Download PASSAGE OF HIGH-FREQUENCY SIGNALS THROUGH POWER

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.
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Maxwell Institute for Mathematical Sciences, and School of Mathematical and Computer Sciences, Heriot-Watt University, Riccarton, Edinburgh, EH14 4AS.
E-mail : [email protected]
URL: http://http://www.macs.hw.ac.uk/departments/mathematics.htm