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INTERNATIONAL JOURNAL OF NUMERICAL ANALYSIS AND MODELING, SERIES B c 2012 Institute for Scientific Computing and Information Volume 3, Number 3, Pages 224–241 PASSAGE OF HIGH-FREQUENCY SIGNALS THROUGH POWER TRANSFORMERS. A. A. LACEY Abstract. We consider the possibility of passing high-frequency signals past power transformers forming part of an electrical grid. We first model a transformer, including its laminated core, to obtain asymptotic behaviour of currents and voltages in the secondary circuit. Having got this we are able to determine the effects of different by-pass mechanisms which might be tried to get the high-frequency signal from the primary to the secondary circuit. Key words. Homogenisation, thin layers, high frequency. 1. Introduction A possible cheap method – hopefully requiring little extra hardware and wiring – of transferring data between, on the one hand, individual consumers of electricity, and on the other, one central data-processing/interrogation point, is to transmit a high-frequency signal along power lines. Unfortunately, there are a number of barriers to such a signal, in the form of transformers used to step down the voltage of the power supply. The transformers used for reducing the voltages utilised by the main transmission lines are not a major problem, as there are relatively few of them and installing special equipment to get high-frequency signals past them might be commercially viable. However, there are many more transformers employed in local sub-stations so a cheap and simple way of ensuring signals get past these is needed, if the technique is to be economical. We start, in Section 2, by looking at basic electromagnetic theory applying in a power or distribution transformer. In particular, we see one reason why it is observed, [3], [9], [10], [22], that at “low” frequencies it is seen that power loss by the transformer (power input into the primary winding less power output by the secondary coil) is small, order of frequency squared, while for “high” frequencies there is higher power loss, of order one, and power output decaying as a power of frequency. To get the correct sizes of losses, the laminated structure of the transformer’s magnetic core must be modelled. Typically, the core consists of alternating layers of a conducting ferro-magnetic and an insulating non-magnetic material. The width of the layers is order 0.4 mm, compared with a macroscopic length scale of 1 m. This means that averaged equations can be derived. In building our model we take, for simplicity, the ferro-magnetic core to behave as a simple material so that magnetic induction is proportional to magnetic field; non-linear behaviour, such as saturation, hysteresis and kinetic effects are all disregarded. Note that power losses discussed in the present paper result largely from eddy currents alone. (It should be noted, however, that in practice hysteresis produces most of the power losses, [18].) We see that the laminated structure of the core keeps (as is well known) power losses due to eddy currents low at a normal mains frequency of 50 Hz, but there is a large power loss at the desired frequency of the signal. Received by the editors April 2011 and, in revised form, June 2012. 2000 Mathematics Subject Classification. 35Q60, 41A60, 78A48, 78A99, 78M40. 224 HIGH-FREQUENCY SIGNALS AND TRANSFORMERS 225 The main treatment of the laminated core is based on homogenisation, using the method of multiple-scales (see, for example, [8]), to obtain an averaged model for this particular structure. This particular problem seems not to have been fully solved in the substantial literature on homogenisation, much of it rigorous, see [19] and [20] for quite general problems, and [1], [2], [4], [5], [6], [11], [12], [17], [21] which consider electro-magnetic fields in various types of heterogeneous media. Of particular note are [13], which looks at layered materials, one of which is a perfect insulator, and [7], which discusses the relationship between the small size of included materials and other small parameters which can appear in particular problems. In the case of present interest, we shall be concerned with the balances between small layer size and high frequency of the electric currents. Other limiting parameters which arise briefly in this work are the high ratios of electrical conductivities and of magnetic permeabilities between the insulating and the ferro-magnetic materials. Section 2 re-derives, using the method of multiple scales, the fast and slow spatial dependencies of the electromagnetic field, found in [13], in a distinguished