8\,\omega^3\,C\,L^2-32\,\omega\,L\right)\,Z_{0} , \left(64-32\,\omega^2\,C\,L\right)\,Z_{0}^2\right)
tau_g(%omega) := (16 ω2 C2 L+32 C) Z03+(4 ω4 C2 L3−8 ω2 C L2+32 L) Z016 ω2 C2 Z04+(8 ω4 C2 L2−32 ω2 C L+64) Z02+ω6 C2 L4−8 ω4 C L3+16 ω2 L2{{\left(16\,\omega^2\,C^2\,L+32\,C\right)\,Z_{0}^3+\left(4\,\omega^4\,C^2\,L^3-8\,\omega^2\,C\,L^2+32\,L\right)\,Z_{0}}\over{16\,\omega^2\,C^2\,Z_{0}^4+\left(8\,\omega^4\,C^2\,L^2-32\,\omega^2\,C\,L+64\right)\,Z_{0}^2+\omega^6\,C^2\,L^4-8\,\omega^4\,C\,L^3+16\,\omega^2\,L^2}}
Abs_S22(%omega) := ∣4 ω C Z02+ω3 C L2−4 ω L∣(4 ω C Z02−ω3 C L2+4 ω L)2+(8 Z0−4 ω2 C L Z0)2{{\left| 4\,\omega\,C\,Z_{0}^2+\omega^3\,C\,L^2-4\,\omega\,L\right| }\over{\sqrt{\left(4\,\omega\,C\,Z_{0}^2-\omega^3\,C\,L^2+4\,\omega\,L\right)^2+\left(8\,Z_{0}-4\,\omega^2\,C\,L\,Z_{0}\right)^2}}}
Arg_S22(%omega) := atan2((16 ω3 C2 L−32 ω C) Z03+(4 ω5 C2 L3−24 ω3 C L2+32 ω L) Z0,−16 ω2 C2 Z04+ω6 C2 L4−8 ω4 C L3+16 ω2 L2){\rm atan2}\left(\left(16\,\omega^3\,C^2\,L-32\,\omega\,C\right)\,Z_{0}^3+\left(4\,\omega^5\,C^2\,L^3-24\,\omega^3\,C\,L^2+32\,\omega\,L\right)\,Z_{0} , -16\,\omega^2\,C^2\,Z_{0}^4+\omega^6\,C^2\,L^4-8\,\omega^4\,C\,L^3+16\,\omega^2\,L^2\right)
Z_IN(s) := C L2 s3+2 C L Z0 s2+4 L s+4 Z02 C L s2+4 C Z0 s+4{{C\,L^2\,s^3+2\,C\,L\,Z_{0}\,s^2+4\,L\,s+4\,Z_{0}}\over{2\,C\,L\,s^2+4\,C\,Z_{0}\,s+4}}
R_IN(%omega) := 4 Z0∣4 ω2 C2 Z02+ω4 C2 L2−4 ω2 C L+4∣{{4\,Z_{0}}\over{\left| 4\,\omega^2\,C^2\,Z_{0}^2+\omega^4\,C^2\,L^2-4\,\omega^2\,C\,L+4\right| }}
X_IN(%omega) := 8 ω3 C2 L Z02−16 ω C Z02+2 ω5 C2 L3−12 ω3 C L2+16 ω L∣16 ω2 C2 Z02+4 ω4 C2 L2−16 ω2 C L+16∣{{8\,\omega^3\,C^2\,L\,Z_{0}^2-16\,\omega\,C\,Z_{0}^2+2\,\omega^5\,C^2\,L^3-12\,\omega^3\,C\,L^2+16\,\omega\,L}\over{\left| 16\,\omega^2\,C^2\,Z_{0}^2+4\,\omega^4\,C^2\,L^2-16\,\omega^2\,C\,L+16\right| }}
Z_OUT(s) := C L2 s3+2 C L Z0 s2+4 L s+4 Z02 C L s2+4 C Z0 s+4{{C\,L^2\,s^3+2\,C\,L\,Z_{0}\,s^2+4\,L\,s+4\,Z_{0}}\over{2\,C\,L\,s^2+4\,C\,Z_{0}\,s+4}}
R_OUT(%omega) := 4 Z0∣4 ω2 C2 Z02+ω4 C2 L2−4 ω2 C L+4∣{{4\,Z_{0}}\over{\left| 4\,\omega^2\,C^2\,Z_{0}^2+\omega^4\,C^2\,L^2-4\,\omega^2\,C\,L+4\right| }}
