Electromagnetic induction, part 2. David W Knight

3.3. Electromagnetic Induction: Part 2.
Contents:

10. Mutual induction.
11. Reciprocity.

<<< Work in progress>>>
.

10. Mutual induction:
The extraordinary property of Faraday's way of thinking about electromagnetic induction is that it is relativistic. Increasing the current increases the number of flux lines, and reducing it vice versa. Moving a loop through a field of external origin causes flux lines to switch from being on the outside of the loop to being on the inside, and vice versa. Either way, the number of flux linkages changes and a voltage is induced; and there is no distinction between a moving field and a changing field. This strange correspondence fascinated both Maxwell and Einstein, and led to a mathematical formulation which Faraday, by his own confession, did not fully understand. Still Faraday could see what was going on in his mind's eye, and we reject his wonderful metaphor in favour of a more abstract view at our peril. We need a formal and rigorous general approach of course; but there is no substitute for having a good idea of what the answer will be before tackling some complicated mathematical problem.
Here, of course, we focus on the effect of a changing field (transformers), rather than the effect of a moving field (generators); but regardless of how we fix our attention (and given the complicated nature of the three-dimensional problem) it must seem, at first, that we are in theoretical deep water. It transpires however that, by using the concept of flux linkage, the matter of incorporating transformers into circuit theory becomes a straightforward topological problem; i.e., a matter of interconnectedness, like two-terminal network theory itself. By using a topological argument, we can capture the magnetic interaction between any pair of inductors in a single parameter; and although we will still need to use field theory to calculate that parameter from first principles, there will be many situations in which we will be content just to measure it.

The general mechanism whereby a time varying current in one coil induces a voltage across the ends of another is illustrated on the right. Here we can see that some, but not all, of the flux links with the pick-up coil; and it should also be obvious that the induced voltage will depend on the proximity of the coils. The law which governs the relationship between inter-coil distance and output voltage is dependent on the geometry of the field, and so will be complicated; but if the system is linear (which it will be to a good approximation in the absence of hard magnetic materials), then the complexity of the field problem cannot add complexity to the circuit-analysis problem. In other words; for any given frequency component in the input current, there will only be one frequency component in the output voltage. This means that, for a given geometry, the relationship between I1 and V2 at a particular frequency is governed by a mathematical constant.

     Recall that the self-inductance of a coil is defined as the number of flux linkages per unit current. It follows that, in the case where current in one coil produces flux linkages in another; there is a parameter, having units of inductance, which will allow us to calculate the induced voltage. That parameter, not surprisingly, is called the mutual inductance, and is defined as: 'the number of flux linkages in coil 2 per unit of current in coil 1' ; i.e.:
M = Λ21 / I1        [Henrys]
Notice that the current is written as a pseudoscalar. As before, it is necessary to use instantaneous voltages and currents when relating circuit parameters to field parameters; and the actual phase relationships can be incorporated later.
     Now, referring again to the diagram above, consider what will happen if the pick-up coil is rotated through 90° about its centre point in the plane of the diagram. The flux which links with one half of the coil will then be in the opposite sense to that which links the other half; and there will be found a critical point at which the total number of linkages is zero. Individual linkages can be positive or negative. Hence, at the critical point, M will be zero, and there will be no output.
       If the pick up coil is rotated through a full 180°, the polarity of the voltage V2 will be reversed. Notice however, that there will be no intermediate output phases during the course of the rotation; the voltage will simply diminish with constant phase until the 90° point is reached, and then reappear inverted. Hence mutual inductance is pseudoscalar; it can be positive or negative, but it has no intermediate directions because it is conceived as the sum of positive and negative linkages per unit current. A surprising corollary of this observation is that M must be given by a two-valued function; and in view of the parameters from which it has to be derived, it must be related to the geometric mean of the inductances of the two coils. Hence we can deduce the general form:
M = k √(L1 L2)        [Henrys]
where k is a dimensionless factor, called the coupling coefficient, which we will shortly derive from the field topology. Notice that, if we take the self-inductances as given, k provides a dimensionless alternative to mutual inductance, and will therefore turn out to be a convenient analysis parameter.
     Before we declare unequivocally that M can be either positive or negative however, we must be mindful of convention. When analysing circuits which have a fixed physical configuration, it is good practice to mark the start of a winding with a dot. Hence, upon 180° rotation of the pick-up coil (or reversal of the winding direction, which amounts to the same thing), the dot jumps from one end of the winding to the other. In that case, if the circuit-analysis arrow adjacent to V2 is always drawn pointing towards the dot, then M will remain unchanged. Hence M can usually be made positive by considering the phasing of the windings; and this is generally a sensible idea. There is one situation in which M must be allowed to change polarity however, and that is when dealing with variometers; a variometer being a pair of coils with a control shaft allowing one to be rotated relative to the other (see A3.1).

