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3.3. Electromagnetic Induction: Part 1.
Contents:
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1. Introduction. 2. Electricity. 3. Fields. 4. Magnetic dipoles. 5. Flux linkage. |
6. Magnetic induction. 7. Electromagnetic induction. 8. Self induction. 9. Polarity of the induced voltage. . |
 A brief introduction
to transformers is given in [AC Theory, 41]. Familiarity with the
basic properties of ideal transformers is here assumed. |
| Glossary: |
| Quantity |
|
Definition | Units |
| Poynting vector |
|
= E × H | [Watts / metre²] |
| Electric field |
|
[Volts / metre] | |
| Magnetic field |
|
[Ampere turns / metre], [A turns / m] | |
| Field strength |
|
= F /  |
[A turns / m] |
|
Magnetomotive force, MMF, Ampere turns |
|
= Ñ I | [Ampere turns], [A turns] |
| Instantaneous current |
|
[Amperes], [A] | |
|
Flux density, Magnetic induction |
B |
= μH = μH = Φ/A |
[Tesla], [T], [gauss ×104], [Wb/m²] |
| Permeability |
|
= μ0 μr |
[Henrys / metre], [H/m] [Webers/Ampere metre], [Wb/A m] |
| Magnetic flux |
|
= A B | [Webers], [Wb], [maxwells ×108] |
| Flux linkage |
|
= Ñ Φ | [Weber turns], [Wb.turns] |
| Effective No. of turns |
|
= N √kH | - |
| Number of turns |
|
- | |
| Linkage efficiency |
|
- | |
|
Current sheet linkage effy. √(Nagaoka's coefficient) |
|
- | |
| Path area |
|
[m²] | |
| Path length |
|
[m] | |
| Self inductance |
|
= Λ / I = Ѳ μ A / |
[Wb turns / A], [Henrys], [H] |
|
Inductance factor (reciprocal reluctance) |
|
= μ A / |
[Wb / A turns], [H / turns²] |
| Coupling coefficient |
|
= Φ12 / Φ1 = M / √(L1 L2) |
- |
| Mutual inductance |
|
= Λ12 / I1 = k √(L1 L2) |
[H] |
| Frequently encountered cgs magnetic units: |
|
|
cgs unit | Conversion | |
| Field strength |
|
Ørsted, Oe | 1 Oe = 1000 / 4π A turns/m |
| Flux |
|
maxwell (line) | 1 maxwell = 10-8 Wb |
|
Flux density (magnetic induction) |
|
gauss |
1 gauss = 10-4 T 1 Tesla = 10 000 gauss |
|
1. Introduction: As can be seen from the preceding articles in this section, it is possible to come to an understanding of inductors without giving much thought to the magnetic fields which give rise to their properties. Such neglect of fundamentals is no longer possible when we come to the subject or transformers however (or electrical machines for that matter, but they are not the subject of this essay). For an ability to analyse circuits involving transformers of any type (not just ideal ones), or to quantify the effects of spurious magnetic coupling, it becomes necessary to relate the physics of fields to the mathematical conventions of circuit analysis. The principles involved are fairly straightforward; but, for reasons which warrant careful consideration by anyone attempting to teach the subject, the reconciliation appears to be an area of some difficulty. A rigorous approach to the subject of electricity involves starting with Maxwell's equations and working out everything from there. Obviously, there is room for a more accessible theory when dealing with lumped-element circuits, and the mathematics of phasors fulfils that requirement. There is a caveat however; which is that the definitions and conventions of phasor analysis need to be consistent with the underlying electromagnetic phenomena. That this issue receives little consideration can be confirmed by examining the various elementary textbooks which purport to cover the subject of magnetic induction. A knowledgeable reader who cares to follow the derivations on offer will soon discover inconsistencies (and may experience a disconcerting reminder of what it felt like to be a beginner). One problem lies with the definition of flux linkage, which falls victim to casual amendment when texts originally written using cgs (centimetre, gram, second) units are updated to use SI (metre, kilogram, second) units. In particular, in searching through a number of sources, it was found that none of the SI publications mentioned field non-uniformity corrections (Nagaoka's coefficient, etc.). This omission precludes the giving of self-consistent definitions for magnetic flux, inductance and reluctance (reciprocal inductance factor), and so confronts the reader with symbols which have the strange habit of changing their meanings according to context. Notwithstanding the questionable definition of basic quantities, there is also a strong tradition of confusion over phase relationships and algebraic signs. This problem arises for two reasons. Firstly: flux linkage is habitually transformed as a magnitude, but is used to derive quantities which require a sign [26]. Secondly: the concept of back EMF (which arises from use of the compensation theorem) is not applicable to closed conservative systems such as component models. Such vector inconsistencies can be seen in diagrams which are obviously incorrectly labelled; but more insidiously, give rise to formulae which, if used to derive the all-important impedance transformation rule of transformers, produce the complex conjugate of the true result. The contortions involved in escaping from such blunders are various and horrid, including dropping signs and working only with reactance magnitudes, or the mathematical abomination of treating complex quantities as reals. Then there are issues which do not relate to mathematical error, but are deleterious to understanding nonetheless. The main one is that there is no consensus regarding whether 'turns' belong in the system of magnetic units. The upshot is that some authors delete 'turns' wherever it occurs, presumably to divert students from the vice of including numerical factors in dimensional analysis. It seems probable however, that readers will be able to grasp the idea that turns do not have the fundamental status of metres, kilograms and seconds; and that people may be prepared to risk perdition in the interests of clarifying (say) the difference between Webers and Weber turns. A critique of the way in which magnetic induction is typically presented could go on to considerable length, but we will confine ourselves here to one last cause for complaint. That is the tendency to assume that there is no such thing as a transformer having active networks connected to more than one winding. The disease usually begins with a statement to the effect that: "a transformer has a primary winding and one or more secondary windings"; despite the fact that the transformer itself has no opinion on such matters. The designation of windings comes purely from external circuit considerations, and imposes restrictions on the subsequent analysis. Hence, for a mastery of transformers which displays ambition beyond the ability to design mains power supplies, the theory needs to emphasise the essentially reciprocal nature of inductive coupling. This is not to say that there is any objection to the use of the terms 'primary' and 'secondary'; but that we can only say: 'when a transformer is wired conventionally (i.e., with ports chosen to have DC continuity), and provided that there is only one port connected to an active network; the winding connected to the active network may be called the primary, and all of the others can be called secondaries'. If there are active networks connected to more than one winding, then the windings are neither primary nor secondary, and the transformer is called a 'hybrid'. The list of issues given above will perhaps go some way towards explaining how transformers come to be regarded as incomprehensible beyond the 'ideal case with lumped parasitics' model. The challenge, of course, is to cover the subject in a manner which addresses those issues. This warrants a somewhat oblique approach, not least because it must convince those who have been taught things differently. In particular, in addition to giving the essential information, it seems necessary to demonstrate the following points:  The definition
of inductance in documents using the SI system requires flux
linkages to be counted in a manner which is equivalent to the
method used by Maxwell and Faraday. Circuit analysis
parameters derived from field vectors are pseudoscalar, not scalar;
i.e., they change sign when the direction of the parent vector
is reversed. The compensation
'theorem' is not conservative. It can be used to eliminate separable
energy transfer processes, but its application to closed systems
violates the principle of conservation of energy. The extraordinary
range or technical possibilities offered by the transformer can
be appreciated by thinking of it as a device which allows the
impedance of one inductor to be modified (almost arbitrarily)
by the time-varying magnetic field produced by another inductor.Some calculus, of course, is unavoidable in dealing with a subject such as this; but the transition from fields to phasors, with pseudoscalars intact, is nowhere near as arduous as an approach which tries to resolve mathematical inconsistencies by making use of dubious physical arguments. Before we move on to the details however, we will address the most glaring oversight in nearly every discussion of magnetically-coupled devices, which is that the information will be incomprehensible to anyone who does not have a Maxwellian understanding of electricity. |
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2. Electricity: The phasor approach to circuit analysis can be seen as a set of techniques which allows the empirical laws of DC electricity to be adapted to deal with AC electricity. Phasor analysis manages to be both straightforward and extraordinarily useful, but it has the notable disadvantage that it can encourage bad habits of thought. In particular, it allows the conception that electricity is a fluid which travels through conductive materials; a physical picture which might have seemed promising in the early days of electrical discovery, but which has subsequently turned out to be wrong. The idea that electrical energy is carried by electrons flowing through wires cannot even account for DC electricity. The reason has to do with the fact that electrons drift through conductors in DC circuits with an average velocity of the order of a few mm/sec, and electrons have very little mass. This means that the transfer of momentum due to electron movements is insufficient to account for the power which a generator delivers to a load. Phasor theory does not concern itself with such issues; but even when strict energy-accounting principles are not applied, accepted techniques of circuit analysis remain inconsistent with electron transport concepts. The obvious way in which phasor analysis departs from the primitive DC theory is that it involves drawing current direction arrows which point through capacitors. Such arrows serve to give algebraic signs to the circuit parameters, just as they do in DC circuit analysis; but they can no longer be imagined to represent migrating mass. Thus, since AC theory is more general than DC theory (and includes DC theory as a special case), we must conclude that an 'electric current' is not as envisaged when the term was coined. It was explained in chapter 1 that 'true' power in electrical circuits is a scalar quantity, i.e., it is represented by a positive real number and cannot be made negative by swapping generator terminals. It still carries a sense of direction however, this being the thermodynamic principle that, on average, energy flows from a source to a sink. Power is, of course, a product of voltage and current; and so it follows that the voltages and currents used in circuit analysis must be defined in such a way that power flows from generator to load. This rule dictates the interpretation of the arrow conventions used in the diagrams below: |
|
In the left-hand diagram, it is assumed that electric potential
increases on moving from a reference point to some other part
of the circuit. In this case, the reference is taken to be the
ground potential; but the choice is, in principle, arbitrary.
