Wireless Notes
Learn free space propagation with Friis transmission equation, FSPL formula, isotropic radiator, EIRP calculation, link budget example, and limitations explained with solved problems for engineering students.
Free space propagation is an ideal condition where a radio wave propagates without any obstacles, reflections, or absorption – signal weakens only due to pure distance.
🌌 What is Free Space Propagation?
Free space propagation is an idealized scenario where: - No obstacles (buildings, trees, ground) - No reflection, diffraction, or scattering - No atmospheric absorption - Signal weakens only with distance (spreading loss)
┌─────────────────────────────────────────────────────────────────────┐
│ FREE SPACE PROPAGATION │
│ │
│ Signal spreads equally in all directions │
│ │
│ ╱ ╱ ╱ │
│ ╱ ╱ ╱ │
│ ╱ ╱ ╱ │
│ 📡 TX ───────╱──╱──╱─────────────────── 📡 RX │
│ ╲ ╲ ╲ │
│ ╲ ╲ ╲ │
│ ╲ ╲ ╲ │
│ │
│ Power density decreases as 1/d² (inverse square law) │
│ Like a balloon expanding – same energy over larger area │
│ │
│ At distance d: Power density = Pt / (4πd²) │
│ Double the distance → 1/4 the power → 6 dB more loss │
└─────────────────────────────────────────────────────────────────────┘
Conditions for Free Space:
- Line of Sight (LOS) between TX and RX
- No nearby reflecting surfaces
- No obstructions in Fresnel zone
- Clear atmosphere (no rain/fog)
Where it applies:
- Satellite communication
- Microwave point-to-point links (tower to tower)
- Deep space communication
- Open rural areas (approximately)
📐 Inverse Square Law
The inverse square law states that when a signal radiates from a source, power density is inversely proportional to the square of the distance.
Power Density at distance d
S(d) = Pt × Gt / (4πd²) [W/m²]
Where
Pt = Transmitted power (watts)
Gt = Transmit antenna gain
d = Distance (meters)
4πd² = Surface area of sphere at distance d
Visual Understanding:
| TX ●────── | 1 m² | Power per m² = Pt/12.57 |
|---|---|---|
| TX ●─────── | 4 m² | Power per m² = Pt/50.27 (1/4th!) |
📡 Friis Transmission Equation
The Friis equation calculates the complete received power – including antenna gains.
┌──────────────────────────────────────────────────────┐
│ │
│ Pr = Pt × Gt × Gr × (λ / 4πd)² │
│ │
│ In dB: │
│ Pr(dBm) = Pt(dBm) + Gt(dBi) + Gr(dBi) - FSPL(dB)│
│ │
│ Where: │
│ Pr = Received power │
│ Pt = Transmitted power │
│ Gt = Transmit antenna gain │
│ Gr = Receive antenna gain │
│ λ = Wavelength (c/f) │
│ d = Distance between antennas │
│ │
└──────────────────────────────────────────────────────┘
Key Observations:
- Received power ∝ 1/d² (inverse square)
- Received power ∝ λ² (lower frequency = more power received)
- Received power ∝ Gt × Gr (antenna gains help)
📉 Free Space Path Loss (FSPL)
FSPL is the signal loss that occurs purely due to distance and frequency (no obstacles).
┌──────────────────────────────────────────────────────┐
│ │
│ FSPL = (4πd/λ)² = (4πdf/c)² │
│ │
│ FSPL (dB) = 20log₁₀(4πd/λ) │
│ = 20log₁₀(d) + 20log₁₀(f) + 32.44 │
│ (d in km, f in MHz) │
│ │
│ Or: FSPL (dB) = 20log₁₀(d) + 20log₁₀(f) - 27.55│
│ (d in meters, f in MHz) │
│ │
└──────────────────────────────────────────────────────┘
FSPL Quick Reference Table:
| Frequency | 100m | 1 km | 10 km | 100 km |
|---|---|---|---|---|
| 900 MHz | 71.5 dB | 91.5 dB | 111.5 dB | 131.5 dB |
| 2.4 GHz | 80.0 dB | 100.0 dB | 120.0 dB | 140.0 dB |
| 5 GHz | 86.3 dB | 106.3 dB | 126.3 dB | 146.3 dB |
| 28 GHz | 101.3 dB | 121.3 dB | 141.3 dB | 161.3 dB |
| 60 GHz | 108.0 dB | 128.0 dB | 148.0 dB | 168.0 dB |
Rules of Thumb:
- Double distance → +6 dB loss
- Double frequency → +6 dB loss
- 10× distance → +20 dB loss
- 10× frequency → +20 dB loss
📻 Isotropic Radiator
An isotropic radiator is a theoretical antenna that radiates equally in all directions (point source). It does not exist in reality but is used as a reference.
