---
title: "Microwave link: 90 km, 10 GHz, 40 dB antennas"
subject: "Propagation and Antenna"
chapter: 5
key: "semester_6/propagation_and_antenna:microwave-link-90km-10ghz"
source: https://www.bctengineer.com/q/semester_6/propagation_and_antenna/microwave-link-90km-10ghz
site: BCT Engineering
---

# Microwave link: 90 km, 10 GHz, 40 dB antennas

Propagation and Antenna · Chapter 5: Propagation between Antennas

- 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_6/propagation_and_antenna/microwave-link-90km-10ghz
- Whole chapter: https://www.bctengineer.com/md/semester_6/propagation_and_antenna/chapter-5

## Question

Microwave link is assumed to be free space condition. The antenna gains are each 40 dB, the frequency is 10 GHz and the path length is 90 km. Calculate the transmission path loss and received power for transmitted power of 10 kW.

## Short answer

λ = 0.03 m. Free space loss LFS = 20 log(4πd/λ) = 151.53 dB; net loss after 40 dB gains is 71.53 dB. Received power Pr = 40 dBW + 40 + 40 − 151.53 = −31.53 dBW (0.704 mW).

## Answer

Under free space conditions the Friis equation gives the path loss and received power.

#### Given

- Gt = Gr = 40 dB, f = 10 GHz, d = 90 km, Pt = 10 kW

#### Step 1: Wavelength

```text
λ = c/f = 3×10⁸ / 10¹⁰ = 0.03 m
```

#### Step 2: Free space (transmission path) loss

```text
4πd/λ = 4π × 9×10⁴ / 0.03 = 3.770×10⁷
LFS   = 20 log(3.770×10⁷) = 151.53 dB

Check: 32.44 + 20 log 90 + 20 log 10000
     = 32.44 + 39.08 + 80 = 151.53 dB
```

Net loss after the antenna gains: 151.53 − 40 − 40 = 71.53 dB.

#### Step 3: Received power

```text
Pt = 10 log(10⁴) = 40 dBW
Pr = Pt + Gt + Gr − LFS
   = 40 + 40 + 40 − 151.53
   = −31.53 dBW = −1.53 dBm
Pr = 10^(−3.153) W ≈ 7.04×10⁻⁴ W
```

**Answer: Path loss = 151.53 dB (71.53 dB including antenna gains); received power ≈ −31.53 dBW = 0.704 mW.**

## Check yourself

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

1. The free space loss at 10 GHz over 90 km is:
   A. 71.53 dB
   B. 131.53 dB
   C. 191.53 dB
   D. 151.53 dB
   Answer: D. 32.44 + 39.08 + 80 = 151.53 dB. (Key point: Path loss)

2. 10 kW expressed in dBW is:
   A. 10 dBW
   B. 30 dBW
   C. 40 dBW
   D. 70 dBW
   Answer: C. 10 log(10^4) = 40 dBW. (Key point: Transmit power in dBW)

3. Which dB expression gives the received power?
   A. Pt − LFS only
   B. Pt + Gt + Gr − LFS
   C. Pt − Gt − Gr − LFS
   D. Pt + Gt + Gr + LFS
   Answer: B. Gains add and path loss subtracts: 40+40+40−151.53. (Key point: Link equation)

4. The received power is about:
   A. +31.53 dBW
   B. −1.53 dBW
   C. −31.53 dBW (0.704 mW)
   D. −71.53 dBW
   Answer: C. −31.53 dBW = −1.53 dBm ≈ 0.704 mW. (Key point: Result)

## Related in Propagation and Antenna

- [Free space path loss: 6 GHz, 10,500 km](https://www.bctengineer.com/question.md?key=semester_6%2Fpropagation_and_antenna%3Afspl-derivation-6ghz-10500km) — Propagation and Antenna, chapter 5
- [Received power: 50 km, 10 GHz link](https://www.bctengineer.com/question.md?key=semester_6%2Fpropagation_and_antenna%3Areceived-power-50km-10ghz) — Propagation and Antenna, chapter 5, asked 2×
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- [Received power: 15.5 km, 3.4 GHz, dish receiver](https://www.bctengineer.com/question.md?key=semester_6%2Fpropagation_and_antenna%3Amax-received-power-15-5km) — Propagation and Antenna, chapter 5
- [Received power: 0.5 km, 1 GHz circuit](https://www.bctengineer.com/question.md?key=semester_6%2Fpropagation_and_antenna%3Areceived-power-0-5km-1ghz) — Propagation and Antenna, chapter 5, asked 2×
- [Friis equation and free space path loss](https://www.bctengineer.com/question.md?key=semester_6%2Fpropagation_and_antenna%3Afriis-path-loss-free-space) — Propagation and Antenna, chapter 5, asked 2×

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