Although we will shortly consider the general case in which the coupling between coils is incomplete; it is instructive to begin with the special limiting case, never quite achievable in practice, in which all of the flux from one coil links with the turns of another. This, of course, is the condition which defines an ideal transformer. One way to approximate complete linkage (neglecting internal inductance) is to use twisted bifilar winding (as in the example given in the previous section); but that is only possible when both coils have the same number of turns (and the distributed capacitance between the wires complicates matters at high frequencies). To link the flux (almost) completely between coils having arbitrary numbers of turns, a magnetic circuit of low reluctance (i.e., a transformer core) is required. For circuit analysis purposes however, we are not concerned with how it is done; provided that the models we end up with can be corrected for the deficiencies of practical devices.

The logical step which will enable us to deduce the input-output relationships (transfer functions) for any system of coupled inductors in the lumped-component limit (i.e., when conductor lengths and the distances between coils are small in comparison to wavelength), lies in the observation that flux linkage is a relationship between the flux through the path and the turns in a particular coil. In other words; flux linkage does not figure directly in the coupling between separate circuits. Hence the flux on a path through several coils is notionally de-linked from the originating coil and re-linked to the others. When we apply this idea to the simple ideal transformer case, the generalisation will soon become apparent.
Note that, in the working to follow, we will assume that all coils are wound with wire which has no resistance. This is permissible in the limit that there are no capacitive currents (i.e., all current follows the wires), because conductor resistance, like internal inductance, is associated with physical processes in the body of the wire, and is therefore excluded from the magnetic interaction (to a very good approximation at least). Conductor losses can therefore be included in circuit analysis as a separate lumped component in series with the wire. Magnetic losses are a separate matter, and can be handled by defining permeability (and hence inductance) as complex; but for the present purpose we will assume that magnetic media are ideal.

10a. Mutual inductance, ideal case.
The diagram below gives a topological representation of the coupling which takes place in an ideal transformer; i.e., the spatial distribution of the field is ignored, and only the route taken by the magnetic flux as it threads through the coils is considered. Information about path length and area has not been discarded however, because it is wrapped up in the inductances and turns numbers, i.e., in the parameters L1, N1, L2 and N2. Notice incidentally, that in the ideal case, the effective number of turns in a coil is the same as the actual number of turns, and so here we use Ns without hats.

The self-induced voltage across the coil L1 is equal to the instantaneous applied voltage V1. It is also equal to the rate of change of flux linkages in L1 per unit time. Hence:
V1 = ∂Λ1/∂t = N1 ∂Φ/∂t
Thus the rate of change of flux in the core, i.e., the actual flux after de-linking from L1 is:
∂Φ/∂t = V1 / N1
The instantaneous voltage induced across the coil L2 is given by the rate of change of its flux linkages per unit time. Hence:
V2 = ∂Λ21/∂t = N2 ∂Φ/∂t
Thus:
V2 = V1 N2 / N1

This, of course, is the classic turns-ratio rule for the ideal transformer; and so offers no surprises, except perhaps for the simplicity of its derivation. Now, mindful that the back voltage is equal to the applied voltage, the inductance L1 is defined in the relationship:
V1 = L1 ∂I1/∂t
Using this as an analogy, we can propose a definition for mutual inductance having the same form; i.e.:
V2 = M ∂I1/∂t
According to this definition, the voltage ratio is:
V2 / V1 = M / L1
The transformer however is a reciprocal network. We can swap the input and the output; and in terms of the analysis, the only effect will be to swap the subscripts 1 and 2 (a proof of this proposition will be given later, but for the time being we will simply accept it). Hence:
V1 / V2 = M / L2
Thus:
M / L1 = L2 / M
i.e.:
M = √(L1 L2)
Recall, from the earlier discussion, that the expression for mutual inductance was expected to be of the form:
M = k √(L1 L2)
Hence, the coupling coefficient, k, for an ideal transformer is 1 (or -1 if we point one of the voltage arrows away from the dot).

