43  Path Loss & Propagation

Key Concepts
  • Free-Space Path Loss (FSPL): The attenuation of radio signal power as it spreads over a spherical wavefront in the absence of obstacles or reflections
  • FSPL Formula: FSPL (dB) = 20 log₁₀(d) + 20 log₁₀(f) + 20 log₁₀(4π/c) ≈ 20 log₁₀(d) + 20 log₁₀(f) − 147.55 (d in metres, f in Hz)
  • Inverse Square Law: Signal power decreases proportionally to the square of distance; doubling distance increases FSPL by 6 dB
  • Frequency Dependence: Higher frequencies experience more path loss at the same distance; 2.4 GHz suffers ~6 dB more FSPL than 900 MHz at equal range
  • dBm: Power measured in decibels relative to 1 milliwatt; used to express transmitted and received signal power
  • Received Signal Strength (RSS): The power of the signal at the receiver, calculated as Transmit Power (dBm) + Antenna Gains (dBi) − FSPL (dB)
  • Path Loss Exponent: The exponent n in the more general path loss model (PL ∝ dⁿ); n=2 in free space, n=2.7–3.5 in buildings

43.1 In 60 Seconds

Free Space Path Loss (FSPL) quantifies how radio signals weaken with distance: doubling the distance costs 6 dB, and doubling the frequency costs another 6 dB. The formula FSPL(dB) = 20log(d) + 20log(f) + 32.45 gives theoretical loss (d in km, f in MHz), while the log-distance model adds a path loss exponent (n=2 for free space, n=3-4 for urban, n=4-6 for indoor) to match real environments. This is the foundation for estimating Wi-Fi range, LoRa coverage, and BLE beacon placement.

43.2 Learning Objectives

By the end of this chapter, you will be able to:

  • Calculate free space path loss (FSPL) for various frequencies and distances
  • Apply the FSPL formula using both MHz/km and GHz/m conventions
  • Apply the log-distance path loss model for different environments
  • Interpret path loss exponents to characterize propagation environments
  • Estimate received signal strength from transmit power and path loss

As a wireless signal travels through the air, it gets weaker with distance, just like a shout fades as you walk away from the person shouting. Path loss is the technical term for this weakening. Understanding path loss helps you figure out how far apart IoT sensors can be while still communicating reliably.

“Why can’t I reach the gateway anymore?” Sammy the Sensor asked sadly. “It is only twice as far away as before!” Max the Microcontroller pulled out a chart. “That is path loss, Sammy. Every time you double the distance, your signal loses 6 dB of strength. And if you also double the frequency, that is another 6 dB gone!”

“Think of it like shouting across a field,” Lila the LED suggested. “Stand close and your friend hears you perfectly. Walk to the other end and they can barely hear a whisper. The sound does not disappear – it just spreads out in all directions, getting thinner.”

Bella the Battery explained further. “In a perfect open space with nothing blocking the signal, we can calculate exactly how much it weakens using the Free Space Path Loss formula. But in the real world – inside buildings, in cities, through forests – the signal fades much faster because walls and trees absorb it.”

“So that is why our building deployment failed!” said Sammy. “The spec said 100-meter range, but that was for open space. Inside, with all those walls, we only got 30 meters. Always calculate for your ACTUAL environment, not the ideal one!”

43.3 Introduction

Understanding wireless signal propagation is critical for IoT deployments. Whether placing Wi-Fi access points, estimating LoRa range, or debugging BLE beacon issues, engineers must understand how radio waves behave in real environments.

Time: ~15 min | Difficulty: Intermediate | P07.C15.U05a

43.4 Why Propagation Matters for IoT

Real-World Problem: A smart building deploys 200 Zigbee temperature sensors with gateway placement based on theoretical 100m range specifications. After installation, 40% of sensors fail to connect. Post-mortem analysis reveals walls attenuate signals by 8-15 dB, reducing effective range to 30-50m.

Cost Impact:

  • Gateway hardware: $500 each
  • Initial deployment: 4 gateways ($2,000)
  • Required gateways: 12 ($6,000)
  • Wasted hardware + reinstallation labor: $15,000+ overrun

Root Cause: Assuming free-space path loss without accounting for real-world attenuation through building materials.

43.5 Free Space Path Loss (FSPL)

Free Space Path Loss describes signal attenuation in ideal conditions (no obstacles, line-of-sight).

