Section 3

Fundamentals and Units of Measurement

Glen Ballou

3.1. UNITS OF MEASUREMENT

Measurements are the method we use to define all things in life. A dimension is any measurable extent such as length, thickness, or weight. A measurement system is any group of related unit names that state the quantity of properties for the items we see, taste, hear, smell, or touch.

A unit of measurement is the size of a quantity in the terms of which that quantity is measured or expressed, for instance, inches, miles, centimeters, and meters.

The laws of physics, which includes sound, are defined through dimensional equations that are defined from their units of measurements of mass, length, and time. For instance,

A Sound Engineer's Guide to Audio Test and Measurement
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where,

L is length,

W is width,

D is distance,

T is time.

A physical quantity is specified by a number and a unit: for instance, 16 ft or 5 m.

3.1.1 Si System

The SI system (from the French Système International d'Unités) is the accepted international modernized metric system of measurement. It is used worldwide with the exception of a few countries including the United States of America.

The SI system has the following advantages:

1. It is internationally accepted.

2. All values, except time, are decimal multiples or submultiples of the basic unit.

3. It is easy to use.

4. It is easy to teach.

5. It improves international trade and understanding.

6. It is coherent. All derived units are formed by multiplying and dividing other units without introducing any numerical conversion factor except one.

7. It is consistent. Each physical quantity has only one primary unit associated with it.

When using the SI system, exponents or symbol prefixes are commonly used. Table 3.1 is a chart of the accepted name of the number, its exponential form, symbol, and prefix name. (Note: because of their size, the numbers from sextillion to centillion have not been shown in numerical form and symbols and prefix names have not been established for these numbers.)

3.1.2 Fundamental Quantities

There are seven fundamental quantities in physics: length, mass, time, intensity of electric current, temperature, luminous intensity, and molecular substance. Two supplementary quantities are plane angle and solid angle.

3.1.3 Derived Quantities

Derived quantities are those defined in terms of the seven fundamental quantities, for instance, speed = length/time. There are sixteen derived quantities with names of their own: energy (work, quantity of heat), force, pressure, power, electric charge, electric potential difference (voltage), electric resistance, electric conductance, electric capacitance, electric inductance, frequency, magnetic flux, magnetic flux density, luminous flux, illuminance, and customary temperature. Following are thirteen additional derived quantities that carry the units of the original units that are combined. They are area, volume, density, velocity, acceleration, angular velocity, angular acceleration, kinematic viscosity, dynamic viscosity, electric field strength, magnetomotive force, magnetic field strength, and luminance.

3.1.4 Definition of the Quantities

The quantities will be defined in SI units, and their U.S. customary unit equivalent values will also be given.

Length (L). Length is the measure of how long something is from end to end. The meter (abbreviated m) is the SI unit of length. (Note: in the United States the spelling “meter” is retained, while most other countries use the spelling “metre.”) The meter is the 1 650 763.73 wavelengths, in vacuum, of the radiation corresponding to the unperturbed transition between energy level 2P₁₀ and 5D5 of the krypton-86 atom. The result is an orange-red line with a wavelength of 6 057.802 × 10^–10 meters. The meter is equivalent to 39.370 079 inches.

Mass (M). Mass is the measure of the inertia of a particle. The mass of a body is defined by the equation:

where:

A_s is the acceleration of the standard mass M_s,

a is the acceleration of the unknown mass, M, when the two bodies interact.

The kilogram (kg) is the unit of mass. This is the only base or derived unit in the SI system that contains a prefix. Multiples are formed by attaching prefixes to the word gram. Small masses may be described in grams (g) or milligrams (mg) and large masses in megagrams. Note the term tonnes is sometimes used for the metric ton or megagram, but this term is not recommended.

The present international definition of the kilogram is the mass of a special cylinder of platinum iridium alloy maintained at the International Bureau of Weights and Measures, Sevres, France. One kilogram is equal to 2.204 622 6 avoirdupois pounds (lb). A liter of pure water at standard temperature and pressure has a mass of 1 kg ± one part in 10⁴.

Mass of a body is often revealed by its weight, which the gravitational attraction of the earth gives to that body.