10b. Windings in series, ideal case.
When two windings on the same transformer are placed in series, a single inductance is obtained; but due to the coupling between the coils, the total inductance is not the same as the sum of the two separate inductances. The reason for this, of course, is that the flux from coil 1 links with the turns of coil 2, and vice versa. This situation is represented topologically in the diagram below. The two fluxes are, of course, superposed (and therefore indistinguishable) in the actual field; but it is perfectly legitimate to consider them separately for accounting purposes. Now (noting that both coils carry the same current) we can determine the inductance, by inspection, as the total number of flux linkages per unit current; i.e.:

L = [ Λ1 + Λ2 + Φ1 N2 + Φ2 N1 ] / I
The terms are, in order: the flux linkage in coil 1 due to its own current; the flux linkage in coil 2 likewise; the linkages in coil 2 due to the flux from coil 1; and the linkages in coil 1 due to the flux from coil 2. We can also make the following substitutions:
L1 = Λ1 / I , L2 = Λ2 / I , Λ1 / N1 = Φ1 , Λ2 / N2 = Φ2
Hence:
L = L1 + L2 + L1 (N2 / N1) + L2 (N1 / N2)
But, from the previous section, we have:
N2 / N1 = M / L1 = L2 / M
Hence
L = L1 + L2 + 2M

Now consider what happens when the connections to (say) L2 are reversed. In that case, the number of turns N2 becomes negative and the direction arrow for the flux Φ2 is reversed (i.e., Φ2 becomes negative also). Thus the cross linkages, Φ1N2 and Φ2N1, both become negative. Hence:
L = L1 + L2 - 2M
The two fluxes in superposition tend to cancel. The amount of energy which can be stored in the field for a given current is reduced, and so the inductance is reduced. Thus there are two configurations for coupled inductors in series, known as series aiding, and series opposition; and in general, the expression for the total inductance is:
L = L1 + L2 ± 2M
One property of this expression which is not immediately obvious is that the total inductance is always positive (assuming that L1 and L2 are positive); even when the coils are in opposition and the coil deemed to have negative turns has more turns than the other. This can be seen by considering the requirement for complete inductance cancellation (which is only possible in the ideal case), i.e. L=0 and so:
L1 + L2 = 2M
Using the substitution: M=√(L1 L2) , we get:
L1 + L2 = 2 √(L1 L2)
and squaring both sides gives:
(L1 + L2)² = L1² + L2² + 2 L1 L2 = 4 L1 L2
Hence:
L1² + L2² = 2 L1 L2
L1² + L2² - 2 L1 L2 = 0
i.e.:
(L1 - L2)² = 0
Complete cancellation can only occur when L1= L2, and so the mutual inductance term cannot be greater than the sum of the separate inductances. This constraint also sets an upper limit on the inductance which can be obtained when the coils are series aiding, i.e., when
L = L1 + L2 + 2√(L1 L2) and L1 = L2 then L = 4L1
What this says is that when the number of turns in a fully-linked coil is doubled, then the inductance is increased by a factor of 4. This rule, of course, is also captured in the expression for L in terms of the transformmer-core inductance factor:
L = AL N² where AL = μ Ae /

e
Earlier we gave the inductance for two windings in series as:
L = L1 + L2 + L1 (N2 / N1) + L2 (N1 / N2)
Putting this in terms of the inductance factor we have:
L = AL [ N1² + N2² + N1² (N2 / N1) + N2² (N1 / N2) ]
i.e.:
L = AL ( N1² + N2² + 2N1N2 )
which factorises:
L = AL ( N1 + N2)²
This gives a slightly broader interpretation of the inductance factor formula because, in the series opposition case, we can consider the quantity N1+N2 to represent the effective number of turns as the sum of positive and negative turns; i.e.:
L = AL Ñ² where Ñ = N1 + N2
with N1 (say) positive and representing turns wound clockwise around a flux which recedes when the current is positive; and N2 negative and representing turns wound anticlockwise around the same flux. No matter whether the negative number is greater in magnitude than the positive number however; the inductance is always positive because it is proportional to the square of the total.