FSPL Formula:

\[L_{FSPL}(dB) = 20\log_{10}(d) + 20\log_{10}(f) + 32.45\]

Where:

  • L = Path loss in decibels (dB)
  • d = Distance in kilometers
  • f = Frequency in MHz

Alternative Formula (meters and GHz):

\[L_{FSPL}(dB) = 20\log_{10}(d) + 20\log_{10}(f) + 92.45\]

Where:

  • d = Distance in meters
  • f = Frequency in GHz
Path loss formula showing FSPL in dB equals 20 log d plus 20 log f plus 92.45, demonstrating frequency-dependent signal attenuation
Figure 43.1: Path loss is frequency dependent - higher frequencies experience greater attenuation
Try It: FSPL Calculator

43.6 Worked Example: Wi-Fi 2.4 GHz Range Calculation

Scenario: ESP32 Wi-Fi module transmitting at 2.4 GHz. What is the path loss at 50 meters?

Given:

  • Distance: d = 50 meters
  • Frequency: f = 2.4 GHz
  • Transmit power: 20 dBm (100 mW - typical ESP32)
  • Receiver sensitivity: -90 dBm (typical Wi-Fi)

Step 1: Calculate FSPL (using km/MHz formula)

Convert distance: 50 m = 0.05 km. Frequency: 2400 MHz.

\[L_{FSPL} = 20\log_{10}(0.05) + 20\log_{10}(2400) + 32.45\] \[L_{FSPL} = 20(-1.301) + 20(3.380) + 32.45\] \[L_{FSPL} = -26.02 + 67.60 + 32.45\] \[L_{FSPL} = 74.03 \text{ dB}\]

Common Formula Pitfall: The two FSPL formula versions use different units. Using meters with the km formula (or GHz with the MHz formula) produces wildly wrong results. Always verify: which constant am I using? 32.45 requires km and MHz. 92.45 requires meters and GHz.

Step 2: Calculate received signal strength

\[P_{RX} = P_{TX} - L_{FSPL}\] \[P_{RX} = 20 \text{ dBm} - 74.03 \text{ dB}\] \[P_{RX} = -54.03 \text{ dBm}\]

Step 3: Compare to sensitivity threshold

  • Received signal: -54.03 dBm
  • Sensitivity threshold: -90 dBm
  • Link margin: -54.03 - (-90) = 35.97 dB (excellent!)

Interpretation: At 50 m in free space, Wi-Fi has ~36 dB of margin. This margin accommodates real-world losses: add 15 dB for a concrete wall, 5 dB for furniture, and 6 dB for fading, and you still have 10 dB of margin left. This is why Wi-Fi works across a house but fails reliably across a warehouse.

The “doubling distance costs 6 dB” rule comes directly from the FSPL formula. Since path loss includes \(20\log_{10}(d)\), doubling distance means:

\[L_{\text{new}} - L_{\text{old}} = 20\log_{10}(2d) - 20\log_{10}(d)\]

\[= 20[\log_{10}(2) + \log_{10}(d)] - 20\log_{10}(d)\]

\[= 20\log_{10}(2) = 20 \times 0.301 = 6.02 \text{ dB}\]

For our ESP32 example at 50 m with -54 dBm received, doubling to 100 m adds 6 dB loss:

\[P_{RX}(100m) = -54 - 6 = -60 \text{ dBm}\]

Still above the -90 dBm sensitivity with 30 dB margin. Doubling again to 200 m: -66 dBm (24 dB margin). At 400 m: -72 dBm (18 dB margin). This shows why Wi-Fi can reach ~300-400 m in free space before the link fails.

43.7 Worked Example: LoRa Long-Range Link Budget

Scenario: LoRaWAN sensor in rural farm environment at 915 MHz. What is maximum range?

Given:

  • Frequency: f = 915 MHz
  • Transmit power: 14 dBm (25 mW - regulatory limit US 915 MHz)
  • Receiver sensitivity: -137 dBm (LoRa SF12)
  • Required link margin: 10 dB (for fading)

Step 1: Calculate maximum allowable path loss

\[L_{max} = P_{TX} - P_{RX,min} - \text{Margin}\] \[L_{max} = 14 \text{ dBm} - (-137 \text{ dBm}) - 10 \text{ dB}\] \[L_{max} = 141 \text{ dB}\]

Step 2: Solve FSPL formula for distance

\[L_{FSPL} = 20\log_{10}(d) + 20\log_{10}(f) + 32.45 = 141 \text{ dB}\] \[20\log_{10}(d) = 141 - 20\log_{10}(915) - 32.45\] \[20\log_{10}(d) = 141 - 59.23 - 32.45\] \[20\log_{10}(d) = 49.32\] \[\log_{10}(d) = 2.466\] \[d = 10^{2.466} = 292.4 \text{ km}\]

Reality Check: This is theoretical free-space range! Real-world factors reduce this significantly:

Environment Typical Range Loss Factor
Rural open field 10-15 km Earth curvature, vegetation
Suburban 2-5 km Buildings, trees
Urban 1-2 km Dense buildings, multipath
Indoor 200-500 m Walls, floors, metal

Actual achievable range: 10-15 km in rural deployment (far less than theoretical 292 km!).

