Download Fundamentals of magnetic field

Fundamentals of magnetic field
The forces between static electric charges are transmitted via electric field (Coulomb's law).
Forces between moving charges (current carrying lines) also appear, those are transmitted via
magnetic field.
Charges moving with constant speed (direct current) cause constant magnetic field, while
charges moving with variable speed (accelerating or slowing) cause variable magnetic field.
In the case of moving wire in a magnetic field or when the magnetic field changes around a
wire, a physical force acts to the charges of wire separating them by polarity, which results an
electric field, an induced voltage.
The magnetic field
Consider two straight long parallel wires in vacuum (or air) of small cross-section compared
to their length. If the charges in the wires move with constant speed (constant currents are
flowing) the forces between the wires are constant. Using notation F1=F2=F the value of these
forces are expressed as
IIl
F = k 1 2 (N).
a
I1
F1
I2
F2
l
a
Forces between current carrying straight conductors
 VAs 
If I1=I2=1 A and l=a=1 m, then F = 2 ⋅ 10 −7 N  =
,

m 
Vs
Vs 4π 10 −7 µ 0
=
=
, here µ 0 = 4π 10−7
the magnetic permeAm
2π
2π
Am
ability of vacuum. This formula may be used to define the current of 1 A.
Using the value of permeability µ0 for calculation of force:
µ IIl
F = 0 1 2 (N).
2π a
The direction of the forces are attractive in the case of unidirectional currents and they are
repulsive in the case of opposite direction of currents.
Description and understanding the magnetic field is simpler when dc current assumed.
Interpret force F2 in the figure as follows: the moving charges of current I1 create special state
of the space (the magnetic field) around the wire which field acts on the charges moving inside the second wire carrying current I2.
consequently k = 2 ⋅ 10 −7
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H1
I1
B1
Magnetic field around a long straight conductor carrying current I1
The first space variable vector which describes the magnetic field is the magnetic field intensity H . In homogeneous substance the magnetic field intensity (field strength) H1 from
current I1 defined as:
I
H1 = 1 , the force F2 to the wire 2 carrying current I2 expressed as:
2π a
F2=H1µ0I2l.
In homogeneous and ferromagnetic substance the calculation of the magnetic field intensity
H is more complicated, the Ampère's excitation law has to be used.
The magnetic field intensity H is a vector, its direction in each point of the space is the same
as the north (N) side of compass needle. Around a single wire the direction of field intensity
is the same as a right-thread screw turns. The SI unit of magnetic field intensity
A
[H ] = .
m
Magnetic field intensity in each point of the space is illustrated by directed lines. These lines
form closed paths, they do not arise and do not end.
H1 B1
I1
F2
H1 B1
I2
I1
I2
F2
Force to a current carrying wire in the field of another conductor
Consider a wire of length l carrying current I in a field of magnetic field intensity H . The
force exerted F = µ 0 lI × H , where the direction of I is the same as the movement of positive charges inside the conductor. In the case shown on the figure
F2 = µ 0 lI 2 × H1 .
2
Fundamentals of magnetic field
Example
A conductor carries 1 A, the magnetic field intensity from 1 m of the conductor is
A
H = 0.159 .
m
A
The magnitude of the force on a wire being in a magnetic field of H = 1 , carrying 1 A is
m
N
F = 2π 10 −7 .
m
The other space variable vector describing the magnetic field is the magnetic flux density B .
The magnitude of flux density depends on the substance in the space, its SI unit in honour of
Tesla's1 scientific activity
Vs
[B] = T = tesla = m 2 .
At magnetic field strength H
B = µ 0µ r H ,
here µr – the relative permeability, a substance specific, non-dimensional factor.
The relative permeability is often not a constant, its value may depend on the magnetic field
intensity and also on the initial magnetic conditions.
Example
The magnetic flux density at magnetic field intensity H = 1
A
in free space B=4π10-7 T.
m
The direction of flux density vector B is usually coincides with that of field intensity H : the
direction of the compass needle to the north pole at each point of the space. The field lines are
directed from the south pole to north inside a magnet (e.g. inside a compass needle) and from
the north to south outside the magnet. Consequently, magnetic flux-lines leave the magnet at
the north pole and go on towards the south. The compass needle in ordinary use directed to
the geographical north pole of earth.