limit of thin layers and high ratio of conductivities. Extending what has been done previously done in the literature, we then use these results to obtain the inductances for transformers for various limiting cases of interest. In Section 3 we use the results of the internal modelling in considering the current flow when a power supply is connected to the primary coil and a load is connected to the secondary. It is clear that, without any extra device linking the two sides of the transformer, there is negligible transmission of any high-frequency electromagnetic signal across the transformer. Connecting some sort of impedance (in the simplest cases, just a resistor, capacitor or inductor) across the transformer to link the primary and secondary circuits in such a way as not to change the performance at low, mains, frequencies, is seen not to significantly enhance the transmission of high frequencies. The possible changed internal behaviour of the transformer windings at high frequencies is briefly looked at in the Discussion, Section 4. 2. Modelling the Magnetic Core 2.1. Basic Case. We start by considering a single piece of iron or steel, surrounded by an air (or other insulating) gap, which in turn is surrounded by a layer carrying an electric current. This surface current density represents the current being carried by the wires in the coil. For simplicity a two-dimensional situation, as in Fig. 1, is taken. With this simplified geometry, the electric field lies in the x1 – x2 plane, while the magnetic field is in the normal, x3 , direction. We can then write E = (E1 , E2 , 0) = E(x1 , x2 , t) for the electric field and represent the magnetic induction B = (0, 0, B) by the scalar quantity B(x1 , x2 , t). Maxwell’s equations, ∂B B ∂E ρ (2.1) ∇×E =− , ∇× =ǫ + j , ∇·E = , ∇·B = 0 , ∂t µ ∂t ǫ then reduce to ∂E1 ∂E2 ∂B − = , (2.2) ∂x2 ∂x1 ∂t ∂ ∂x2 B µ = j1 , ∂ ∂x1 B µ = −j2 , as the last of (2.1) automatically holds, the third is not used, and we can neglect the √ ǫ ∂E/∂t term for frequencies much less than 1/(w ǫ0 µ0 ) = O(3 × 108 s−1 ). Note 226 A. A. LACEY x2 2w1 Air gap, Di 2w2 Ferro-magnetic core Dm ∂Dm Current J per unit length in the third (x3) dimension ∂Di x1 Figure 1. The basic layout of a magnetic core, Dm , surrounded by a layer carrying an electric current. An insulating region, Di , separates the two. The macroscopic length scale, w, is typically of order 1 m. that ρ is charge density, ǫ is electrical permittivity with ǫ0 its value in free space, µ is magnetic permeability with µ0 its value in free space, and w is a measure of the size of the cross-section, such as w1 or w2 as shown in Fig. 1. The current density is j = (j1 , j2 , 0), lying in the x1 – x2 plane, and is given by Ohm’s law, (2.3) j1 = σE1 , j2 = σE2 , with σ the electrical conductivity. In the insulating region Di , σ is negligible so ∂B ∂B = = 0 and B = B(t) in Di . ∂x1 ∂x2 With only small magnetic field outside the transformer, assuming it is designed well and there is little leakage, B = µ0 J on the outer surface of the insulating region, ∂Di , which is where the layer of windings carries a surface current density J per unit length. Then (2.4) B = Bi (t) ≡ µ0 J(t) in Di . At ∂Dm , the interface between the magnetic core and the insulator, the tangential components of the magnetic field H (as opposed to magnetic induction B) are continuous so µm (2.5) Bm = Bi (t) = µm J(t) at ∂Dm . µ0 Here Bm denotes the magnetic induction within the core. Eliminating electric field E and eddy current j, Bm satisfies the heat equation (2.6) ∂B 1 = ∇2 B , ∂t σm µm HIGH-FREQUENCY SIGNALS AND TRANSFORMERS 227 with ∇2 now being the two-dimensional Laplacian, and writing σ = σm in the ferro-magnet. The ferro-magnet conductivity, σm , and magnetic permeability, µm , are here assumed to be constant. The EMF induced in one loop of the outer boundary is then I Z Z d dJ ∂Bm E ·dx = − B ·n dS = − µ0 |Di | + dS , (2.7) E= dt D dt Dm ∂t ∂Di where |Di | is the cross-sectional area of Di (and Dm is the region occupied by the core). Taking the length scale of the transformer in the third dimension to be ℓ, with N1 turns in the primary coil and N2 turns in the secondary, the voltage E2 induced in the secondary by a varying current I1 (t) in the primary is Z N1 N2 dI1 ∂Bm (2.8) E2 = − µ0 |Di | − N2 dS . ℓ dt Dm ∂t Here Bm (x1 , x2 , t) is the magnetic induction produced by the current density J1 (t) = N1 I1 (t)/ℓ in the primary windings. With a “slowly varying” current, so that (2.6) can be approximated by ∇2 B = 0 and Bm = µm J = µm N1 I1 /ℓ, (2.8) becomes dI1 dI1 N1 N2 (µ0 |Di | + µm |Dm |) = −M , ℓ dt dt with M being the (real) mutual