Moulton  Lectures

 

On

Electro-Acoustics

 

 

 

Presented by: Dave L Moulton

 

 

 

 

                                    Location:  Thales Acoustics Harrow UK

                           Date:  16-October-2002

 

Content

 

This third lecture looks into the types of graphical units and sound level interpretations commonly used by electro-acoustics engineers.

 

Frequency response graphs of electro-acoustic performance for earphones, microphones and headset attenuation needs to be presented so that the ordinate scale (y-axis) has acoustic related units and the abscissa (x-axis) relates to frequency.

 

Generally for earphone and noise level measurements the ordinate is a function of the acoustic intensity of the sound emitted from the noise source or from the earphone. So the start point for this lecture is the definition of acoustic intensity.

 

On With the Lecture

 

Acoustic Intensity  I

 

Acoustic Intensity I of a sound wave is defined as the average rate of change of energy transmitted per unit area in the direction of wave propagation.

 

Intensity is measured in Watts per Metre squared  (Wm^-2).

 

Intensity can also be expressed in terms of pressure P in the following way:

 

 

Where Z_a is the Acoustic impedance.  This is analogous to the definition of electrical power, the only difference being that in acoustics we talk about the power per unit area whereas in the electrical system we only need consider the power.

 

Threshold of Hearing

The lowest audio intensity perceivable by humans is internationally known as the intensity at the threshold of hearing:

 

I_TH = 10^-12 Wm^-2

 

From [3.1] it is also clear that we can write:

 

For normal air density at 25°C 

 

P_TH = 20´10^-6 Pascals,    often written as 20mPa

 

 

Addition Of Non-Coherent Intensities.

Sound Intensities from non-coherent sources (different frequencies) can be added together to give a resultant sound intensity I_T.

 

 

Thus

 

 

Sound Pressure Level  L_P.

Sound Pressure Level is probably the most commonly used acoustic measurement unit and is defined as follows:

 

or

 

 

Where  I is the measured acoustic intensity and P the measured pressure. L_P  is a decibel ratio and has units called dBSPL.

 

For example a sound pressure P of 1 Pascal (Typical output from a sound calibrator) has an L_P value of:

 

 

A 1 Pascal sound calibrator is often called a 94dBSPL calibrator.

 

Typical examples of the dBSPL scale are shown below in table 3.1.

 

 

Table 3.1

 

Multiple Fixed Sound Pressure Levels

Consider a source of noise which can be detected and displayed on an analyzer which displays the sound pressure levels  at each of the contributing frequencies.  The analyzer may display the spectral noise content as shown in figure 3.1 below:

 

 

Figure 3.1

 

 

What is the Total Sound Pressure Level?

All of the frequency components above contribute to the total Sound Pressure Level L_Ptotal  of the noise.  We have to be careful not to assume that we can add up all of the individual sound pressure levels to get the total level. Remember that the Sound Pressure Level is a Logarithmic ratio of Intensity. Referring back to equation [3.3] we can add intensities and get a total intensity I_total and then calculate  L_Ptotal. 

 

Thus

 

To calculate I_total we need to sum the individual intensities at each frequency. This can easily be achieved by taking the Antilog of the individual L_P and then adding them all together.

 

Thus

 

 

Now in general we could assume that the acoustic noise can be broken down into a defined number of frequency components, lets say that there are ‘k’ components. We can now write equation [3.8] in the more general form:

 

 

Which means we can now express our total Sound Pressure Level L_Ptotal as follows:

 

 

Note:  from the definition of  L_Ptotal  in equation [3.7], I_TH will cancel.

 

Here are some typical examples:

 

 

 

Time Varying Intensity Spectra

 

In normal life we experience sound pressure levels that are the cumulative result of time varying intensities across a broad band of audio frequencies. Equation [3.10] describes the case where the sound intensity level does not change with time.