S
N
S
H
B
N
Direction of the magnetic field, the definition
Inside particular materials – the ferromagnetic materials – the magnetic flux density is significantly increased in contrast to free space. Simple and illustrative explanation of this phenomenon the contribution of the molecular magnets (or circular currents) of such materials to
1
Tesla, Nikola (1856-1942) engineer, investigator, Serbian origin
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the flux density of external magnetic field. The relative permeability µr expresses the ratio of
the flux density in comparison with that in free space; 1 ≤ µr ≤ 103-106.
The value of µr is usually determining by measurements or complicated calculations.
The magnetic flux density also illustrated by directed lines.
The force F acting to a piece of current-carrying wire with length l and current I in arbitrary
material substance can be expressed as: F = lI × B .
In the case according to the figure F2 = lI 2 × B1 .
Example
The force to a current-carrying wire with 1 A inside a magnetic field of 1 T is F = 1
N
.
m
The magnetic flux of an area is a scalar value, defined as the surface integral of the flux density to the area of interest:
Φ = ∫ BdA , in homogeneous field Φ=BA, the SI unit in honour of Weber's2 scientific activity
A
[Φ]=Wb =weber=Vs.
The magnetic field is often visualised as lines of magnetic flux that form closed paths. The
lines are close together where the magnetic field is strong and farther apart where the field is
weaker. By convention the flux lines leave the north-seeking end (N) of a magnet (e.g. compass needle) and enter its south-seeking end (S).
Example
The flux through an area of 1 m2 in a perpendicular homogeneous field of 1 T is 1 Wb.
The Ampère's3 law of excitation
The most important rule for calculation of magnetic circuit
The line integral of the magnetic field intensity vector H around a closed path is equal to the
aggregated current through area A enclosed by that path, this aggregated current termed the
excitation Θ of the area A.
∫ Hdl = ∫ JdA = Θ .
A
If the path studied goes along separate parts of homogeneous field intensity than the integral
of the left side of equation is simplified as a sum. Whereas if the charge flow concentrated in
wires than the integral of the right side of equation is simplified as a sum: ∑ H i l i = ∑ I j .
i
j
In the case of constant permeability µr the law of excitation may also be written as:
B
1
∫ Hdl = ∫ µ dl = µ ∫ Bdl = I , or ∫ Bdl = µ I , here µ=µ0µr.
Example
Consider a current-carrying wire with current 1 A, in distance a from it the field intensity:
I
H=
.
2π a
2
3
Weber, Wilhelm Eduard (1804-1891) German physicist
Ampère, Andrè-Marie (1775-1836) French physicist, mathematician, chemist
4
Fundamentals of magnetic field
If the surrounding substance is non-ferromagnetic and the closed path investigated is a concentric circle with radius a and the direction of integral agrees to the direction of field intenI
I
dl =
2π a = I .
sity H than ∫ Hd l =
∫
2π a
2π a
Similar result is obtained when (in non-ferromagnetic substance) the path is closing via different arcs according to the next figure:
l4
l2
r1
H
r2
I
l3
l1
Demonstration of the law of excitation
along l1 the field intensity H1 =
I
,
2π r1
along l3 and l4 the field intensity H is perpendicular to the direction of path of integral, that
is why the a scalar product Hdl = 0 ,
I
along l2 the field intensity H 2 =
,
2π r2
3
3 
2π r1 = I 
4 
1 4
l1
Hd l = I .
I 1
1 ∫

∫ H 2 dl = 2π r2 4 2π r2 = 4 I 
l2

I
∫ H dl = 2π r
1
Knowing the existing or desired field intensity the excitation producing it may be calculated.
The integral along a path with H = const. Hdl = Hdl . If the path considered consists of
pieces with H = const. then
∫ Hdl = ∑ H l
i i
=Θ .
i
Visualisation of magnetic field (flux-lines)
Current loop
I
B
Flux-lines of a current loop (turn)
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Solenoidal and toroidal coil
Since the length of a solenoidal coil l is much greater than its diameter d, i.e. l » d. the field
inside the coil may be considered uniform and outside it may be neglected. The same idea
used for a toroidal coil, if average diameter Da » d.
At these coils the single turns are in series, they have the same current. Applying the
Ampère's law of excitation Θ=Hl=NI, where N – the number of turns (number of wires, number of currents).
l
d
d
Da
Magnetic field of solenoidal and toroidal coils
At a given direction of current flow the direction of the magnetic field produced in a coil depends on the direction of the twist of turns.
