Moulton Lectures
On
Electro-Acoustics
|
Lecture
26 Fundamental Electro-Magnetics |
Presented
by:
Dave
L Moulton
Location: Thales Acoustics Harrow UK
Date: 10-March-2004
Background
We now move onto the subject of electro-acoustic
transducers. These devices are generally described as sound radiators or sound
detectors. Typical examples of sound radiators are Loudspeakers and Earphones,
whereas a very common example of a sound detector is a microphone. The
technology behind electro-acoustic transducers varies depending on the
transducer type and its intended usage. Before we can fully understand the
operation of transducers we must first have a basic understanding of the
fundamental physical laws that govern the operation of such devices and enable
us to model their performance. Transducers such as earphones and microphones
generally look like relatively simple devices, however the physics behind their
operation is far from simple. To model the performance generally requires
working in three domains Electrical, Mechanical and Acoustic. Each of these
domains are linked by transformers. In lecture 1 we looked at transforming between the acoustic and
mechanical domains and in lecture 6 I
applied this transformation to derive a mathematical model for a passive
hearing protector.
My intention is to do a series of lectures that
describing the operation of
electromagnetic earphones and microphones. These are some of the most common types of transducer used in the
field of communications audio acoustics.
Content
I will focus
this lecture on the link between the two domains which convert electrical
energy to mechanical motion. This lecture will cover the following areas:
·
Describe the three domains as applied to dynamic
transducers.
·
Describe the concept of magnetic flux.
·
Show the law that links the electrical domain to the
mechanical domain.
·
Explain Ampère’s law
·
Show how
Ampère’s law can be used.
On With The Lecture
Firstly let us consider the simplified block diagram of an earphone transducer. Figure 26.1 shows the domain conversions necessary to convert an electrical input into an acoustic output.
Figure 26.1
Notice that an earphone transducer utilizes the physics of three domains. The first domain is the Electrical, the second is the Mechanical and the third is the Acoustic. Thus an earphone by its very nature converts electrical input signals containing speech information into the mechanical movement of a diaphragm. For the purposes of this lecture we will assume a perfectly linear system, thus diaphragm motion can be considered to move in sympathy with the electrical speech amplitude. The diaphragm movement causes air to be vibrated in sympathy with the speech amplitude resulting in an acoustic wave transmission. We as humans detect the acoustic disturbance of air via our ears and perceive it as audible speech.
The three domains are linked together by the two transformers T1 and T2. We are already familiar with transformer T2, as this is the one that converts mechanical force ‘F’ into acoustic pressure ‘P’, This was covered in Lecture 1. But as a reminder equation 26.1 shows the fundamental relationship between force and pressure, and figure 26.2 shows the equivalent domain transformer.
Where ‘A’ is the area over which the pressure acts.
Figure 26.2
So what about the link between the electrical domain and the mechanical domain?
To establish this link we need to understand the fundamentals of Electromagnetics. By its very name Electromagnetics is the physics that relates electrical voltages and currents to their magnetic equivalents. You may be wondering what magnetics has to do with mechanics. To answer this, one must consider the type earphone technology under discussion, I have chosen a technology that is common to the majority of loudspeakers and earphones found in the audio communications industry. This is where mechanical force is generated as a result of a magnetic circuit, such magnetic circuits exist in conventional moving coil, central armature and rocking armature earphone transducers. The details of these types of transducer will be discussed in a later lecture.
Let us start by reminding ourselves about the concept of flux. As an example Figure 26.3 shows the magnetic flux lines emanating from a magnet and the electric flux lines between two oppositely charged plates. We will adopt the convention that magnetic flux flows from the North side (N) of the magnet towards the South side (S). In the case of the Electric system, the convention is that the flux flows from the positively charged plate to the negatively charged plate.
Figure 26.3
For reasons of mathematical representation, flux is normally depicted as a series of lines and described by the symbol f. Magnetic flux has its own unit known as the Weber (Wb). In reality, flux is a continuous medium with varying density and does not normally flow in discrete lines. The lines are purely a means of visualizing the flux and do not in any way invalidate the mathematics.
One quantity that is very important and often used to describe the amount of flux is the magnetic flux density (usually symbolized as B). Pictorially B represents the number of flux line per unit area and can be described mathematically as:
Where ‘A’ is the area. In this case area is a vector and has direction, thus ‘B’ must also be a vector and flux f is a scalar. Figure 26.4 shows the flux density at different points in a magnetic field. Flux density ‘B’ also has its own unit called the Tesla (T).
Note: a magnetic field is simply a region where magnetic flux exists.
Notice that for the same area ‘A’ the flux density B1 is greater than B2, thus more lines of f pass through the same area at B1 than B2.
Figure 26.4
Another important relationship is the one that links the magnetic flux density ‘B’ to the magnetic field strength ‘H’. This relationship is described in equation 26.3 below. We will come back to this in a future lecture. Needless to say plots of B and H can be used to show some very important characteristics of magnetic materials.