X_OUT(%omega) := 8 ω3 C2 L Z02−16 ω C Z02+2 ω5 C2 L3−12 ω3 C L2+16 ω L∣16 ω2 C2 Z02+4 ω4 C2 L2−16 ω2 C L+16∣{{8\,\omega^3\,C^2\,L\,Z_{0}^2-16\,\omega\,C\,Z_{0}^2+2\,\omega^5\,C^2\,L^3-12\,\omega^3\,C\,L^2+16\,\omega\,L}\over{\left| 16\,\omega^2\,C^2\,Z_{0}^2+4\,\omega^4\,C^2\,L^2-16\,\omega^2\,C\,L+16\right| }}
Stability(S);
00
Delta(%omega) := 256 ω4 C4 Z08+(512 i ω5 C4 L−1024 i ω3 C3) Z07+(1024 ω4 C3 L−256 ω6 C4 L2) Z06+(128 i ω7 C4 L3−768 i ω5 C3 L2+3072 i ω3 C2 L−4096 i ω C) Z05+(−160 ω8 C4 L4+1280 ω6 C3 L3−4608 ω4 C2 L2+8192 ω2 C L−4096) Z04+(−32 i ω9 C4 L5+320 i ω7 C3 L4−1536 i ω5 C2 L3+4096 i ω3 C L2−4096 i ω L) Z03+(−16 ω10 C4 L6+192 ω8 C3 L5−768 ω6 C2 L4+1024 ω4 C L3) Z02+(−8 i ω11 C4 L7+112 i ω9 C3 L6−576 i ω7 C2 L5+1280 i ω5 C L4−1024 i ω3 L3) Z0+ω12 C4 L8−16 ω10 C3 L7+96 ω8 C2 L6−256 ω6 C L5+256 ω4 L4256 ω4 C4 Z08+(1024 i ω5 C4 L−2048 i ω3 C3) Z07+(−1792 ω6 C4 L2+7168 ω4 C3 L−6144 ω2 C2) Z06+(−1792 i ω7 C4 L3+10752 i ω5 C3 L2−18432 i ω3 C2 L+8192 i ω C) Z05+(1120 ω8 C4 L4−8960 ω6 C3 L3+23040 ω4 C2 L2−20480 ω2 C L+4096) Z04+(448 i ω9 C4 L5−4480 i ω7 C3 L4+15360 i ω5 C2 L3−20480 i ω3 C L2+8192 i ω L) Z03+(−112 ω10 C4 L6+1344 ω8 C3 L5−5760 ω6 C2 L4+10240 ω4 C L3−6144 ω2 L2) Z02+(−16 i ω11 C4 L7+224 i ω9 C3 L6−1152 i ω7 C2 L5+2560 i ω5 C L4−2048 i ω3 L3) Z0+ω12 C4 L8−16 ω10 C3 L7+96 ω8 C2 L6−256 ω6 C L5+256 ω4 L4{{256\,\omega^4\,C^4\,Z_{0}^8+\left(512\,i\,\omega^5\,C^4\,L-1024\,i\,\omega^3\,C^3\right)\,Z_{0}^7+\left(1024\,\omega^4\,C^3\,L-256\,\omega^6\,C^4\,L^2\right)\,Z_{0}^6+\left(128\,i\,\omega^7\,C^4\,L^3-768\,i\,\omega^5\,C^3\,L^2+3072\,i\,\omega^3\,C^2\,L-4096\,i\,\omega\,C\right)\,Z_{0}^5+\left(-160\,\omega^8\,C^4\,L^4+1280\,\omega^6\,C^3\,L^3-4608\,\omega^4\,C^2\,L^2+8192\,\omega^2\,C\,L-4096\right)\,Z_{0}^4+\left(-32\,i\,\omega^9\,C^4\,L^5+320\,i\,\omega^7\,C^3\,L^4-1536\,i\,\omega^5\,C^2\,L^3+4096\,i\,\omega^3\,C\,L^2-4096\,i\,\omega\,L\right)\,Z_{0}^3+\left(-16\,\omega^{10}\,C^4\,L^6+192\,\omega^8\,C^3\,L^5-768\,\omega^6\,C^2\,L^4+1024\,\omega^4\,C\,L^3\right)\,Z_{0}^2+\left(-8\,i\,\omega^{11}\,C^4\,L^7+112\,i\,\omega^9\,C^3\,L^6-576\,i\,\omega^7\,C^2\,L^5+1280\,i\,\omega^5\,C\,L^4-1024\,i\,\omega^3\,L^3\right)\,Z_{0}+\omega^{12}\,C^4\,L^8-16\,\omega^{10}\,C^3\,L^7+96\,\omega^8\,C^2\,L^6-256\,\omega^6\,C\,L^5+256\,\omega^4\,L^4}\over{256\,\omega^4\,C^4\,Z_{0}^8+\left(1024\,i\,\omega^5\,C^4\,L-2048\,i\,\omega^3\,C^3\