In a conductor carrying alternating current,
if currents are flowing through one or more
other nearby conductors, such as within a
closely wound coil of wire, the distribution
of current within the first conductor will
be constrained to smaller regions. The resulting
current crowding is termed the proximity effect.
This crowding gives an increase in the effective
resistance of the circuit, which increases
with frequency.
== Explanation ==
A changing magnetic field will influence the
distribution of an electric current flowing
within an electrical conductor, by electromagnetic
induction. When an alternating current (AC)
flows through a conductor, it creates an associated
alternating magnetic field around it. The
alternating magnetic field induces eddy currents
in adjacent conductors, altering the overall
distribution of current flowing through them.
The result is that the current is concentrated
in the areas of the conductor farthest away
from nearby conductors carrying current in
the same direction.
The proximity effect can significantly increase
the AC resistance of adjacent conductors when
compared to its resistance to a DC current.
The effect increases with frequency. At higher
frequencies, the AC resistance of a conductor
can easily exceed ten times its DC resistance.
== Example ==
For example, if two wires carrying the same
alternating current lie parallel to one another,
as would be found in a coil used in an inductor
or transformer, the magnetic field of one
wire will induce longitudinal eddy currents
in the adjacent wire, that flow in long loops
along the wire, in the same direction as the
main current on the side of the wire facing
away from the other wire, and back in the
opposite direction on the side of the wire
facing the other wire. Thus the eddy current
will reinforce the main current on the side
facing away from the first wire, and oppose
the main current on the side facing the first
wire. The net effect is to redistribute the
current in the cross section of the wire into
a thin strip on the side facing away from
the other wire. Since the current is concentrated
into a smaller area of the wire, the resistance
is increased.
Similarly, in two adjacent conductors carrying
alternating currents flowing in opposite directions,
such as are found in power cables and pairs
of bus bars, the current in each conductor
is concentrated into a strip on the side facing
the other conductor.
== Effects ==
The additional resistance increases power
losses which, in power circuits, can generate
undesirable heating. Proximity and skin effect
significantly complicate the design of efficient
transformers and inductors operating at high
frequencies, used for example in switched-mode
power supplies.
In radio frequency tuned circuits used in
radio equipment, proximity and skin effect
losses in the inductor reduce the Q factor,
broadening the bandwidth. To minimize this,
special construction is used in radio frequency
inductors. The winding is usually limited
to a single layer, and often the turns are
spaced apart to separate the conductors. In
multilayer coils, the successive layers are
wound in a crisscross pattern to avoid having
wires lying parallel to one another; these
are sometimes referred to as "basket-weave"
or "honeycomb" coils. Since the current flows
on the surface of the conductor, high frequency
coils are sometimes silver-plated, or made
of litz wire.
== Dowell method for determination of losses
==
This one-dimensional method for transformers
assumes the wires have rectangular cross-section,
but can be applied approximately to circular
wire by treating it as square with the same
cross-sectional area.
The windings are divided into 'portions',
each portion being a group of layers which
contains one position of zero MMF. For a transformer
with a separate primary and secondary winding,
each winding is a portion. For a transformer
with interleaved (or sectionalised) windings,
the innermost and outermost sections are each
one portion, while the other sections are
each divided into two portions at the point
where zero m.m.f occurs.
The total resistance of a portion is given
by
R
A
C
=
R
D
C
(
R
e
(
M
)
+
(
m
2
−
1
)
R
e
(
D
)
3
)
{\displaystyle R_{AC}=R_{DC}{\bigg (}Re(M)+{\frac
{(m^{2}-1)Re(D)}{3}}{\bigg )}}
RDC is the DC resistance of the portion
Re(.) is the real part of the expression in
brackets
m number of layers in the portion, this should
be an integer
M
=
α
h
coth
⁡
(
α
h
)
{\displaystyle M=\alpha h\coth(\alpha h)\,}
D
=
2
α
h
tanh
⁡
(
α
h
/
2
)
{\displaystyle D=2\alpha h\tanh(\alpha h/2)\,}
α
=
j
ω
μ
0
η
ρ
{\displaystyle \alpha ={\sqrt {\frac {j\omega
\mu _{0}\eta }{\rho }}}}
ω
{\displaystyle \omega }
Angular frequency of the current
ρ
{\displaystyle \rho }
resistivity of the conductor material
η
=
N
l
a
b
{\displaystyle \eta =N_{l}{\frac {a}{b}}}
Nl number of turns per layer
a width of a 
square conductor
b width of the winding window
h height of a square conductor
== Squared-field-derivative method ==
This can be used for round wire or litz wire
transformers or inductors with multiple windings
of arbitrary geometry with arbitrary current
waveforms in each winding. The diameter of
each strand should be less than 2 δ. It also
assumes the magnetic field is perpendicular
to the axis of the wire, which is the case
in most designs.