inductance E2 = − (2.9) M = N1 N2 (µ0 |Di | + µm |Dm |)/ℓ . Similarly, the two self inductances for the primary and secondary coils are (2.10) L1 = N12 (µ0 |Di | + µm |Dm |)/ℓ , L2 = N22 (µ0 |Di | + µm |Dm |)/ℓ , respectively. These assume that the transformer is “perfect”, with M 2 = L1 L2 . In practice, this will not hold but the inequality M 2 ≤ L1 L2 will be satisfied. For a sinusoidal alternating current, we can write I1 (t) = Re{Iˆ1 eiωt }, with Iˆ1 a complex constant. Similarly setting B = Re{B̂(x1 , x2 )eiωt }, and so on, (2.6) leads to the complex Helmholtz equation (2.11) with ∇2 B̂ = iωσm µm B̂ in Dm , N1 ˆ I1 on ∂Dm . ℓ We may then write B̂ = (µm N1 /ℓ)Iˆ1 H̃, where H̃ is given by B̂ = µm (2.12) ∇2 H̃ = iωσm µm H̃ in Dm , H̃ = 1 on ∂Dm . The resulting complex EMF produced in the secondary coil is Z iωN1 N2 Eˆ2 = − µ0 |Di | + µm H̃ dS Iˆ1 = −iωM (ω)Iˆ1 , ℓ Dm with complex, and frequency-dependent, mutual inductance Z N1 N2 N1 N2 (2.13) M (ω) = µ0 |Di | + µm H̃ dS Iˆ1 = L̃(ω) , ℓ ℓ Dm Z (2.14) where L̃(ω) = µ0 |Di | + µm H̃ dS . Dm 228 A. A. LACEY The self inductances are L1 = N12 L̃/ℓ and L2 = N22 L̃/ℓ. An indication of the efficiency, or inefficiency, of the transformer, corresponding to how much power is lost through Ohmic heating produced by the eddy currents induced in the core, can be got by looking at just the primary windings – we can imagine an infinite resistance in the secondary circuit so that the current in the secondary coil vanishes: I2 ≡ 0. For primary current I1 (t) and an induced EMF E1 (t) across the primary coil, the instantaneous power supplied to the transformer is −E1 (t)I1 (t). Using standard alternating current so that I1 = Re{Iˆ1 eiωt } and E1 = Re{Ê1 eiωt } = Re{−iωL1 eiωt }, the power supplied is P1 (t) = ω|Iˆ1 |2 Re{ei(φ+ωt) } × Re{iL1 ei(φ+ωt) } = −ω|Iˆ1 |2 cos(φ + ωt) × [Im{L1 } cos(φ + ωt) + Re{L1 } sin(φ + ωt)] , where |Iˆ1 | is the amplitude of the current and φ = Arg {Iˆ1 } is its phase. Hence the average rate of power loss is Z ωµm N12 ω ˆ 2 Im{H̃} dS . (2.15) P̄1 = − |I1 | Im{L1 } = − 2 2ℓ Dm (Note that this power dissipation can also be got by looking at Ohmic heating in a cross-section: I I ℓ B ∂ B −E1 I1 = −N1 × JE ·dx = ℓ ds N1 ∂Di µ ∂Di µσ ∂n Z B B ∇(B/µ) B ∇ ·∇ σ + ∇· dS µ µ µ σ D Z Z B ∂E1 ∂E2 1 ∂ 2 =ℓ E ·j + − dS = ℓ E ·j + B dS . ) µ ∂x2 ∂x1 2µ ∂t D D =ℓ More completely, with sinusoidal currents I1 , I2 flowing through the two windings, and producing EMFs E1 , E2 across them (see the idealised, simple diagram in Fig.2), E1 = −ωRe{(L1 Iˆ1 + M Iˆ2 )ieiωt }, E2 = −ωRe{(M Iˆ1 + L2 Iˆ2 )ieiωt }. The mean power input and output are then i ωh (2.16) P̄1 = −E1 I1 = − Im{L1 }|Iˆ1 |2 + Re{M }Im{Iˆ1∗ Iˆ2 } + Im{M }Re{Iˆ1∗ Iˆ2 } , 2 (2.17) where (2.18) P̄2 = E2 I2 = ∗ i ωh Im{L2 }|Iˆ2 |2 + Re{M }Im{Iˆ2∗ Iˆ1 } + Im{M }Re{Iˆ2∗ Iˆ1 } , 2 denotes complex conjugate, and the mean power loss is i ωh P̄1 − P̄2 = − Im{L1 }|Iˆ1 |2 + 2Im{M }Re{Iˆ1 Iˆ2∗ } + Im{L2 }|Iˆ2 |2 . 2 The effectiveness of the transformer in transferring power from the primary winding to its destination in the the secondary circuit might, alternatively, be measured by how much power, P̄2 , is output by the secondary windings. HIGH-FREQUENCY SIGNALS AND TRANSFORMERS 229 Core I2 Vi2 I1 Vo2 Vi1 Vo1 Figure 2. Currents in the two windings shown for N1 = N2 = 1 (single loops). Induced EMFs are E1 = Vi1 − Vo1 , E1 = Vi2 − Vo2 . 2.2. High and low frequencies. The key problem for the transformer’s performance, (2.12), can be scaled by writing x = wy, where w is a typical distance in the cross-section. Then (2.19) ∇2 H̃ = iw2 ωσm µm H̃ = iη 2 H̃ in D̃m , H̃ = 1 on ∂ D̃m , where D̃m the region in y space corresponding to Dm , ∂ D̃m is its boundary, ∇2 is now the two-dimensional Laplacian with respect to y, and the non-dimensional parameter η is given by √ (2.20) η = w ωσm µm . By “low” frequencies, we then mean η 2 = w2 ωσm µm ≪ 1 . (2.21) In such cases, H̃ ∼ 1 − iη 2 W , where ∇2 W + 1 = 0 in D̃m , W = 0 on ∂ D̃m . Then the inductances for the transformer are given by (2.22) L1 = N12 L̃/ℓ , L2 = N22 L̃/ℓ , where now, using µm ≫ µ0 , L̃ = (2.23) ∼ µ0 |Di | + µm Z Dm M = N1 N2 L̃/ℓ , H̃ dS ∼ µm Z µm |Dm | − iµm w2 (w2 ωσm µm ) H̃ dS Dm Z W dS . D̃m Correspondingly, the power loss behaves as 1 N 2I 2 2 4 2 µm w ω σm , 2 ℓ for typical current I and typical number of windings N (see [3], [9], [10], [22]). Note that in extreme cases, extra care should be taken in the interpretation of (2.24). For instance, with a large resistance in the secondary circuit, the power loss will be large in comparison with N 2 I22 µ2m w4 ω 2 /ℓ. (2.24) P̄1 − P̄2 ∼ 230 A. A. LACEY Likewise, “high” frequency means (2.25) η 2 = w2 ωσm µm ≫ 1 . The core’s eddy currents are now confined to a boundary layer of width 1/η near the boundary ∂ D̃m so, in dimensionless variables, η in D̃m , (2.26) H̃ ∼ exp − √ (1 + i)νy 2 where νy is the non-dimensional distance from ∂ D̃m . Returning to dimensional variables, r ωσm µm (2.27) H̃ ∼ exp − (1 + i)νx in Dm , 2 with νx the distance into the core, so r Z ωσm µm (1 + i)νx dS L̃ ∼ µ0 |Di | + µm exp − 2 Dm r µm ∼ µ0 |Di | + |∂Dm | (1 − i) 2ωσm r µm (2.28) |∂Dm | (1 − i) , ∼ 2ωσm assuming that µm ≫ ωσm µ20 |Di |2 / |∂Dm |2 . Input and output power, and power loss now all behave as r N 2 I 2 ωµm (2.29) P̄1 , P̄2 , P̄1 − P̄2 ∼ . 