---
title: "Two-port network model and parameters"
subject: "RF and Microwave Engineering"
chapter: 3
key: "semester_7/rf_and_microwave_engineering:two-port-network-model"
source: https://www.bctengineer.com/q/semester_7/rf_and_microwave_engineering/two-port-network-model
site: BCT Engineering
---

# Two-port network model and parameters

RF and Microwave Engineering · Chapter 3: RF and Microwave Network Theory and Analysis

- Asked in: 2071 Bhadra, 10 marks
- Video: [Introduction to Two-Port Networks](https://www.youtube.com/watch?v=ru2ItcD6unI) — ALL ABOUT ELECTRONICS
- 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/two-port-network-model
- Whole chapter: https://www.bctengineer.com/md/semester_7/rf_and_microwave_engineering/chapter-3

## Question

Design a two-port network model and derive the required parameters.

## Short answer

A two-port network is represented by Z, Y, h, ABCD, or S parameters relating port voltages, currents, or waves. At microwave frequencies S parameters are used: S11 and S22 are reflection coefficients, S21 and S12 transmission coefficients, measured with matched loads. S is derived from Z by [S]=(Z-Z0U)(Z+Z0U)^-1.

## Answer

A **two-port network** is any linear circuit with an input port (1) and an output port (2), such as an amplifier, filter, attenuator or transmission line section. It is modeled as a "black box" described only by the relations between the port voltages, currents or waves.

```text
      I1 ->                     <- I2
  o---------+-------------+---------o
  +         |             |         +
  V1        |  Two-port   |        V2
  -         |  network    |         -
  o---------+-------------+---------o
   port 1                     port 2
```

Four variables (V1, I1, V2, I2) exist; any two can be written in terms of the other two, giving the parameter sets below.

#### 1. Impedance (Z) parameters
```text
 V1 = Z11·I1 + Z12·I2
 V2 = Z21·I1 + Z22·I2
```
Z11 = V1/I1 at I2 = 0 (output open); Z21 = V2/I1 at I2 = 0; Z12 = V1/I2 at I1 = 0; Z22 = V2/I2 at I1 = 0. Units: ohm. Reciprocal → Z12 = Z21.

#### 2. Admittance (Y) parameters
```text
 I1 = Y11·V1 + Y12·V2
 I2 = Y21·V1 + Y22·V2
```
Measured with the opposite port **short-circuited** (V = 0). [Y] = [Z]⁻¹. Units: siemens.

#### 3. Hybrid (h) parameters
```text
 V1 = h11·I1 + h12·V2
 I2 = h21·I1 + h22·V2
```
h11 = input impedance (output shorted), h21 = current gain, h12 = reverse voltage ratio (input open), h22 = output admittance. Used for BJT models.

#### 4. Transmission (ABCD) parameters
```text
 V1 = A·V2 + B·I2'      (I2' = -I2, current leaving port 2)
 I1 = C·V2 + D·I2'
```
Cascaded networks multiply: [ABCD]total = [ABCD]1 · [ABCD]2. Reciprocal → AD − BC = 1. Example: a series impedance Z has A = 1, B = Z, C = 0, D = 1.

#### 5. Scattering (S) parameters – the microwave model
At microwave frequencies the opens and shorts needed above are hard to make, so the network is described by waves. With reference impedance Z0:
```text
 an = (Vn + Z0·In)/(2√Z0)   incident wave
 bn = (Vn - Z0·In)/(2√Z0)   reflected wave

 b1 = S11·a1 + S12·a2
 b2 = S21·a1 + S22·a2
```
- S11 = b1/a1 at a2 = 0 (port 2 matched): input reflection coefficient.
- S21 = b2/a1 at a2 = 0: forward transmission coefficient.
- S22 = b2/a2 at a1 = 0: output reflection coefficient.
- S12 = b1/a2 at a1 = 0: reverse transmission coefficient.

**Derivation of S from Z:** substitute V = √Z0(a + b) and I = (a − b)/√Z0 into V = Z·I:
```text
 Z0(a + b) = Z(a - b)
 (Z + Z0·U) b = (Z - Z0·U) a
 [S] = (Z - Z0·U)(Z + Z0·U)^-1
```
For a two-port with ΔZ = (Z11 + Z0)(Z22 + Z0) − Z12·Z21:
```text
 S11 = [(Z11 - Z0)(Z22 + Z0) - Z12·Z21]/ΔZ
 S21 = 2·Z21·Z0/ΔZ
```

#### Properties used in design
| Property | Z / Y form | S form |
|---|---|---|
| Reciprocal | Z12 = Z21 | S12 = S21 |
| Lossless | Re(Zij) = 0 | [S]*ᵀ[S] = U |
| Symmetric | Z11 = Z22 | S11 = S22 |
| Matched | – | S11 = S22 = 0 |

**Useful derived quantities:** input reflection Γin = S11 + S12·S21·ΓL/(1 − S22·ΓL); return loss = −20 log|S11|; insertion loss = −20 log|S21|; transducer gain (matched) = |S21|².

**Example:** a series impedance Z between two Z0 lines gives S11 = S22 = Z/(Z + 2Z0) and S21 = S12 = 2Z0/(Z + 2Z0); for Z = 50 Ω and Z0 = 50 Ω, S11 = 1/3 and S21 = 2/3.

## Check yourself

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

1. Z11 = V1/I1 is measured with
   A. Port 2 short-circuited
   B. Port 2 open-circuited (I2 = 0)
   C. Port 2 matched to Z0
   D. Port 1 open
   Answer: B. Z parameters are open-circuit parameters, so I2 = 0 for Z11 and Z21. Y parameters use shorts. (Key point: Condition for Z parameters)

2. How do ABCD matrices of cascaded networks combine?
   A. They are added
   B. They are multiplied: [ABCD]total = [ABCD]1 x [ABCD]2
   C. They are inverted
   D. They are averaged
   Answer: B. The ABCD form is chosen for cascades because the total matrix is the product of the individual ones. (Key point: ABCD cascade)

3. Why is the S-parameter model used at microwave frequencies instead of Z, Y, h?
   A. Opens and shorts needed by those sets are hard to make at microwave frequencies, whereas matched waves are easy
   B. S-parameters are real numbers
   C. Z, Y, h do not exist for linear networks
   D. S-parameters need no reference impedance
   Answer: A. Microwave work uses incident and reflected waves with matched loads, avoiding troublesome open and short terminations. (Key point: Why S-parameters)

4. A series impedance Z = 50 ohm between two 50 ohm lines has S11 and S21 equal to
   A. S11 = 1/3, S21 = 2/3
   B. S11 = 2/3, S21 = 1/3
   C. S11 = 1/2, S21 = 1/2
   D. S11 = 0, S21 = 1
   Answer: A. S11 = Z/(Z+2Z0) = 50/150 = 1/3 and S21 = 2Z0/(Z+2Z0) = 100/150 = 2/3. (Key point: Series impedance example)

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