hello:

Problem Background
   I am using Maple to perform a symbolic derivation for analyzing a coupled stripline directional coupler (as illustrated in the figure) The objective is to start from the known 4-port admittance matrix Y4 of a single coupled-line section(figure (a)) and derive the resulting 4-port admittance matrix Y_T for the composite structure shown in figure (b).

Diagram Explanation (Refer to Attached Figure)

• Figure (a) represents a standard 4-port coupled-line segment, with admittance matrix Y4. The port ordering is labeled in green as (1,2,3,4).

• Figure (b) shows a composite network constructed by combining two identical Y4 structures, where:

  • The right block are renumbered as red labels (5–8), which differ from the default Y4 port ordering.

    • The middle section includes two shorted node pairs that merge the internal connections:

      • Ports 2 and 6 are connected to form internal node A. so we get equtions:I2=-I6,V2=V6

      • Ports 4 and 8 are connected to form internal node B.so we get equtions:I4=-I8,V4=V8

Using a matrix-based linear algebra approach, I aim to compute the effective 4-port admittance matrix Y_T for the final structure shown in Figure (b), with the active ports being 1, 3, 5, and 7.

The even-mode and odd-mode responses are expressed symbolically. From these, the basic admittance elements are defined:

• Y0, Ym1, Ym2, Ym3 using Yoe, Yoo, and θ.

Construct the 4×4 admittance matrix Y4 for the left-side coupled-line segment, using port ordering (1,2,3,4) as shown in Figure (a).

The Secondary (right-side) coupled-line block uses a different port order (Ports 5–6–7–8 correspond to the original 2–1–4–3).
Therefore, a permutation matrix P is introduced to reorder the ports as:

(1,2,3,4) → (2,1,4,3)
The reordered admittance matrix Y4_R is obtained via:

Y4_R := P . Y4 . Transpose(P)

Using BlockDiagonalMatrix or equivalent logic, we assemble the composite 8×8 admittance matrix Y8, representing:

[1 2 3 4 | 5 6 7 8]
The left half corresponds to Y4, and the right half to Y4_R.

According to the physical configuration:

• Ports 2 and 6 are shorted to form Node A

• Ports 4 and 8 are shorted to form Node B

We define a node-merging matrix S (6×8) to map the 8-port structure into a reduced 6-port structure using:

[1, 3, 7, 5, A(=2+6), B(=4+8)]
Apply the transformation:

Y6 := S . Y8 . Transpose(S)

From the reduced matrix Y6, extract:

• External port indices: E = [1, 2, 3, 4] → Ports 1, 3, 7, 5

• Internal node indices: I = [5, 6] → Nodes A, B

Perform Kron reduction to eliminate internal nodes and obtain the final reduced matrix:

Y_T := Y_EE - Y_EI . Inverse(Y_II) . Y_IE

When I try to execute:

Y8 := BlockDiagonalMatrix([Y4, Y4_R])
Maple does not return a usable or well-formed 8×8 matrix

1. How to correctly construct the block-diagonal matrix Y8 := BlockDiagonalMatrix([Y4, Y4_R]) so that it's recognized as an 8×8 matrix in Maple?

2. Validate whether the full logic flow from Y4 → port mapping → block matrix → merging → Kron reduction is mathematically sound and correctly implemented, and get the final 4 port Y parameter matrix.

3. If possible, please suggest any more stable or simplified alternative matrix operations for symbolic Kron reduction involving port reordering and node merging.

here is the maple file:

Download CoupleLine_WH2.mw

thank in advance for your help.

Dear all:

    I am using maple2022 to learn Fourier Series package - Maple Application Center (maplesoft.com) . the author provide the package here:

cgi.math.muni.cz/kriz/xsrot/fourierseries/  I use the lastest version: Package FourierTrigSeries version 0.41 

there is no .mla so I do not know how to use this package.(the package is a little old). 

please enlight me if you know how to install this old package.

but the author provide the code inside fourierseries-structs.mws 

here is what I do:

1. first delete some lines which is not useful(which I shared in the attachment as FourierTrigSeries.mw)

and remain everything inside module and save as .mw (maybe the orginal .mws is a old version of maple files)

2. export as .FourierTrigSeries.mpl (also attached)

3. use the examples.mws (the code also from author)

I got error as:

the package can be recognized, but the first line wil give some error which is out of my knowledge field.

everything mentioned above is here, including the original author's files.

fouriertrigseries_0_41en_mapleprime.zip

Could you please have a look. your idea is valuable to me.

PS: this package is not available The Fourier Series package for Maple - Maple Application Center (maplesoft.com) whenI try to learn  Teaching Fourier Series with Maple II 

if anyone keep it before, please share it.

thanks for your help.

rockyicer 

dear all:

    here I try to repeat the results as follows:

 (23) is my target

my 2 solving processes are included in the attachment.

using LinearSolve(A, b) and regular solve command can not generate results as (23)

Please take a look.

question_DHT.mw

thanks for your help.

best regards

Dear all:

    I am trying to use the subs command to replace 5 varibles with other specific 5 varibles.

to be simple, here is issue.