right)\,Z_{0}^7+\left(-1792\,\omega^6\,C^4\,L^2+7168\,\omega^4\,C^3\,L-6144\,\omega^2\,C^2\right)\,Z_{0}^6+\left(-1792\,i\,\omega^7\,C^4\,L^3+10752\,i\,\omega^5\,C^3\,L^2-18432\,i\,\omega^3\,C^2\,L+8192\,i\,\omega\,C\right)\,Z_{0}^5+\left(1120\,\omega^8\,C^4\,L^4-8960\,\omega^6\,C^3\,L^3+23040\,\omega^4\,C^2\,L^2-20480\,\omega^2\,C\,L+4096\right)\,Z_{0}^4+\left(448\,i\,\omega^9\,C^4\,L^5-4480\,i\,\omega^7\,C^3\,L^4+15360\,i\,\omega^5\,C^2\,L^3-20480\,i\,\omega^3\,C\,L^2+8192\,i\,\omega\,L\right)\,Z_{0}^3+\left(-112\,\omega^{10}\,C^4\,L^6+1344\,\omega^8\,C^3\,L^5-5760\,\omega^6\,C^2\,L^4+10240\,\omega^4\,C\,L^3-6144\,\omega^2\,L^2\right)\,Z_{0}^2+\left(-16\,i\,\omega^{11}\,C^4\,L^7+224\,i\,\omega^9\,C^3\,L^6-1152\,i\,\omega^7\,C^2\,L^5+2560\,i\,\omega^5\,C\,L^4-2048\,i\,\omega^3\,L^3\right)\,Z_{0}+\omega^{12}\,C^4\,L^8-16\,\omega^{10}\,C^3\,L^7+96\,\omega^8\,C^2\,L^6-256\,\omega^6\,C\,L^5+256\,\omega^4\,L^4}}
MSG(%omega) := 11
K(%omega) := 11
mu(%omega) := 11
muprime(%omega) := 11
U(%omega) := 00

Characteristic Impedance, Z_iT, and Propagation Constant, %gamma

solve(ev(S(s)[1,1])=0,Z_0);
[Z0=−L C L s2+42 C,Z0=L C L s2+42 C]\left[ Z_{0}=-{{\sqrt{L}\,\sqrt{C\,L\,s^2+4}}\over{2\,\sqrt{C}}} , Z_{0}={{\sqrt{L}\,\sqrt{C\,L\,s^2+4}}\over{2\,\sqrt{C}}} \right]
define(Z_iT(s),rhs(last(%)));
ZiT(s):=L C L s2+42 C\mathit{Z}_{iT}\left(s\right):={{\sqrt{L}\,\sqrt{C\,L\,s^2+4}}\over{2\,\sqrt{C}}}
doallmxops:true$
solve(determinant(ev(ABCD(%i*%omega))-pow_e_gamma*ident(2))=0,pow_e_gamma);
[powe,γ=−ω C L ω2 C L−4+ω2 C L−22,powe,γ=ω C L ω2 C L−4−ω2 C L+22]\left[ \mathit{pow}_{e,\gamma}=-{{\omega\,\sqrt{C}\,\sqrt{L}\,\sqrt{\omega^2\,C\,L-4}+\omega^2\,C\,L-2}\over{2}} , \mathit{pow}_{e,\gamma}={{\omega\,\sqrt{C}\,\sqrt{L}\,\sqrt{\omega^2\,C\,L-4}-\omega^2\,C\,L+2}\over{2}} \right]
define(pow_e_gamma(%omega),rhs(last(%)));
powe,γ(ω):=ω C L ω2 C L−4−ω2 C L+22\mathit{pow}_{e,\gamma}\left(\omega\right):={{\omega\,\sqrt{C}\,\sqrt{L}\,\sqrt{\omega^2\,C\,L-4}-\omega^2\,C\,L+2}\over{2}}

A (Attenuation per Section), %phi (Phase Shift per Section) and tau_g (Group Delay per Section)

define(A(%omega),realpart(ev(log(pow_e_gamma(%omega)))));
define(%phi(%omega),imagpart(ev(log(pow_e_gamma(%omega)))));