Hence the voltage arrow shows the assumed direction of increase
from zero to a finite potential, as defined purely for the purpose
of circuit analysis. Once that definition has been made, the
direction of the current is automatically fixed according to
the convention that power is positive when flowing into a positive
resistance. When a generator is connected to a pure resistance, the voltage is in phase with the current. In that case, since both phasors point in the same direction and the choice of reference phase is arbitrary; the imaginary parts of both the voltage and the current can be set to zero (this is how AC theory maps onto DC theory). Effectively, the phasors have dropped a dimension and turned into one-dimensional objects; but note that they have not turned into scalars. They belong instead to a special class of vectors called pseudoscalars, which are distinct from scalars in that they change sign under the operation of 180° rotation. This special property is illustrated in the middle diagram, where a redefinition of the voltage via phase reversal has forced a reversal in the definition of current direction. The outcome, of course, is that power remains positive, the direction of energy flow being dictated not by the chosen signs of the voltage and current, but by the sign of the resistance. The right-hand diagram represents the more general situation in which a generator applies a voltage to an impedance. Now, although power will be transferred provided that the impedance has a resistive component; we also have the theoretical possibility that the impedance may be purely reactive, in which case the net power transfer will be zero. We nevertheless adopt the relative sign conventions for voltage and current on the basis that power will flow into the impedance if the impedance has a positive resistive component. The choice remains correct because the mechanism which decides how power will flow is built into the definition of impedance itself. That mechanism, of course, depends on the properties of the 90° rotation operator j. If the load impedance is purely reactive we have: Z = jX where the reactance, X, is (by convention) positive if inductive and negative if capacitive. Thus: I = V/(jX) = -jV/X A pure reactance rotates the phase of I by ±90° relative to V. The effect of this rotation is that (depending on the instantaneous signs of the voltage and current) sometimes power flows from generator to load, and sometimes it flows from load to generator; and the net power transfer averaged over a whole cycle of the AC waveform is zero. In general, the average power delivered to an impedance is given by the scalar (dot) product of voltage and current, i.e.: P = V•I = |V| |I| Cosφ where Cosφ is known as the power-factor and, for an impedance Z = R + jX φ = Arctan(X/R) The dot product, according to generally accepted convention, only ever gives a positive result. Hence its use may now seem slightly paradoxical given that we have admitted that instantaneous power can flow the wrong way. We could try to wriggle out of that problem by saying that the formula only relates to average power, but actually, it has to be admitted that the scalar product is not quite as scalar as it might seem. It will give a negative answer if the power-factor is negative. |
| Recall that the impedance of a passive network can only ever lie in the right-hand side of the Z-plane. No true passive network can contain a negative resistance because energy produced from nothing violates the classical principle of causality. There is however, nothing to prevent energy from flowing out of an active network; and so we could, should we happen to be feeling sufficiently misanthropic, devise a theory of circuits using negative resistances instead of generators. |
 Z-plane |
|
In fact, negative resistances do
crop-up in circuit analysis from time to time. They occur when
a network 'unexpectedly' turns out to be active. This can happen,
for example, when modelling antenna systems with multiple feed-points,
the negative resistance at one input being due to energy originating
from another input. The same can occur in any set of mutually
coupled two-terminal elements (such as a transformer); but in
that case, the situation is not unexpected, and we can analyse
the problem more sensibly by redefining auxiliary energy sources
as generators. Still, it has to be said that negative resistance is not forbidden. It occurs when a two-terminal network defined as an impedance has energy flowing into it from outside the system under consideration (provided, of course, that the energy input exceeds the amount required to cancel the true resistance of the network). The resulting impedance then lies in the left-hand side of the Z-plane, and the power in relation to that impedance is negative. Thus it might seem that the V I scalar product rule for power has let us down in terms of generality; but in fact, there is an interpretation which would be incorrect for general n-dimensional vector theory, but which must be valid in two-dimensional phasor theory because it resolves the power-flow paradox. The dot product of general vector theory is obtained using the acute angle (θ say) between two vectors lying in n-space. There is also however an obtuse angle (180 - θ) which suggests an alternative solution but is always ignored by convention. Now recall that: Cos(180 - θ) = -Cosθ The obtuse-angle dot product is the negative of the ordinary dot product. Hence, in phasor theory, we must conclude that the V I dot product should be taken using the actual phase angle φ of the impedance Z=V/I. Then, if |φ| is greater than 90°, as will occur if the resistive component of Z is negative, the power 'flowing into' Z will be negative as required. Thus it transpires that power, in phasor theory, although normally considered to be positive, is in fact pseudoscalar. It is not a true vector, but neither is it a magnitude, and it can become negative in some circumstances. Armed with that information, we can now return to the question: "what is an electric current?"; but we cannot give a satisfying answer just yet. The best we can say so far is that 'current is that which has to be defined in conjunction with voltage in order to account for electrical power'. It transpires that, from within the horizons of phasor theory, we do not know what electricity is. This means that the phasor theory, and the DC theory it supersedes, are both incomplete. The reason why phasor theory cannot answer fundamental questions, is that it is a projection or 'degenerate form' of a higher-dimensional theory. Just what has been lost in projection can be deduced by asking the question: "where exactly on a circuit diagram are the spatial dimensions and coordinates of the components?" It is easy to forget that circuit theory is purely topological. It assumes that all spatial dimensions tend to zero, which is why it is only valid when the wavelength at the analysis frequency is much greater than the length of any energy transmission path. The parent theory which does answer the question "what is electricity?" is, of course, James Clerk Maxwell's electromagnetic theory. This was presented in early form in 1864, and in a more mature form in 1873. To a 19th Century audience expecting some kind of confirmation of the 'fluid in transit' idea, the change in perception was radical, and for some, impossible to digest. Maxwell established that electricity is an invisible form of light. The difference between it and visible light is determined only by the frequency; and the empirical rules of electrical circuit building turn out to be those which persuade light of very long wavelength to cling to the outside surfaces of electrical conductors. Maxwell's extraordinary contribution to physics was to postulate the existence of a type of electric current (which he called "displacement current") which needs no charges in order to flow. It is often said that he made this deduction by noticing that the then known laws of electricity were in violation of the 'principle of local conservation of charge'; but that is a modern re-interpretation. A more plausible route to the discovery lies in the fact that Maxwell was pioneer and advocate of a technique known as dimensional analysis. He had, at his disposal, fairly good measurements of the permeability of free space, μ0 (obtained from the force between current-carrying conductors) and the permittivity of free space, ε0 (obtained from the force between charged bodies). What he is likely to have noticed first therefore, is that the reciprocal of the geometric mean of μ0 and ε0 has dimensions of velocity, and that the velocity so obtained is the same as the speed of light, i.e.:
Maxwell produced a complete set of equations which describe the strengths and directions of lines of electric and magnetic force at any point in relation to an electrical system, i.e., a description of the electromagnetic field. With the inclusion of the displacement current to maintain conservation of charge; it then becomes possible to delete all terms relating to physical matter and still have energy present. This energy exists by virtue of continuous exchange between the electric and magnetic fields, exactly according to the ordinary laws of electricity; i.e., a changing magnetic field produces an electric field, and a changing electric field produces a magnetic field; the two fields being represented as vectors which point at right angles. In the absence of matter however, the cyclical variation of the electric field is always exactly in phase with the variation of the magnetic field (from the viewpoint of a stationary observer), and to keep the energy exchange process in balance, the energy must propagate in a direction at right angles to both fields at velocity c. On the matter of this phase relationship incidentally, it may be helpful to identify the source of a common misconception. In the last years of his life, Maxwell had only one student of note; and that was John Ambrose Fleming, now controversially associated with the invention of the thermionic diode. In 1908, Fleming published his 'Elementary Manual of Radio Telegraphy', in which he described the propagating wave as having the maximum of one field corresponding to the zero of the other [28]. This blunder, from an influential writer who had sat at the feet of the great Maxwell, is an enduring source of confusion. The offending passage was corrected in the second edition of the book, but the damage was done and the error found its way into other books to muddle understanding for generations. The reality is that the energy travels at the speed of light and so, if we could observe the fields from the viewpoint of the energy, they would indeed be 90° out of phase. To an ordinary observer however, the electric and magnetic components of a radiation field are perfectly in phase, as is required to make the impedance of free space resistive (i.e., Fleming's picture of the electromagnetic wave was not consistent with the concept of radiation resistance). The important inference is that propagating energy will be found to have its electric field in phase with its magnetic field; and the velocity is a constant independent of the physical frame of reference. This is the 'relativistic' property of Maxwell's equations, which eventually led Einstein to the Special Theory of Relativity. For our purposes however, we simply note that this is analogous to the requirement that the voltage must be in phase with the current at any point in a circuit where energy dissipation is the only process occurring. The analogy becomes even clearer when we note that the units of the electric field E are Volts per metre, and the fundamental units of the magnetic field H are Amperes per metre. In a degenerate version of electromagnetism, the spatial dimensions are all replaced by unity; so that the electric field is replaced by voltage, and the magnetic field is replaced by current. The product of voltage and current has units of energy per unit of time, or power. The product of E and H has units of power per unit area, which is a measure of illumination. Hence, an electrical circuit is a structure which causes electromagnetic energy of very long wavelength to follow wires and converge upon resistances. Another part of the story we already know, of course, is that if the structure becomes large in relation to wavelength, then some of the electricity escapes and propagates off into space. The question we have yet to answer however is: "if the free electrons in the wires do not carry the energy, then what exactly do they do?" The answer is that they modify the refractive index of space in the vicinity of conductors, causing energy in transit to be steered into the region just outside, and to some extent slightly inside, the conducting surface. A problem which we may have now, is that electricity is only the second most badly taught subject in pre-university physics. The worst taught subject is that of optical refraction, which is generally explained in terms of light slowing down. The speed of light in vacuum is the same for all non-accelerating observers, and since matter consists mainly of empty space, there is no other conceivable medium. The apparent velocity can be less than or greater than c however, as a result of interference effects. Light is actually steered or 'refracted' by partially cancelling it in one place and augmenting it in another. The steering fields for electrical conduction are mainly provided by the free electrons in the conductors. What happens is that the electrons oscillate in response to the alternating electric field of a passing electromagnetic wave. This constitutes absorption of energy, but a collection of free charges will re-radiate most of that energy almost immediately, albeit with a slight shift of phase. The process of absorption and near-simultaneous re-emission is called 'scattering'. Once an EM wave enters a region rich in scattering objects, it very quickly loses its identity and is replaced by the sum or 'superposition' of all of the scattered waves. It is the superposition of incident and scattered waves which describes the way in which energy flows through the system; a resultant wave (which can be thought of as a contiguous vector path traced through the overall field) having the ability to negotiate bends, and appearing to travel at a velocity (called the wave velocity or the phase velocity) which differs from c. Not all of the energy is scattered back incidentally. Some of it is absorbed because the moving electrons can donate energy to other processes occurring in the body of the conductor; i.e., there is some loss due to 'resistance', but that is the price we pay for the steering service. So now we have a picture of electricity which is very different from the 'magical fluid' idea which prevails in the popular imagination. We need to make a circuit in order to obtain energy from a generator, not because something must flow from one terminal to the other, but because energy transport requires the combination of an electric field and a magnetic field. In the case of household electricity; the utility company provides the electric field, and the consumer provides the magnetic field by closing the circuit. A further possible manipulation is then to correct the power factor, so that the maximum amount of energy flows to the desired destination. It is also possible to send energy back into to distribution system, by using a local generator to shift the load impedance into the left-hand side of the Z-plane. Through the logical contortions involved in the process of discovery, a scientific culture is bound to develop many terms which all turn out to mean 'light'. Thus we have: 'gamma rays', 'X-rays', 'radio', 'electromagnetic energy', 'electricity'; and several others. The distinction between electricity and light should be regarded primarily as a technological matter; i.e., the nomenclature changes at the shifting boundary at which it becomes difficult to manufacture structures which are small in comparison to wavelength. The scientific importance of electricity is that it leads to the understanding that light can be decomposed into electric and magnetic fields, which have a quasi-independent existence in the sense that they can be separately detected and manipulated. The technology of electricity is that which allows the relative phases of the two fields from a given energy source to be controlled, so that energy can be made to flow in some desired manner. |
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3. Fields: The 'electrons moving through wires' conception of electricity causes magnetism to be regarded as a peripheral phenomenon; which means that an understanding of magnetism usually requires a simultaneous re-appraisal of everything which has previously been learned about electricity. Hence the need to make the point that, if voltage is to be regarded as electric potential, then current must be regarded as magnetic potential; and the corresponding fields have equal status in the transmission of energy. The next step therefore is to visualise electricity in terms of its fields. In physics, a field is defined as 'a physical quantity which can take on different values at different points in space and time. The field itself is independent of the system of co-ordinates (it stays the same regardless of how we choose to define a point within it) and its value at a given point may be described by a scalar, a vector, or, in general, a tensor (which is a mathematical object having an arbitrary number of variable attributes, not just magnitude and direction). If the medium is homogeneous and isotropic (i.e., on average, the same throughout and the same in all directions), the electric and magnetic fields can be treated as vectors; i.e., they can be defined by stating the magnitude and direction at any given point. It is usual to conceptualise vector fields using the analogy of fluid motion; i.e., each point in the field is assigned an arrow indicating the direction of flow, and a number indicating the flow rate. Thus we find ourselves using the old-fashioned terms "electric flux" and "magnetic flux"; although it must be stressed that an electric or magnetic field, on its own, does not represent a transportation process. Instead, the E and H fields are represented as vectors so that their interaction can be represented by an operation called 'vector multiplication', which produces a new vector at right angles to the original two. It is this vector product (or 'cross product') which represents a true flux, it being the illumination field with units of energy per unit time per unit area. Notice, incidentally, that this implies that the E and H fields can be treated together as a tensor field; which is known as the Maxwell bivector or electromagnetic tensor. For many purposes however, it is sufficient to deal with the E and H fields separately; albeit while keeping the notion of energy flow in mind. The electric and magnetic fields can be visualised using Faraday's "lines of force"; as demonstrated by the diagrams below: |
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The left-hand diagram is a two-dimensional representation of
the electric field as it might exist between two charged particles
or spheres, or between two conductors seen in cross-section.