Try It: LoRa Range Estimator

43.8 Worked Example: Comparing Protocols at the Same Distance

Understanding FSPL explains why different protocols have such different range specifications. Here is a side-by-side comparison of four IoT technologies at 100 meters line-of-sight.

Protocol Frequency TX Power FSPL at 100 m RX Power Sensitivity Link Margin
Wi-Fi 2,400 MHz 20 dBm 80.0 dB -60.0 dBm -90 dBm 30.0 dB
BLE 2,400 MHz 0 dBm 80.0 dB -80.0 dBm -95 dBm 15.0 dB
Zigbee 2,400 MHz 4 dBm 80.0 dB -76.0 dBm -100 dBm 24.0 dB
LoRa SF7 915 MHz 14 dBm 71.7 dB -57.7 dBm -124 dBm 66.3 dB
LoRa SF12 915 MHz 14 dBm 71.7 dB -57.7 dBm -137 dBm 79.3 dB

The ~8.4 dB FSPL advantage of 915 MHz vs 2.4 GHz, combined with LoRa’s extreme receiver sensitivity (-137 dBm at SF12 vs -90 dBm for Wi-Fi), gives LoRa a 49 dB advantage in link margin. Each 6 dB doubles the range, so 49 dB translates to roughly 280x the distance – explaining how LoRa reaches 10+ km where Wi-Fi reaches 100 m.

43.9 Log-Distance Path Loss Model

Real environments do not follow free-space propagation. The Log-Distance Path Loss Model accounts for obstacles and environment.

Log-Distance Formula:

\[L(d) = L_0 + 10n\log_{10}\left(\frac{d}{d_0}\right) + X_\sigma\]

Where:

  • L(d) = Path loss at distance d (dB)
  • L_0 = Path loss at reference distance d_0 (typically 1 m)
  • n = Path loss exponent (environment-dependent)
  • d_0 = Reference distance (typically 1 m)
  • X_sigma = Gaussian random variable for shadowing (dB)

Path Loss Exponents by Environment:

Environment Path Loss Exponent (n) Interpretation
Free space n = 2.0 Ideal conditions
Urban cellular n = 2.7 to 3.5 Buildings, reflections
Indoor office n = 2.5 to 3.0 Cubicles, furniture
Indoor factory n = 2.0 to 3.0 Open floor vs machinery
Indoor residential n = 3.0 to 4.0 Walls, floors
Obstructed urban n = 4.0 to 6.0 Dense buildings, no LOS

Key Insight: Higher path loss exponent = signal degrades faster with distance.

Try It: Log-Distance Path Loss Calculator

43.9.1 Worked Example: Indoor Office Propagation at 2.4 GHz

Scenario: An ESP32 Wi-Fi sensor is deployed 25 meters from an access point inside an office building (n = 3.0). The path crosses two plasterboard walls. What is the expected received signal strength?

Given:

  • Frequency: 2.4 GHz
  • Distance: 25 m
  • Path loss exponent: n = 3.0 (indoor office)
  • Reference path loss at 1 m (L_0): 40 dB (typical for 2.4 GHz)
  • Wall attenuation: 4 dB per plasterboard wall, 2 walls = 8 dB
  • Shadowing margin (X_sigma): 6 dB (95% reliability)
  • Transmit power: 20 dBm

Step 1: Calculate log-distance path loss

\[L(25m) = 40 + 10 \times 3.0 \times \log_{10}(25/1) + 8 + 6\] \[L(25m) = 40 + 30 \times 1.398 + 14\] \[L(25m) = 40 + 41.9 + 14 = 95.9 \text{ dB}\]

Step 2: Calculate received power

\[P_{RX} = 20 - 95.9 = -75.9 \text{ dBm}\]

Step 3: Compare to receiver sensitivity

  • Wi-Fi receiver sensitivity: -90 dBm
  • Link margin: -75.9 - (-90) = 14.1 dB (adequate but thin)

Design implication: With only 14 dB of margin, adding a third wall (another 4 dB loss) and a person walking through the path (3-5 dB body loss) could reduce the margin to 5-7 dB, causing intermittent connectivity. For reliable coverage, either reduce the sensor-to-AP distance to 15 m or place the AP to avoid wall crossings.