A(ω):=log⁡(sin⁡2(atan2(0,ω2 C L−4)2) ω2 C L ∣ω2 C L−4∣+(cos⁡(atan2(0,ω2 C L−4)2) ω C L ∣ω2 C L−4∣−ω2 C L+2)22)A\left(\omega\right):=\log \left({{\sqrt{\sin ^2\left({{{\rm atan2}\left(0 , \omega^2\,C\,L-4\right)}\over{2}}\right)\,\omega^2\,C\,L\,\left| \omega^2\,C\,L-4\right| +\left(\cos \left({{{\rm atan2}\left(0 , \omega^2\,C\,L-4\right)}\over{2}}\right)\,\omega\,\sqrt{C}\,\sqrt{L}\,\sqrt{\left| \omega^2\,C\,L-4\right| }-\omega^2\,C\,L+2\right)^2}}\over{2}}\right)ϕ(ω):=atan2(sin⁡(atan2(0,ω2 C L−4)2) ω C L ∣ω2 C L−4∣,cos⁡(atan2(0,ω2 C L−4)2) ω C L ∣ω2 C L−4∣−ω2 C L+2)\phi\left(\omega\right):={\rm atan2}\left(\sin \left({{{\rm atan2}\left(0 , \omega^2\,C\,L-4\right)}\over{2}}\right)\,\omega\,\sqrt{C}\,\sqrt{L}\,\sqrt{\left| \omega^2\,C\,L-4\right| } , \cos \left({{{\rm atan2}\left(0 , \omega^2\,C\,L-4\right)}\over{2}}\right)\,\omega\,\sqrt{C}\,\sqrt{L}\,\sqrt{\left| \omega^2\,C\,L-4\right| }-\omega^2\,C\,L+2\right)
%omega_c:2/sqrt(L*C);
/* Exactly AT cutoff the argument of the logarithm is real and negative, so
   atan2(0,0) is undefined and evaluating A/%phi there raises an error
   (it did in 0.2.4 too).  Evaluate inside the passband and inside the
   stopband instead, which is the point being made anyway: A = 0 below
   cutoff, A > 0 above it. */
A(%omega_c/2);
%phi(%omega_c/2);
A(2*%omega_c);
%phi(2*%omega_c);
2C L{{2}\over{\sqrt{C}\,\sqrt{L}}}00π3{{\pi}\over{3}}log⁡(14−8 32)\log \left({{14-8\,\sqrt{3}}\over{2}}\right)π\pi
C:5e-15$
L:75e-12$
Z_0:50$
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 CONSTANT_K_SECTION.wxm
Plot generated by CONSTANT_K_SECTION.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 CONSTANT_K_SECTION.wxm
Plot generated by CONSTANT_K_SECTION.wxm
Plot generated by CONSTANT_K_SECTION.wxm
Plot generated by CONSTANT_K_SECTION.wxm
Plot generated by CONSTANT_K_SECTION.wxm
Plot generated by CONSTANT_K_SECTION.wxm
Plot generated by CONSTANT_K_SECTION.wxm
Plot generated by CONSTANT_K_SECTION.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 CONSTANT_K_SECTION.wxm
Plot generated by CONSTANT_K_SECTION.wxm
Plot generated by CONSTANT_K_SECTION.wxm
Plot generated by CONSTANT_K_SECTION.wxm
Plot generated by CONSTANT_K_SECTION.wxm
load("coma.mac")$
step_response(1/ABCD(s)[1,1]);
true\mathbf{true}