10c. Mutual inductance, general case.
The simple flux-linkage counting principle is easily extended to the general (i.e., non-ideal) case. Starting with the unloaded transformer, we can say that the driven coil (which we may refer to as the 'primary' when it is the only coil connected to an active network) produces a flux:
Φ = Λ1 / Ñ1
Of this however, only a proportion, kΦ (say), threads through the turns of L2, and the remainder, (1-k)Φ, is private to L1. Notice here that the symbol, k, given to the communal proportion of the total flux, is the same as for the coupling factor discussed earlier. We have, as yet, no grounds for making such an association, but what follows is by way of a proof that they are the same. Also notice that the communal flux is shown, in the diagram below, as threading through all of the (effective) turns of L2. What happens in reality is that the whole of the flux from L1 threads through some of the turns of L2; but from a flux-linkage counting point-of-view, the two conceptions are interchangeable. It is the transpositional equivalence between 'all of some' and 'some of all' which lies behind the principle of reciprocity in the coupling between pairs of inductors. What it means is that, if the coupling is expressed in a dimensionless manner, then the coupling from coil 1 to coil 2 will be the same as the coupling from coil 2 to coil 1.
Now, as before, we relate the voltage driving L1 to the rate of change of flux linkages, but this time we obtain the time-varying flux from the effective number of turns; i.e.:

V1 = ∂Λ1/∂t = Ñ1 ∂Φ/∂t
Thus, bearing in mind that only a proportion (k) of the flux threads L2, we determine the induced voltage:
V2 = ∂Λ21/∂t = Ñ2 k ∂Φ/∂t
Hence:
V2 / V1 = k Ñ2 / Ñ1
Also, defining mutual inductance as before, we have:
V2 = M ∂I1/∂t and V1 = L1 ∂I1/∂t
Hence:
M / L1 = k Ñ2 / Ñ1
and by the principle of reciprocity:
M / L2 = k Ñ1 / Ñ2
Hence:
M / (L1 k) = L2 k / M
and so the general expression for mutual inductance is:
M = ± k √(L1 L2)

The coupling coefficient is the proportion of the flux produced by a current in one of the coils which is shared by both coils. The general expression for M can carry either algebraic sign, because it is derived from the real two-valued function: √(k² L1 L2); but it gives no information on how to make the choice. For a complete definition of M, we need to use Maxwell's corkscrew rule (or phasing dots) to resolve the ambiguity.
The voltage ratio, which is equal to the turns ratio in the case of an ideal transformer, is in general (and as shown above) proportional to the ratio of the effective numbers of turns. Recall that Ñ = √kH, and in the case of a current-sheet solenoid, kH = kL (Nagaoka's coefficient). Hence, for pairs of current-sheet solenoids in proximity:
V2 / V1 = k (N2 / N1) √(kL2 / kL1)
kL depends only on the length / diameter ratio of the solenoid. It does not change with the number of turns, provided that the winding does not change in length (or diameter) when turns are added or subtracted. Hence, to a good first-order approximation (i.e., the current-sheet approximation) the voltage induced across a solenoid depends on the turns ratio, the distance between the coils, and the overall shapes of the coils. It does not depend strongly on the turn spacings and wire diameters, provided that the wire diameter is small in comparison to the coil diameter. Even so, the calculation of M (or k) from field considerations is somewhat involved. For those interesed in doing so, the subject is covered in detail in F W Grover's monograph "Inductance Calculations" [Grover 1946] and in the (public domain) NBS science paper [Sci 169] on which the later book is based.