2ℓ 2σm Perhaps more importantly for such a √ high-frequency case, the impedance for the transformer grows as ω L̃, here order ω, while that due to other parts of the secondary circuit can be expected to grow as ω. With a specified power in the √ primary circuit remote√from the transformer, I1 is order 1/ ω and, from (2.29), output power is also 1/ ω. This should be expected because of the well known skin effect. (With specified remote voltage, and an inductance-dominated impedance in the primary circuit growing like ω, these powers would be only O(ω −3/2 ), see below.) Note that for a transformer of size 1 m with a purely iron or steel core, so that µm ∼ O(10−3 NA−2 ) and σm ∼ O(107 Ω−1 m−2 ), and normal mains frequency, so that ω ∼ O(300s−1 ), η ∼ O(1.7 × 103 ) ≫ 1. Such a transformer would be useless: due to the high conductivity, the skin depth is small, of order 1 mm, and significant eddy currents occur. 2.3. The laminated magnetic core. To reduce the effective conductivity of the transformer’s core, eddy currents must be blocked, so a laminated structure, with alternating layers of ferro-magnetic and insulating material, is employed. A periodic arrangement with a width, say 2h, of order 0.4 mm (somewhat smaller than the skin depth at mains frequency), is assumed here. The present analysis extends [13]. The configuration is now roughly as sketched in Fig. 3. For simplicity the typical cross-section scale is taken to be the half-width: w = w1 . A blow-up of the interior of the core is shown in Fig. 4. For the present, we allow a small, but not necessarily zero, electrical conductivity σi for the insulating material. For convenience, we shall consider both the magnetic HIGH-FREQUENCY SIGNALS AND TRANSFORMERS 231 Insulating material Ferro-magnetic material 2w2 x2 x1 J, current density in windings 2w1 Figure 3. Cross-section of a transformer with a laminated core. The lengths of the sides of the rectangular cross-section, w1 and w2 , are taken to be of a similar size, say order w. Insulator Magnetic conductor µ = µm, σ = σm Insulator 2αh µ = µ i , σ = σi Magnetic conductor 2h Figure 4. Interior of a laminated core. intensity H as well as the magnetic induction B = µH. As before, B = (0, 0, B) and H = (0, 0, H), while to avoid the use of too many suffices, we write the dimensionless, two-dimensional, position vector y = x/w as y = (y, z) and the electric field as E = (F, G, 0). The key parts of Maxwell’s equations, (2.2), then become ∂F ∂G ∂H ∂H ∂H − = wµ , = wσF , = −wσG . ∂z ∂y ∂t ∂z ∂y The usual continuity equations at an interface with normal n and tangents e are (2.30) [H ·e] = [B ·n] = [E ·e] = [σE ·n] = 0 so, at the boundaries z =const. between laminations, (2.31) [H] = 0 , [F ] = 0 , [σG] = 0 . (Here we use the notation [·] for the jump of a quantity at some point.) There is now a small parameter δ = h/w (= 2 × 10−4 for h = 0.2 mm, w = 1 mm) and, because of the rapid variations in the z direction, we write Z = z/δ and 232 A. A. LACEY use the method of multiple scales (see [8]): F , G and H are regarded as functions of y, z, Z and t. For harmonic dependence on time t, with angular frequency ω, we again use complex versions of the quantities. Scaling magnetic intensity with ˆ iωt }), we write external surface current density (J = Re{Je H = Re{JˆH̃eiωt } . To get a balance in the second of (2.30) in the magnetic material, we put ˆ m h)F̃ eiωt } , F = Re{(J/σ and, for a balance between the terms on the left-hand side of (2.30), we write 2 iωt ˆ G = Re{(Jw/σ }. m h )G̃e The new variables depend upon y, z and Z; for example, G̃ = G̃(y, z, Z). The equations become: (1) In the magnetic material, (2.32) ∂ F̃ ∂ F̃ ∂ G̃ +δ − = iΛH̃ , ∂Z ∂z ∂y ∂ H̃ ∂ H̃ +δ = F̃ , ∂Z ∂z δ2 ∂ H̃ = −G̃ , ∂y where Λ = h2 ωµm σm . (2) In the insulating material, (2.33) ∂ F̃ ∂ F̃ ∂ G̃ +δ − = iδmΛH̃ , ∂Z ∂z ∂y ∂ H̃ ∂ H̃ +δ = δ 2 sF̃ , ∂Z ∂z δ2 ∂ H̃ = −sG̃ , ∂y where m = µ0 /(µm δ) = wµ0 /hµm and s = σi /(δ 2 σm ) = w2 σi /h2 σm . (3) At the interfaces, (2.34) [H̃] = 0 , [F̃ ] = 0 , G̃m = sδ 2 G̃i , where the suffices m and i refer to the values in the magnetic and insulating materials respectively. For a transformer of interest with normal mains frequency, Λ is typically around 2 × 10−1 and m is of size 3. We therefore here take the distinguished limit of both Λ and m being of order one. For completeness we assume, for the present, that s is also of order one. (This would mean that σi is of size δ 2 σm , i.e. of size 4 × 10−8 × 107 Ω−1 m−1 = 0.4 Ω−1 m−1 . In reality, the insulator can be expected to have a conductivity which is much lower than this.) From the last of (2.34), it is appropriate to rescale G in the conductor and write G̃m = δ 2 Ĝ so that this interface condition becomes (2.35) Ĝ = sG̃ , with Ĝ now referring to the insulator value only. Note that this rescaling indicates that, excluding end effects, the electric current across the conducting lamellæ is small. The equations in the conducting magnetic material are now replaced by (2.36) ∂ F̃ ∂ F̃ ∂ G̃ +δ − = iΛH̃ , ∂Z ∂z ∂y ∂ H̃ ∂ H̃ +δ = F̃ , ∂Z ∂z ∂ H̃ = −Ĝ . ∂y HIGH-FREQUENCY SIGNALS AND TRANSFORMERS 233 Seeking asymptotic approximations to the solutions in the form F̃ ∼ F̃0 + δ F̃1 + . . . , Ĝ ∼ Ĝ0 + δ Ĝ1 + . . . , H̃ ∼ H̃0 + δ H̃1 + . . . , for δ → 0, we first see that ∂ H̃0 =0 ∂Z in the insulator, so here H̃0 = H̃i (y, z) for some function H̃i of only y and z. ∂ H̃0 ∂ F̃0 = F̃0 and = H̃0 , while H̃0 = H̃i on a layer’s In the ferro-magnet, ∂Z ∂Z surfaces, which we may write as Z = ±1. Then (2.37) H̃0 = H̃i cosh kZ , cosh k F̃0 = k H̃i sinh kZ , cosh k with k = Λ1/2 eiπ/4 . Also in a magnetic layer, (2.38) Ĝ0 = − ∂ H̃0 ∂ H̃i cosh kZ =− × . ∂y ∂y cosh k Returning to the insulating parts, G̃0 = − 1 ∂ H̃i , which automatically satisfies s ∂y the interface condition (2.35). Also ∂ G̃0 1 ∂ 2 H̃i ∂ F̃0 = =− . ∂Z ∂y s ∂y 2 (2.39) Averaging, with respect to Z, over a period of the core and eliminating secular terms, (2.40) 0= ∂ F̃0 α ∂ 2 H̃i = (k tanh k)H̃i − , ∂Z s ∂y 2 where α is the ratio of the widths of the insulating and magnetic layers (see Fig. 4). Hence we get, to leading order, the dimensionless averaged problem sk tanh k ∂ 2 H̃i = H̃i for − 1 < y < 1 , with H̃i = 1 on y = ±1 , (2.41) ∂y 2 α so (2.42) H̃i ∼ cosh Ay cosh A with A2 = ks tanh k and k = Λ1/2 eiπ/4 . α It should be noted that there can be boundary layers near the sides y = ±1 and z = ±w2 /w1 . It can also be observed that taking Λ → ∞ (“high” frequency), k → ∞, and A ∼ eiπ/8 Λ1/4 s1/2 /α1/2 , while for Λ → 0 (“low” frequency), k → 0, and A ∼ eiπ/4 Λ1/2 s1/2 /α1/2 . In dimensional terms, cosh(Ax1 /w) cosh(k(x2 − Xn )/h) ˆ iωt B = Re{B̂eiωt } ∼ Re µm × × Je cosh A cosh k in a magnetic layer n lying in Xn − h < x2 < Xn + h, say. The magnetic induction in the insulating layers is much smaller. (See [13].) The complex inductance is then 234 A. A. LACEY given by Z L̃ = B̃ dS/Jˆ D ∼ µm × no. of layers × Z w −w cosh(Ax1 /w) dx1 × cosh A 4µm w1 w2 tanh A tanh k , (1 + α)kA (2.43) ∼ Z h −h cosh(kx2 /h) dx2 cosh k with w = w1 , w2 the side lengths of the cross-section (see Fig. 3). 4µm w1 w2 tanh k Notice that taking s → 0, so that A → 0 and L ∼ × , will 1+α k give the same inductance as (2.14) with the µ0 term neglected, on solving (2.12) in the thin layers of ferro-magnet. In particular, taking r s = 0 and high-frequency, 4µm w1 w2 µm Λ ≫ 1, L ∼ ; this is just (2.28), L ∼ |∂Dm | e−iπ/4 , on using (1 + α)k ωσm √ k = h ωµm σm and |∂Dm | ∼ 4w1 w2 /(h(1 + α)). Likewise, taking s = 0 and low-frequency, Λ ≪ 1, L ∼ 4µm w1 w2 (1 + α) 1− k2 3 ∼ 4µm |Dm |− i(4w1 w2 µm )(h2 ωσm µm ) , 3(1 + α) which is just (2.23), on using |Dm | = 4w1 w2 /(1+α), noting that there are w2 /(h(1+ α)) layers contributing to the integral in (2.23), replacing η 2 by Λ, and using, in R the integral H̃ dS for each layer, W = 12 (1 − Z 2 ), with x2 = hZ varying from −h to h and x1 from −w1 to w1 . For the “low-frequency” case, Λ → 0, with s > 0, 4µm w1 w2 k → 0 , A → 0 , and L ∼ , 1+α which is simply the leading term in (2.23). The “high-frequency” case, Λ → ∞, now gives k → ∞, (2.44) A ∼ eiπ/8 Λ1/4 s1/2 /α1/2 = eiπ/8 (ωµm σm )1/4 (hs/α)1/2 → ∞ , 4µm w1 w2 w1 w2 α 1/2 L∼ ∼ 4e−3iπ/8 (1 + α)Ak 1 + α h3 s µm (ωσm )3 1/4 and , which appears very different from (2.28) because of the influence of the non-zero dimensionless conductivity s. (Note that: 1. Here, “high” or “low” frequency means Λ = h2 ωµm σm , not w2 ωµσ, is large or small; for ω ≈ 3 × 102 s−1 , µm ≈ 2 × 10−3 NA−2 , σm ≈ 107 Λ−1 m−1 , h ≈ 2 × 10−4 m, and w ≈ 1m, we get h2 ωµm σm ≈ 0.25 but w2 ωµm σm ≈ 6×106 . 