I
I
B
I
B
Magnetic field of right-thread and left-thread coils
Force to a current-carrying wire in magnetic field
In homogeneous magnetic field according to the formula for force:
F = lI × B .
B
F
B
I
I
F
Illustration of force to a current-carrying wire in homogeneous magnetic field
6
I
Fundamentals of magnetic field
"Useful" magnetic field and leakage
If consider linked coils (like the coils of transformers or the stator-rotor coils of rotating electrical machines) only a portion of produced by one coil magnetic field is linking to the other
coil, the rest of the field is „leaking”. The last part termed as flux leakage or magnetic field
leakage. The measure of leakage is defined with coefficient of leakage σ:
φ
σ = l (0 ≤ σ ≤ 1),
φt
where φt – the total flux, while φl the flux leakage.
In certain cases the magnetic leakage is important, e.g. the leakage reactance can limit the
short circuit current.
The law of flux refraction (the boundary conditions)
The field intensity H and the flux density B pass a boundary layer of substances with different permeability different way.
Refraction of flux density vector
Let us examine the conditions at the boundary between two materials of permeability µ1 and
µ2. Consider the flux lines crossing the boundary as shown in figure with angle of incidence
α1 and angle of refraction of α2.
The flux magnitude through an elemental area dA approached from each side of the boundary
must be identical, assuming that no magnetic flux emerges from surface as dA →0. Since the
flux-lines are closed, the overall fluxes in the two substances are identical:
Φ = ∫ BdA =B1ndA=B1cosα1dA=B2cosα2dA= B2ndA,
dA
that is the normal component of flux density vector B remains unchanged, it crosses a boundary continuously.
According to the excitation law the line integral of the magnetic field intensity around a
closed path of width dl is equal to zero if no excitation (no current flowing) on either side.
Around the closed path ∫ Hdl = 0 since no current linked:
∫ Hdl = H1tdl- H2tdl= H1sinα1dl- H2sinα2dl=0, hence H1t=H2t,
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that is the tangential component of field intensity vector H remains unchanged, it crosses a
boundary continuously.
Refraction of field intensity vector
The tangential component of flux density vector B and the normal component of field intensity vector H are changing through the boundary layer.
From the statements above
H1sinα1 = H2sinα2, or substituting flux density for field intensity:
B1
B2

sin α 1 =
sin α 2 
sin α 1
sin α 2
tgα 1 µ r1
=
⇒
=
.
µ 0 µ r1
µ 0µ r 2

cos
cos
µ
µ
α
µ
µ
α
tg
α
µ
r
r
r
0
1
1
0
2
2
2
2

B1 cos α 1 = B2 cos α 2 
The flux-lines at iron-air border-layer
Suppose that µr1»µr2 (e.g. at the iron-air boundary, where µriron =106, µrair =1), than
tgα1»tgα2, α1»α2, i.e. α1~ 90° while α2~ 0. Consequently the magnetic flux emerges into air
normal to the surface of iron with approximately infinitely permeability, the flux-lines leave
the iron at right angles.
The Faraday's4 induction law
This law is one of the most important statement of electrical engineering, discovery of the
phenomenon described in it made (and make) possible the generation and public use of electrical energy.
4
Faraday, Michael (1791-1867) English physicist
8
Fundamentals of magnetic field
Consider a wire loop (a single turn of a coil), if in any case the flux enclosed by that loop is
changing, it produces electrical field, a voltage appears (inducing) in the loop. The magnitude
of induced voltage ui(t) is proportional to the flux change φ(t) in the time unit:
dφ (t )
ui (t ) =
.
dt
The flux change occurs either because the magnetic field is changing with time (transformer
inductance) or because the wire loop is moving relative to a magnetic field (motional inductance). The Faraday's induction law describes both phenomena.
a) If a wire loop is fixed and the flux is varying with time because the excitation current or the
magnetic circuit is changing the phenomena called transformer inductance.
b) In the case of motional inductance a conductor (or wire loop) during its displacement
„crosses” the magnetic flux lines i.e. the motion has a component perpendicular to the fluxlines.
The main point of induction is that the change of magnetic field causes electrical field.