Where m0 is known as the permeability of free space (vacuum) and has the constant value of 4p´10-7Hm-1. mr is know as the relative permeability of the material and can vary considerably over a range of magnetic materials.
In lecture 4 we discussed electrical, mechanical and acoustic analogies. In the same way we also have an analogous link to a magnetic circuit.
Figure 26.5
In a magnetic circuit the through variable is represented by the flux. The across variable is represented by the Magneto Motive Force (MMF). The magnetic resistance is more commonly known as magnetic Reluctance.
Magneto Motive Force has the units of Amps, and is often referred to in terms of Ampere turns. Thus the MMF generated by a coil is the current flowing in the wires of the coil multiplied by the number of turns of the coil.
We can now look at the link between electrical current ‘I’ and mechanical force ‘F’ generated from a magnetic system. If we consider a length of wire ‘l’ carrying a current ‘I’ placed at an angle ‘q’ to a magnetic field ‘B’. Then the wire will experience a force given by the mathematical law:
Pictorially this looks like:
Figure 26.6
You may recognize equation [26.4] as being the cross product between B and lI. Thus:
The best way to remember the direction of the force on a wire is to use what is commonly known as Flemings left hand rule.
Figure 26.7
So we now have the rule that links the electrical domain to the mechanical domain. In the majority of transducers that utilize magnetic circuits to produce the mechanical force the magnetic field is often directed radial to a coil of wire, thus q is often 90° which simplifies equation [26.5] down to:
So our domain transformer looks like that shown in figure 26.8 below:
Figure 26.8
The transformation process turns the mechanical velocity into the across variable and the mechanical force into the through variable. Thus our block model shown in figure 26.1 for an earphone transducer can be improved to:
Figure 26.8
I now want to introduce you very important law of electro-magnetics. This is known as ‘Ampères circuital law’ and is described by equation [26.7] below:
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At first glance Ampère’s circuital law may look a bit frightening, but in fact it is very straightforward and can be described in words as:
The Circulation of a B-field around a current
carrying wire = m0mr multiplied by
the total current encircled.
In equation [26.7] the integral on the left hand side is a
line integral which sums the dot product of the B-field with the tangential
component 
to the encircling
path ‘C’ and an incremental length dl. This is shown in figure
26.9, where the red lines are the tangential components of the B-field to the
point of intersection with the enclosed path ‘C’ . For example at point p
on the enclosed path the tangential component 
is multiplied by B
and the incremental path length Dl, thus:
If we sum all such tangential components around the closed path we get an expression for the total circulation of the B-field around the closed path ‘C’, thus:
Figure 26.9
If we now look at the right hand side of equation [26.7] we see that it describes the total enclosed current. In this case ‘J’ represents the current density (current per unit area) and the integral simply sums the multiplications of ‘J’ and increments of area ‘dA’. In this case the total summed area is the one that is enclosed by the path ‘C’. As an example let us use Ampère’s law to work out an expression for the magnetic flux density at a distance r from a straight wire carrying a current I. The situation is shown in figure 26.10 below:
Figure 26.10
We are interested in calculating a value for B at a distance r from the wire. If we start by considering the left hand side of Ampère’s circuital law it should be clear that our path is a circle and the angle between the B-field and a tangential component to the circular path is in fact 0º. Thus the left hand side can be re-expressed as:
Now let us consider right hand side of Ampère’s law. In this case the area enclosed by the path ‘C’ is 2πr2. The Current density ‘J’ is defined as the current ‘I’ divided by 2πr2, thus the total enclosed current is simply I.
Bringing both sides of the equation together we get:
Thus :
One more example. Let us use Ampère’s law to calculate the B-field inside of a long solenoid. The solenoid can have a length l, and N turns of wire each carrying a current I. A diagram of our solenoid is shown in figure 26.11.
Note the direction of the magnetic field with respect to the direction of the current and the orientation of the coil turns. One method that I use to remember the direction of the B-field around a wire is to think of the word ‘CLOCK IN’. If the current were to flow ‘IN’ to the page then the B-field will circulate CLOCKwise around the wire. See figure 26.10.
Figure 26.11
Now let us look at a section through the solenoid and create an artificial path (ABCD) around which we can integrate.
Figure 26.12
If we now apply Ampère’s circuital law to the path ‘ABCD’ we immediately see that the total enclosed current is in fact NI. We have now dealt with the right hand side of equation [26.7], thus:
All we have to do now is integrate our way around the closed path ‘ABCD’. We can do this integration along each of the four straight line paths independently. Thus:
Now we can evaluate each integral in turn.
Along edges AB and CD the B-field can be considered to be perpendicular to the line. Thus the angle of intersection is 90º making the integrals go to zero. Thus:
and
Along the line DA we can consider the B-field to be very small compared to that which exists inside the solenoid, so we can neglect its contribution, Thus:
This leaves us with the remaining line BC inside the solenoid, which simply equates to the length of the solenoid multiplied by the internal B-field. Thus:
Equating both sides of Ampère’s equation results in:
Giving the B-field inside the solenoid as:
End of Lecture