There is, of course, a line for any arbitrarily chosen point
in the field, but the drawing shows only a few of them to give
the idea. The important property is that the field lines emerge
almost perpendicular to a conducting surface and follow curved
curved paths between points of high and low potential. Note,
incidentally, that although the field lines are often called
'lines of force', the field does not have the Newtonian dimensions
of force (mass × acceleration). The lines represent force
only in the loose sense, that a positively charged particle in
the field will be repelled by the (+) electrode and attracted
to the (-) electrode and will be accelerated in the direction
of the field vector. The middle diagram depicts the magnetic field around a wire when the instantaneous current of moving charges (which by convention flows from + to -) is flowing away from the observer. The right hand diagram shows the field when charges are moving towards the observer. The direction of charge migration is indicated by a cross or a dot drawn inside the conductor, the cross representing the tail fins of an arrow moving away, and the dot representing the point of an arrow approaching. The convention that the magnetic field lines rotate in a clockwise direction when positive charges are moving away is known as 'Maxwell's corkscrew rule'. Electrons, of course, move contrary to the arrow convention, but their existence was not known in Maxwell's lifetime. Once again, the dimensions of the field (Amperes per metre) do not correspond to Newtonian force, but the direction of the 'magnetic flux' indicates the sense of the force which will be exerted on a compass needle. The North-seeking pole of a magnet will be repelled by field lines coming towards it and attracted by the field lines going away from it. Hence, if the current is large enough to overcome the Earth's magnetic field, a compass placed near the wire will point in the direction of the field vector. The diagram below shows how the electric and magnetic fields combine to make electromagnetic energy. In this case we consider only a single electric field line (and all of the others are left to the imagination). The key at the top left gives the direction of the vector cross product: P = E × H where P is a field representing the energy per unit time flowing through an infinitesimal area, and is known as the Poynting Vector. Notice incidentally, that the terms 'field' and 'vector' tend to be used interchangeably. Strictly, a vector is a list of numbers (e.g., magnitude in each co-ordinate direction), whereas a vector field is a list of equations which give a vector when the co-ordinates of a point are put in. |
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The presence of an impedance in the circuit gives rise to a potential
difference, which causes electric field lines to curve from one
side of the impedance to the other. The diagram shows an instant
when both E and H are positive according
to the convention of the vector cross product, in which case
energy flows into the impedance. If the impedance is purely reactive,
the energy will be stored. If the impedance is purely resistive
(a black object in optical terms), then the energy will be converted
into heat or transduced into useful work. If the impedance represents
the terminals of another network, then the energy will be transported
away to a place where its fate can also be determined using fields. The rules of phasor theory now follow from Maxwell's discovery that E is in phase with H when energy is propagating towards an absorbing boundary. Consider first, what happens when the impedance is a pure resistance. In that case, V is in phase with I, and so E is in phase with H. Hence E and H change sign together, and P always points towards the impedance when its magnitude is greater than zero. When the impedance is a pure reactance however, V and I are exactly 90° out of phase. This means that E and H have the same sign for exactly 50% of a generator cycle, and opposite signs for the other 50% of the cycle. Hence the direction of P reverses cyclically, and the average power flowing into the impedance is exactly the same as the average power flowing out. Intermediates between pure resistance and pure reactance correspond, of course, to the relative amounts of time that P spends pointing towards or away from the impedance. On the issue of where the electrical energy is located relative to the connecting wires, note the earlier comment that electric field lines emerge almost perpendicular to a conducting surface. They only bend away from the perpendicular when there is a potential difference parallel to the surface. With the magnetic field lines encircling the wire, this means that the Poynting vector lies parallel to the wire provided that there is no potential difference on moving along the wire. In fact, a detailed analysis shows that the Poynting vector, considered as the average of a large number of microscopic energy transfer processes, is tilted very slightly towards a conductor on the outside (due to the small voltage drop caused by the internal resistance), and points almost directly inwards once under the surface. This means that the electrical energy is located almost completely outside the wire, and the small amount which flows inwards only does so to be absorbed. So that is how electricity is visualised using fields; but it has to be said that there are subtleties. One peculiar consequence; on which we depend when solving routine electromagnetic problems, but which also relates to the fundamental rules of physics; is known as the principle of continuity of energy (not to be confused with the principle of conservation of energy). Maxwell's equations imply that energy does not simply disappear from one part of the Universe and reappear in another; it flows through space in a definable way and maintains its integrity (i.e, it is conserved in transit). This was deduced independently and almost simultaneously in 1884 by John Henry Poynting (after whom the Poynting vector is named) and Oliver Heaviside [25]. The underlying reason for continuity was, of course, not understood at the time; but nowadays we associate electromagnetic energy with photons, i.e., tiny units which can be emitted or absorbed, but not modified. The principle of continuity dictates that electric and magnetic fields from different sources do not combine to produce electromagnetic radiation. If they did, the Universe would turn in to a fireball; but then again, perhaps it has already done that, and has subsequently inflated to attain a density at which such interference can no longer occur. The practical significance of energy continuity however, is that it allows us to consider different energy transport processes separately insofar as a system is linear (i.e., does not convert energy at one frequency into energy at other frequencies). Hence, when we model the fields around an electrical system, we know that there are other fields present (such as the Earth's magnetic field), but they can usually be ignored. Thus the fields imagined or depicted relate only to the process under consideration; and this allows us to tackle complicated problems by breaking them down into separate parts. Without continuity of energy, we would not be able to understand the Universe (quite apart from the inconvenience of being blasted apart while trying to do so). |
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4. Magnetic dipoles: It was allowed to pass without comment in the previous section, but it should have been obvious, that the spatial characteristics of electric and magnetic fields are very different. An electric field line may start from a point (a positive charge say) and terminate on another point (a negative charge say). It transpires however that magnetic field lines cannot do that, which means that they can only be envisaged as as continuous loops. This peculiar principle is embodied in Maxwell's equations as the statement that 'the divergence of the magnetic field is zero', i.e., it doesn't spread out from or converge onto points. The reason is that there are no magnetic charges, i.e., there are no tiny North poles and South poles swarming around on their own. The last statement warrants some qualification however, there being considerable lore attached to 'magnetic monopoles'. The version of Maxwell's equations used nowadays is actually due to Oliver Heaviside [25]. Heaviside reduced an original twenty equations to the four which relate the electric and magnetic fields, but he disliked the asymmetry which allowed electric fields to diverge while magnetic fields could not. Therefore he added a term to allow for the possibility of magnetic charges (which he called 'magnetons'), even though it could always be set to zero in the problems he studied. Later on, this idea was followed-up by Paul Dirac, who went on to formulate a relativistic version of quantum mechanics (and thereby predicted the existence of anti-matter) and showed that the existence of magnetic monopoles could explain the quantisation of electric charge (i.e., that charge is not infinitely divisible, but comes in discrete amounts). Monopoles figure in some of the various attempts to formulate a grand unification theory of physics, i.e., they might have existed shortly after the Big Bang; but they do not necessarily exist now, and have so-far never been isolated experimentally. In the absence of charges on which to land, magnetic field lines can only form loops. Thus, when we imagine the field lines coming out of the North pole of a magnet and looping around to the South pole, we don't imagine that there are monopoles crowded at the poles like the charges on a capacitor plate. The act of cutting a bar-magnet in two produces two new magnets, rather than separate poles, and so we must conclude that the lines entering at one pole travel through the body of the magnet and emerge at the other. Which brings us to the subject of magnetic materials and the origins of their properties. The classification of materials as either diamagnetic (μ ≤ μ0) or paramagnetic (μ > μ0) was introduced in Chapter 2; the ferromagnetism which confers the ability to store large amounts of energy in the magnetic field being a giant form of paramagnetism. The ferromagnetic materials can also be further divided into 'hard' and 'soft', the former becoming permanently polarised during exposure to a strong magnetic field, an the latter losing its magnetism upon removal. Recall that dielectric materials, which lack free charges, can be polarised by an electric field. Polarisation depends on the existence or creation of electric dipoles, which correspond to molecules or crystalline unit cells which have asymmetric internal charge distribution. Similarly, there are polarisable magnetic dipoles within materials; the difference being that there is no metaphorical knife sharp enough allow us to cut the smallest magnetic dipole in half. The reason is that magnetic dipoles are not associated with pairs of monopoles, but with the spins and orbital behaviour of the charged sub-atomic particles themselves. A magnetic field of material origin is always associated with an electric current, and the spin of a charged particle can be taken to constitute a current. Normally we say that protons and electrons "have spin", rather than simply saying that they spin; because a fundamental particle is a small package of electromagnetic energy, and it is unrealistic to envisage it spinning in the same way as a macroscopic object. A charged particle is essentially a quantity of energy (E=mc²), trapped in some manner involving a non-divergent energy flow, and having a patent