If a mass is weighed on the moon, its mass would be the same as on earth, but its weight would be less due to the small amount of gravity.

where,

W is the weight,

g is the acceleration due to gravity.

Time (t). Time is the period between two events or the point or period during which something exists, happens, etc.

The second (s) is the unit of time. Time is the one dimension that does not have powers of ten multipliers in the SI system. Short periods of time can be described in milliseconds (ms) and microseconds (µs). Longer periods of time are expressed in minutes (1 min = 60 s) and hours (1 h = 3600 s). Still longer periods of time are the day, week, month, and year. The present international definition of the second is the time duration of 9, 192, 631, 770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the atom of caesium 133. It is also defined as 1/86, 400 of the mean solar day.

Current (I). Current is the rate of flow of electrons. The ampere (A) is the unit of measure for current. Small currents are measured in milliamperes (mA) and microamperes (µA), and large currents are in kiloamperes (kA). The international definition of the ampere is the constant current that, if maintained in two straight parallel conductors of infinite length and negligible cross-sectional area and placed exactly 1 m apart in a vacuum, will produce between them a force of 2 × 10^–7 N/m2 of length.

A simple definition of one ampere of current is the intensity of current flow through a 1 ohm resistance under a pressure of 1 volt of potential difference.

Temperature (T). Temperature is the degree of hotness or coldness of anything. The kelvin (K) is the unit of temperature. The kelvin is 1/273.16 of the thermodynamic temperature of the triple point of pure water. Note: the term degree (°) is not used with the term kelvin as it is with other temperature scales.

Ordinary temperature measurements are made with the celsius scale on which water freezes at 0°C and boils at 100°C. A change of 1°C is equal to a change of 1 kelvin, therefore 0°C = 273.15 K: 0°C = 32°F.

Luminous Intensity (I_L). Luminous intensity is the luminous flux emitted per unit solid angle by a point source in a given direction. The candela (cd) is the unit of luminous intensity. One candela will produce a luminous flux of 1 lumen within a solid angle of 1 steradian.

The international definition of the candela is the luminous intensity, perpendicular to the surface, of 1/600 000 m² of a black body at the temperature of freezing platinum under a pressure of 101 325 N/m² (pascals).

Molecular Substance (n). Molecular substance is the amount of substance of a system that contains as many elementary entities as there are atoms in 0.012 kg of carbon 12.

The mole is the unit of molecular substance. One mole of any substance is the gram molecular weight of the material. For example, 1 mole of water (H₂O) weighs 18.016 g.

Plane Angle (α). The plane angle is formed between two straight lines or surfaces that meet. The radian (rad) is the unit of plane angles. One radian is the angle formed between two radii of a circle and subtended by an arc whose length is equal to the radius. There are 2π radians in 360°.

Ordinary measurements are still made in degrees. The degree can be divided into minutes and seconds or into tenths and hundredths of a degree. For small angles, the latter is most useful.

Solid Angle (A). A solid angle subtends three dimensions. The solid angle is measured by the area, subtended (by projection) on a sphere of unit radius by the ratio of the area A, intercepted on a sphere of radius r to the square of the radius (A/r²).

The steradian (sr) is the unit of solid angle. The steradian is the solid angle at the center of a sphere that subtends an area on the spherical surface, which is equal to that of a square whose sides are equal to the radius of the sphere.

Energy (E). Energy is the property of a system that is a measure of its ability to do work. There are two main forms of energy— potential energy and kinetic energy.

1. Potential energy (U) is the energy possessed by a body or system by virtue of position and is equal to the work done in changing the system from some standard configuration to its present state. Potential energy is calculated with the equation:

2. Kinetic energy (T) is the energy possessed by virtue of motion and is equal to the work that would be required to bring the body to rest. A body undergoing translational motion with velocity, v, has a kinetic energy given by:

The joule (J) is the unit of energy. The mechanical definition is the work done when the force of 1 newton is applied for a distance of 1 m in the direction of its application, or 1 Nm. The electrical unit of energy is the kilowatt-hour (kWh), which is equal to 3.6 × 10⁶ J.

In physics, the unit of energy is the electron volt (eV), which is equal to (1.602 10 ± 0.000 07) × 10^–19 J.

Force (F). Force is any action that changes, or tends to change, a body's state of rest or uniform motion in a straight line.

