Moulton Lectures
On
Electro-Acoustics
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Lecture
27 Fundamental Electro-Magnetics 2 |
Presented
by:
Dave
L Moulton
Location: Racal Acoustics Limited (UK)
Date: 21-March-2004
Background
In lecture 26, I gave a very simple introduction to
the concept of magnetic flux density ‘B’, I also related flux density to
mechanical force ‘F’. In order to understand the operation of transducers
we now need to take our study of electro-magnetics a little further. In this
lecture I want to look at the importance of magnetic circuits and the relevance
of air gaps in a magnetic circuit.
Content
I will focus
this lecture on the concept of magnetic Reluctance and magnetic circuits. The
lecture will have the following approach.
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Explain the concept of Magneto Motive Force MMF.
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Show how the magnetic circuit is analogous to other
domain circuits..
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Explain the effect of an air gap in a magnetic circuit.
·
Explain the concept of a magnetic load line
On With The Lecture
Firstly let us remind ourselves of Ampère’s circuital law
Now let us consider the situation where we have a magnetic field created by a magnet and not by a current carrying wire. This can be described by the magnetic system shown in figure 27.1 below:
Figure 27.1
This system can be represented in the form of a magnetic circuit shown below in figure 27.2. In this circuit the source of magnetic flux is the permanent magnet, and the path of the flux is directed through the metal yoke. The yoke is made from a magnetic material with a relative permeability greater than 1 ( μy >1). This type of material is generally termed paramagnetic and is able to concentrate lines of magnetic flux within itself.
Figure 27.2
The Magnetic Resistance ‘R ‘ is more commonly known as the Magnetic Reluctance, which is related to the MMF and flux in a similar way to the laws for electrical, acoustical and mechanical circuits.
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(electrical)
(acoustical)
(mechanical)
There is however a very important characteristic of [27.2] which makes it significantly different to the other three equations. In the case of [27.3], [27.4] and [27.5] for a given V,P and F, increasing the Resistance Re, Ra and Rm respectively results in a reduction in the equivalent currents I, Uv and v. This is mainly due to the effect of heat energy dissipation in the three different resistance mediums. In the case of the magnetic circuit, for a given MMF the flux Φ remains constant regardless of changes to the reluctance R. This may seem strange, but it is a direct consequence of the physical laws which describe magnetic fields.
In a magnetic circuit the Reluctance does not result in any form of energy loss in the system.
Gauss’s Law shown in equation [27.6] indicates that the divergence of a magnetic field is always zero. This basically means that lines of magnetic flux are continuous and never terminate anywhere. Magnetic flux lines have no beginning and no end, they always form a completely closed path.
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Gauss’s law can be described in figure 27.3 below:
Figure 27.3
Now let us go back to Ampère’s circuital law [27.1] and try to apply it to the magnetic circuit shown in figure 27.2.
One significant thing to notice here is that the source of magnetic flux is not the result of a current in a wire. Thus in this case the right hand side of equation [27.1] reduces to zero.
It is normal to re-write equation [27.7] in terms of the H-field, thus:
In the case of the magnet we can consider the lines of flux within it to run parallel to it physical length.
Figure 27.4
In the case of the yoke assembly we can consider the flux lines to also run parallel to each of it sides, resulting in the flux lines running parallel to the effective length of the yoke ‘ ly ‘. Thus the H-field within the yoke ‘Hy’ will also run parallel to the sides.
We can now apply Ampère’s circuital law to the entire circuit, giving:
Notice that each of the terms on the RHS of [27.9] must have the same units as the defining equation on the LHS. So the product of H and l have units of Amps (ref lecture 26). Thus the product Hl represents the Magneto Motive Force (MMF) for the magnet and yoke assembly.
Thus:
and
Gauss’s law tells us that for the circuit shown in figure 27.2, Φm = Φy, thus:
Note: The magnet also has some internal Reluctance. This is true for all magnetic materials. Thus our magnetic circuit can be improved to show this Reluctance.
Figure 27.5
At this point in the lecture you may be wondering about the significance of the magnetic Reluctance. So far it only appears to be acting as a constant of proportionality relating the MMF to the Flux.
The Reluctance does in fact have a very significant affect on the flux density ‘B’ of a magnetic field. This will become more clear as we continue through this lecture.
