---
title: "Identify device from S-matrix with ±S14 entries"
subject: "RF and Microwave Engineering"
chapter: 4
key: "semester_7/rf_and_microwave_engineering:identify-device-s-matrix-hybrid"
source: https://www.bctengineer.com/q/semester_7/rf_and_microwave_engineering/identify-device-s-matrix-hybrid
site: BCT Engineering
---

# Identify device from S-matrix with ±S14 entries

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

- Asked in: 2081 Bhadra, 5 marks; 2074 Bhadra, 8 marks (asked 2 times)
- Video: [Magic Hybrid T-junction | S matrix | Microwave Engineering | Lec-66](https://www.youtube.com/watch?v=FTlHyfiJSiA) — Education 4u
- 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/identify-device-s-matrix-hybrid
- Whole chapter: https://www.bctengineer.com/md/semester_7/rf_and_microwave_engineering/chapter-4

## Question

Identify and explain, along with a neat diagram, the properties of the passive microwave device described by the following S-Matrix: S = [S11 S12 S13 S14; S12 S22 S13 −S14; S13 S13 0 0; S14 −S14 0 0].

## Short answer

The matrix describes a magic tee. Ports 1 and 2 are collinear, port 3 H-arm (S13=S23), port 4 E-arm (S14=-S24). Under lossless matched conditions, S13=S14=1/√2, S11=S22=S12=0, giving all ports matched and E/H isolation. It functions as a sum/difference network.

## Answer

#### Identification
```text
       | S11   S12   S13   S14 |
 [S] = | S12   S22   S13  -S14 |
       | S13   S13   0     0   |
       | S14  -S14   0     0   |
```
- Port 3 receives equal signals **in phase** from ports 1 and 2 (S31 = S32 = S13): port 3 behaves as the **H-arm**.
- Port 4 receives equal signals **180° out of phase** from ports 1 and 2 (S41 = −S42 = S14): port 4 behaves as the **E-arm**.
- S34 = S43 = 0: the E and H arms are isolated; S33 = S44 = 0: those arms are matched.
- The matrix is symmetric, so the device is reciprocal.

These are the features of a **magic tee (hybrid tee, E-H tee)** with ports 1 and 2 as the collinear arms.

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

#### Finding the ideal values (lossless condition)
For a lossless junction, [S]*ᵀ[S] = I.
```text
 Row 3: |S13|² + |S13|² = 1  ->  S13 = 1/√2
 Row 4: |S14|² + |S14|² = 1  ->  S14 = 1/√2
 Row 1: |S11|² + |S12|² + 1/2 + 1/2 = 1
        -> S11 = 0 and S12 = 0
 Row 2: |S12|² + |S22|² + 1/2 + 1/2 = 1  ->  S22 = 0
```
So matching the E and H arms automatically matches ports 1 and 2 and isolates them:
```text
              | 0   0   1   1 |
 [S] = (1/√2) | 0   0   1  -1 |
              | 1   1   0   0 |
              | 1  -1   0   0 |
```

#### Properties
1. **Input at H-arm (3):** b1 = b2 = a3/√2 – equal in-phase split; b4 = 0.
2. **Input at E-arm (4):** b1 = a4/√2, b2 = −a4/√2 – equal split, 180° apart; b3 = 0.
3. **Input at port 1:** b3 = b4 = a1/√2; port 2 gets nothing (S21 = 0), so collinear arms are isolated.
4. **Equal in-phase inputs at 1 and 2:** b3 = √2·a (all power at H-arm), b4 = 0. Equal anti-phase inputs: all power at E-arm.
5. All ports matched, lossless and reciprocal.

#### Uses
Balanced mixers, radar duplexers, power combiners/dividers, impedance bridges and monopulse comparators.

## Check yourself

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

1. The given S-matrix describes which device?
   A. Circulator
   B. Magic tee
   C. Directional coupler
   D. Isolator
   Answer: B. Port 3 gets equal in-phase signals from ports 1 and 2 (H-arm), port 4 gets them out of phase (E-arm), and S34 = 0. (Key point: Identification)

2. Row 3 gives |S13|^2 + |S13|^2 = 1. What is S13?
   A. 1
   B. 1/2
   C. 1/sqrt(2)
   D. sqrt(2)
   Answer: C. 2|S13|^2 = 1 gives |S13| = 1/sqrt(2). (Key point: Lossless condition)

3. Applying the lossless condition to row 1 gives
   A. S11 = 0 and S12 = 0, so ports 1 and 2 are matched and isolated
   B. S11 = 1 and S12 = 0
   C. S11 = S12 = 1/sqrt(2)
   D. S11 = 0 and S12 = 1
   Answer: A. |S11|^2 + |S12|^2 + 1/2 + 1/2 = 1 forces both to zero. (Key point: Resulting S11, S12)

4. With input at port 1 only, power goes to
   A. Port 2 only
   B. Ports 3 and 4 equally, none to port 2
   C. Port 3 only
   D. All ports equally
   Answer: B. S31 = S41 = 1/sqrt(2) and S21 = 0, so ports 1 and 2 are isolated. (Key point: Input at port 1)

## Related in RF and Microwave Engineering

- [Identify device from S-matrix with S22 term](https://www.bctengineer.com/question.md?key=semester_7%2Frf_and_microwave_engineering%3Aidentify-device-s-matrix-s22) — RF and Microwave Engineering, chapter 4
- [Identify device from numeric 4×4 S-matrix](https://www.bctengineer.com/question.md?key=semester_7%2Frf_and_microwave_engineering%3Aidentify-device-s-matrix-numeric) — RF and Microwave Engineering, chapter 4
- [Passive device to split power into half](https://www.bctengineer.com/question.md?key=semester_7%2Frf_and_microwave_engineering%3Apower-splitter-device) — RF and Microwave Engineering, chapter 4