Using the term of induced voltage a phenomenon in magnetic field may be replaced with a
phenomenon in electrical circuit.
In real equipment the two types of inductance (transformer and motional) often appear simultaneously (eg. in rotating electrical machines).
Important notice: If the space contains both static electric and changing magnetic fields, the
electric field become non-conservative, because the line integral is no more path-independent,
since exist such closed paths which enclose changing magnetic field and the integral along
such paths is not zero. In this case the electric potential as scalar descriptor is unusable.
In a closed loop the induced voltage produces current according to the resistance of loop. The
voltage on the resistance is balancing the induced voltage if no other voltage source in the
loop. The Kirchhoff's voltage law for dc circuits: ∑ R j I j + ∑ U ik = 0 , have to be extended
j
∑ R I + ∑U
j
j
j
ik
k
k
+ ∑ U bn = 0 ,
n
here Ui – is induced voltage, Ub – is non induced voltage (e.g. one of galvanic source). (For
sinusoidally changing alternating current the voltage law is valid for phasors and impedances
are taken into account instead of resistors.)
Transformer induction
The reference direction of the flux changing and that of the charge-separating electrical field
intensity are according to the figure,
U i = − Ed l .
dφ
>0
dt
E
-
+
Ui
Reference directions for transformer induction
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The induced voltage depends not of the magnitude of magnetic flux but the magnitude and the
direction of the derivative of magnetic flux.
φ
Ui
φ
Ui
+
+
-
φ
φ
dφ
>0
dt
dφ
<0
dt
t
t
Polarity of the induced voltage at different directions of flux and derivative of flux (φ > 0)
Ui
Ui
+
+
-
φ
φ
φ
φ
t
dφ
>0
dt
t
dφ
<0
dt
Polarity of the induced voltage at different directions of flux and derivative of flux (φ < 0)
10
Fundamentals of magnetic field
The flux linkage
Usually the changing flux is encircled by a coil of N series turns (and the turns exciting in the
same direction) so the induced voltages of the single turns are aggregated for the coil. If each
turn of wire linked with flux of the same magnitude then the resultant induced voltage
dφ (t )
ui (t ) = N
.
dt
The sum of the fluxes linked with the single turns gives the resultant flux linkage ψ=Nφ
which can be used for calculation of the resultant induced voltage:
dψ (t )
ui (t ) =
.
dt
The physical unit of the flux linkage Ψ is the same as that of the flux Φ: [Ψ]=Wb=Vs.
Lenz's5 law
As follows form the conservation of energy principle the currents and forces produced by
induction have such an effect, which is decreasing the process generating them.
dφ
has such polarity that produces a current i through an external
The induced voltage U i =
dt
resistance which opposing the original change of flux linkage, decreasing the inducing effect.
The force acting to a current-carrying wire moving in magnetic field brakes the movement.
In other words the magnetic field produced in the process of induction acts for conservation
of the initial state.
This principle appears in the phenomenon of self induction.
φ
dφ
>0
dt
i
R Ui
-
i
+
Magnetic effect of the current produced by induced voltage
Motional induced voltage
Motional induced voltage is generating in a moving wire because a force of interaction appears between the static magnetic field and the charges travelling with the wire. (For currentcarrying conductors the produced force expressed as: F = lI × B . This force act to the
charges which forward it to the conductor.)
Define a current I ∗ which is not a „real” one, but helps to calculate a force since the movement of charges. The force F ∗ according to I ∗ :
5
Lenz (Lenc), Heinrich Friedrich Emil (1804-1865) physicist, German origin
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h
Q
h
Q × B = Qv × B , because I ∗ =
and v = .
t
t
t
Inside a conductor, moving in a homogeneous magnetic field B with speed v perpendicular
both to it's own direction and to B , magnetic force acts to the charges in the conductor. This
force separates the charges inside the conductor thus produces an electric field E . The direction of the electric field is the same as the force acting to the positive charges.
F ∗ = hI ∗ × B =
B
l
F∗
I∗
+Q
+Q
dh
E
-
+
+
v
F
ui(t)
i (t)
R
-
h
A possible illustration of motional inductance
F∗
E=
= v × B . Due to this electric field the charges accumulating by polarity at the ends of
Q
conductor, which results the appearance of induced voltage. The voltage ui between the ends
of a conductor with length l in homogeneous magnetic field expressed as
ui = − E l = − v × B l = lB × v , if the reference of voltage is from charges (+) to charges (-).