Contrast with free-space prediction: Using the log-distance model with n=2.0, no walls, and no shadowing gives a path loss of only 68 dB and a received power of -48 dBm – a 28 dB overestimate that would lead to deploying sensors far beyond reliable range.

43.9.2 Why Does Frequency Affect Path Loss?

A common question is: why do higher frequencies lose more signal to free space? The signal energy does not actually change – the difference is in the antenna’s ability to capture it.

Radio waves spread spherically from the transmitter. At any distance, the signal energy is distributed over the surface of a sphere. A receiving antenna captures a fraction of that sphere proportional to its effective aperture (area). For a simple dipole antenna, the effective aperture is proportional to the square of the wavelength:

\[A_{eff} = \frac{\lambda^2}{4\pi}\]

At 915 MHz, the wavelength is 33 cm, giving an aperture of ~86 cm squared. At 2.4 GHz, the wavelength shrinks to 12.5 cm, giving an aperture of ~12 cm squared – 7x smaller. The same energy arrives at the receiver, but the antenna captures 7x less of it. This is why the FSPL formula includes the frequency term: it accounts for this aperture difference, not actual energy absorption by the air.

Practical consequence: Sub-GHz protocols (LoRa at 868/915 MHz, Sigfox at 868 MHz) achieve longer range than 2.4 GHz protocols (Wi-Fi, BLE, Zigbee) partly because their antennas capture a larger fraction of the arriving signal, not because higher-frequency signals lose energy faster in free space.

Line chart comparing path loss versus distance for three environments: free space with n=2.0 showing slowest signal degradation, indoor office with n=3.0 showing moderate degradation, and dense urban with n=5.0 showing rapid signal attenuation, illustrating how environment affects wireless range
Figure 43.2: Signal strength degradation comparing free space, indoor office, and dense urban environments

Common Pitfalls

FSPL models free-space propagation without obstacles. Indoor paths with walls, floors, and furniture have path loss exponents of 2.7–3.5, causing 10–20 dB more attenuation than FSPL predicts. Fix: use the log-distance path loss model with an empirically measured path loss exponent for indoor deployments.

dB is a dimensionless ratio (gain or loss). dBm is an absolute power level. Adding 20 dB gain to 30 dBm gives 50 dBm — not 50 dB. Fix: always track units explicitly; start with dBm, add/subtract dB gains and losses, finish with dBm received power.

Designers often assume that a higher-frequency radio is more capable, but higher frequency means more path loss at the same distance. Fix: calculate FSPL at your deployment frequency, not at a reference frequency, before comparing range specifications.

43.10 Summary

  • Free Space Path Loss (FSPL) provides the baseline for wireless range calculations using the formula: FSPL = 20log(d) + 20log(f) + 32.45 (d in km, f in MHz)
  • Higher frequencies experience more path loss – a 2.4 GHz signal loses ~8.4 dB more than a 915 MHz signal at the same distance
  • Path loss exponent (n) characterizes the environment: n=2 for free space, n=3-4 for indoor, n=4-6 for obstructed urban
  • Theoretical range far exceeds practical range – real deployments must account for environmental losses
  • Link budget analysis determines if a wireless connection will work by comparing received power to receiver sensitivity

43.11 What’s Next

Topic Chapter Description
Material Attenuation & RSSI Material Attenuation and RSSI How building materials (concrete, glass, metal) absorb and reflect signals, and how RSSI measurements are used for localization
Link Budget Design Link Budget Analysis Systematic calculation of transmit power, antenna gain, cable loss, and receiver sensitivity to determine coverage
Antenna Fundamentals Antenna Gain and Patterns How antenna gain, directivity, and radiation patterns affect wireless range and coverage planning
Multipath Fading Multipath and Fading Effects How reflections and interference create fading, and techniques such as MIMO and frequency diversity that mitigate it
Wi-Fi Range Planning Wi-Fi Deployment Planning Applying FSPL and log-distance models to plan access point placement for indoor Wi-Fi coverage
LoRaWAN Range & Spreading Factors LoRaWAN Link Budget Using spreading factor selection and link budget analysis to maximize LoRa coverage in rural and urban deployments