--- title: "Numerical: λ/4 monopole and two-ray received power" subject: "Wireless Communication" chapter: 3 key: "semester_7/wireless_communication:monopole-two-ray-received-power" source: https://www.bctengineer.com/q/semester_7/wireless_communication/monopole-two-ray-received-power site: BCT Engineering --- # Numerical: λ/4 monopole and two-ray received power Wireless Communication · Chapter 3: Radio Wave Propagation in Mobile Networks - Asked in: 2071 Bhadra, 6 marks - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_7/wireless_communication/monopole-two-ray-received-power - Whole chapter: https://www.bctengineer.com/md/semester_7/wireless_communication/chapter-3 ## Question A mobile is located 5 km away from a base station and a vertical λ/4 monopole antenna with a gain of 2.55 dB to receive cellular radio signals. The electric field at 1 km from the transmitter is measured to be 10⁻³ V/m. The carrier frequency used for this system is 900 MHz. a) Find the length and effective aperture of the receiving antenna. b) Find the received power at the mobile using two ray ground reflection model assuming the height of the transmitting antenna is 50 m and the receiving antenna is 1.5 m above ground. ## Short answer The wavelength is 0.3333 m, so the quarter-wave monopole length is 0.0833 m and its effective aperture is 0.0159 m^2. Using the two-ray ground reflection model, the received field is 113.1 microV/m and received power is about 5.40 x 10^-13 W, which is -122.68 dBW or -92.68 dBm. ## Answer **Given:** d = 5 km, f = 900 MHz, G_r = 2.55 dB, E₀ = 10⁻³ V/m at d₀ = 1 km, h_t = 50 m, h_r = 1.5 m. #### a) Length and effective aperture of the antenna Wavelength: ```text λ = c / f = (3 × 10⁸) / (900 × 10⁶) = 0.3333 m ``` Length of a quarter-wave monopole: ```text L = λ / 4 = 0.3333 / 4 = 0.0833 m (8.33 cm) ``` Gain in linear form: ```text G = 10^(2.55/10) = 1.799 ``` Effective aperture: ```text A_e = G λ² / (4π) = 1.799 × (0.3333)² / (4π) = 1.799 × 0.1111 / 12.566 = 0.0159 m² ``` #### b) Received power using two-ray ground reflection model First check that the far-distance approximation is valid: ```text 20 h_t h_r / λ = 20 × 50 × 1.5 / 0.3333 = 4500 m d = 5000 m > 4500 m → approximation valid ``` For large d, the total received field is: ```text E_TOT(d) ≈ (2 E₀ d₀ / d) × (2π h_t h_r / (λ d)) ``` Substitute: ```text 2 E₀ d₀ / d = 2 × 10⁻³ × 1000 / 5000 = 4 × 10⁻⁴ V/m 2π h_t h_r / (λ d) = 2π × 50 × 1.5 / (0.3333 × 5000) = 471.24 / 1666.7 = 0.2827 E_TOT = 4 × 10⁻⁴ × 0.2827 = 1.131 × 10⁻⁴ V/m = 113.1 µV/m ``` Received power (power density × effective aperture), with free-space impedance 120π Ω: ```text P_r = (|E|² / 120π) × A_e = ((1.131 × 10⁻⁴)² / 376.99) × 0.0159 = (3.393 × 10⁻¹¹) × 0.0159 = 5.40 × 10⁻¹³ W ``` In decibels: ```text P_r = 10 log(5.40 × 10⁻¹³) = −122.68 dBW = −92.68 dBm ``` **Answer:** L = 0.0833 m, A_e = 0.0159 m², E = 113.1 µV/m and P_r ≈ 5.40 × 10⁻¹³ W (−122.68 dBW, or −92.68 dBm). ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. What is the length of a quarter-wave monopole at 900 MHz? A. 0.333 m B. 0.167 m C. 0.0833 m D. 0.0417 m Answer: C. lambda = 0.3333 m, so L = lambda/4 = 0.0833 m. (Key point: Antenna length) 2. Which formula gives the effective aperture used here? A. A_e = G lambda^2 / (4 pi) B. A_e = 4 pi / (G lambda^2) C. A_e = G lambda / (4 pi) D. A_e = lambda^2 / G Answer: A. A_e = G lambda^2/(4 pi), which gives 0.0159 m^2 for G = 1.799. (Key point: Effective aperture) 3. Why is the large-d two-ray approximation valid here? A. d = 5 km exceeds 20 h_t h_r / lambda = 4500 m B. d is less than h_t C. The frequency is 900 MHz D. The gain is 2.55 dB Answer: A. The approximation needs d greater than 20 h_t h_r / lambda; 5000 m > 4500 m. (Key point: Validity of far approximation) 4. How is received power obtained from the total field E? A. P_r = E^2 x A_e B. P_r = (|E|^2 / 120 pi) x A_e C. P_r = E / A_e D. P_r = (|E|^2 x 120 pi) / A_e Answer: B. Power density is |E|^2/(120 pi) and multiplying by effective aperture gives received power, about 5.40e-13 W. 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