The newton (N) is the unit of force and is that force which, when applied to a body having a mass of 1 kg, gives it an acceleration of 1 m/s². One newton equals 1 J/m, 1 kg(m)/s², 10⁵ dynes, and 0.224 809 lb force.

Pressure. Pressure is the force (in a fluid) exerted per unit area on an infinitesimal plane situated at the point. In a fluid at rest, the pressure at any point is the same in all directions. A fluid is any material substance which in static equilibrium cannot exert tangential force across a surface but can exert only pressure. Liquids and gases are fluids.

The pascal (Pa) is the unit of pressure. The pascal is equal to the newton per square meter (N/m²).

Power (W). Power is the rate at which energy is expended or work is done. The watt (W) is the unit of power and is the power that generates energy at the rate of 1 J/s.

Electric Charge (Q). Electric charge is the quantity of electricity or electrons that flows past a point in a period of time. The coulomb (C) is the unit of electric charge and is the quantity of electricity moved in 1 second by a current of 1 ampere. The coulomb is also defined as 6.24196 × 10¹⁸ electronic charges.

Electric Potential Difference (V). Often called electromotive force (emf) and voltage (V), electric potential difference is the line integral of the electric field strength between two points. The volt (V) is the unit of electric potential. The volt is the potential difference that will cause a current flow of 1 A between two points in a circuit when the power dissipated between those two points is 1 W.

A simpler definition would be to say a potential difference of 1 V will drive a current of 1 A through a resistance of 1 Ω.

Electric Resistance (R). Electric resistance is the property of conductors that, depending in their dimensions, material, and temperature, determines the current produced by a given difference of potential. It is also that property of a substance that impedes current and results in the dissipation of power in the form of heat.

The ohm (Ω) is the unit of resistance and is the resistance that will limit the current flow to 1 A when a potential difference of 1 V is applied to it.

Electric Conductance (G). Electric conductance is the reciprocal of resistance. The siemens (S) is the unit of electric conductance. A passive device that has a conductance of 1 S will allow a current flow of 1 A when 1 V potential is applied to it.

Electric Capacitance (C). Electric capacitance is the property of an isolated conductor or set of conductors and insulators to store electric charge. The farad (F) is the unit of electric capacitance and is defined as the capacitance that exhibits a potential difference of 1 V when it holds a charge of 1 C.

where:

C is the electric charge in coulombs,

V is the electric potential difference in volts,

A is the current in amperes,

S is the conductance in siemens.

Electric Inductance (L). Electric inductance is the property that opposes any change in the existing current. Inductance is only present when the current is changing. The henry (H) is the unit of inductance and is the inductance of a circuit in which an electro-motive force of 1 V is developed by a current change of 1 A/s.

Frequency (f). Frequency is the number of recurrences of a periodic phenomenon in a unit of time. The hertz (Hz) is the unit of frequency and is equal to one cycle per second, 1 Hz = 1 cps. Frequency is often measured in hertz (Hz), kilohertz (kHz), and megahertz (MHz).

Sound Intensity (W/m²). Sound intensity is the rate of flow of sound energy through a unit area normal to the direction of flow. For a sinusoidally varying sound wave the intensity I is related to the sound pressure p and the density β of the medium by

where:

c is the velocity of sound.

The watt per square meter (W/m²) is the unit of sound intensity.

Magnetic Flux (ø). Magnetic flux is a measure of the total size of a magnetic field. The weber (Wb) is the unit of magnetic flux, and is the amount of flux that produces an electromotive force of 1 V in a one-turn conductor as it reduces uniformly to zero in 1 s.

Magnetic Flux Density (ß). The magnetic flux density is the flux passing through the unit area of a magnetic field in the direction at right angles to the magnetic force. The vector product of the magnetic flux density and the current in a conductor gives the force per unit length of the conductor.

The tesla (T) is the unit of magnetic flux density and is defined as a density of 1 Wb/m².

Luminous Flux (Φ_v). Luminous flux is the rate of flow of radiant energy as evaluated by the luminous sensation that it produces. The lumen (lm) is the unit of luminous flux, which is the amount of luminous flux emitted by a uniform point source whose intensity is 1 steradian.

where:

cd is the luminous intensity in candelas,

sr is the solid angle in steradians.

