---
title: "Received power: 50 km, 10 GHz link"
subject: "Propagation and Antenna"
chapter: 5
key: "semester_6/propagation_and_antenna:received-power-50km-10ghz"
source: https://www.bctengineer.com/q/semester_6/propagation_and_antenna/received-power-50km-10ghz
site: BCT Engineering
---

# Received power: 50 km, 10 GHz link

Propagation and Antenna · Chapter 5: Propagation between Antennas

- Asked in: 2079 Chaitra, 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/received-power-50km-10ghz
- Whole chapter: https://www.bctengineer.com/md/semester_6/propagation_and_antenna/chapter-5

## Question

Find the received power in dB at a distance of 50 km over a free space with 10 GHz frequency consisting of transmitting antenna with 25 dB gain and a receiving gain of 20 dB. The power radiated by transmitting antenna is 150 dB.

## Short answer

For f=10 GHz, λ=0.03 m; d=50 km gives LFS=146.42 dB. With Pt=150 W=21.76 dBW, Gt=25 dB, Gr=20 dB, Pr=21.76+25+20-146.42=-79.66 dBW=-49.66 dBm≈10.8 nW.

## Answer

We use the Friis equation in dB form. The radiated power is taken as **150 W** (the "150 dB" in the question is read as 150 W, the usual value in this problem).

#### Given

- d = 50 km = 5×10⁴ m, f = 10 GHz
- Gt = 25 dB, Gr = 20 dB, Pt = 150 W

#### Step 1: Wavelength

```text
λ = c/f = 3×10⁸ / 10×10⁹ = 0.03 m
```

#### Step 2: Free space loss

```text
4πd/λ   = 4π × 5×10⁴ / 0.03 = 2.094×10⁷
LFS(dB) = 20 log(2.094×10⁷) = 146.42 dB
```

(Check: 32.44 + 20 log 50 + 20 log 10000 = 32.44 + 33.98 + 80 = 146.42 dB.)

#### Step 3: Transmitted power in dB

```text
Pt = 10 log 150 = 21.76 dBW
```

#### Step 4: Received power

```text
Pr = Pt + Gt + Gr − LFS
   = 21.76 + 25 + 20 − 146.42
   = −79.66 dBW
   = −49.66 dBm
Pr = 10^(−7.966) W ≈ 1.08×10⁻⁸ W = 10.8 nW
```

**Answer: Pr ≈ −79.66 dBW (−49.66 dBm), i.e. about 10.8 nW.**

(If the transmitted power were literally 150 dBW, the result would simply be 150 + 45 − 146.42 = 48.58 dBW; such a power is unrealistic, so 150 W is used.)

## Check yourself

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

1. The wavelength at 10 GHz is:
   A. 3 m
   B. 0.3 m
   C. 0.03 m
   D. 0.003 m
   Answer: C. lambda = 3e8/10e9 = 0.03 m. (Key point: Wavelength)

2. LFS at 50 km and 10 GHz is:
   A. 146.42 dB
   B. 126.42 dB
   C. 166.42 dB
   D. 32.44 dB
   Answer: A. 20 log(4 pi x 5e4/0.03) = 146.42 dB. (Key point: Free space loss)

3. Treating Pt as 150 W, Pt in dBW is:
   A. 150 dBW
   B. 21.76 dBW
   C. 15 dBW
   D. 50 dBW
   Answer: B. 10 log 150 = 21.76 dBW. (Key point: Transmit power)

4. Pr = Pt + Gt + Gr - LFS equals:
   A. -79.66 dBW
   B. -49.66 dBW
   C. -146.42 dBW
   D. 48.58 dBW
   Answer: A. 21.76 + 25 + 20 - 146.42 = -79.66 dBW (-49.66 dBm). (Key point: Received power)

## Close variants set in other exams

- [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×

## Related in Propagation and Antenna

- [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
- [Microwave link: 90 km, 10 GHz, 40 dB antennas](https://www.bctengineer.com/question.md?key=semester_6%2Fpropagation_and_antenna%3Amicrowave-link-90km-10ghz) — Propagation and Antenna, chapter 5
- [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: 100 MHz, 0.75 km](https://www.bctengineer.com/question.md?key=semester_6%2Fpropagation_and_antenna%3Areceived-power-100mhz-0-75km) — Propagation and Antenna, chapter 5
- [10 GHz LOS link loss and Friis derivation](https://www.bctengineer.com/question.md?key=semester_6%2Fpropagation_and_antenna%3Alos-link-10ghz-100km-friis) — Propagation and Antenna, chapter 5
- [Receiving antenna gain](https://www.bctengineer.com/question.md?key=semester_6%2Fpropagation_and_antenna%3Areceiving-antenna-gain) — Propagation and Antenna, chapter 2

## Related in other subjects

- [Numerical: free-space received power (165 W, 325 MHz)](https://www.bctengineer.com/question.md?key=semester_7%2Fwireless_communication%3Afree-space-received-power-165w) — Wireless Communication, chapter 3
- [Numerical: 10 km link feasibility with Okumura](https://www.bctengineer.com/question.md?key=semester_7%2Fwireless_communication%3Aokumura-link-feasibility) — Wireless Communication, chapter 3
- [Numerical: 50 W in dBm/dBW and received power](https://www.bctengineer.com/question.md?key=semester_7%2Fwireless_communication%3Atransmit-power-dbm-free-space) — Wireless Communication, chapter 3
- [Numerical: received power with cable and mismatch loss](https://www.bctengineer.com/question.md?key=semester_7%2Fwireless_communication%3Areceived-power-coax-mismatch) — Wireless Communication, chapter 3