Understanding RF Math

In the world of wireless design, an engineer must strike a delicate balance between coverage and performance. A high-speed wireless network doesn’t happen by accident. It’s the outcome of careful design and consideration where the art and science of Radio Frequency (RF) mathematics come into play. This knowledge enables the optimal tuning of a network, ensuring that signals are clear, interference is minimised, and users experience consistent, high-quality connectivity.

In this post, I will discuss the fundamental concepts of RF math that play a vital role in the successful deployment and management of wireless networks and explore the units of measurement that underlie these principles.

Measuring RF Power
The Inverse Square Law
- Rule of 3s and 10s
  - Rule of 3s
  - Rule of 10s
Conclusion

Measuring RF Power

Measuring RF power is essential to wireless communication and network engineering. It crosses the boundaries of physics, engineering, and technology. Whether it’s for tuning the transmit power of a Wireless Access Point (AP) or assessing a wireless network, understanding how to measure RF power accurately is essential.

The units of measurement for measuring RF power can be categorised into two main types: actual units of power and relative units of comparison.

Actual units of power represent fixed values, something quantifiable and definite. For example, imagine a full water bottle that contains 500 millilitres. This is an actual value that you can touch and measure.

Relative units, on the other hand, represent a comparison between two similar values. If you had another bottle and said it was half the size of the full bottle, you’d be giving a comparative measurement. Knowing the actual size of one bottle lets you determine the size of the other.

Units of power give us firm values for assessing the transmit and receive amplitude of an AP, while units of comparison allow us to gauge the changes in AP power from one location to another or evaluate the effects of different components introduced to the system, e.g. antennas or antenna cables.

Units of Power

Watt

The International System of Units (SI) defines a Watt (W) as the basic unit of power, equal to one joule per second. This unit quantifies the rate of energy transfer or energy conversion. In the context of Wi-Fi, you can use it to measure the strength of the signal an Access Point (AP) transmits.

If an AP is transmitting a signal at 1 Watt at a specific voltage (e.g. 12 Volts), it uses 1 joule of energy every second to send that signal. Volts describe the “pressure” or “force” pushing the energy through, and the Watts tell you how fast that energy is being used. Together, they give you a complete picture of how the transmitter in the AP works.

Milliwatt

A milliwatt (mW) is a unit of power that is 1/1,000th of a Watt. This is a vital measurement in Wi-Fi equipment since most APs and client devices transmit power levels between 1 mW and 100 mW. Transmit powers of 1W and above are usually only permitted for outdoor use, such as wirelessly connecting buildings or providing wide-area coverage. Outdoor Wi-Fi often requires higher power levels to overcome distances and obstacles like trees or buildings. However, such high power levels come with regulatory constraints and considerations to prevent interference with other systems, so understanding these rules and how to comply with them is essential for anyone working with outdoor Wi-Fi deployments.

Wi-Fi vendors usually allow you to adjust an AP’s transmit power settings. Some vendors express the transmit power as the intentional radiator (IR), while others might use equivalent isotropically radiated power (EIRP). Some vendors express this using mW or dBm, while others represent it as a percentage of the maximum transmit power for a particular AP model.

Units of Comparison

Decibels

Decibels (dB) measure the power ratio between two values using a logarithmic scale to express significant variations of power in a more manageable form. People commonly use this concept in wireless networking to indicate differences in signal strength.

We often use decibels to quantify the strength of signals that a transmitter generates, as well as the gain or loss of antennas and cables and the ratio between different signal levels. Because decibels offer a convenient way to express large ranges of values on a manageable scale, they are popular in wireless networking.

The term “decibel” is derived from two components:

Deci – a prefix that comes from the Latin word “decimus”, meaning tenth. In the metric system, “deci” represents a factor of one-tenth or 0.1.
Bel – named after Alexander Graham Bell, the inventor of the telephone. The “bel” serves as a measurement unit for denoting the ratio between two power levels, calculated as the base-10 logarithm of this ratio.

Using decibels (dB) to measure wireless signals is advantageous due to the logarithmic nature of the dB scale, enabling a compact representation of the wide range of signal strengths encountered in wireless communication.

Consider an example where an AP emits a signal with an output power of 50 mW while a mobile device further away receives the signal at a greatly reduced power of just 1 mW. This drop in power is due to various factors like distance and obstacles, and the total power loss is a significant 49 mW. While saying “we lost 49 mW” is accurate, this doesn’t readily convey how significant this loss is relative to the initial power. It’s a simple subtraction that leaves you calculating ratios or percentages to understand the true impact.