Luminous Flux Density(E_v). The luminous flux density is the luminous flux incident on a given surface per unit area. It is sometimes called illumination or intensity of illumination. At any point on a surface, the illumination is given by:

The lux (lx) is the unit of luminous flux density, which is the density of radiant flux of lm/m²,

Displacement. Displacement is a change in position or the distance moved by a given particle of a system from its position of rest, when acted on by a disturbing force.

Speed/Velocity. Speed is the rate of increase of distance traveling by a body. Average speed is found by the equation:

where:

S is the speed,

l is the length or distance,

t is the time to travel.

Speed is a scalar quantity as it is not referenced to direction. Instantaneous speed = dl/dt. Velocity is the rate of increase of distance traversed by a body in a particular direction.

Velocity is a vector quantity as both speed and direction are indicated. The l/t can often be the same for the velocity and speed of an object. However, when speed is given, the direction of movement is not known. If a body describes a circular path and each successive equal distance along the path is described in equal times, the speed would be constant but the velocity would constantly change due to the change in direction.

Weight. Weight is the force exerted on a mass by the gravitational pull of the planet, star, moon, etc., that the mass is near. The weight experienced on earth is due to the earth's gravitational pull, which is 9.806 65 m/s², and causes an object to accelerate toward earth at a rate of 9.806 65 m/s² or 32 ft/s².

The weight of a mass M is M(g). If M is in kg and g in m/s2, the weight would be in newtons (N). Weight in the U.S. system is in pounds (lb).

Acceleration. Acceleration is the rate of change in velocity or the rate of increase or decrease in velocity with time. Acceleration is expressed in meters per second squared (m/s²), or ft/s² in the U.S. system.

Amplitude. Amplitude is the magnitude of variation in a changing quantity from its zero value. Amplitude should always be modified with adjectives such as peak, rms, maximum, instantaneous, etc.

Wavelength (M). In a periodic wave, the distance between two points of the corresponding phase of two consecutive cycles is the wavelength. Wavelength is related to the velocity of propagation (c) and frequency (f) by the equation:

The wavelength of a wave traveling in air at sea level and standard temperature and pressure (STP) is:

or

For instance, the length of a 1000 Hz wave would be 0.33 m, or 1.09 ft.

Phase. Phase is the fraction of the whole period that has elapsed, measured from a fixed datum. A sinusoidal quantity may be expressed as a rotating vector OA. When rotated a full 360 degrees, it represents a sine wave. At any position around the circle, OX is equal in length but said to be X degrees out of phase with OA.

It may also be stated that the phase difference between OA and OX is α. When particles in periodic motion due to the passage of a wave are moving in the same direction with the same relative displacement, they are said to be in phase. Particles in a wave front are in the same phase of vibration when the distance between consecutive wave fronts is equal to the wavelength. The phase difference of two particles at distances X₁ and X₂ is:

Periodic waves, having the same frequency and waveform, are said to be in phase if they reach corresponding amplitudes simultaneously.

Phase Angle. The angle between two vectors representing two periodic functions that have the same frequency is the phase angle. Phase angle can also be considered the difference, in degrees, between corresponding stages of the progress of two cycle operations.

Phase Difference (ø). Phase difference is the difference in electrical degrees or time, between two waves having the same frequency and referenced to the same point in time.

Phase Shift. Any change that occurs in the phase of one quantity or in the phase difference between two or more quantities is the phase shift.

Phase Velocity. The phase velocity is when a point of constant phase is propagated in a progressive sinusoidal wave.

Temperature. Temperature is the measure of the amount of coldness or hotness. While kelvin is the SI standard, temperature is commonly referenced as °C (degrees Celsius) or °F (degrees Fahrenheit).

The lower fixed point (the ice point) is the temperature of a mixture of pure ice and water exposed to the air at standard atmospheric pressure.

The upper fixed point (the steam point) is the temperature of steam from pure water boiling at standard atmospheric pressure.

In the Celsius scale, named after Anders Celsius (1701–1744) and originally called Centigrade, the fixed points are 0°C and 100°C. This scale is used in the SI system.

The Fahrenheit scale, named after Gabriel Daniel Fahrenheit in 1714, has the fixed points at 32°F and 212°F.

