---
title: "Magic tee diagram and S-parameter derivation"
subject: "RF and Microwave Engineering"
chapter: 4
key: "semester_7/rf_and_microwave_engineering:magic-tee-s-parameter-derivation"
source: https://www.bctengineer.com/q/semester_7/rf_and_microwave_engineering/magic-tee-s-parameter-derivation
site: BCT Engineering
---

# Magic tee diagram and S-parameter derivation

RF and Microwave Engineering · Chapter 4: RF and Microwave Components and Devices

- Asked in: 2080 Bhadra, 4 marks
- Video: [Magic Tee (Basics, Working, Internal structure, S Matrix & Applications) Explained in Microwave](https://www.youtube.com/watch?v=if30sXaKWkY) — Engineering Funda
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_7/rf_and_microwave_engineering/magic-tee-s-parameter-derivation
- Whole chapter: https://www.bctengineer.com/md/semester_7/rf_and_microwave_engineering/chapter-4

## Question

Draw a neat diagram of a magic tee and derive its S-parameter.

## Short answer

A magic tee combines E-plane and H-plane tees with ports 1 and 2 collinear, port 3 H-arm, and port 4 E-arm. Using symmetry S13=S23, S14=-S24, isolation S34=0, matching S33=S44=0, reciprocity, and unitarity gives S=(1/√2)[[0,0,1,1],[0,0,1,-1],[1,1,0,0],[1,-1,0,0]].

## Answer

A **magic tee** is an E-plane tee and an H-plane tee joined at the same point. Ports 1 and 2 are collinear arms, port 3 is the H-arm and port 4 the E-arm.

```text
              E-arm (4)
                | |
     -----------+ +-----------
 (1) ===========+   +=========== (2)
     -----------+ +-----------
               / /
              / /  H-arm (3)
```

#### Derivation of the S-matrix
1. **H-arm property:** input at port 3 splits equally and in phase to 1 and 2: S13 = S23.
2. **E-arm property:** input at port 4 splits equally and in anti-phase: S14 = −S24.
3. **Isolation of E and H arms** (by geometry/symmetry): S34 = S43 = 0.
4. **Matching:** E and H arms matched with posts: S33 = S44 = 0.
5. **Reciprocity:** Sᵢⱼ = Sⱼᵢ.

So far:
```text
       | S11  S12  S13  S14 |
 [S] = | S12  S22  S13 -S14 |
       | S13  S13   0    0  |
       | S14 -S14   0    0  |
```
6. **Lossless (unitary) condition:**
```text
 Row 3: 2|S13|² = 1           ->  S13 = 1/√2 (real, by choice of reference plane)
 Row 4: 2|S14|² = 1           ->  S14 = 1/√2
 Row 1: |S11|² + |S12|² + 1 = 1 -> S11 = S12 = 0
 Row 2: |S12|² + |S22|² + 1 = 1 -> S22 = 0
```
Result:
```text
              | 0   0   1   1 |
 [S] = (1/√2) | 0   0   1  -1 |
              | 1   1   0   0 |
              | 1  -1   0   0 |
```
All ports are matched, ports 1–2 and 3–4 are isolated, and power fed into any port divides equally between the two ports that are not isolated from it.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. In a magic tee, the E-arm input splits equally with
   A. in phase, while the H-arm splits anti-phase
   B. equal phase for both arms
   C. 180° phase difference, while the H-arm input splits in phase
   D. 180° for both arms
   Answer: C. S14 = -S24 for the E-arm and S13 = S23 for the H-arm. (Key point: Arm properties)

2. Which relation reflects E-H isolation?
   A. S12 = 1
   B. S13 = S14
   C. S33 = 1
   D. S34 = S43 = 0
   Answer: D. E and H arms are decoupled by symmetry. (Key point: Isolation)

3. Row 3 gives 2|S13|^2 = 1, so
   A. S13 = 1/sqrt2
   B. S13 = 1/2
   C. S13 = 1
   D. S13 = sqrt2
   Answer: A. Equal power split between ports 1 and 2 requires |S13|^2 = 1/2. (Key point: Unitary condition)

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