2. For sufficiently large frequency, the µ0 contribution to inductance cannot be neglected, and, because of σi being non-zero, eddy currents will be confined to a boundary layer near the windings; a result like (2.28) will be recovered.) Examining the effect of having s positive more carefully, we first look at Λ ≪ 1, so that |k| ≪ 1, (1/k) tanh k ∼ 1 − 13 k 2 + . . . , 1/2 s 1/2 ks 1 A= tanh k ∼k 1 − k2 + . . . α α 6 (2.45) and L̃ ∼ 4µm w1 w2 i s × 1 − h2 ωµm σm 1 + + ... . 1+α 3 α HIGH-FREQUENCY SIGNALS AND TRANSFORMERS 235 There is an extra dissipative term, that in s/α, compared with the expression in (2.23). This arises from Joule heating produced by the currents in the “insulating” material. Taking Λ to be of order one, we only consider the case of s being small. Then A is again small so tanh A 1 ks tanh k ∼ 1 − A2 + · · · ∼ 1 − + ... A 3 3α 4µm w1 w2 tanh k ks tanh k × 1− + ... (2.46) and L̃ ∼ 1+α k 3α for k = Λ1/2 eiπ/4 . It is observed that even in this case the power loss term (which behaves like Im{k −1 tanh k}) becomes comparable with output (which behaves like Re{k −1 tanh k}). For Λ large, three cases should be considered: (a) s ≪ Λ−1/2 ; (b) the distinguished limit of s comparable with Λ−1/2 ; and (c) s ≫ Λ−1/2 . p ks/α is again small although k is large and 1/2 4µm w1 w2 1 µm −iπ/4 4µm w1 w2 L̃ ∼ × iπ/4 =e . 1+α (1 + α)h ωσm e h(ωµm σm )1/2 For (a), Λ ≪ s−2 , A ∼ (2.47) This corresponds exactly to (2.28) since the perimeter of the cross-section of the laminations is |∂Dm | ∼ 4w1 w2 /(1 + α)h. For (b), we may write S = sΛ1/2 , which is order one, then A ∼ eiπ/8 S 1/2 /α1/2 is order one, and, from (2.43), ! 1/2 1/2 1/2 4w1 w2 µ α S m (2.48) L̃ ∼ × e−3iπ/8 tanh eiπ/8 . (1 + α)h ωσm S α p For (c), Λ ≫ s−2 , A ∼ ks/α is now large, s 1/4 A∼ h1/2 (ωµm σm )1/2 eiπ/8 , and α 1/4 α 1/2 µ 4µm w1 w2 4w1 w2 m (2.49) L̃ ∼ ∼ × × e−3iπ/8 , (1 + α)kA (1 + α) h3 s (ωσm )3 recovering (2.44). In all of this, we have been neglecting the contribution to the inductance which arises from the magnetic inductance in the insulator, where the magnetic permeability, µ0 , is much less than that in the ferro-magnet, µm : µ0 /µm ≈ 10−3 . For the insulator to make a significant contribution, the skin depth would have to be no more than the layer width, h, times µ0 /µm , i.e. 1/|k| ≤ O(µ0 /µm ) , or, equivalently, 106 ≈ (µm /µ0 )2 ≤ O(Λ) . For cases of interest, mains frequency ω ≈ 300 s−1 gives Λ ≈ 1/5, while a high signal frequency, say ω ≈ 106 s−1 , gives Λ ≈ 600. The insulator’s contribution to the inductance is clearly negligible for both. 236 A. A. LACEY For the first particular case of interest, mains frequency ωp ≈ 300s−1 gives dimensionless frequency Λ = Λp ≈ 0.2, which might be considered order one – or possibly small, which would mean that 4µm w1 w2 i L̃p ≡ L̃(ωp ) ≈ 1 − ωp µm σm . 1+α 3 This can be compared with the value for a purely iron core, with a non-dimensional frequency of η 2 ≈ 6 × 106 , r µm × 4w(1 − i) . L̃pp ≈ 2ωp The latter is very much smaller (|L̃p /L̃pp | ≈ (1 + α)w(ωp µm σm )1/2 ≈ 4 × 103 ) and has a significant imaginary (loss) part. The laminar structure of the core is seen to both block the eddy currents, thereby reducing power loss, and allow the magnetic field to penetrate more of the iron, so increasing the core’s inductance. With a much higher frequency, say ωs ≈ 106 s−1 , for the intended signal, Λ = Λs ≈ 600 and, since s is small (typically between 10−9 and 10−16 , taking σi in the range 10−8 Λ−1 m−1 to 10−16 Λ−1 m−1 ), (2.47) applies: 1/2 4µm w1 w2 −iπ/4 µm L̃s ≡ L̃(ωs ) ≈ e . (1 + α)h σm ωs We now see that there is a significant imaginary (loss) part and the size of the inductance is only a small fraction of that at mains frequency: L̃ 1 s = Λ−1/2 ≈ 0.04 . ∼ s L̃p (ωs µm σm )1/2 h This indicates poor performance, at high frequencies, of transformers, which are designed for low frequencies, and that there will be need for some sort of “by-pass” to get a high-frequency signal across a transformer. 3. Circuits Containing the Transformer For the purposes of passing a high-frequency signal from one place to another across a transformer, we regard the circuit from which the signal originates as the “primary” and that to which it is being transferred as the “secondary”. A simple representation of the parts of the circuit of interest is then shown in Fig. 5. The complex forms of the mutual and self inductances of the transformer are taken, from the preceding section, to be (3.1) L1 (ω) = l1 L̃(ω) , L2 (ω) = l2 L̃(ω) , M (ω) = mL̃(ω) , with l1 , l2 , m real positive constants satisfying l1 l2 < m2 (the inequality is taken as strict to allow for “leakage”) and the complex function L̃ of frequency ω behaving as (3.2) (3.3) L̃(ω) → L0 L̃(ω) ∼ L∞ e −iπ/4 ω −1/2 as ω → 0 , as ω → ∞ , where L0 and L∞ are both real and positive. The load shown in Fig. 5 will generally be considered as a resistor and inductance in series, Z2 (ω) = iωLc + Rc , but other loads can be used. In all cases we expect HIGH-FREQUENCY SIGNALS AND TRANSFORMERS V = V2 V = V1 I1 237 I2 Z1 V = Vs L1 Primary V =0 M L2 Secondary I1 Z2 Load I2 Figure 5. Simple circuit diagram showing the basic layout. The load, with impedance Z2 , might be regarded as some combination (series and parallel) of inductances, resistors and capacitors. that Z2 (ω) will be order ω for ω → ∞, since the (long) transmission lines will be surrounded by a non-magnetic dielectric material. Taking the primary current I1 (t) to be given: I1 (t) = Re{Iˆ1 eiωt } with Iˆ1 a complex constant, for the current in the secondary circuit, we then have iω(M Iˆ1 + L2 Iˆ2 ) + R2 Iˆ2 + Z2 Iˆ2 = 0 , where the internal resistance R2 in the secondary windings has been included, so Iˆ2 = −iωM Î1 /(iωL2 + R2 + Z2 ) . (3.4) Using M ∼ mω −1/2 e−iπ/4 L∞ and L2 ∼ l2 ω −1/2 e−iπ/4 L∞ as ω → ∞, Iˆ2 ∼ ω 1/2 mL∞ Iˆ1 /(Z2 e3iπ/4 − ω 1/2 l2 L∞ ) . (3.5) With an imposed voltage V̂s at a distance some distance from the transformer, V̂s = (Z1 + R1 + iωL1 )Iˆ1 + iωM Iˆ2 so (3.6) Iˆ1 = (Z1 + R1 + R2 )V̂s ; (Z1 + R1 )(Z2 + R2 ) + iω(L1 (Z2 + R2 ) + L2 (Z1 + R1 )) + ω 2 (M 2 − L1 L2 ) here the internal resistance R1 in the primary windings has been included, as has the impedance Z1 in the transmission lines linking signal source and transformer. This impedance is again expected to be of order ω for large ω. Then |Iˆ1 | = O(ω −1 ) for ω → ∞, while, from (3.5), |Iˆ2 | = O(ω −3/2 ). The mean power into the transformer is now P̄1 = 1 1 Re{Iˆ1∗ V̂1 } = Re{Iˆ1∗ [(iωL1 + R1 )Iˆ1 + iωM Iˆ2 ]} , 2 2 where V1 = Re{V̂1 eiωt } is the voltage drop across the primary windings of the transformer, so (3.7) i 1h P̄1 = R1 |Iˆ1 |2 − ωIm{L1 }|Iˆ1 |2 − ωIm{M Iˆ1∗ Iˆ2 } = O(ω −3/2 ) for ω → ∞ . 2 238 A. A. LACEY Likewise, for V2 = Re{V̂2 eiωt } the potential across the secondary windings of the transformer, the mean output power is 1 1 1 P̄2 = − Re{Iˆ2∗ V̂2 } = Re{Iˆ2∗ Z2 Iˆ2 } = − Re{Iˆ2∗ [(iωL2 + R2 )Iˆ2 + iωM Iˆ1 ]} 2 2 2 i 1h 2 2 (3.8) = −R2 |Iˆ2 | + ωIm{L2 }|Iˆ2 | + ωIm{M Iˆ2∗ Iˆ1 } = O(ω −2 ) for ω → ∞ 2 and the mean power loss is P̄l = (3.9) = (3.10) = P̄1 − P̄2 i 1h R1 |Iˆ1 |2 + R2 |Iˆ2 |2 − ω Im{L1 }|Iˆ1 |2 + 2Im{M }Re{Iˆ1∗ Iˆ2 } + Im{L2 }|Iˆ2 |2 2 O(ω −3/2 ) for ω → ∞ . We see that P̄2 /P̄1 = O(ω −1/2 ) → 0 as ω → ∞. With most of the power input into the transformer being lost at high frequencies, possibilities of a simple by-pass mechanism, represented by impedance ZA (ω) in Fig. 6, must be considered. The extra components in the circuitry should not significantly affect the performance of the transformer at low, mains, frequency. For V = V1 I1 ZA M L1 R1 I1 V = V2 I2 Z2 L2 R2 I3 I2 V =0 Figure 6. A transformer with a by-pass impedance, ZA . easier transmission of power and current at higher frequencies than low frequencies, a capacitor might be an obvious choice, in which case ZA = 1/iωC. Then, on writing, S = R1 L2 + R2 L1 + Z2 (L1 + L2 − 2M ) + L2 ZA , (3.11) − Iˆ2 ω 2 (L1 L2 − M 2 ) − iω(R1 L2 + R2 L1 + M ZA ) − R1 R2 = 2 ω (L1 L2 − M 2 ) − iωS − ZA (R2 + Z2 ) − Z2 (R1 + R2 ) Iˆ1 which is again of order ω −1/2 for ω → ∞. The “by-pass current”, I3 = Re{Iˆ3 eiωt } which flows through the capacitor is given by (3.12) (iω(L1 + L2 − 2M ) + ZA + R1 + R2 )Iˆ3 = (iω(M − L1 ) − R1 )Iˆ1 − (iω(M − L2 ) − R2 )Iˆ2 so that |Iˆ3 /Iˆ1 | is order one for |ZA | ≤ O(ω 1/2 ) (such as for the obvious capacitor, or a resistor) but only order ω 1/2 /|ZA | for |ZA | growing faster than ω 1/2 (as would be the case for a by-pass coil). HIGH-FREQUENCY SIGNALS AND TRANSFORMERS 239 The primary current I1 is still limited by the transmission inductance so that I1 will be of order 1/ω and the mean power input to transformer and by-pass is P̄1 = 1 1 Re{Iˆ1∗ V̂1 } = Re{Iˆ1∗ [(iωL1 + R1 )(Iˆ1 + Iˆ3 ) + iωM (Iˆ2 − Iˆ3 )]} 2 2 and is again of order ω −3/2 . The secondary current, from (3.11), is once more additionally restricted by the load impedance Z2 (ω) (still assumed to be of order ω for large ω) and gives mean output power into the secondary circuit 1 1 P̄2 = − Re{Iˆ2∗ V̂2 } = Re{Iˆ2∗ Z2 Iˆ2 } , 2 2 which is again of order ω −2 for ω → ∞. We see that whether ZA is, for high frequencies, large (if it behaves as an inductor), order one (as for a resistor), or small (as given by a capacitor), the by-pass does not alter the sizes of the powers carrying the signal. It appears that even if a significant current (and power) by-passes the transformer through ZA , because of the high impedance of the secondary windings, O(ω|L2 |) = O(ω 1/2 ), being much less than that of the transmission lines and load, |Z2 | = O(ω), very little goes into the secondary circuit. The output/input power ratio for the transformer is always O(ω −1/2 ), as might be expected from the skin effect. 