To interchange between °C and °F, use the following equations:

The absolute temperature scale operates from absolute zero of temperature. Absolute zero is the point where a body cannot be further cooled because all the available thermal energy is extracted.

Absolute zero is 0 kelvin (0 K) or 0° Rankine (0°R). The Kelvin scale, named after Lord Kelvin (1850), is the standard in the SI system and is related to °C.

0°C = 273.15 K

The Rankine scale is related to the Fahrenheit system.

32°F = 459.67°R

The velocity of sound is affected by temperature. As the temperature increases, the velocity increases. The approximate formula is

where:

T is the temperature in °C.

where:

T is the temperature in °F.

Another simpler equation to determine the velocity of sound is:

Things that can affect the speed of sound are the sound wave going through a temperature barrier or going through a stream of air such as from an air conditioner. In either case, the wave is deflected the same way that light is refracted in glass.

Pressure and altitude do not affect the speed of sound because at sea level the molecules bombard each other, slowing down their speed. At upper altitudes they are farther apart so they do not bombard each other as often, so they reach their destination at the same time.

Thevenin's Theorem. Thevenin's Theorem is a method used for reducing complicated networks to a simple circuit consisting of a voltage source and a series impedance. The theorem is applicable to both ac and dc circuits under steady-state conditions.

The theorem states: the current in a terminating impedance connected to any network is the same as if the network were replaced by a generator with a voltage equal to the open-circuit voltage of the network, and whose impedance is the impedance seen by the termination looking back into the network. All generators in the network are replaced with impedance equal to the internal impedances of the generators.

Kirchhoff's Laws. The laws of Kirchhoff can be used for both dc and ac circuits. When used in ac analysis, phase must also be taken into consideration.

Kirchhoff's Voltage Law (KVL). Kirchhoff's voltage law states that the sum of the branch voltages for any closed loop is zero at any time. Stated another way, for any closed loop, the sum of the voltage drops equal the sum of the voltage rises at any time.

FIGURE 3.1 Kirchhoff's voltage law.

In the laws of Kirchhoff, individual electric circuit elements are connected according to some wiring plan or schematic. In any closed loop, the voltage drops must be equal to the voltage rises. For example, in the dc circuit of Fig.3.1, V₁ is the voltage source or rise such as a battery and V₂, V₃, V₄, and V₅ are voltage drops (possibly across resistors) so:

or:

In an ac circuit, phase must be taken into consideration, therefore, the voltage would be:

where:

Kirchhoff's Current Law (KCL). Kirchhoff's current law states that the sum of the branch currents leaving any node must equal the sum of the branch currents entering that node at any time.

Stated another way, the sum of all branch currents incident at any node is zero.

FIGURE 3.2 Kirchhoff's current law.

In Fig. 3.2 the connection on node current in a dc circuit is equal to 0 and is equal to the sum of currents I₁, I₂, I₃, I₄, and I₅ or:

or:

The current throughout the circuit is also a function of the current from the power source (V₁) and the current through all of the branch circuits.

In an ac circuit, phase must be taken into consideration, therefore, the current would be:

where:

e^jwt is cos At + jsin At or Euler's identity.

Ohm's Law. Ohm's Law states that the ratio of applied voltage to the resultant current is a constant at every instant and that this ratio is defined to be the resistance.

If the voltage is expressed in volts and the current in amperes, the resistance is expressed in ohms. In equation form it is:

or:

where:

e and i are instantaneous voltage and current,

V and I are constant voltage and current,

R is the resistance.

Through the use of Ohm's Law, the relationship between voltage, current, resistance or impedance, and power can be calculated.

Power is the rate of doing work and can be expressed in terms of potential difference between two points (voltage) and the rate of flow required to transform the potential energy from one point to the other (current). If the voltage is in volts or J/C and the current is in amperes or C/s, the product is joules per second or watts:

or:

where:

J is energy in joules,

C is electric charge in coulombs.

Fig. 3.3 is a wheel chart that relates current, voltage, resistance or impedance, and power. The power factor (PF) is cos I where I is the phase angle between e and i. A power factor is required in ac circuits.