10d. Coupled inductors in series, general case:
Having established the physical meaning of the coupling coefficient, determining the inductance of a series combination of coupled inductors is now straightforward. It is, of course, the total number of linkages per unit current; i.e.:

L = [ Λ1 + Λ2 + Λ21 + Λ12 ] / I
where the last two terms are respectively: the linkages in coil 2 due to the flux from coil 1; and vice versa. Now identifying:
Λ21 = Ñ2 Φ21 where Φ21 = k Φ1 and Φ1 = Λ1 / Ñ1
and vice versa, we have:
L = L1 + L2 + [ (Ñ2 / Ñ1) k Λ1 + (Ñ1 / Ñ2) k Λ2 } / I
i.e.:
L = L1 + L2 + (Ñ2 / Ñ1) k L1 + (Ñ1 / Ñ2) k L2
but, from the working in the previous section:
M = (Ñ2 / Ñ1) k L1 = (Ñ1 / Ñ2) k L2

Hence:
L = L1 + L2 + 2M
Also, if we swap the connections to one of the coils, we make both the turns in that coil and the flux from that coil negative; causing both of the mutual inductance terms to change sign. Thus, in general:
L = L1 + L2 ± 2M
The expression is the same as for the ideal case, the difference being that the magnitude of the mutual inductance is not at its theoretical maximum. An equivalent expression is:
L = L1 + L2 ± 2k √(L1 L2)
which gives the ideal case when k=1.

10e. Measuring M:
One matter which must be understood is that a pair of coupled inductors with a coupling coefficient falling considerably short of unity does not constitute a failed transformer design. On the contrary, there are many situations in which a small value of k is desirable; particularly in the design of bandpass filters. An important application (which we will examine later) lies in the flat-topped bandpass characteristic which can be obtained by using a pair of loosely-coupled LC parallel resonators. This, of course, is the basis of the ubiquitous superheterodyne IF (intermediate-frequency) transformer.
In contrast to the usefulness of loosely coupled transformers, there are fewer obvious applications for series-connected pairs of loosely-coupled inductors. Instead, the utility of the configuration lies in the fact that it can be used to measure the mutual inductance of a pair of coils which are intended for use as a transformer. If the coils are connected series aiding, then the inductance is:
L+ = L1 + L2 + 2M
And when the coils are connected in series opposition, the inductance is:
L- = L1 + L2 - 2M
Subtracting the latter from the former gives:
L+ - L- = 4M
If the inductances of the coils are measured separately, k can also be determined; i.e.:
k = M / √(L1 L2)

11. Reciprocity.
If a pair of coupled inductors is used as a transformer; it will be found, in general, that the output voltage falls short of that predicted by using the effective turns ratio (i.e., Ñ2/Ñ1). Assuming that any measurement data are corrected for the effects of wire resistance and other circuit parasitics; the shortfall is given by the coupling coefficient k ; i.e.:
V2 / V1 = k Ñ2 / Ñ1
If we swap the roles of the two windings, i.e., we connect coil 2 to the generator and measure the voltage across coil 1, we will find that the shortfall is given by exactly the same factor; i.e.:
V1' / V2' = k Ñ1 / Ñ2
This is what is meant by the principle of reciprocity as applied to transformers.
Now notice that in the second case, the voltages have been given primes (i.e., marked with apostrophes). That is because they are not the same voltages as in the first case. To understand this point, consider a 1:1 transformer made from current-sheet coils both having the same value of Nagaoka's coefficient; i.e., Ñ1=Ñ2. Using coil 1 as the primary we have:
V2 = V1 k
and using coil 2 as primary:
V1' = V2' k
In the first instance, V2 is less than V1. In the second instance V1' is less than V2'. A shortfall in using the transformer one way does not translate into an anti-shortfall when the windings are swapped.
The concept of 'effective turns number', Ñ, was introduced earlier as a logical necessity in ensuring that flux-linkages are counted correctly. It is involved in the definition of inductance twice, once in the determination of MMF, and once in the counting of linkages to the resulting flux. Hence the inductance of a coil is given by an expression:
L = Ñ² μ A /

This can be turned into an expression involving actual turns, thus:
L = N² kH μ A /

from which we deduce the identity of the linkage efficiency parameter:
√kH = Ñ / N
We can also agregate the path geometry and permeability parameters into a single inductance factor; i.e.:
AL = μ A /