This voltage is an induced voltage, produced by the charge-separating electric field E, often
mentioned as electromotive force (emf).
dφ
∫ Edl = − dt .
The induced voltage causes a (real) current in closed circuit. The interaction of the current I
and the magnetic field B produces a physical force F against the movement, according to the
Lenz's law. (Another explanation: the density of flux lines increasing in the direction of
movement.) This mean that in case of closed circuit the movement of wire requires continuous force, energy.
There are two forces discussed:
- a force acts to the charges travelling with the conductor, the consequence of which is
the charge-separating effect of an electric field E and the induced voltage Ui,
- due to the current caused by this induced voltage a force acts to the conductor.
The direction of the two forces are different.
Flux linkage of a long straight wire of finite dimension
The magnetic field produced by low-frequency alternating current or direct current flowing
through a long circular conductor will be not only external to the conductor but also exists
within the conductor. The internal flux will link only a fraction of the current, this linkage
must therefore be treated separately from external.
Consider such a conductor in free space of radius rc, carrying a current I.
12
Fundamentals of magnetic field
External flux
The external magnetic field may be calculated approximately by the same way as that of a
conductor with infinitely small size (one dimensional conductor).
Assuming uniform current distribution within the conductor, the current density J or the current I:
I
I
J=
= 2 , or I = ∫ JdAc = Jrc2π ,
Ac rc π
where rc – the radius of conductor, Ac – the cross-section of conductor.
In range of radius a>rc the field intensity He of the external magnetic field
I
H e (a ) =
, while a≥rc.
2π a
µ I
The external flux density in free space is Be (a ) = µ 0 H e (a ) = 0 .
2π a
The annulus external flux dφe enclosed by an area dA=lda determined by annulus da through
length l of conductor (into the plane of the paper) is
µ I
dφ e (a ) = Be (a )dA = 0 lda .
2π a
Since the single wire is considered as one turn the flux linkage is equal to the flux dψe=dφe,
the total external flux is
R
R
µ Il da µ 0 Il R
ψ e = ∫ dφ = 0 ∫
=
ln ,
2
π
a
2
π
rc
rc
rc
where – R is a sufficient distance to give zero field (theoretically R→∞).
H
Hi
He
a
R
rc
a
The magnetic field intensity vs. distance
The external inductivity derived from the external flux:
ψ
φ
µ l R
Le = e = e = 0 ln .
2π
I
I
rc
13
da
VIVEM111 Alternating current systems
Example
In air µ=µ0, then Le =
2014
µ0l
= 0.5l ⋅ 10 −7 (H), or Lb= 0.05 µH/m.
8π
Internal flux
The conductor is considered to be made up of an infinite number of parallel conducting elements and the current distribution is uniform (the current density is constant throughout the
cross-section).
The current flowing inside radius a<rc calculated as a fraction of current I:
a2
I a = Ja 2π = I 2 for a≤rc.
rc
The internal magnetising force Hi is due only to the current Ia within the radius a
a2
a
I
H i (a ) = a = I
=
, a≤rc.
2
2π a
2π arc
2π rc2
(At the surface of the conductor the formulas for external and internal fields give the same
results.)
a
,
The internal flux density Bi (a ) = µ H i (a ) = µ I
2π rc2
where µ – the total permeability of the conductor material, for non-ferromagnetic materials
usually considered as µ=µ0.
The internal flux for annulus area dA:
µ Ia
dφ i (a ) = BidA =
lda .
2π rc2
If the whole cross section of the conductor is considered as one turn, then the flux dφi links
only a turn Na fraction of 1, proportional to the cross-section for radius a≤rc
a2
Na = 2 .
rv
Hence the elemental internal flux linkage for a≤rc
µ Ia 3
a 2 µ Ia
dψ i (a ) = N a dφ i (a ) = 2
l
da
=
lda .
rc 2π rc2
2π rc4
The total internal flux linkage:
r
µ Il c 3
µ Il
ψi =
.
a da =
4 ∫
2π rc 0
8π
The inductivity derived from the internal flux: Li =
Example
if µ=µ0, then Li =
ψ i µl
=
I
8π
µ0l
= 0.5l ⋅ 10 −7 (H), or Li= 0.05 µH/m.