3.2. RADIO FREQUENCY SPECTRUM

The radio frequency spectrum of 30 Hz– 3,000,000 MHz (3 × 1012 Hz) is divided into the various bands shown in Table 3.2.

FIGURE 3.3 Power, voltage, current wheel.

3.3. DECIBEL (DB)

Decibels are a logarithmic ratio of two numbers. The decibel is derived from two power levels and is also used to show voltage ratios indirectly (by relating voltage to power). The equations or decibels are

Fig. 3.4 shows the relationship between the power, decibels, and voltage. In the illustration, “dBm” is the decibels referenced to 1 mW.

Table 3.3 shows the relationship between decibel, current, voltage, and power ratios.

Volume unit (VU) meters measure decibels that are related to a 600 Ω impedance, O VU is actually +4 dBm. When measuring decibels referenced to 1 mW at any other impedance than 600 Ω, use:

Example: The dBm for a 32 Ω load is:

This can also be determined by using the graph in Fig. 3.5.

To find the logarithm of a number to some other base than the base 10 and 2.718, use:

A number is equal to a base raised to its logarithm:

therefore:

The natural log is a number divided by the natural log of the base equals the logarithm.

FIGURE 3.4 Relationship between power, dBm, and voltage.

FIGURE 3.5 Relationship between VU and dBm at various impedances.

Example: Find the logarithm of the number 2 to the base 10:

In information theory work, logarithms to the base 2 are quite commonly employed. To find the log₂ of 26:

To prove this, raise 2 to the 4.70 power:

2^4.70 = 26

3.4. SOUND PRESSURE LEVEL

The sound pressure level (SPL) is related to acoustic pressure as seen in Fig. 3.6.

FIGURE 3.6 Sound pressure level versus acoustic pressure.

3.5. SOUND SYSTEM QUANTITIES AND DESIGN FORMULAS

Various quantities used for sound system design are defined as follows:

D₁. D₁ is the distance between the microphone and the loudspeaker, Fig. 3.7.

D₂. D₂ is the distance between the loudspeaker and the farthest listener, Fig. 3.7.

D₀. D₀ is the distance between the talker (sound source) and the farthest listener, Fig. 3.7.

FIGURE 3.7 Definitions of sound system dimensions.

D_s. D_s is the distance between the talker (sound source) and the microphone, Fig. 3.7.

D_L. D_L is the limiting distance and is equal to 3.16 D_c for 15%Alcons in a room with a reverberation time of 1.6 s. This means that D₂ cannot be any longer than D_L if Alcons is to be kept at 15% or less. As the RT₆₀ increases or the required %Alcons decreases, D₂ becomes less than D_L.

EAD. The equivalent acoustic distance (EAD) is the maximum distance from the talker that produces adequate loudness of the unamplified voice. Often an EAD of 8 ft is used in quiet surroundings as it is the distance at which communications can be understood comfortably. Once the EAD has been determined, the sound system is designed to produce that level at every seat in the audience.

D_c. D_c Critical distance (D_c) is the point in a room where the direct sound and reverberant sound are equal. D_c is found by the equation:

where:

Q is the directivity of the sound source,

R is the room constant,

M is the critical distance modifier for absorption coefficient,

N is the modifier for direct-to-reverberant speaker coverage.

It can also be found with the equation:

M. The critical distance modifier (M) corrects for the effect of a different absorption coefficient within the path of the loudspeaker's coverage pattern.

N. The critical distance modifier (N) corrects for multiple sound sources. N is the number describing the ratio of acoustic power going to the reverberant sound field without supplying direct sound versus the acoustic power going from the loudspeakers providing direct sound to a given listener position.

%Alcons. The English language is made up of consonants and vowels. The consonants are the harsh letters that determine words. If the consonants of words are understood, the sentences or phrases will be understood. V. M. A. Peutz and W. Klein of Holland developed and published equations for the % articulation loss of consonants (%Alcons). The equation is:

where:

Q is the directivity of the sound source,

V is the volume of the enclosure,

M is the critical distance modifier for absorption,

N is the critical distance modifier for multiple sources,

D₂ is the distance between the loudspeaker and the farthest listener.

When D_c ≥ D_L, then %Alcons = 9 RT₆₀.