 

A more realistic representation of sound intensity at a particular frequency would be to average the intensity over a given time frame.  For example we could look at the time averaged intensity at a frequency of 1kHz over a period of 10 seconds. The average intensity can be represented mathematical as:

 

 

Where T is the total averaging time (in this case 10 secs), and Dt is the time increment (in this case 1 sec)

 

This can be re-expressed in a more convenient form as:

 

 

Where I_Tn  represents the intensity at frequency index ‘n’ averaged over time T.

 

In the case of [3.11] the expression would look like:

 

 

Note that [3.13] represents the time averaged intensity at one particular frequency occurring at index ‘n’.  If we have multiple frequencies all identified by index (n=0,1,2,3,4……n), then our total time average intensity will be:

 

 

The Total Time averaged Sound Pressure level can be represented as:

 

 

Expression [3.15] is also known as the loudness equivalent level of the noise spectrum, normally denoted as L_eq.

 

Thus:

 

or

 

Fortunately audio analyzers are able to do the time averaging at each frequency, so we only have to deal with calculating equation [3.17], which can be written more practically as:

 

 

Where L_PTn is the time averaged sound pressure level at frequency index ‘n’.

 

Perceived Exposure Level

Equation [3.17]  represents the total time averaged exposure level, but not the perceived exposure level.  To get the  perceived level we need to take into account the non-linear response of the human ear to frequencies and Intensity levels.

 

For example at a low intensity level occurring at 63Hz  the average human ear is 26dB less sensitive than at 1kHz.  This means that 26dB more power is required at 63Hz to achieve the same perceived loudness level as at 1kHz.

 

A graph of the intensity responses of the human ear is shown below:

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

The 3 curves shown are known as weighting curves.  The A-weighting curve tends to be used for low intensities (Less than 10^-4Wm^-2) and is also commonly used as the weighting when quoting health and safety noise exposure levels. The C-weighting curve tends to apply for the very high deafening intensity levels.

 

For  health and safety exposure levels the A-weighting is normally applied to the Intensity measurements, resulting in an  L_eq(A) level.

 

 

 

This calculation is widely used to determine noise exposure levels. There are other modifications to this equation which relate to the rate of growth of the noise level. However, we won’t go into that during this lecture.

 

Typically the Health and Safety Executive (HSE) stipulate that in an 8-hour day the recommended noise exposure level L_eq(A) should not exceed 85dBA. The absolute maximum exposure level should not go beyond 90dBA. This is currently the  mandatory requirement of the European Health and Safety at work Legislation.

 

Octave Bands

In audio acoustics we tend to talk about frequencies in terms of Octaves or fractions of an Octave.  What this really means is that the audio frequency band, usually 200Hz to 4kHz, can be broken down into sub bands. See example below:

 

 

It is important to notice that the banded spectra is represented by a series of upright rectangles, each with a center frequency f_n . The level of the flat top of each rectangle is the logarithmic sum of all the discrete frequency levels within each rectangle bandwidth. The calculation to achieve each individual  flat top level is exactly the same as that used in equation [3.17].

 

The frequencies f₀ , f₁ , f₂ , …… f₇  are the center frequencies of each rectangle.

 

There is an Octave relationship between each frequency. 

 

In the case of single Octave spectra each center frequency is twice the one before it.

 

 

In general each octave center frequency relative to an initial start frequency  f ₀  can be determined from:

 

 

Octave Bandwidths

If each Octave band can be represented as a rectangle with a centre frequency, then each octave must have its own bandwidth (upper and Lower frequency limit).  This is easily determined using the half index.  See below.

 

Higher Resolution

Often in acoustics we need to achieve a higher resolution than single octave bands. The reason for this is the filtering affect of the octave level calculation.  For example if we had a frequency response with a sharp trough occurring over a very narrow band, then the octave resolution would miss this information.

 

Audio analyzers, in particular the B&K 2012, will provide resolution as high as a 96^th of an octave. In the acoustics lab we tend to use 24^th octave resolution for earphone and microphone responses and 3^rd octave resolution for attenuation measurements.

 

The calculations of the center frequencies and bandwidths for any octave resolution R can be found as follows:

 

Center frequency:  

 

Upper frequency:  

Lower frequency:  

 

 

 

 

End of Lecture