Now let us move on and consider the situation were we have a magnetic system similar to that shown in figure 27.1, but this time we introduce an air gap.
Figure 27.6
Now let us derive a relationship which shows the parameters that define the Reluctance of the air gap.
We know from Gauss’s law that the total flux in the system is constant, thus the flux produced by the magnet is the same as the flux across the air gap.
Thus:
Thus:
Leading to:
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So we see from [27.16] that the Reluctance is determined by the physical dimensions of the air gap. The same approach can also be used to determine the Reluctance of the yoke, thus:
Notice that the Reluctance reduces as the cross sectional area increases. For a given flux Φ as A increases the magnetic flux density B reduces:
Reluctance is one of those strange physical parameters that cannot be thought of in quite the same way as a conventional resistance.
The next question we have to ask ourselves is: why are we so interested in the physics of an air gap?
In our case the answer to this is quite simple. If a magnetic field exists across an air gap then there will be a resultant mechanical force also acting across the gap. This has great significance in the fundamental design and operation of a lot of electromagnetic transducers, such as balanced armature and central armature earphones and microphones. A more detailed study of the rocking armature transducer will be covered in the next lecture. Let us first consider the equivalent electrical circuit of the air gap system shown in figure 27.7
Figure 27.7
The force that exists across an air gap is directly proportional to the magnetic flux density ‘B’ across the gap. Thus:
The force attempts to pull the two ends of the gap together, but in this case it is prevented from doing so by the much stronger mechanical restoring force of the yoke material.
Figure 27.8
A designer of electromagnetic transducers will be very interested in knowing the static B-field across a gap. I use the word static in this case because the magnetic force is often offset against the mechanical tension in a transducer diaphragm. Setting the static magnetic and mechanical forces is often very deterministic of the sensitivity, stability and linearity of an acoustic transducer.
Figure 27.9 shows a very simplistic interpretation of a moving armature transducer. Notice that the static force is determined by the flux produced in the magnet and the dynamic force is determined by the alternating flux produced in the current carrying coil.
Note: in this case the static conditions are set by balancing the inertial force of the armature and the magnetic attractive force across the gap with the mechanical restoring force in the diaphragm.
Figure 27.9
I now want to give an insight into a method that can be used to determine the static flux density across the air gap. Firstly we must refer back to our circuit shown in figure 27.7. Using Ampère’s circuital law we can write:
Normally the reluctance of the air gap is significantly greater than the reluctance of the yoke material, thus we can make the valid approximation:
Thus we can re-write [27.20] as follows:
Thus:
We also know from Gauss’s law that the flux produced by the magnet is the same as the flux across the gap, thus
Rearranging:
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So if we know the flux density produced by the magnet we can determine the flux density across the gap.
To know the flux density produced by the magnet we need to look at the hysteresis curve associated with the magnet material. This can be determined from the BH relationship for the magnet.
In this relationship the relative permeability of the magnet is itself a very non-linear function of Hm. The hysteresis curve in figure 27.10 below shows the degree of non-linearity. This curve shows all the possible values of Bm for given levels of magnetizing field strength Hm. This curve is specific to the magnet alone, however there is a way of using the curve to determine the equivalent flux density left in the magnet when the external H-field is removed, this remanent field Bmr is a function of the dimensions of the magnet and air gap. This method results in a system load line which intersects the hysteresis curve at the appropriate value for Bmr.
Figure 27.10
We derive the system load line in the following way:
First we re-write equation [27.23] as:
Combining with [27.25] we get:
Leading to:
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This gives us a linear equation relating the B-field of the
magnet to the H-field as a function of the physical dimension parameters of the
magnet and the air gap. For practical purposes, most
non-ferromagnetic substances (such as wood, plastic, glass, bone, copper
aluminium, air and water) have a permeability almost equal to μ0;
that is, their relative permeability is 1.0.
Thus we can simplify [27.29] to:
We can now plot this linear function on to the BH curve and look at where the two intersect. The point of intersection will tell us the appropriate value of Bmr. This method gives a graphical insight into how the Bmr changes with both the magnet and gap dimensions for a given BH curve.
The resulting flux density across the gap can be calculated as:
Figure 27.11
End of Lecture