FSM. The feedback stability margin (FSM) is required to ensure that a sound reinforcement system will not ring. A room and sound system, when approaching feedback, gives the effect of a long reverberation time. A room, for instance, with an RT₆₀ of 3 s could easily have an apparent RT₆₀ of 6–12 s when the sound system approaches feedback. To ensure that this long reverberation time does not happen, a feedback stability margin of 6 dB is added into the needed acoustic gain equation.

NOM. The number of open microphones (NOM) affects the gain of a sound reinforcement system. The system gain will be reduced by the following equation:

Every time the number of microphones doubles, the gain from the previous microphones is halved, as the total gain is the gain of all the microphones added together.

NAG. The needed acoustic gain (NAG) is required to produce the same level at the farthest listener as at the EAD. NAG in its simplest form is:

NAG, however, is also affected by the number of open microphones (NOM) in the system. Each time the NOM doubles, the NAG increases 3 dB. Finally, a 6 dB feedback stability margin (FSM) is added into the NAG formula to ensure that the system never approaches feedback. The final equation for NAG is:

where:

PAG. The potential acoustic gain (PAG) of a sound system is:

where:

ΔD₀, ΔD₁, ΔD₂, are found as in NAG.

Q. The directivity factor (Q) of a transducer used for sound emission is the ratio of sound pressure squared, at some fixed distance and specified direction, to the mean sound pressure squared at the same distance averaged over all directions from the transducer. The distance must be great enough so that the sound appears to diverge spherically from the effective acoustic center of the source. Unless otherwise specified, the reference direction is understood to be that of maximum response.

Geometric Q can be found by using the following equations:

1. For rectangular coverage between 0 degrees and 180 degrees:

2. For angles between 180 degrees and 360 degrees when one angle is 180 degrees, and the other angle is some value between 180 degrees and 360 degrees:

3. For conical coverage:

c_∠. C_∠ is the included angle of the coverage pattern. Normally C_∠ is expressed as an angle between the –6 dB points in the coverage pattern.

EPR. EPR is the electrical power required to produce the desired SPL at a specific point in the coverage area. It is found by the equation:

a. The absorption coefficient (a) of a material or surface is the ratio of absorbed sound to reflected sound or incident sound:

If all sound was reflected, a would be 0. If all sound were absorbed, a would be 1.

ā. The average absorption coefficient (ā) for all the surfaces together and is found by:

MFP. The mean-free path (MFP) is the average distance between reflections in a space. MFP is found by:

where:

V is the space volume,

S is the space surface area.

ΔD_x. ΔD_x is an arbitrary level change associated with the specific distance from the Hopkins-Stryker equation so that:

In semirever berant rooms, Peutz describes ΔD_x as:

**200 for SI units

where:

h is the ceiling height.

SNR. SNR is the acoustical signal-to-noise ratio. The signal-to-noise ratio required for intelligibility is:

SPL. SPL is the sound pressure level in dB-SPL re 0.00002 N/m2.
SPL is also called L_p.

Max Program Level. Max program level is the maximum program level attainable at a specific point from the available input power. Max program level is:

L_sens. Loudspeaker sensitivity (L_sens) is the on-axis SPL output of the loudspeaker with a specified power input and at a specified distance. The most common L_sens are at 4 ft, 1 W and 1 m, and 1 W.

Sa. Sa is the total absorption in sabines of all the surface areas times their absorption.

dB-SPL_T. The dB-SPL_T is the talker's or sound source's sound pressure level.

dB-SPL_D. The dB-SPL_D is the desired sound pressure level.

dB-SPL. The dB-SPL is the sound pressure level in decibels.

EIN.EIN is the equivalent input noise.

where:

BW is the bandwidth,

Z is the impedance.

Thermal Noise. Thermal noise is the noise produced in any resistance, including standard resistors. Any resistance that is at a temperature above absolute zero generates noise due to the thermal agitation of free electrons in the material. The magnitude of the noise can be calculated from the resistance, absolute temperature, and equivalent noise bandwidth of the measuring system. A completely noise-free amplifier whose input is connected to its equivalent source resistance will have noise in its output equal to the product of amplification and source resistor noise. This noise is said to be the theoretical minimum.

FIGURE 3.8 Thermal noise graph.