Find values of the B field due to each winding
individually. This can be done using a simple
magnetostatic FEA model where each winding
is represented as a region of constant current
density, ignoring individual turns and litz
strands.
Produce a matrix, D, from these fields. D
is a function of the geometry and is independent
of the current waveforms.
D
=
γ
1
⟨
[
|
B
→
1
^
|
2
B
→
1
^
⋅
B
→
2
^
B
→
2
^
⋅
B
→
1
^
|
B
→
2
^
|
2
]
⟩
1
+
γ
2
⟨
[
|
B
→
1
^
|
2
B
→
1
^
⋅
B
→
2
^
B
→
2
^
⋅
B
→
1
^
|
B
→
2
^
|
2
]
⟩
2
{\displaystyle \mathbf {D} =\gamma _{1}\left\langle
{\begin{bmatrix}\left|{\hat {{\vec {B}}_{1}}}\right|^{2}&{\hat
{{\vec {B}}_{1}}}\cdot {\hat {{\vec {B}}_{2}}}\\{\hat
{{\vec {B}}_{2}}}\cdot {\hat {{\vec {B}}_{1}}}&\left|{\hat
{{\vec {B}}_{2}}}\right|^{2}\end{bmatrix}}\right\rangle
_{1}+\gamma _{2}\left\langle {\begin{bmatrix}\left|{\hat
{{\vec {B}}_{1}}}\right|^{2}&{\hat {{\vec
{B}}_{1}}}\cdot {\hat {{\vec {B}}_{2}}}\\{\hat
{{\vec {B}}_{2}}}\cdot {\hat {{\vec {B}}_{1}}}&\left|{\hat
{{\vec {B}}_{2}}}\right|^{2}\end{bmatrix}}\right\rangle
_{2}}
B
→
j
^
{\displaystyle {\hat {{\vec {B}}_{j}}}}
is the field due to a unit current in winding
j
j is the spatial average over the region
of winding j
γ
j
=
π
N
j
l
t
,
j
d
c
,
j
4
64
ρ
c
{\displaystyle \gamma _{j}={\frac {\pi N_{j}l_{t,j}d_{c,j}^{4}}{64\rho
_{c}}}}
N
j
{\displaystyle N_{j}}
is the number of turns in winding j, for litz
wire this is the product of the number of
turns and the strands per turn.
l
t
,
j
{\displaystyle l_{t,j}}
is the average length of a turn
d
c
,
j
{\displaystyle d_{c,j}}
is the wire or strand diameter
ρ
c
{\displaystyle \rho _{c}}
is the resistivity of the wireAC power loss
in all windings can be found using D, and
expressions for the instantaneous current
in each winding:
P
=
[
d
i
1
d
t
d
i
2
d
t
]
D
[
d
i
1
d
t
d
i
2
d
t
]
¯
{\displaystyle P={\overline {{\begin{bmatrix}{\frac
{di_{1}}{dt}}{\frac {di_{2}}{dt}}\end{bmatrix}}\mathbf
{D} {\begin{bmatrix}{\frac {di_{1}}{dt}}\\{\frac
{di_{2}}{dt}}\end{bmatrix}}}}}
Total winding power loss is then found by
combining this value with the DC loss,
I
r
m
s
2
×
R
D
C
{\displaystyle I_{rms}^{2}\times R_{DC}}
The method can be generalized to multiple
windings.
== See also ==
Skin effect
== External links ==
Skin Effect, Proximity Effect, and Litz Wire
Electromagnetic effects
Skin and Proximity Effects and HiFi Cables
