---
title: "Identify device from S-matrix with S22 term"
subject: "RF and Microwave Engineering"
chapter: 4
key: "semester_7/rf_and_microwave_engineering:identify-device-s-matrix-s22"
source: https://www.bctengineer.com/q/semester_7/rf_and_microwave_engineering/identify-device-s-matrix-s22
site: BCT Engineering
---

# Identify device from S-matrix with S22 term

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

- Asked in: 2075 Bhadra, 6 marks
- 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-s22
- Whole chapter: https://www.bctengineer.com/md/semester_7/rf_and_microwave_engineering/chapter-4

## Question

Which of the passive microwave device is explained by this S-matrix? Judge the condition and explain its characteristics. [S] = [S11 0 S13 S14; 0 S22 −S13 S14; S13 −S13 0 0; S14 S14 0 0].

## Short answer

It is a magic tee with ports 1 and 2 collinear, port 3 E-arm, port 4 H-arm. Losslessness forces S11 = S22 = 0, giving the ideal magic tee S-matrix. It is matched, reciprocal, and separates sum and difference signals.

## Answer

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

The device is a **magic tee (hybrid E-H tee)** with ports 1 and 2 as the collinear arms.

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

#### Judging the condition (lossless junction)
For a lossless device [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 + |S13|² + |S14|² = 1
        |S11|² + 1/2 + 1/2 = 1  -> S11 = 0
 Row 2: similarly            -> S22 = 0
```
So the given form is consistent only when **S11 = S22 = 0**: matching the E and H arms automatically matches the collinear arms. The device is then the ideal magic tee:
```text
              | 0   0   1   1 |
 [S] = (1/√2) | 0   0  -1   1 |
              | 1  -1   0   0 |
              | 1   1   0   0 |
```
If S11, S22 ≠ 0 in practice, the junction is lossy or imperfectly matched.

#### Characteristics
- **H-arm input (4):** equal in-phase split to ports 1 and 2; no output at E-arm.
- **E-arm input (3):** equal split to 1 and 2 with 180° phase difference; no output at H-arm.
- **Input at port 1:** half to each of E and H arms; port 2 isolated.
- **Sum/difference action:** in-phase inputs at 1 and 2 add at the H-arm; anti-phase inputs add at the E-arm.
- Matched, lossless, reciprocal. Used in balanced mixers, duplexers, impedance bridges and monopulse comparators.

## Check yourself

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

1. The matrix describes
   A. a circulator
   B. a directional coupler
   C. a magic tee (hybrid E-H tee)
   D. an isolator
   Answer: C. Isolated pairs and E/H arm properties identify it. (Key point: Identification)

2. Port 3 is the E-arm because
   A. it receives in-phase signals
   B. it is isolated from port 1
   C. it is matched only if S11 ≠ 0
   D. it receives equal anti-phase signals from ports 1 and 2 (S31 = -S32)
   Answer: D. Anti-phase indicates difference arm. (Key point: E-arm)

3. Applying lossless conditions gives
   A. |S13| = |S14| = 1/sqrt2 and S11 = S22 = 0
   B. S11 = S22 = 1
   C. |S13| = 1
   D. S11 = 1/2
   Answer: A. Rows 3 and 4 fix the magnitudes and row 1 then forces S11 = 0. (Key point: Lossless condition)

## Related in RF and Microwave Engineering

- [Identify device from S-matrix with ±S14 entries](https://www.bctengineer.com/question.md?key=semester_7%2Frf_and_microwave_engineering%3Aidentify-device-s-matrix-hybrid) — RF and Microwave Engineering, chapter 4, asked 2×
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