4. Discussion In Sec. 2 we used standard asymptotic methods to fully derive the frequencydependent inductance (2.43), accounting for small but non-zero conductivity in the insulating layers; the same cosh dependency as appears in [13] is (unsurprisingly) seen. We then used (2.43) to derive inductances in a number of new limiting cases. Sec. 3 found the asymptotic behaviour for power transmission across transformers for high frequencies, given the results of Sec. 2. Although the calculations are elementary, these results appear to be novel. It is clear from Sec. 2 that the laminar structure of power transformers’ cores acts to stop significant eddy currents and increase penetration of magnetic fields, hence allowing them to operate efficiently and effectively at mains frequencies. However, at higher frequencies the smaller skin thickness means that induced currents cause significant power loss and the magnetic field reaches only a small fraction of the magnetic core, so the transformers are then inefficient, with much smaller inductances. This suggests considerable problems for the proposed scheme for signal transmission through standard transformers. (Air cores are known to be better than ferro-magnetic cores in transformers used for very high frequencies.) Simple by-pass mechanisms, as considered in Sec. 3, appear to be rather ineffective at very high frequencies because of the low impedances of the windings compared with the remainder of the secondary circuit. This might suggest that an alternative scheme which could be considered would be to connect an extra, air-cored, transformer in series with the original, power, transformer. Such a new transformer’s inductances would remain approximately constant, even at high frequencies. 240 A. A. LACEY Clearly, this could be impractical in the scope of the modifications needed to the power-supply equipment. Another drawback with such a design is the sizes of the added inductances. Not only should they be small enough compared with the original transformer, to avoid changing the power supply, but they must be large compared with the inductances elsewhere in the circuits – indicating the need for very large devices. One point not addressed so far is possible capacitor-like effects within the windings: the small spacing between successive loops will give rise to a capacitance, negligible at mains frequencies but possibly significant at higher ones. A model for this might be got by another homogenisation procedure, but a simple reasonable representation could have each set of windings acting as a coil and capacitor in parallel, see Fig. 7. (Some discussion of homogenising windings can be found in [14], [15], and [16]) Resistances are considered in series with the inductances and not the I1 I2 L1 M L2 C1 C2 R1 R2 Figure 7. Representation of a transformer to account for capacitance effects at high frequencies. capacitances as the latter link one winding to the next, acting as short cut for the current. Including the capacitances has the effect of replacing terms (iωLk + Rk ) by (iωLk + Rk )/(1 + iωCk Rk − ω 2 Lk Ck ), k = 1, 2, in the circuit calculations. The transformer impedances are now seen to behave as 1/iωCk , of order 1/ω, for high frequencies: most of the primary current (which is still of size 1/ω) travels through the capacitance so the effectiveness of the transformer is further reduced. To be more precise, the current flowing around the windings will be −1 ω × ω −2 1/(R1 + iωL1 ) ˆ = O ω −5/2 , O I1 × = O C1 L1 iωC1 the total induced EMF in the secondary windings is O ω −5/2 × ωM = O ω −2 , and the indicated current in the secondary circuit is then O ω −2 Z2 = O ω −3 . The power transmitted is O(ω −5 ). The “blocking” effect of the high inductance in the secondary circuit remains and including a by-pass to the transformer would still appear to be ineffective. We must emphasise that having such decay depends crucially on having inductorlike impedance in the circuits. If Z1 or Z2 was only of order one for high frequencies, the transmitted power would not decay so quickly. HIGH-FREQUENCY SIGNALS AND TRANSFORMERS 241 Finally, we must note that hysteresis and kinetic effects, the former being believed to be at least important as induced currents regarding power losses, have still to be included in the transformer model. References [1] M. Artola & M. 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