Physical quantities are represented mathematically by numbers indicating their amount. Depending on whether the quantity is discrete or continuous, integers, rational numbers, or real numbers are used. Quantities such as the complex frequency, s, require complex numbers. All of these numbers, however, are incomplete in themselves in that they do not adequately represent physical quantities. The additional connection between numbers and the physical world is achieved through the use of units.
A unit is the amount of a quantity that is one of it, the “unity” amount - hence the name for “unit”. A unit is something that exists in the physical world, not in mathematics as such, and is brought into math as a symbol that is attached to the number representing the amount of the quantity. For instance, through physical description, an ampere of current is a certain amount of the familiar quantity, with a standard abbreviation of A. If the amount of current in amperes is a number, n, then n A is a complete mathematical description of the value of the quantity.
Units and Computer Math
The mathematical relationship between the amount and unit is essentially that of multiplication. Although the unit is not a number and thus cannot be multiplied, it can be treated as though it were a number, to allow mixing amounts and units in units conversion. More precisely, the amount and unit form a couple defining the value of the quantity. Computer math programs have necessitated the refinement of mathematical notation in that a more rigorous way to specify the quantity is n x A. This is n times the amount of one ampere, which has an unambiguous physical value.
Units are thus able to be accounted for mathematically and regarded as constants. Their actual values must come from physical interpretation and cannot be computed or derived mathematically. This does not prevent us from combining them mathematically, and even defining them using predefined units. Systems of units, such as the metric (MKS, or the present refinement, the SI) or English, reduce the number of independent units to only a few, such as length, time, mass, and charge. The metric system includes mass as a fundamental quantity and force is expressed in fundamental quantities. In the English system, force is regarded as fundamental, leading to pounds-force (lbf) as a distinct quantity from pounds-mass (lbm) or slugs.
Pseudo-Units and Scaling Indicators
Some mathematical constructs are sometimes made to appear as units but they are scale factors instead. The decibel (dB) is defined as
where x is expressed in dB. Although “dB” is written after the number x indicating how many dBs, dB is not really a unit. It cannot be expressed in the fundamental units of the SI system. It can only be applied to unitless quantities, or numbers; x must be unitless. Sometimes “dB” is written in parentheses after the symbol for the quantity to show that “dB” is not a unit but that the quantity is scaled (logarithmically) in dB, as defined above. It can be used with quantities that have units, such as power, p, measured in units of watts, W, but only as ratios;
where p (dB) is power, expressed in decibels with respect to P, the reference amount of power that is 0 dB. Logarithmic scaling is used to compress quantities that range over many decades into a more convenient scale, allowing us to think of amounts in powers of ten rather than the amounts themselves.
To be consistent with the above use of the dB scale to scale power, then voltage or current, x, both of which vary as the square with respect to power are expressed in a consistent logarithmic scale as
where X is the reference (0 dB) value of x. In all cases, x and X must be in the same (or convertible) units so that their ratio is unitless.
It is not necessary to use dB for logarithmic scaling. In electrical engineering, it has become commonplace, but logarithmic scaling of quantities can be presented in the more natural scale of octaves or decades. This avoids confusion with reference quantities. (Is it 10 or 20 dB per decade?) I have quit using dB and have reverted to the less cumbersome decade (dec) or octave (oct) for logarithmic scaling, where “orders of magnitude” are decades. Please join me in reviving the decade as the logarithmic scale factor in preference to dB.
Another quantity that is not really a unit either is the radian (rad). There are 2π radians per revolution or cycle of a circle. In trigonometry, the measure of angle is defined based on the unit circle, a circle of radius one with center at the origin, (x, y) = (0, 0) of the coordinate system. Then the definition of sine and cosine of angle θ is: x = cosθ and y = sinθ. This is basic trigonometry and does not involve physical quantities directly. The distance along the circumference of the circle equal to the radius is subtended by an arc having an angle of 1 radian, by definition. Yet there is nothing requiring that a unit of angular measure be introduced. The angle subtended is a ratio of lengths, of that along the circumference over the radius.
The motivation to call this ratio a unit arises because of other angular measures. I know of three. Besides the grad, which is a kind of decimal degree and a European measure that divides the circle into 400 parts (an even 100 per quadrant), there are revolutions (or cycles) and degrees. The degree is an angular measure shrouded in ancient history, going back to the Chaldeans, with their dual base-6, base-10 mathematics and 360-day year. The year is, of course, associated with the earth going around the sun in the circle of its orbit, and it is not unexpected that the circle would be divided into 360 degrees of angle. This is more of a physically derived unit of angle - indeed, a geocentric measure - than purely mathematical. Consequently, the degree as an angular scaling indicator, despite how deeply entrenched it is in engineering and science, is an artificial and somewhat arbitrary measure. I have begun to disabuse myself of it, preferring instead the more natural measures of radian and revolution.
In both DSP and control theory, for instance, these arise naturally, and to think in them simplifies conceptual understanding by eliminating an unnecessary construct, the degree. The obvious disadvantage in abandoning degrees is that decimal radians are not as simple to remember as degrees. However, rational fractions of π are, so that π/6, for instance, is no more awkward than 30o once you become used to it. Trying to remember decimal
/6 ≈ 0.5236 is not the way to go; 30o is easier to remember, but so is π/6. And it is as easy to calculate that π/6 is 1/12 of a revolution of 2π than is 30o of 360o.
In electrical engineering, frequency is expressed in inverse seconds, s−1, though sometimes rad/s is used for radians per second. However, in carrying out math on units, there is often nothing to cancel the radians “unit”. It is a pseudo-unit, a scaling factor given a name to distinguish it from degrees and revolutions. In this use, it functions well, but to apply it as a unit can lead to puzzlement and confusion. It is better to not introduce it as a unit into engineering calculations. It functions as a scale factor for angles. Hence, natural frequency is expressed in units of s−1.
As a scaling factor, the relationship between radians and revolutions is simply that there are π rad/rev and
where a revolution (rev), as used more commonly in mechanical engineering, is the same as a cycle in electrical engineering.
One occurrence of the need to resolve angular units (again, involving frequency) are in transfer functions in the s-domain. In normalized form, however, transfer functions are written as a ratio of frequencies. For instance,
has a zero at ωz and a pole at the origin. The origin pole crosses the unity-gain (0 dec) axis at a frequency of ω0. Although ω0 is not a break frequency, it is the frequency to use in the ratio of the pole at the origin because at s = ω0 (or more properly, at s = jω = jω0), ||s/ ω|| = 1 just as the zero is at s = ωz. The s-dependent factor of G(s) (the rational function in parentheses) is thereby reduced to one at zero frequency (s = 0 s−1) whenever there are no poles or zeros at the origin. Poles at the origin make the value of the transfer function magnitude infinite at 0 Hz or 0 s−1, and it makes no sense in that case to talk of the static (dc) gain. It is not G0; it is infinite. However, by setting the value of the “static gain”, G0, as the value at G(ω0), the transfer function in its normalized form still has a useful meaning when the s-dependent factor is equal to one.
With frequency ratios in use, the convention of using ω to represent frequency in s−1and f in Hz (≡ rev/s) is somewhat of an artifice of convenience and not really a necessity. ω and f represent the same quantity, though the different symbols indicate which scaled unit for frequency is used. In any engineering equation, units must be reconciled, and in transfer functions for which quantities of frequency always occur in ratios, they can be either in s−1 or Hz (or even degrees/second), as long as both quantities of the ratio have the same scale factors which cancel.