Question: Kron Reduction from 8 port Yp matrix to 4 port Yp matrix

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

Modeling Objective

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.

Technical Procedure Breakdown

Step 1: Define the Basic Admittance Terms

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.

Step 2: Build the Original 4-Port Matrix Y4

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).

Step 3: Port Reordering for the Right Coupler

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)

Step 4: Construct the 8×8 Overall Admittance Matrix Y8

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.

Step 5: Port Shorting (Node Merging)

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)

Step 6: Submatrix Extraction and Kron Reduction

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

❗Current Issue / Question

When I try to execute:

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

❓What I need help with:

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.