monopolar electric field, and a patent dipolar magnetic field. Spinning charges can be imagined as tiny bar magnets; or better still, since they are able to change their orientations, as tiny compass needles. The difference is that, while a compass needle can point in any direction to align itself with an externally applied field, the orientations of microscopic dipoles are quantised, i.e., limited in terms of possible orientation relative to an external field. The ordinary charges (protons and electrons) can have spin quantum numbers of ±½, which means that they can align their dipoles with an external field, or against it, but there are no intermediate states. Diamagnetism is associated with paired spins, and paramagnetism is associated with un-paired spins. Spin pairing can be understood by considering what happens when two bar-magnets are placed side by side, with North poles adjacent to South poles. The fields from the two magnets form closed loops, and the combination ceases to interact with external fields and becomes diamagnetic (i.e., a collection of such paired magnets is, on average, slightly less permeable than free space). Hence, among the vast numbers of spinning particles in materials, most are paired, and paramagnetism arises from the relatively small number of un-paired particles which can exist in some types of material. The materials of primary interest for the construction of inductive devices are, of course, ferromagnetic. Ferromagnetism occurs when the spins of un-paired electrons become aligned over relatively long distances, like strings of bar magnets attached North to South. These regions of constructive alignment are called 'magnetic domains', and are capable of internal re-arrangement to change the orientation of the overall magnetic dipole. In a soft ferromagnetic material, in the absence of an externally applied field, the domains are randomly orientated and there is no overall magnetism. When a field is applied however, the domains are progressively forced into alignment and energy is stored, to be returned when the external field decreases in strength and the material relaxes. Hence, the presence of ferromagnetic material vastly increases the magnetic energy which can be stored in a given volume of space, and the macroscopic average relative permeability of the volume is consequently much greater than 1. In a hard ferromagnetic material, there exists an energy barrier with respect to domain realignment. Hence, if a field strong enough to overcome the barrier is applied, energy is absorbed, but not all of it is returned when the field is reduced. The domains become permanently aligned to some extent over very long range, and so the material obtains a permanent magnetic field. Hard materials are, of course, not suitable for ordinary inductor cores, because they behave in a highly non-linear fashion. It is important to be aware however, that no ferromagnetic material is completely soft; there will always be some residual magnetism after removal of an external field, and so coils with magnetic cores are not perfect linear devices. |
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5. Flux Linkage: When the term 'linkage' is used in the context of magnetism, the metaphor is meant to be interpreted in the strong sense; i.e., a link is a loop interlocked with another loop, as in a chain. Particularly, the term 'flux linkage' refers to the fact that the magnetic flux lines which form around a current-carrying conductor can also loop around other conductors, this being part of the explanation for the phenomenon of electromagnetic induction. Hence, flux linkage is involved in the process whereby electrical energy supplied to one current loop can be dissipated in a resistance in series with a separate loop, the resulting structure, called a transformer-coupled network, being analogous to a chain. Before we consider inter-circuit linkages however, we need to consider the linkages which occur in a single circuit, particularly when that circuit is coiled into a set of overlapping loops all carrying the same conduction current. |
| A flux linkage can be considered to exist whenever a loop of magnetic flux completely surrounds a current. The diagram on the right represents the flux surrounding a pair of wires (seen in cross section) which are both carrying the same current (i.e., such as might occur in a two-turn coil). Notice first, that if there is any appreciable depth of current penetration into the body of the conductor (i.e., skin depth), then there can be lines of flux below the conductor surface. These lines, which are associated with internal inductance, do not encircle all of the current, and so contribute less than a whole flux linkage. On the outside, at or very close to the surface, there are lines which encircle the current only once and so correspond to single flux linkages. At greater distances however, the field is best considered as a superposition of the lines from both (or, in general, many) turns, and it takes the form of loops which enclose more than one turn. A flux loop which girdles the current N times is said to contribute N flux linkages. |
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The reason why we evoke the concept
of flux linkage, is that it gives us a simple way of defining
inductance. To understand it however, we must use Faraday's way
of thinking about magnetic energy storage. Faraday imagined a
conductor to be surrounded by a flux of lines (sometimes also
called tubes) each having equal weight. In other words, every
one of his lines contributed a unit of flux (later called a maxwell);
so that regions where the field is strong have many lines, and
regions where the field is weak have few lines. Also, lines are
created by putting energy into the field, and disappear when
energy is taken out. Taken in conjunction with the observation
that the inductance of a coil is proportional to N² (where
N is the number of turns); it transpires that the increase in
the amount of magnetic energy which can be stored, for a given
current, when a conducting loop is coiled, can be determined
by counting the flux linkages. Thus, according to Faraday's interpretation,
coiling increases inductance because it allows field lines to
link the current more than once. Hence the definition: Inductance = flux linkages per unit of current. Notice that we have not bothered to include a constant of proportionality. The units are to be defined so that such a constant will not be needed. Nowadays, given wider familiarity with vectors; we are a lot happier to think of magnetic flux in an abstract way. It is sufficient to assign a magnitude and a direction to describe a point in a vector field; but the magnitude which describes the intensity of the magnetic flux can nevertheless be envisaged as the number of maxwells (i.e., lines) enclosed by an infinitesimal area perpendicular to the flux. Hence there is a perfect correspondence between Faraday's and the modern conceptions of flux density (i.e., flux per unit area), provided that we define the relationship correctly. Herein lies a difficulty however, which is that Faraday and Maxwell saw the determination of inductance as a counting problem, whereas later writers saw it as a field integration problem. Neglect of detail can then result in a set of field parameter definitions which are only approximate. We should note that the phrase 'approximate definition' is an oxymoron. A typical attempt at defining inductance is based on the idea that the flux links the current once in a single loop circuit, and that it links the current N times in an N-turn coil. As we have seen however, neither proposition is strictly true. The flux associated with internal inductance contributes slightly less than 1 flux linkage per line; and of the flux associated with external inductance, not all of it will link with all of the turns. Thus, except in the special case of a single turn loop at very high frequency (when the skin depth has gone to zero), the assumption of maximum possible linkage will lead to error. To solve that problem, we need to define a linkage efficiency parameter, i.e., a correction factor (<1) by which the number of turns can be multiplied in order to ensure that the linkages have been counted correctly. We will start by calling the number of turns so corrected the effective number of turns, and give it the symbol Ñ. Thus: 0 < Ñ < N The correction parameter is then Ñ/N, which, for reasons which will shortly become clear, can also be written as √kH , i.e.: Ñ = N √kH Where kH is a quantity which, in general, depends on the geometry of the system and the properties of the materials involved; but which has already been derived for a solenoid coil in section 3.1-11. For a current-sheet solenoid kH = kL, where kL is Nagaoka's coefficient. |
| One way in which to maximise linkage efficiency is to provide a magnetic path of low reluctance; i.e., a path of high permeability, short length, and reasonably large cross section. This can be achieved by (say) winding wire onto a ferrite bead. There will still be internal inductance, and there will still be external flux which only links the current once; and in a coil of many turns, there will also be flux which links some but not all of the turns; but the amount of energy which can be stored in the magnetic material will be so great, in comparison to that which can be stored in the wire and the surrounding medium, that it will make other contributions almost negligible. That is why the inductance of a coil wound on a closed transformer core is almost entirely dependent on the number of turns and the core inductance factor (AL). The 'reluctance' of a transformer core, incidentally, is defined as 1/AL. Reluctance in a magnetic circuit is analogous to resistance in an electrical circuit, and so AL is analogous to conductance. |
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| Notice that the possible range of Ñ (effective turns) was stated above to lie between 0 and N. We might expect a lower limit of 1 (or very slightly less, due to the partial linking associated with internal inductance), but the linking can be considerably less than 1 when the flux from one part of the loop cancels the flux in another part of the loop; i.e.; linking can be constructive or destructive. In a single turn loop, linkage is maximised by maximising the loop area; i.e., by getting a particular point in the conductor to be as far away as possible from any other point which is not carrying current in exactly the same spatial direction. Hence we need a loop shape criterion when defining linkage; and so it is assumed that the external flux of a single turn loop links the current once (i.e., maximally) when the area is at its maximum; i.e., when the loop is circular. |
| In a multi-turn coil, the effective number of turns (and hence the number of flux linkages) can be reduced by changing the winding direction at some point during the making of the coil. This is often done for the purpose of minimising inductance; i.e., to cancel linkages so that Ñ is as close to zero as is possible. This is (for example) the reason for the hairpin bifilar pattern used in some types of wirewound resistor (see right). It follows that if we start counting upwards when winding a coil, and then change direction and start counting downwards; the effective number of turns will eventually become negative (at least, notionally). The definition of a particular winding direction as positive or negative is not completely arbitrary, because it affects the relative directions of the flux lines on the inside and outside of the coil. Hence the handedness of a helix (i.e., left-handed or right-handed) plays a part in determining the orientation of the overall magnetic field. The usual choice however, is to define the effective turns number as positive (i.e., to treat it as a magnitude) and use Maxwell's corkscrew rule to determine the field polarity. This has the effect of fixing the polarity of the overall flux solely according to the sign of the current; i.e.; since inductance is defined as flux linkage per unit current (and is by convention positive), then flux linkage is positive when current is positive. |