Hence:
L = Ñ² AL = N² kH AL
The use of inductance factors is familiar practice when dealing with transformer cores; but here we generalise it to any coil, with the proviso that it is the effective number of turns, not the actual number, which is required when determining inductance. The square of the linkage efficiency (i.e., kH) is associated with the turns, not with the inductance factor; because AL is a measure of magnetic conductivity. In other words, we argue that kH cannot be made to disappear by factoring it into AL, because the reluctance of a path is not affected by the efficiency with which the flux from a particular tangle of wire is injected into it.
     So far, so good; but many readers will be aware that linkage inefficiency is not discussed (or even recognised as a possibility) in standard teaching references. Also absent is any attempt at proving the principle of reciprocity. The latter is so fundamental to our understanding of transformers, that the lack of its confirmation from first principles must be regarded as a glaring ommission; and yet the issue is never addressed. These two shortcomings in the traditional way of developing the subject are, in fact, related. As will now be demonstrated; it transpires that the principle of reciprocity cannot be proved unless linkage efficiency is taken into account.
     In the discussion to follow, we will make use of a theoretical model which is topologically equivalent to the general case of a pair of coupled inductors. We will then have two choices in the matter of how to derive the principle of reciprocity; either by using actual turns, or by using effective turns. In the former (pathological) case; we will arrive at a mathematical paradox; i.e., a statement, obtained without algebraic error, but which is obviously untrue. In general, a paradox can occur whenever a problem is addressed using an incorrect set of starting assumptions. In this case, the resolution lies in finding a correct definition for the overall magnetic path, and hence a measure of the extent to which the turns in a particular coil are linked to flux on that path. Thus it will be shown that the use of effective turns, instead of actual turns, is required for reasons of mathematical consistency. Once the path and the linkages are correctly defined, proof of the principle of reciprocity is trivial.
     In order to simplify the working, we will make use of a relationship which exists between magnetic flux and the generalised inductance factor. Recall that inductance is defined as the number of flux linkages per unit current. Thus we have a list of alternatives:
L = Ñ² AL = Λ / I = Ñ Φ / I
where the effective number of turns, in this instance, is that of the coil carrying the current I. Hence:
Ñ² AL = Ñ Φ / I
i.e.:
Φ = Ñ AL I
Also, the rate of change of flux on a path, from which the voltage induced in a pickup coil can be obtained (by multiplying by the effective number of turns in the pickup coil), is given by differentiation; i.e.:
∂Φ/∂t = Ñ AL ∂I/∂t
The collection of the physical path parameters into a single coefficient, AL, constitutes the mapping from the field to the topological representation.
     In the previous section, the magnetic flux associated with a pair of coupled inductors was separated into four parts. These were: the flux which is private to coil 1; the flux which is private to coil 2; the flux from coil 1 which links with the turns of coil 2; and the flux from coil 2 which links with the turns of coil 1. In the context of conventional (tightly-coupled) transformer theory, the two private fluxes are known as leakage fluxes; i.e., they are the fluxes which escape from involvement in the coupling. The magnetic system so defined is also meant to account for all of the flux; i.e., any flux line which fails to link with the turns of both coils belongs, by definition, to the leakage flux of the coil which produced it. In this way, the four fluxes are divided between three magnetic paths; these being: the communal path and the two leakage paths.
     The general situation, as just described, is captured in the topological model shown below. One obvious feature of this model, is that it can also be taken to represent a structure consisting of two coils wound on three transformer cores; the behaviour of the system being considered in the limit where the magnetic flux outside the cores (or paths) is negligible. Thus we have an ideal transformer made non-ideal by the provision of leakage paths; an arrangement which is exactly analogous to the way in which coupling occurs in practical devices.
     A strange property of the model is that; if the reluctance is finite on a given path, but elsewhere infinite; then the flux from a particular coil (neglecting internal inductance) has no choice but to occupy the paths provided for it. Hence there can be no linkage inefficiency in the coupling between (say) coil 1 and paths a and b. Thus we can (notionally) calculate the inductance L1 from the actual turns and the inductance factors ALa and ALb.
     The overall inductance factor which results when a coil is wound around a pair of transformer cores is easily derived from first principles. The applied voltage is equal to the back voltage, which is in turn equal to the rate of change of linkages per unit time. If the flux is linked to every turn, then the total number of linkages is simply NΦ; and since there are two paths, the flux is divided into two parts. Hence, for coil 1 we can write:

V1 = ∂Λ1/∂t = N1 ∂Φ1/∂t
= N1 ∂(Φ1a + Φ1b)/∂t = L1 ∂I1/∂t
but earlier we derived the relationship:
Φ = Ñ AL I
where, in this case Ñ=N, and so:
L1 ∂I1/∂t = N1 ∂(N1 ALa I + N1 ALb I)/∂t
i.e.:
L1 = N1² (ALa + ALb)
We can also use an identical method (but with different subsripts) to derive the inductance of coil 2. Thus:
L2 = N2² (ALb + ALc)

When a coil is wound around several paths, the overall inductance factor is simply the sum of the individual inductance factors. This can also be deduced by analogy with admittance theory; because we know that the overall conductance of a set of parallel conductances is obtained by addition.
     So now; having (perhaps) convinced ourselves that we can deal with the reciprocity problem by using actual turns, we can examine the implications of what we have obtained so far. We know from earlier work that k exists in the following relationships:
V2 / V1 = k Ñ2 / Ñ1 = M / L1     and     V1' / V2' = k Ñ1 / Ñ2 = M / L2
Also, since we have successfully defined the inductances using actual turns, it might seem appropriate to use actual turns here (and had we followed the traditional line of development we would have no choice). Thus:
k N2 / N1 = M / L1     and     k N1 / N2 = M / L2
Eliminating M we get:
k L1 N2 / N1 = k L2 N1 / N2
We cannot determine k using this expression, but we can substitute for L1 and L2. Thus:
(ALa + ALb) N1² N2 / N1 = (ALb + ALc) N2² N1 / N2
which boils down to the statement:
(ALa + ALb) = (ALb + ALc)               × Not true!
ALa does not have to be the same as ALc. This may not be so obvious when trying to imagine the three-dimensional fields; but we can easily build a practical version of the model using toroidal inductor cores, and if resistance and internal inductance are treated as separate lumped elements in series with the windings, the experiment can be made to agree with the theory to within a few parts per thousand. We are therefore at liberty to choose radically different AL values for the two leakage paths. This means that there can be only one situation in which we can make ALa and ALc the same by definition, and that is when ALa=ALc=0 ; i.e., the assumption of perfectly efficient linkage is only true when there are no leakage paths. Such a condition, of course, applies only to ideal transformers.
     So where did we go wrong? The answer is that the inductances of the coils were obtained correctly, but it was done in the wrong way. If a transformer is not ideal; then flux linking is by definition inefficient, because there is a magnetic sub-path in the system to which the flux from a particular coil cannot link. In the case of coil 1 (say); the turns are wound around paths a and b, but they cannot produce any flux on path c. The inductance therefore falls short of the maximum possible for the overall path. Now observe that if the turns of a coil were to be wound around all three paths (as can be achieved in practice by winding a coil on a stack of toroidal inductor cores), then the inductance factor for that coil would be to sum of the inductance factors for the three paths. Hence we can define the inductance factor for the overall path as:
AL = ALa + ALb + ALc
Now, mindful of the linkage inefficiency, we can properly write expressions for the inductance L1 as follows:
L1 = Ñ1² AL = N1² kH1 AL = Ñ1² (ALa + ALb + ALc)
But earlier we obtained another expression which also gives the correct inductance:
L1 = N1² (ALa + ALb)
combining the two versions we have:
N1² (ALa + ALb) = N1² kH AL
i.e:
kH1 = (ALa + ALb) / (ALa + ALb + ALc)
A similar argument applies to coil 2, to give:
kH2 = (ALb + ALc) / (ALa + ALb + ALc)
We can immediately check that this reasoning is mathematically consistent by testing it in the voltage ratio relations used earlier; i.e. since:
V2 / V1 = k Ñ2 / Ñ1 = M / L1     and     V1' / V2' = k Ñ1 / Ñ2 = M / L2
then
k L1 Ñ2 / Ñ1 = k L2 Ñ1 / Ñ2
and substituting for the inductances gives:
AL Ñ1² Ñ2 / Ñ1 = AL Ñ2² Ñ1 / Ñ2
i.e., M is now the same regardless of which winding is used as the primary.
     We have now, effectively, demonstrated that the transformer is a reciprocal network when the magnetic path is properly defined; but the proof will be more convincing if we derive expressions for k in the two possible ways, i.e., when coil 1 is used as the primary, and when coil 2 is used as the primary. The voltage induced in coil 2 by an alternating current in coil 1 is:
V2 = ∂Λ21/∂t = Ñ2 ∂Φ21/∂t = Ñ2 k ∂Φ1/∂t = M ∂I1/∂t
but
k Φ1 = Φ1b = Ñ1 ALb I1
hence:
Ñ2 Ñ1 ALb ∂I1/∂t = M ∂I1/∂t
Also we have, from the voltage ratio:
V2 / V1 = k Ñ2 / Ñ1 = M / L1
hence:
M = k L1 Ñ2 / Ñ1 = Ñ1 Ñ2 ALb
i.e.:
k Ñ1² AL Ñ2 / Ñ1 = Ñ1 Ñ2 ALb
and so:
k = ALb / AL = ALb / (ALa + ALb + ALc)
It is trivial to repeat the derivation using coil 2 as the primary, and the result is exactly the same. Therefore the transformer is a reciprocal network (QED). We can also check this result against the standard form:
M = k √(L1 L2) = k √( AL Ñ1² AL Ñ2² ) = Ñ1 Ñ2 AL k = Ñ1 Ñ2 AL ALb / AL
The mutual inductance expressed using actual turns is:
M = ALb N1 N2 √( kH1 kH2 ) = ALb N1 N2 √[ (ALa + ALb) (ALb + ALc) / AL² ]
This is, unfortunately, not a practical starting point for the calculation of the mutual inductance of pairs of air-cored coils; but the model used does suggest a way of using toroidal inductor cores to make loosely-coupled transformers for bandpass filters, the advantage being minimal stray magnetism (i.e., no need for a screening can).