8π
14
Fundamentals of magnetic field
As mentioned above the current Ia in the segment of conductor inside an arbitrary radius a≤rc
a2
is only the fraction of the total current I: I a = I 2 , whereas the same segment of the conducrc
tor is linked with the total internal flux φi. Hence the internal inductivity of a segment of conductor derived from the internal flux:
ψ
r2
l i (a ) = i2 = Li c2 for a≤rc.
a
a
I 2
rc
li
Li
a
rc
Change of internal inductivity
The inductance of the conducting elements nearer the centre is greater then that of those near
the outside.
When the current is alternating the inductive reactance of the elements near the centre is
greater then that of those near the outside, and hence more current flows in the elements near
outside.
Skin effect
According to complex, complicated calculations in a circular conductor the current density
decreasing from the surface to the centre. A depth of penetration δ used to define distance at
which the current density decreases to 1/e of its value at the outer surface.
In the case if the radius of the conductor rc> (3-5)δ, the current density can be considered
uniform up to the depth of penetration and zero within the radius rc-δ.
The influence of skin effect is obviously more pronounced at larger inductive reactance –
which varies with the angular frequency of the current ω, and with the inductance. The inductance varies with the permeability of the material of the conductor µ, consequently the larger
the permeability, the smaller the depth of penetration.
On the other hand, the greater the resistivity of the conductor ρ, the smaller will be the effect
of the variation in inductive reactance in causing a non-uniform current distribution. The relationship between the depth of penetration δ and ω, µ and ρ expressed in formula
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VIVEM111 Alternating current systems
δ=
2ρ
2
=
ωµ
2πµ 0
2014
ρ
1
=
fµ r
πµ 0
ρ
ρ
= 503,3
.
fµ r
fµ r
The equation for the depth of penetration above may be applied to flat sheets (rc→∞).
The same considerations apply to the penetrations of alternating magnetic flux into conductors.
The proximity effect
Similar phenomenon occurs when two conductors carrying alternating currents are in each
other's magnetic fields, there will also be a redistribution of currents. The currents are
crowding to those parts of the conductors, which link the least amount of flux and therefore
have less inductivity.
The illustration of proximity effect with skin effect, the current flow identical
Example
The values for copper: ρCu=1.78⋅10-8 Ωm, µrCu=1,
the depth of penetration at frequency f=50 Hz δCu=9.49 mm (8.66 mm at f=60 Hz).
The values for aluminium: ρAl=2.9⋅10-8 Ωm, µrAl=1,
the depth of penetration at frequency f=50 Hz δAl=12 mm (10.95 mm at f=60 Hz).
The values for iron: ρFe=13⋅10-8 Ωm, µrFe=5000,
the depth of penetration at frequency f=50 Hz δFe=0.35 mm (0.32 mm at f=60 Hz).
δ (m)
0.07
0.06
0.05
0.04
0.03
Al
0.02
Cu
0.01
f (Hz)
0
0
20
40
50
60
80
The depth of penetration vs. frequency
16
100
Fundamentals of magnetic field
Skin effect in ferromagnetic surrounding
The phenomena in a conductor surrounded by ferromagnetic material are basically the same
as the skin and proximity effects in any system of conductors with alternating current in them.
The figure shows in cross section of a deep, narrow bar of a squirrel-cage rotor and further
shows the general character of the slot-leakage field produced by the current in the bar within
this slot, in particular when the depth of the slot exceeds its width.
If the rotor iron has infinite permeability, all the leakage-flux lines would close in paths below
the slot, as shown. If imagine the bar to consist of layers, which are electrically in parallel, the
leakage inductance of the bottom layers is greater than that of the top layers because the bottom layer is linked by more leakage flux. Consequently, the alternating current in the lowreactance upper layers will be greater than that in the high-reactance lower layers. The current
will be forced toward the top of the slot, in addition the current in the upper layers will lead
the current in the lower ones. The non uniform current distribution results in an increase in
the effective resistance and a smaller decrease in the effective leakage inductance of the bar.
a)
Magnetic field of a current-carrying wire in a
deep slot
b)
Representative shapes of slots in squirrelcage rotor
a) double cage b) deep-bar
Since the distortion in current distribution depends on an inductive effect, the effective resistance is function of the frequency (i.e. of the slip) and also function of the depth of the bar and
of the permeability and resistivity of the bar material.