- Radiates equally in all directions (spherical pattern)
- Gain = 0 dBi (reference)
- All real antennas compared to it (dBi = dB relative to isotropic)
- Used in FSPL calculations as baseline
⚡ EIRP (Effective Isotropic Radiated Power) {#eirp}
EIRP indicates how much power would be needed from an isotropic antenna to produce the same signal strength.
A 100 mW transmitter + 15 dBi antenna = equivalent to transmitting from a 3.16 W isotropic antenna. Antenna gain effectively multiplies power (in that direction).
💰 Link Budget Example
Link budget is the complete calculation of whether the signal will reach the receiver with enough power or not.
Satellite Communication Example:
| Parameter | Value |
|---|---|
| Transmit Power (Pt) | +40 dBm (10 W) |
| TX Antenna Gain (Gt) | +45 dBi (large dish) |
| TX Cable Loss | -2 dB |
| EIRP | +83 dBm |
| Distance (GEO) | 36,000 km |
| Frequency | 14 GHz (Ku-band uplink) |
| FSPL | -207.1 dB |
| Atmospheric Loss | -0.5 dB |
| Rain Attenuation | -3 dB (margin) |
| RX Antenna Gain (Gr) | +35 dBi (satellite) |
| System Noise Temp | 27 dBK (500K) |
| Received Power | -94.6 dBm |
| Receiver Sensitivity | -110 dBm |
| Link Margin | 15.4 dB ✅ (sufficient) |
⚠️ Limitations of Free Space Model
In the real world, pure free space conditions rarely exist:
| Limitation | Real World Effect |
|---|---|
| No reflections assumed | Ground, buildings reflect signals |
| No obstacles | Trees, walls, hills block |
| No atmosphere | Rain, fog attenuate (>10 GHz) |
| No multipath | Multiple paths cause fading |
| LOS required | Often NLoS in practice |
| Flat terrain assumed | Earth curvature limits range |
When Free Space Model is Valid:
- Satellite links ✅ (closest to free space)
- Microwave tower-to-tower ✅ (clear LOS, elevated)
- Open rural areas ≈ (approximately)
- Indoor short range ≈ (before first wall)
- Urban environments ❌ (use empirical models)
📝 Summary
| Concept | Formula | Key Point |
|---|---|---|
| Power density | S = Pt×Gt/(4πd²) | Spreads over sphere |
| Friis equation | Pr = Pt×Gt×Gr×(λ/4πd)² | Complete link equation |
| FSPL (dB) | 20log(d)+20log(f)+32.44 | d in km, f in MHz |
| Distance effect | Double d → +6 dB loss | Inverse square law |
| Frequency effect | Double f → +6 dB loss | Higher freq = more loss |
| EIRP | Pt + Gt (dBm+dBi) | Effective radiated power |
| Link budget | EIRP - FSPL + Gr ≥ Sensitivity | Must exceed threshold |
❓ FAQ
Q: Frequency badhne se path loss kyun badhti hai? A: Actually, the signal spreads equally in space. At higher frequencies, the antenna's effective aperture (capture area) is smaller, so less power is collected. This is a mathematical effect, not actual extra loss.
Q: When should we use the free space model? A: For satellite communication, clear LOS microwave links, and initial rough calculations. For urban/indoor environments, use empirical models (Hata, COST 231).
Q: FSPL mein antenna gain include hai? A: No. FSPL is only propagation loss. For complete received power, use the Friis equation: Pr = Pt + Gt + Gr - FSPL.
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