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6. Magnetic induction: The term 'electromotive force' or 'EMF' is sometimes used when referring to a voltage produced by the action of a generator; i.e., an EMF is a voltage associated with a source of energy. Similarly, 'magnetomotive' force or 'MMF', is proportional to the current supplied by a generator; but, due to the phenomenon of flux linkage, MMF is only equal to current when the circuit is composed of a single (i.e., non-overlapping) circular loop. In electrical systems, power and energy, and also the illumination fields 'power per unit area' and 'energy per unit area' are always proportional to the square of an EMF, the square of an MMF, or to a product of EMF and MMF. This is evident from the elementary formulae: P=V²/R, P=I²R, P=IV, E=(½)CV², E=(½)LI², and so on. Thus, both EMF and MMF have units which are proportional to the square root of energy. We can also observe that the energy stored in an inductor is proportional to the inductance, and inductance is proportional to N²; from which it follows that, if MMF is a function of N, then it must be directly proportional to N. The act of winding a current loop into a coil must be deemed to increase the MMF, because it increases the magnetic contribution to stored energy without the need for an increase in current. Thus, given that we have deduced the proportionality from dimensional requirements, MMF has units of Ampere turns. (and is often referred to as 'Ampere turns'). It is also typically defined as the current multiplied by the number of turns; but that last step is questionable. Flux linkage is a logically-consistent explanation for the increase in MMF which results from coiling, and we know that there will always be flux which cannot link with every turn. Hence we define MMF as: F = Ñ I where Ñ is the effective number of turns, as discussed earlier. Now notice that the symbol for current (I) has been written un-bold, whereas in AC circuit analysis we would normally write it in bold script, to indicate that it is a phasor. The reason is that we are presently engaged in the business of relating circuit parameters to the actual magnetic field; and the field strength is strictly proportional to the instantaneous current. It is also the case that, for the purpose of obtaining a complete description of the field, the current has only magnitude and one of two possible directions; i.e., by reversing it, we reverse the direction of all of the flux loops (and thereby reverse the poles in the case of an electromagnet), but we cannot otherwise alter the relative distribution of the field. Hence the current, as defined for the purpose of mapping from electromagnetics to circuit theory, is pseudoscalar, just as it is in DC circuit analysis. We can convert it back into a phasor later, because by then we will have defined all of our new magnetic parameters and the generalisation will not affect them; but for now, we must consider only instantaneous current (or DC). The units of electric field strength are [Volts / metre]. Similarly, the units of magnetic field strength are [Ampere turns / metre]. Hence the intensity at some point in the magnetic field is given by a vector (H) which has a magnitude proportional to the MMF, divided by some distance over which the MMF is considered to act. That distance is known as the path length, and is given the symbol (sometimes with subscripts, depending
on the context). The path length in a single-loop circuit is
the average length of a flux loop, which is difficult to calculate;
but for solenoids and closed transformer cores, the situation
is more straightforward. In the case of a solenoid, it is the
length (or height) of the cylinder taken from centre-to centre
of the connecting wires. In the case of a transformer core, assuming
that the field outside the core is negligible, it is the average
length of a flux loop in the core body, and is given as the e value
(effective path length) in the manufacturer's datasheet.. The amount of energy which can be stored in the magnetic field around a current loop is proportional to the average permeability of the magnetic path. Hence, according to Faraday's way of thinking; for a given current, the number of field lines is multiplied by the permeability. Hence we can define a new field which is more fundamental and informative than H, i.e.; B = μ H where the vector B is called the magnetic induction or flux density, i.e., it is the number of Faraday lines per unit area at a given point in the field. It should be obvious, that the inductance of a loop is dependent on B, rather than on H alone. To define the complete spatial distribution of the magnetic field, we must solve Maxwell's equations for the system. For the derivation of circuit analysis parameters however, it is not necessary to do that. Instead, we can define an average flux density as the total flux divided by the total path area. Notice that this new quantity is still a field vector. By defining it over the total area however, we imply that its direction is the average direction of all of the field lines in the bundle enclosed by the area. This average direction is always perpendicular to the plane of the loop. Therefore the total flux has only two possible directions (depending on the direction of the current), i.e., it is pseudoscalar, and so we give average flux density the symbol B (un-bold, in italic script, not to be confused with susceptance, B). Hence: B = μ H = Φ / A [Tesla] where H = F / [Amp.
turns / metre]is the average field strength on the path, and Φ is the total magnetic flux associated with a single loop. Notice that Φ is also pseudoscalar (as is H); i.e., it can be positive or negative, and so it is definitely not a magnitude. The SI unit of flux is the Weber [Wb], which is equivalent to 108 lines (or maxwells). The unit of flux density (magnetic induction) is the Tesla [T], or Wb/m², which is equivalent to 104 gauss. The gauss is still commonly used in scientific papers, transformer-core datasheets, etc., and its relationship to the Tesla should therefore be memorised. When a coil is wound on a closed-circuit high-permeability core, the energy stored in the core is so great in comparison to that stored elsewhere, that the path area is, to a good first-order approximation, the same as the average cross-sectional area of the magnetic channel (i.e., the Ae value given in the core's datasheet). For an un-cored loop or coil, the path area is perhaps less-immediately obvious, but the definition is not difficult to understand. |
| Consider a planar (i.e., flat) current loop. However we determine the total flux, the process is equivalent to counting the lines, and each line must be counted only once. Now, noting that each line encircles the current, the number can be determined by counting in the plane of the loop, either on the inside, or on the outside (but not both). Counting the flux on the outside is somewhat problematic, because the field extends (at least notionally) to infinity. Hence the integration is carried out over the area enclosed by the loop; and that area is consequently identified as the path area. |
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While the logic behind the definition
is straightforward however, exact determination of the area is
not so easy. The problem is that practical conductors have finite
(and often substantial) thickness, which means that the boundary
established by the conduction current is necessarily diffuse.
The process of incorporating known physical behaviour into a
model by defining what happens at various points in a co-ordinate
system is known as "establishing the boundary conditions".
In this case, we want to count all of the flux lines exactly
once, and so the relevant boundary condition is established by
defining a perimeter line such that the conduction current flowing
outside this line is exactly equal to the current flowing on
the inside. Widespread practice is then to assume that the boundary
lies at the conductor centre-line; but this is not an accurate
choice for several reasons: Firstly, the conduction path on the
inside of a loop is shorter than on the outside (i.e., the resistance
is less), and so the current will tend to concentrate on the
inside. Secondly, there will be a redistribution of the current
at high frequencies; especially in multi-turn coils, due to the
proximity effect. Lastly, there will be displacement currents
and interfacial effects (i.e., it is not possible to make an
electrical connection without disturbing the magnetic field);
and so, what looks initially like a simple problem turns out
to be rather complicated. Due to the difficulties in establishing the path area, magnetic problems are usually addressed by starting with simplified models. The simplification in question is that of assuming that conductors are filamentary, i.e., of negligible thickness in comparison to other dimensions. Note that a hypothetical filament has no internal inductance, because it is too small to have flux loops inside it. Recall also, that the conductor in a current-sheet solenoid has width, but no radial thickness, and so is filamentary when the solenoid is viewed end-on. The filamentary current model will provide a first (and usually fairly accurate) approximation for the behaviour of a practical magnetic device, and can be further refined by the inclusion of internal inductance and various other correction terms. For further discussion of this matter in relation to solenoids, read the whole of section 3.1. Although the process of determining the path area may involve approximation in practice, it is nevertheless rigorously defined as the area bounded by the median line of the conduction current density; i.e., a line chosen so that the integral of the current density on the inside is equal to the integral of the current density on the outside (see TA3.1). Hence we may take the final step in defining the inductance of a multi-turn coil, by multiplying the total flux through a single turn by the effective number of turns, to obtain a parameter called the 'number of flux linkages' (or 'flux linkage' for short). To this we assign the symbol Λ (Capital "Lambda"), i.e.: Λ = Ñ Φ [Weber turns] Notice that Λ is pseudoscalar, because Φ is pseudoscalar; i,e., all quantities derived from the field are bi-directional vectors. Inductance, the absolute measure of the ability of an electrical device to store energy in the magnetic field, was identified earlier as the number of flux linkages per unit of current. Hence: L = Λ / I [Weber turns / Amp.] The composite unit is known as the Henry, in honour of Joseph Henry (1797 - 1878) who studied magnetic induction at the same time as Faraday, and made comparable scientific contributions. The inductance of a coil or current loop is, by convention, positive. Negative inductance can occur in circuit analysis, but generally in situations involving inductance cancellation; in which case a negative inductance is obtained when the cancellation signal is too great. Thus, in the relationship given above, it can be seen that flux linkage must be allowed to be either positive or negative, because instantaneous current can be either positive or negative. This freedom is required for reasons of mathematical consistency; i.e., failure to allow it constitutes the ludicrous proposition that inductance changes sign when the current is reversed. It also allows us to determine the overall direction of the dipolar field around a coil by using Maxwell's corkscrew rule. The general formula for the inductance of a coil can now be obtained by working backwards through the preceding discussion. Thus: |