Fig. 3.8 provides a quick means for determining the rms value of thermal noise voltage in terms of resistance and circuit bandwidth.

For practical calculations, especially those in which the resistive component is constant across the band-width of interest, use:

where:

f₁ – f₂ is the 3 dB bandwidth,

R is the resistive component of the impedance across which the noise is developed,

T is the absolute temperature in K.

RT₆₀. RT₆₀ is the time required for an interrupted steady-state signal in a space to decay 60 dB. RT₆₀ is normally calculated using one of the following equations: the classic Sabine method, the Norris-Eyring modification of the Sabine equation, and the Fitzroy equation. The Fitzroy equation is best used when the walls in the X, Y, and Z planes have very different absorption materials on them.

Sabine:

Norris-Eyring:

Fitzroy:

where:

V is the room volume,

S is the surface area,

a is the total absorption coefficient,

X is the space length,

Y is the space width,

Z is the space height.

Signal Delay. Signal delay is the time required for a signal, traveling at the speed of sound, to travel from the source to a specified point in space:

where:

SD is the signal delay in milliseconds,

c is the speed of sound.

3.6. ISO NUMBERS

“Preferred Numbers were developed in France by Charles Renard in 1879 because of a need for a rational basis for grading cotton rope. The sizing system that resulted from his work was based upon a geometric series of mass per unit length such that every fifth step of the series increased the size of rope by a factor of ten.” (From the American National Standards for Preferred Numbers). This same system of preferred numbers is used today in acoustics. The one-twelfth, one-sixth, one-third, one-half, two-thirds, and one octave preferred center frequency numbers are not the exact n series number. The exact n series number is found by the equation:

where:

n is the ordinal numbers in the series.

For instance, the third n number for a 40 series would be

The preferred ISO number is 1.18. Table 3.4 is a table of preferred International Standards Organization (ISO) numbers.

3.7. GREEK ALPHABET

The Greek alphabet plays a major role in the language of engineering and sound. Table 3.5 shows the Greek alphabet and the terms that are commonly symbolized by it.

3.8. AUDIO STANDARDS

Audio standards are defined by the Audio Engineering Society (AES), Table 3.6, and the International Electrotechnical Commission (IEC), Table 3.7.

3.9. AUDIO FREQUENCY RANGE

The audio spectrum is usually considered the frequency range between 20 Hz and 20 kHz, Fig. 3.9. In reality, the upper limit of hearing pure tones is between 12 kHz and 18 kHz, depending on the person's age and sex and how well the ears have been trained and protected against loud sounds. Frequencies above 20 kHz cannot be heard as a sound, but the effect created by such frequencies (i.e., rapid rise time) can be heard.

The lower end of the spectrum is more often felt than heard as a pure tone. Frequencies below 20 Hz are difficult to reproduce. Often the reproducer actually reproduces the second harmonic of the frequency, and the brain translates it back to the fundamental.

FIGURE 3.9 Audible frequency range.

3.10 COMMON CONVERSION FACTORS

Conversion from U.S. to SI units can be made by multiplying the U.S. unit by the conversion factors in Table 3.8. To convert from SI units to U.S. units, divide by the conversion factor.

3.11 TECHNICAL ABBREVIATIONS

Many units or terms in engineering have abbreviations accepted either by the U.S. government or by acousticians, audio consultants, and engineers. Table 3.9 is a list of many of these abbreviations. Symbols for multiple and submultiple prefixes are shown in Table 3.1.

3.12 SURFACE AREA AND VOLUME EQUATIONS

To find the surface area and volume of complex areas, the area can often be divided into a series of simpler areas and handled one at a time. Figs. 3.10–3.17 are equations for various and unusual volumes.

FIGURE 3.10 Equations for finding surface areas for complex shapes.

FIGURE 3.11 Equations for finding surface areas for complex shapes.

FIGURE 3.12 Equations for finding surface areas for complex shapes.

FIGURE 3.13 Equations for finding surface areas for complex shapes.

FIGURE 3.14 Equations for finding surface areas for complex shapes.

FIGURE 3.15 Equations for finding surface areas for complex shape

FIGURE 3.16 Equations for finding surface areas for complex shapes.

FIGURE 3.17 Equations for finding surface areas for complex shapes.