However, by using the logarithmic dB scale, you can quantify this power loss more intuitively:

$\text{Power Loss (dB)} = 10 \times \log_{10} \left( \frac{\text{Initial Power}}{\text{Final Power}} \right)$

In this case, with an initial power of 50 mW and a final power of 1 mW, the power loss in dB becomes:

$\text{Power Loss (dB)} = 10 \times \log_{10} \left( \frac{50}{1} \right)$

$\text{Power Loss (dB)} = 10 \times 1.69897 = 16.9897 \, \text{dB}$

The power loss is approximately 16.99 dB, meaning the received power at the mobile device is about 17 dB lower than the original 50 mW from the AP. This dB figure instantly tells a network engineer how significant the loss is without needing to calculate percentages or ratios. It simplifies understanding and comparisons, demonstrating why decibels are helpful in wireless communications.”

Decibels Relative to an Isotropic Radiator (dBi)

dBi is a unit of measurement used to express an antenna’s gain or directional efficiency compared to an isotropic radiator – a theoretical device that radiates energy in all directions equally. Imagine an isotropic radiator as a light bulb that emits light “evenly in all directions”. The dBi value quantifies how well an antenna focuses its energy in a specific direction compared to this ‘light bulb.’ An isotropic radiator’s gain is considered to be 0 dBi because it doesn’t focus energy in any particular direction over another.

So when an antenna has a gain of, say, 6 dBi, it’s like using a flashlight that focuses light (or, in our case, radio waves) 6 decibels more effectively in a specific direction than the isotropic ‘light bulb.’ It also means that the antenna adds 6 dBi of gain to the transmitting signal of an AP. This is particularly useful in scenarios where you want to focus your wireless signal to cover a specific area or reach a particular distance, thus improving the performance of your wireless network.

Antennas are designed with specific patterns that determine how they distribute energy in space. Some antennas focus energy more in one direction, providing higher gain in that direction. Other antennas might have a more omnidirectional pattern, providing equal gain in all directions.

Decibels Relative to a Half-Wave Dipole Antenna (dBd)

dBd measures the gain or directional efficiency of an antenna in comparison to a half-wave dipole antenna, which is a common reference antenna in the world of Wi-Fi.

A half-wave dipole antenna is a simple antenna design that is often used as a baseline for comparison due to its popularity and standardised characteristics. It’s a reference antenna with known gain properties (2.14dBi), particularly in terms of its radiation pattern and efficiency.

When an antenna is rated in dBd, it means its gain is being measured and compared against the performance of a half-wave dipole antenna. For example, if an antenna has a gain of 3 dBd, it has 3 dB more gain than a half-wave dipole, which is equivalent to a gain of 5.14 dBi (2.14 dBi + 3 dB). The dBd value indicates how much better the antenna performs compared to the reference dipole antenna.

dBd is more common in other radio and broadcasting applications, especially in professional and ham radio contexts. It is rare to use dBd for measuring 802.11-based networks, likely because the industry has largely standardized around the use of dBi for antenna measurements.

Decibels Relative to 1 Milliwatt (dBm)

dBm is a unit of measurement that expresses power levels in decibels relative to a fixed reference point of 1 mW, specifying an absolute power level. Engineers commonly use it in wireless networking to measure the power output of devices like APs and client devices.

dBm offers a convenient way to describe power levels because it uses a logarithmic scale that easily compares different power levels. For example, an AP transmitting at a power level of 1 milliwatt is expressed as 0 dBm. If you increase the power level to 10 milliwatts, you would express it as 10 dBm; if you decrease it to 0.1 milliwatts, it becomes -10 dBm.

A positive dBm value represents a power level higher than 1 mW, while a negative dBm value signifies a power level lower than 1 mW.

Why 1 mW? The choice of 1 mW as a reference point has historical and practical reasons:

Ease of Calculations: Decibels simplify the comparison of ratios, and a fixed reference point makes calculations easier, allowing for straightforward comparison and conversion of power levels.
Historical Convention: Early phone networks and telecommunications systems often quantified signal strength in milliwatts, leading to the widespread use of dBm across various disciplines.
Range of Usable Powers: 1 mW sits approximately in the middle of the power range encountered in telecommunications and radio frequency (RF) engineering. This makes it a convenient unit for describing both very low and very high power levels using reasonable numbers.
Standardization: A universally accepted measurement unit simplifies system design and ensures better interoperation.

Converting Watts to Decibels

In the realm of wireless networking, you’ll encounter transmit power levels specified either in Watts or dBm. Seamlessly converting between these two units is essential for accurately comparing and integrating different systems.