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L = Λ / I = Ñ Φ / I = Ñ A B / I = Ñ A μ H / I = Ñ A μ F / (
I )= Ñ A μ Ñ I / ( I
)= μ Ѳ A / |
Flux linkage, |
Λ = Ñ Φ Φ = A B B = μ H H = F / F = Ñ I Ñ = N √kH |
Hence:
For a coil wound on a closed transformer core, where (to a reasonably good approximation) kH→1, this becomes: L = μ N² Ae / e(Where Ae and e are the effective path area and effective
path length). The quantity:AL = μ Ae / e [Henrys
/ turn²]is, of course, the core inductance factor (and its reciprocal is the core reluctance). For a current-sheet solenoid, kH = kL, where kL is the fringing-field or field non-uniformity correction factor known as Nagaoka's coefficient (see 3.1-6). kL is generally less than 1, because some of the field lines come out through the sides of a solenoid, and so fail to link with all of the turns. kL→1 when a solenoid is very long and thin because the flux lines only spread out towards the ends of a helical coil (to loop around on the outside), and if the ends are far apart, the spreading (non-uniform) region is small in comparison to the major part of the enclosed field. |
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Before we move on; it is perhaps worth drawing attention to the
fact that 'turns' have been included in the system of magnetic
units used here. Turns is, of course, just a number, and so has
no unit of measurement. Hence some authors do not include turns
when stating units, and may even go so far as to claim that the
deletion is in aid of mathematical rigour. There is a slight
downside; in that the distinctions between current and MMF, and
flux and flux linkage, are then lost; not helped by the widespread
practice of using the symbol Φ for both flux and flux linkage;
but these privations no doubt help to provide the student with
a challenging learning environment. On a more serious note; those familiar with dimensional analysis will know of it as a technique which allows relationships to be determined by comparing the measurement units of the variables associated with a physical system. The limitation of dimensional analysis is that it cannot determine constants of proportionality, because scale factors (i.e., pure numbers) have no units. The corollary however, is that pseudo-dimensions, such as turns, can have no effect on the validity of dimensional analysis (and may provide valuable additional information). Hence the argument, that the inclusion of numerical parameters is unrigorous, falls down; and the deletion of turns from magnetic units serves no purpose except to spread confusion. |
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7. Electromagnetic induction: It is important to distinguish between the terms magnetic induction and electromagnetic induction. Magnetic induction (flux density) is, literally, the amount of magnetism induced (i.e, forced) into a volume of space. Electromagnetic induction, on the other hand, is the propensity for a time-varying magnetic field to create an electric field (and vice versa). The latter phenomenon was discovered by Faraday, in 1831, after he had spent some six years pondering the riddle: 'why is it that a current produces a magnetic field, but a magnetic field does not produce a current'. With hindsight, we can easily identify the barrier to understanding by inserting the adjectives 'direct' in front of 'current, and 'static' in front of 'magnetic field'. Faraday's final experimental apparatus for this investigation was effectively a transformer (although the term was not coined until many years later); i.e., he used a separate coil and a battery to provide the field. The riddle was solved when he noticed that the galvanometer gave tiny kicks in opposite directions as he switched the battery on and off. Faraday's subsequent interpretation of the induction coil experiment (in terms of flux linkages, of course) became one of the cornerstones of Maxwell's great synthesis of electricity and optics. It was Einstein who gave the full explanation however, by deducing that a magnetic field acquires the character of an electric field when viewed from a moving frame of reference (and vice versa); the degree of mapping from one to the other being proportional to the relative velocity, and becoming absolute at velocity c. This means that there can be no greater relative velocity than c, and so the geometry of space-time had to be redefined as a consequence of Maxwell's equations. Einstein's Special Relativity also tells us that there is no distinction between a changing field and a moving field; and so an electric field can be created from a magnetic field in either way. Nowadays, an electromagnetic induction engine of the changing field variety is called a transformer, and a rotating engine of the moving field variety is called an alternator. Recall that, using Faraday's mental picture of the magnetic field, flux lines (and hence linkages) are created when energy is put into the field, and destroyed when energy is taken out. Hence Faraday's wondrously simple explanation for what turned out to be the seed for the whole of modern physics (here paraphrased): 'The voltage induced across the ends of a coil is proportional to the rate of change of flux linkages per unit of time' (Faraday's Law of induction). Note that the word 'proportional' can be replaced by 'equal' if the system of units is chosen correctly; and that was one of the effects of Maxwell's later unification of the E and H fields. Hence: V = dΛ/dt [Volts] This equation has enormous implications; not least, for our purposes, because it lies behind the theory of AC electricity in general, and inductive devices in particular. |
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8. Self induction: The definition of inductance was given earlier as: L = Λ / I [Henrys] This can be re-written: Λ = L I [Weber turns] Now recall that inductance, here seen as the factor of proportionality in the relationship between flux linkages and current, depends only on the physical geometry of the system and the permeability of the magnetic path. Both permeability and geometry can change with field strength, in the latter case (for example) because the force between parallel current-carrying conductors causes coils to contract in length as the current is increased; but these are small effects, and it is not unrigorous to conceive of an idealised coil, of constant inductance, and then apply corrections when dealing with practical devices. Hence we can differentiate the expression above on the assumption of constant inductance; i.e.: ∂Λ/∂t = L ∂I/∂t (here we use the partial differential symbol, ∂, to indicate that other dependent variables are being held constant). Now, according to Faraday's law, ∂Λ/∂t is a voltage, and so, assigning a symbol to it, we can write: Vb = L ∂I/∂t Some readers may be familiar with a different version of this equation, in which the voltage is given as negative. In fact, there is a reason for choosing a sign at this point, but the argument is a subtle one and we will look at it in detail later. It will turn out that the choice made here (i.e., do nothing) is the correct one. For the time being therefore, simply note that both L and t are defined as positive, and so the sign of the voltage is dictated by the derivative (i.e., by the sign of the gradient of the graph of I vs. t). Hence, an increasing current gives a positive voltage, and vice versa. The expression above tells us that when the current passing through a coil changes, a voltage is induced in the coil itself. This phenomenon is called 'self induction', and was discovered independently by Joseph Henry in 1832 (although it is, of course, a corollary of Faraday's law of 1831). The voltage produced is habitually referred-to as the "back EMF", which is unfortunate because it is not an EMF in the strictest sense. An inductor (not affected by any magnetic field but its own) is a passive electrical device, and no amount of work with a hacksaw will reveal a generator inside it. Hence the self-induced voltage is a reaction force, or potential-difference; and we will refer to it here as the 'back voltage' or 'reaction voltage'. The effect of the reaction voltage is to oppose the change giving rise to it (Lenz's law); which means that when an inductive circuit is connected to a generator of time-varying voltage, there will be a limitation on the current which can flow (this being in addition to the limitation provided by the resistance). By noting that the reaction is in opposition to the driving force, we have cryptically chosen a sign for L∂I/∂t (and, as we shall see, it is positive according to the conventions of circuit analysis). Now suppose that we have an idealised coil of negligible resistance. If we connect it to a battery, the back voltage will prevent the current from rising instantaneously, but the system will eventually reach equilibrium and an enormous constant current will flow. If we connect it to an alternator however, the current will never become constant, and so the back voltage will prevent the current from rising to its DC limit and will remain equal to the instantaneous applied voltage. If the generator produces a sinusoidal output, and the time reference is arbitrary (so that we can assume that V=0 when t=0), the instantaneous driving voltage at time t is given by the expression: V(t) = Vp Sin(2πf t) where V(t) should be read as " V, a function of t " (i.e., all parameters are constant, except for t), and Vp is the peak voltage. Hence, using Faraday's law: Vb = L ∂I/∂t = Vp Sin(2πf t) i.e.: ∂I/∂t = (Vp / L) Sin(2πf t) The instantaneous current can now be found by integrating both sides of this expression with respect to time, i.e.: |
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∂t |
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Sin(2πf t) dt |
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Sin(ax) dx = |
a |
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2πf L |
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Note that the constant of integration (c) in the expression
above must be zero, because it is obvious that when the peak
generator voltage is set to zero, the current will always be
zero. Thus Faraday's law has given us the relationship between
voltage and current for a pure inductance connected to a sine-wave
generator. Notice also the appearance of the quantity 2πfL,
which, since the cosine function is dimensionless, has units
of [Volts / Amp.] or [Ohms]. Thus we attribute the current limitation
to a type of resistance: 'reaction resistance' or 'reactance';
and this quantity is, of course, important enough to warrant
