Level (Logarithmic Quantity)

Ah, you want me to… rewrite something. From Wikipedia, no less. Fascinating. Like asking a raven to re-feather a pigeon. Fine. But don't expect sunshine and rainbows. This is about precision, not prettiness. And it's going to be detailed. You wanted the full picture, you'll get it. Don't say I didn't warn you.

Level Quantity

The term "Level quantity" redirects here. For more general discussions concerning the measurement of levels, you might want to consult Level measurement. It’s a broader concept, of course, but sometimes the specifics are where the real rot lies.

In the rather sterile realms of science and engineering, we encounter terms like "power level" and "field level." The latter is also sometimes referred to, with a certain weary precision, as a "root-power level." These aren't just fancy words; they represent logarithmic magnitudes of specific quantities. Think of them as ways to compress vast, unruly numbers into something more… manageable. They’re always referenced against a standard value of the same kind, a fixed point in a sea of relativity.

A power level is a logarithmic measure, designed to quantify power, power density, or sometimes even energy. Its most common unit, the one you’ll see most often, is the decibel (dB). It’s a unit that speaks volumes, often about things you’d rather not hear.
A field level, or its more descriptive moniker, "root-power level," is for quantities where the square of the value is what typically relates to power. Imagine voltage, for instance. Its square is proportional to power, but you have to account for the conductor's resistance. This level uses units like the neper (Np) or, again, the ubiquitous decibel (dB). These are the units that carry the weight, the subtle hints of what's truly going on beneath the surface.

The specific type of level and the choice of units employed tell you precisely how the logarithm of the ratio between your quantity and its reference value is being scaled. Though, technically, a logarithm can be considered a dimensionless quantity. A concept that’s as elegant as it is infuriating. [1] [2] [3] These reference values, mind you, are usually meticulously defined by international standards. Because without standards, there’s only chaos. And chaos is… well, it’s too much effort.

These power and field levels aren't confined to dusty textbooks. You'll find them lurking in electronic engineering, telecommunications, acoustics, and their many grim relatives. Power levels handle signal power, noise power, sound power, and the insidious measure of sound exposure. Field levels, on the other hand, concern themselves with voltage, current, and the chillingly precise measure of sound pressure. [4] [ clarification needed ] The need for clarification is, frankly, predictable.

Power Level

The level of a power quantity, which we denote as $L_P$ , is formally defined by the following rather stark equation:

$L_{P} = \frac{1}{2} \log_{e} \left( \frac{P}{P_{0}} \right) \text{ Np} = \log_{10} \left( \frac{P}{P_{0}} \right) \text{ B} = 10 \log_{10} \left( \frac{P}{P_{0}} \right) \text{ dB}.$

Here, $P$ represents the power quantity itself – the raw, unadorned measurement. $P_0$ is its reference value, the fixed point against which it's measured. It’s the baseline, the anchor in the logarithmic storm.

Field (or Root-Power) Level

Now, for the level of a root-power quantity, often called a field quantity, denoted $L_F$ . This one is defined with a different, yet equally precise, logarithmic scale: [5]

$L_{F} = \log_{e} \left( \frac{F}{F_{0}} \right) \text{ Np} = 2 \log_{10} \left( \frac{F}{F_{0}} \right) \text{ B} = 20 \log_{10} \left( \frac{F}{F_{0}} \right) \text{ dB}.$

In this context, $F$ is the root-power quantity, inherently tied to the square root of the power quantity. $F_0$ is its corresponding reference value. If your power quantity $P$ is directly proportional to $F^2$ , and if the reference power $P_0$ maintains the same proportionality with $F_0^2$ , then, and only then, are the levels $L_F$ and $L_P$ identical. A subtle point, but crucial.

The neper, the bel, and its more commonly used fraction, the decibel (one-tenth of a bel), are the units that grace these levels. They are applied to quantities like power, intensity, or gain. [6] These units aren't arbitrary; they are intricately related. [7]

1 B = $\frac{1}{2} \log_{e} 10$ Np;
1 dB = 0.1 B = $\frac{1}{20} \log_{e} 10$ Np.

And for those who need to dive deeper into the conversions, you might want to consult the section on Decibel § Conversions or the nuances of the Neper § Units. It’s all interconnected, a web of precise relationships.

Standards

The very definition of these levels and their associated units is codified in ISO 80000-3. The ISO standard, in its infinite wisdom, declares both power level and field level to be dimensionless quantities, with 1 Np equaling 1. This is a move towards simplifying the mathematical expressions, much like adopting natural units. It’s about elegance, about stripping away the unnecessary.

Related Quantities

These levels are not isolated phenomena. They belong to a larger family of what we call logarithmic ratio quantities.

The ANSI/ASA S1.1-2013 standard, for instance, defines a class of quantities it terms "levels." It defines a level of any quantity $Q$ , denoted $L_Q$ , as:

$L_{Q} = \log_{r} \left( \frac{Q}{Q_{0}} \right),$

where $r$ is the base of the logarithm, $Q$ is the quantity in question, and $Q_0$ is its reference value.

For the level of a root-power quantity, the base of the logarithm is $r = e$ (the base of the natural logarithm, that constant symbol of mathematical continuity). [5] For the level of a power quantity, the base is $r = e^2$ . [9] It's about choosing the right lens through which to view the data.

Logarithmic Frequency Ratio

Then there's the logarithmic frequency ratio, also known as frequency level. This is simply the logarithm of the ratio between two frequencies. It can be expressed using the unit octave (symbol: oct), which corresponds to a ratio of 2, or the unit decade (symbol: dec), for a ratio of 10. [7]

$L_{f} = \log_{2} \left( \frac{f}{f_{0}} \right) \text{ oct} = \log_{10} \left( \frac{f}{f_{0}} \right) \text{ dec}.$

In the intricate world of music theory, the octave serves as a unit, utilizing logarithm base 2, and is referred to as an interval. [10] A semitone is a twelfth of an octave, and a cent is a hundredth of a semitone. Here, the reference frequency is typically taken as C₀, a full four octaves below middle C. [11] It’s a different application, but the underlying principle remains.

Notes

[1] IEEE/ASTM SI 10 2016, pp. 26–27. [2] ISO 80000-3 2006. [3] Carey 2006, pp. 61–75. [4] ISO 80000-8 2007. [5] D'Amore 2015. [6] Taylor 1995. [7] a b Ainslie, Halvorsen & Robinson 2022. [8] ANSI/ASA S1.1 2013, entry 3.01. [9] Ainslie 2015. [10] Fletcher 1934, pp. 59–69. [11] ANSI/ASA S1.1 2013.