Skip to main content

Numerical: received power, E-field, flux density, voltage

Short answer

Use Friis equation with lambda=0.333 m: received power is 7.04e-10 W (-61.5 dBm), E-field magnitude 3.87 mV/m, power flux density 3.98e-8 W/m2, and rms voltage at receiver input is 0.375 mV (0.188 mV across matched terminals).

Assume free space propagation, a receiver is located 10 km away from a 50 W transmitter. The carrier frequency is 900 MHz, antenna gain at transmitter and receiver are 1 and 2 respectively. Find: i) The power received at the receiver ii) The magnitude of E-field at the receiver antenna iii) The power flux density iv) The rms voltage applied to the receiver input. The receiver antenna has a purely real impedance of 50 Ω and is matched to the receiver.

Friis free space equation: Pr = Pt·Gt·Gr·λ² / [(4π)²d²]

Given: Pt = 50 W, d = 10 km, f = 900 MHz, Gt = 1, Gr = 2, R = 50 Ω (matched).

λ = c/f = 3×10⁸ / 900×10⁶ = 0.3333 m

i) Received power

Pr = 50 × 1 × 2 × (0.3333)² / [(4π)² × (10⁴)²]
   = 7.04 × 10⁻¹⁰ W
   = 10 log(7.04×10⁻¹⁰) = −91.53 dBW = −61.53 dBm

ii) Magnitude of E-field

Effective aperture of the receiving antenna:

Ae = Gr·λ²/(4π) = 2 × 0.1111 / 12.566 = 0.01768 m²

Since Pr = |E|²·Ae / 120π:

|E| = √(Pr × 120π / Ae)
    = √(7.04×10⁻¹⁰ × 376.99 / 0.01768)
    = 3.87 × 10⁻³ V/m = 3.87 mV/m

iii) Power flux density

Pd = |E|²/120π = Pr/Ae
   = 7.04×10⁻¹⁰ / 0.01768
   = 3.98 × 10⁻⁸ W/m²

iv) RMS voltage at the receiver input

The antenna is modelled as a voltage source V (its open-circuit voltage) in series with Rant = 50 Ω. With a matched receiver, V divides equally between the antenna resistance and the receiver, so Pr = (V/2)²/Rant = V²/(4Rant):

V = √(4 × Rant × Pr) = √(4 × 50 × 7.04×10⁻¹⁰)
  = 3.75 × 10⁻⁴ V = 0.375 mV

This V = 0.375 mV is the value the standard textbook (Rappaport) gives as the rms voltage applied to the receiver input. The voltage actually measured across the matched receiver terminals is V/2 ≈ 0.188 mV.

Answer: Pr = 7.04×10⁻¹⁰ W (−61.5 dBm); |E| = 3.87 mV/m; Pd = 3.98×10⁻⁸ W/m²; Vrms = 0.375 mV (open-circuit antenna voltage; 0.188 mV across the receiver terminals).

✓ Answer checked by Claude Opus 5.5 (high effort)

Related questions

All Wireless Communication chapter 3 questions →