Now I and using the stupid way(manually replace eachstep by my self) which did work but with some small issue such as  simplify or subs not work for replacements

all the steps I did are contained inside here:

ModelExtrinsicParasiticExtraction_QuestionsV2.mw

thanks for your help.

best regards

dear all,

I am trying to use the solve to get symbolic answer.

to be simple, here is issue:

I have 15equations with 16 varibles.

the 15 equation as follows:

first 4 equation set 

equList_Ya :=

[ Va1 Ya11 + Va2 Ya12 + Va3 Ya13 + Va4 Ya14 = Ia1, 

  Va1 Ya21 + Va2 Ya22 + Va3 Ya23 + Va4 Ya24 = Ia2, 

  Va1 Ya31 + Va2 Ya32 + Va3 Ya33 + Va4 Ya34 = Ia3, 

  Va1 Ya41 + Va2 Ya42 + Va3 Ya43 + Va4 Ya44 = Ia4]

2nd 4 equation set

equList_Yb := 

[ Vb1 Yb11 + Vb2 Yb12 + Vb3 Yb13 + Vb4 Yb14 = Ib1, 

  Vb1 Yb21 + Vb2 Yb22 + Vb3 Yb23 + Vb4 Yb24 = Ib2, 

  Vb1 Yb31 + Vb2 Yb32 + Vb3 Yb33 + Vb4 Yb34 = Ib3, 

  Vb1 Yb41 + Vb2 Yb42 + Vb3 Yb43 + Vb4 Yb44 = Ib4]

the 3rd 4 equation set

equList_ConnectCondition :=

[Ia3 = -Ib1, Ia4 = -Ib2, Va3 = Vb1,  Va4 = Vb2]

the 4th 3 equation set

 equList_Y11Condition := [Va2 = 0, Vb3 = 0, Vb4 = 0]

so total equation number is 4+4+4+3=15

equList_Y11all := {Ia3 = -Ib1, Ia4 = -Ib2, Va2 = 0, Va3 = Vb1, Va4 = Vb2, Vb3 = 0, 
 Vb4 = 0, Va1*Ya11 + Va2*Ya12 + Va3*Ya13 + Va4*Ya14 = Ia1,
 Va1*Ya21 + Va2*Ya22 + Va3*Ya23 + Va4*Ya24 = Ia2,
 Va1*Ya31 + Va2*Ya32 + Va3*Ya33 + Va4*Ya34 = Ia3, 
 Va1*Ya41 + Va2*Ya42 + Va3*Ya43 + Va4*Ya44 = Ia4,
 Vb1*Yb11 + Vb2*Yb12 + Vb3*Yb13 + Vb4*Yb14 = Ib1,
 Vb1*Yb21 + Vb2*Yb22 + Vb3*Yb23 + Vb4*Yb24 = Ib2, 
 Vb1*Yb31 + Vb2*Yb32 + Vb3*Yb33 + Vb4*Yb34 = Ib3,
 Vb1*Yb41 + Vb2*Yb42 + Vb3*Yb43 + Vb4*Yb44 = Ib4}

the 16 varible is 

var_Y11 :=

[ Va1, Va2, Va3, Va4,

  Ia1, Ia2, Ia3, Ia4,

  Vb1, Vb2, Vb3, Vb4,

  Ib1, Ib2, Ib3, Ib4]

I hope to get Ia1=f(Va1) to calculate Y11(=Ia1/Va1)

but the solve give nothing:

Y11_all := solve(equList_Y11all, {Ia1}, symbolic = true);                        

Y11_all := NULL

Question1: why the above solve output nothing???

since not work,so I try another way:

polys_Y11 := map(lhs - rhs = 0, equList_Y11all);
    polys_Y11 := {Va2 = 0, Vb3 = 0, Vb4 = 0, Ia3 + Ib1 = 0, 

      Ia4 + Ib2 = 0, Va3 - Vb1 = 0, Va4 - Vb2 = 0, 

      Va1 Ya11 + Va2 Ya12 + Va3 Ya13 + Va4 Ya14 - Ia1 = 0, 

      Va1 Ya21 + Va2 Ya22 + Va3 Ya23 + Va4 Ya24 - Ia2 = 0, 

      Va1 Ya31 + Va2 Ya32 + Va3 Ya33 + Va4 Ya34 - Ia3 = 0, 

      Va1 Ya41 + Va2 Ya42 + Va3 Ya43 + Va4 Ya44 - Ia4 = 0, 

      Vb1 Yb11 + Vb2 Yb12 + Vb3 Yb13 + Vb4 Yb14 - Ib1 = 0, 

      Vb1 Yb21 + Vb2 Yb22 + Vb3 Yb23 + Vb4 Yb24 - Ib2 = 0, 

      Vb1 Yb31 + Vb2 Yb32 + Vb3 Yb33 + Vb4 Yb34 - Ib3 = 0, 

      Vb1 Yb41 + Vb2 Yb42 + Vb3 Yb43 + Vb4 Yb44 - Ib4 = 0}

and solve it with 

what I expected is Ia1=f(Va1) which Va1 as indepentent,but the above solution have no Va1 
Question2: is the (22) give the right answer for Ia1 ? what is the indepent var for the above solution?

here is my full maple worksheet

CoupleLineSeriesConnect_Question.mw

thx for your help.

best regards