12. Magnetic shunt:

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< writing in progress >

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Transformer circuit theory:

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Compensation Theorem:

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In the discussion - , it was stated that the back voltage is actually a potential difference,

the substitution of an EMF in place of a PD is sometimes permissable (and analytically useful) according to a wisdom known as the compensation theorem. This 'theorem' is argued to be a corollary of Kirchhoff's 2nd law, and like Kirchhoff's laws, requires little conscious effort once the principle has been understood. It is simply the observation that, for the purposes of circuit analysis, an impedance can be replaced by a perfect generator (i.e., a generator with zero output impedance) having a voltage across its terminals equal to the voltage drop across the impedance. In other words, if the impedance is Z and the current through it is I, then the voltage across the substitute generator is given by:
V = I Z
Hence there are situations in which it is permissable to substitute EMF in place of PD, and in circuit analysis we often do so without thinking; but there are some serious pitfalls in using this approach.
The first issue is that we cannot replace every generator in a network with an impedance, i.e., we cannot always perform the inverse substitution. This fundamental irreversibility should make us deeply suspicious, and rightly so; because it implies a violation of the principle of conservation of energy every time a substitution is made.

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Caveats:
Component model is a closed conservative system.
Compensation theorem is non-conservative - every application eliminates an energy transfer process and replaces it with an energy delivery process; i.e., reverses the Poynting vector.
Application of comp. theorem to "back EMF" gives the wrong sign for the transferred reactance. Hence complex conjugate is obtained when deriving the Z transformation rule.

Primaries, secondaries, terminals and ports:
It is not unusual to read introductory passages which say something like: "A transformer is a device which has a primary winding and one or more secondary windings." It takes some reading between the lines to grasp the full implication of such a statement, which is: 'we neglect the possibility that there may be active networks connected to several windings, and so confine our skill to that of dealing with only the most trivial of magnetic problems'.

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Bring transformers into phasor theory as multi-port networks with defined transfer functions.

A transformer is a system of coupled inductors with three or more terminals.

A transformer is a system of coupled inductors with two or more ports.

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Hybrids:

No distinction between pri. and sec.
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Conventional transformer.
Leakage inductance.

"The short circuited turn", Thomas Roddam, WW March 1957 p114-117
"The short circuited screen" (Letters to the editor), Thomas Roddam, WW June 1957, p274.
"Effect of a Conducting Shield on the Inductance of an Air-Core Solenoid." Ted L Simpson. IEEE Trans. on Magnetics, Vol. 35, No.1, Jan 1999, p508-515.

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TX to Ae

Ch.3 Contents

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