To convert a power level from watts (W) to dBm, you can use the formula:

$\text{dBm} = 10 \times \log_{10} \left( \text{Power in watts} \times 1000 \right)$

If you have a transmitter emitting at 2 watts, you’d convert it to dBm as follows:

$\text{dBm} = 10 \times \log_{10} \left( 2 \times 1000 \right) = 33\text{dBm}$

Note that the factor of 1000 is used to convert watts to milliwatts, as 1 W = 1000 mW.

To go the other way and convert a power level from dBm to watts, you can use:

$\text{Power in watts} = 10^{\left( \frac{\text{dBm}}{10} \right)} \times 0.001$

Here, the factor of 0.001 converts from mW to W.

If you have a signal strength of 30 dBm and want to find out what that is in watts:

$\text{Power in watts} = 10^{\left( \frac{30}{10} \right)} \times 0.001 = 1\text{W}$

Check out my dBm to Watt conversion table, which is a valuable tool for quickly converting between these two units without needing to perform the calculations manually.

The Inverse Square Law

In simple terms, the Inverse Square Law is a broader principle that explains how the intensity of a signal (light, sound, radio waves, and even gravitational forces) diminishes with the square of the distance from the source. In Wi-Fi, this means that as you move away from an AP or client device, the strength of the signal will drop significantly. This provides a foundational understanding of why Wi-Fi signals get weaker the further you are from the source, an essential insight for network engineers who must design and optimize wireless networks for various environments.

Let’s say you have an AP transmitting at a power $P$ of 1 Watt (or 1000 mW, or 30 dBm), and you want to know how the intensity $I$ changes as you move away from it. Using the Inverse Square Law:

At a distance $r$ of 1 meter, the intensity $I$ would be:

$I = \frac{1}{4\pi(1)^2} = \frac{1}{12.57} = 0.0796 \, \text{W/m}^2$

At a distance $r$ of 2 meters, the intensity $I$ would be:

$I = \frac{1}{4\pi(2)^2} = \frac{1}{50.27} \approx 0.0199 \, \text{W/m}^2$

You’ll notice that the intensity at 2 meters is exactly one-quarter the intensity at 1 meter, which illustrates the Inverse Square Law. In dB terms, this would be a drop of about 6 dB when you double the distance, assuming free-space conditions. It provides a foundational understanding of why Wi-Fi signals get weaker the further you are from the source.

However, while the Inverse Square Law is a useful general rule, wireless networking often involves more complex scenarios. This leads us to a more nuanced formula specifically tailored for telecommunications engineering: Free Space Path Loss (FSPL). Free Space Path Loss is a formula tailored to predict a radio signal’s power loss as it travels over a distance through free space, taking into account its specific frequency (without reflecting or refracting).

Rule of 3s and 10s

The “Rule of 10s and 3s” is a rule of thumb often used in Wi-Fi engineering to estimate signal strength changes due to distance variations. This rule applies to both signal loss and gain:

Rule of 3s

For every 3 dB gain, the output power doubles.
For every 3 dB loss, the output power halves.

Rule of 10s

A 10 dB gain multiplies the output power by a factor of 10.
A 10 dB loss divides the output power by a factor of 10.

Examples

3 dB Gain: When an AP (Access Point) transmits at 100 mW and you add an antenna with a 3 dBi gain, the EIRP doubles to 200 mW.
3 dB Loss: On the other hand, if you connect that same 100 mW AP to a cable causing 3 dB of loss, the signal power at the end of the cable drops to 50 mW, halving the original power.
10 dB Gain: Adding an antenna with a 10 dBi gain to an AP transmitting at 40 mW increases the EIRP to 400 mW, multiplying the original 40 mW by 10.
10 dB Loss: Connecting a 40 mW AP to a cable that causes a 10 dB loss reduces the signal power at the end to 4 mW, dividing the original 40 mW by 10.

Remember, in this context, dBm is a unit of absolute power, and dB represents relative change. For instance, a +10 dBm signal that gains 3 dB would result in a +13 dBm signal.

Conclusion

Whether you’re setting up a home Wi-Fi network or managing a large-scale enterprise system, the concepts discussed in this blog serve as essential tools for troubleshooting and optimizing wireless performance. Remember, the world of wireless communication is filled with variables, and having a strong foundational knowledge allows you to adapt and overcome challenges more effectively.

For those keen to explore further, there are many excellent books, online courses, and forums dedicated to wireless networking. Mastering the basics covered in this blog will serve as a solid stepping stone for your future endeavors in this field.

Thank you for reading, and feel free to share this blog with anyone who might find it useful. If you have any questions or would like to discuss specific challenges you’re facing, leave a comment below. Stay connected!