J
h
J
h
fr=f1
fr=0
Distribution of current density in the bars of double cage rotor
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VIVEM111 Alternating current systems
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The effect can be explained as follows: the particular parts or layers of the wire are linked by
ψ
and the impedance
different magnetic flux, therefore the inductivity l =
i
z = r 2 + (2πf r l) also differ depending on the distance from the top of the slot. To increase
2
the frequency-dependence of the rotor resistance double cage or deep slot are applied.
In a double cage rotor the upper bars (close to the rotor surface) are made from material of
less cross-section and more resistivity (e.g. brass) while lower bars are made from material of
more cross-section and less resistivity (e.g. copper). At standstill or startup (S=1) the frequency of the rotor current is equal to that of the stator current (e.g. fr=50-60 Hz), due to the
skin effect the rotor current flows mainly in the upper cage of more resistance (and less reactance), while around the rated (nominal) speed (e.g. Sn≈0.03-0.05, frn=1.5-3 Hz) the distribution of rotor current determined by the ratio of the resistances of the cages.
By means of construction may be achieved that the effective resistance of rotor coil is more at
startup and less at nominal speed.
J
fr=f1
fr=0
h
Distribution of current density in the wire of a deep slot rotor
The phenomena and its influence is similar in rotor with deep slot, but vary of the current
density J is continuous in function of the distance from the top of the slot.
Exploiting the frequency dependence of the effective rotor resistance some improvement may
be achieved: the starting torque increasing, the speed fall at nominal performance decreasing.
w
w1
Rr(Sn)
Rr(S=0)
T
Tbg
M
Ti1s2
Tl Ts1
Static speed-torque characteristic of induction machine
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Fundamentals of magnetic field
The starting torque Ts2 at the smaller resistance wouldn't be enough to start with load torque
Tl. On the other hand, at the greater resistance the speed fall at load torque Tl would be greater
in steady state. The two characteristics are simplifications, the transition between the characteristics is continuous during accelerating, the vary of leakage inductance is neglected. In the
figure: w1 – synchronous speed, Tbg – breakdown torque (breaking mode), Tbm – breakdown
torque (motor mode), Ts1, Ts1 – starting torque, Tl – load torque, Sn – nominal slip.
Influence of eddy current to the distribution of the magnetic field
Due to the change of the magnetic flux induced emf appears and eddy currents flow in the
conductor. According to Lenz's law the eddy currents oppose the original change of the flux,
the inducing process, producing such a field, which displaces the flux to the surface of the
conductor. The depth of penetration of the magnetic field may be calculated as that of current
density.
Ieddy
dψ
dt
Influence of eddy currents to the distribution of magnetic field
In ferromagnetic parts conducting, controlling magnetic field the eddy currents are not desired
because of the displacing the flux and increasing the losses. The eddy currents can be decreased using lamination of iron or using or ferrite core.
Composed by: Kádár István
March 2014.
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VIVEM111 Alternating current systems
2014
Questions for self-test
1. Explain the force on a current-carrying conductor in magnetic field.
2. Review the variables describing the magnetic field.
3. Define the magnetic field intensity, flux and flux density.
4. Explain the magnetic permeability.
5. Explain the Ampere's law of excitation.
6. Explain the Faraday's induction law Faraday's induction law.
7. Illustrate the flux leakage.
8. Illustrate approximately the magnetic field of current-carrying conductor and conductor
loop.
9. Illustrate approximately the magnetic field of solenoidal and toroidal coils.
10. How approximated the calculation of magnetic field of solenoidal and toroidal coils?
11. Define the flux linkage.
12. Explain the motional induction.
13. Explain the transformer induction.
14. Explain the Lenz's law for motional and transformer induction.
15. Illustrate the magnetic field of current-carrying conductor of finite dimension.
16. Illustrate the current distribution in a conductor of finite dimension.
17. Explain the inductivity of a conductor of finite dimension.
18. Explain the skin effect.
19. Explain the depth of penetration in a current-carrying conductor.
20. How approximated the current distribution with respect to skin effect?
21. Explain the proximity effect.
22. Illustrate the skin effect in ferromagnetic surrounding.
23. Explain the eddy current and it's influence on the distribution of magnetic field.
24. How can be reduced the eddy current?
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