its own symbol, XL. What we can do now with the expressions for instantaneous voltage and current is to multiply them together to determine the instantaneous power flowing between the generator and the coil. Thus: P(t) = -( Vp² / XL ) Sin(2πf t) Cos(2πf t) This can be simplified using the standard trigonometric identity: Sin(a) Cos(b) = (½)[ Sin(a+b) + Sin(a-b) ] i.e.: Sin(x) Cos(x) = (½)Sin(2x) Hence: P(t) = -(½) ( Vp² / XL ) Sin(4πf t) The quantity Vp² is, incidentally, equal to 2Vrms² (i.e. Vp is the RMS voltage multiplied by √2), and so: P(t) = -( Vrms² / XL ) Sin(4πf t) The instantaneous power is a pure sine wave at twice the excitation frequency; and therefore, as advertised earlier, alternates between positive and negative. The average of a sine wave is, of course, zero; and so the overall work carried out by a generator on an idealised inductance is nil. Now consider the phase relationship which exists between the instantaneous values of the voltage and current. We have: V(t) = Vp Sin(2πf t) and I(t) = -(Vp/XL) Cos(2πf t) If the time co-ordinate is chosen so that the voltage is a sine wave in the mathematical sense (i.e., V=0 and changing in the positive direction when t=0), then the current is the negative of a cosine wave. There is a fixed 90° phase difference between current and voltage. Notice also that when t=0, -Cos(2πf t)=-1. When the voltage is at zero, but changing in the positive direction; the current is at its most negative, and will take a further ¼ of a cycle before it reaches zero. Thus the current in an inductance lags the applied voltage; a fact learned in kindergarten of course, but here we see how it comes from Faraday's law. If there is a fixed phase difference between the voltage and current associated with a purely inductive circuit element, then the mathematical relationship can be established without stating the time variations of V and I explicitly. Hence we can dispense with sine and cosine functions, and devise a formal mapping into an algebra which represents the phase difference by means of a unit vector in the +90° direction of a planar co-ordinate system. The choice which puts V 90° ahead of I, assuming that the phase co-ordinate increases in the anticlockwise direction (i.e., adopting the standard trigonometrical convention) is: V / I = j XL The unit vector j converts the relationship into a two-dimensional vector operation, and so V and I are now written in bold typeface. j, of course, transforms algebraically as √-1; and so we pass from Faraday's law to phasor theory. |
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9. Polarity of the induced voltage: In the preceding section, we obtained an expression for the reaction voltage of a passive inductor: Vb = L ∂I/∂t It was then commented that this expression is frequently written with a minus sign. Such ad-hoc amendment was avoided however, and the AC behaviour of inductors fell out of the algebra without mishap. Now we find ourselves in the strange position of having to justify our failure to make a widely repeated mistake. The confusion results from the misapplication of Lenz's law. Faraday's law states that a changing magnetic field will induce a voltage across the ends of a conductor suitably disposed within the field. Maxwell's equations have it that a changing magnetic field produces an electric field, but that is the same thing. If the conductor forms part of a closed circuit; then a current will flow, and may be registered by a fast-responding galvanometer. Lenz's law of 1835 tells us that the current is such that the magnetic field it produces opposes the change giving rise to it. Lenz's law, as most authors are wont to point out, is actually a manifestation of the principle of conservation of energy; the corollary being that work must be done in order to make the current flow. The implication of Lenz's law in relation to inductors is that; if a coil or loop of wire is connected to an alternator, the current which flows will induce a voltage which opposes the applied voltage. Now, since the instantaneous value of this reaction force is always in opposition to the applied force; then, according to many textbooks, the expression for it must be negative. Well, perhaps; but here we need to be extremely careful because, if we make a logical error at this point, our attempts to derive the important results of transformer theory will run into trouble. The problem is that a minus sign seems to imply that the reaction voltage is 180° out-of-phase with the driving voltage; but this view is contrary to the conventions of circuit theory. The issue can be resolved by considering the distinction between an EMF and a potential difference (PD). Except for a special case known as a 'short circuit', every impedance produces a reaction force. We can conceptualise the mechanism by saying that the generator produces a driving or 'electromotive' force, and the load pushes-back by developing a potential difference. Since every action has an equal and opposite reaction (cf. Newton's 3rd law), we may reasonably deduce that: |
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Vemf - Vpd
= 0 Thus Vpd gets its minus sign, but that does not not imply that it is out-of-phase with Vemf . On the contrary, as should be obvious from the diagram on the right, what we are really saying is that: Vemf = Vpd |
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The ambiguity arises because Kirchhoff's
second law can be stated in two ways: either, "the algebraic
sum of the voltages around a circuit is zero"; or, "the
sum of the PDs is equal to the sum of the EMFs". In the
first case, PDs are negative with respect to EMFs. When analysing
circuits however, we draw arrows, which tell us how to combine
the terms in the expressions which describe the circuit behaviour;
in which case the minus sign belongs to the analysis, rather
than to the term itself. It should all come out the same in the
end of course; but it is not unusual to find confusion carried
forward into the mathematical working. This leads to definitions
which later cause the reactive component of an impedance to change
sign on referral from one side of a transformer to another; and
a variety of obfuscatory strategies which serve to cover-up the
error. Some authors drop signs and continue to work with reactance
magnitudes. Others jump to the correct transformation rule without
proof (on the fair bet that readers will be too lazy to check);
or worst of all, treat complex numbers as reals when deriving
the transformation. In the face of such skullduggery, it is not
surprising that students give up and come to regard transformers
as incomprehensible beyond the ideal case. The difference between EMF and PD was once emphasised in the teaching of electricity. The matter lost its weight in the latter third of the 20th Century however, there being some doubt about the importance of distinguishing between two quantities which are always equal. It becomes important in dealing with inductance however, as an antidote to the assumption that the reaction force needs to be treated in a special way. There is nothing wrong with Lenz's law. In 1835 it was a brilliant observation; but the principle of conservation of energy is now axiomatic upon all physical investigation, and while Lenz's law deserves mention, it does not need to be specifically applied in this context. What is meant by that is that, if we measure voltages relative to a common reference, the "back EMF" is actually a PD, and is therefore equal in magnitude and phase to the force giving rise to it. Hence, should we be able to find a way of measuring the back voltage independently, we will find that it is very definitely in phase with the generator voltage. In fact, there is a way of doing that, using an exotic device to which we might apply the snappy title: 'inductor with integral field-probe winding' (although some may be inclined to call it a 'transformer'). The reason why the reaction force produced by an inductor is special, is that the agency giving rise to the force can be intercepted, i.e., back voltage can be observed in a manner which separates it from the driving voltage. If the intention is to measure it accurately, then this can be done by arranging another coil to be affected by the magnetic field in almost exactly the same way as the driven coil. A simple solution is to take two strands of insulated wire and twist them together, then wind several turns of this bifilar pair onto a high-permeability closed-circuit magnetic core, such as a ferrite toroid. One of the windings can then be connected to a signal generator, and the magnitudes and phases of the voltages across the driven and the sensor windings can be compared using a dual-channel oscilloscope. In doing that however, we must keep track of the phasing of the windings, and so we arbitrarily define a lead wire going into one side of the core as the 'start', and the wire coming out of the other side as the 'finish' of the associated winding. We then label the 'starts' with dots on the circuit diagram. |
| Now, provided that the magnetic coupling between the two coils is very tight, it will be found that closing the switch 's' in the circuit on the right will make practically no difference to the relationship between Vgen and Vback. In fact, there will be a small difference due to losses and incomplete flux linkage, but not sufficient to be noticeable within the precision achievable from oscilloscope measurements. |
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So now to the question of why the
voltage (strictly, the open-circuit voltage) across the extra
winding is equal to the back voltage produced by the inductor.
The reason is that the back voltage is due to a changing magnetic
field, and the experiment has been constructed in such a way
that the fields affecting the two windings are practically the
same. Hence, since the two windings in this case are also identical
(they have the same number of turns and occupy almost the same
volume of space), the voltage across the sensor coil is (practically)
equal to the reaction voltage which controls the current flowing
from the generator. We have not learned anything new about magnetism from that; but what we have deduced is fundamental. A transformer is a device which allows the reaction voltage produced by one inductor to be sampled by means of another inductor (without necessarily making any direct electrical connection). To that we can add, that the sampling can be in whole or in part, depending on the degree of magnetic coupling; a matter which we will quantify shortly by introducing the concept of mutual inductance. At the risk of getting a little ahead of ourselves, we can also say that the transformer is a reciprocal network, i.e., what applies to one winding applies to any other. Hence we can use an auxiliary coil to create a time-varying magnetic field which will modify the reaction voltage produced by an inductor. The simplest way in which to do that is to connect an impedance to the second winding; in which case, if the load contains a resistive element, then the reaction will be changed in such a way as to allow energy to flow from the generator to the resistance. Another possibility however, so deleteriously neglected in many textbooks, is to connect an active network to the second winding and thereby use a transformer (say) to cancel an unwanted component of a signal, or use a 'neutralisation' winding to cancel the 'self-capacitance' of an inductor. |
| The more exotic uses of transformers require a clear understanding of the phase relationships between all of the currents and voltages involved. This, as mentioned above, is a general source of confusion; but to begin, we can note that closing the switch in the circuit above converts our bifilar transformer into an inductor wound with two-strand bunch-wire, and we end up with the circuit on the right. What we have now is a 1:1 autotransformer; a network of dubious practical merit, but leaving little doubt regarding the relative phases of Vin and Vout. |
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© D W Knight 2009.
David Knight asserts the right to be recognised as the author of this work.
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