Properties of ferromagnetic materials, magnetic circuits – principle of calculation
Ferromagnetic materials
Several materials represent different macroscopic properties, they give different response to
external magnetic field. The reason for difference is certain different microscopic properties,
like electron shell configuration, movement and spin of electrons, or rather the influence of
external magnetic field on these properties.
In physics dia- para- and ferromagnetic materials are distinguished, while in the practice of
electrical engineering all non-ferromagnetic substances considered as free space with relative
permeability µr=1. Magnitude of the relative permeability of ferromagnetic materials (iron,
nickel, cobalt and their alloys) may be very high even in order 103-106. Ferromagnetic material may be alloyed from non-ferromagnetic substances.
In ferromagnetic materials the relationship between B and H (magnetisation curve) is nonlinear, multivalued, „history” dependent, and often time dependent, its characteristics can not
be described analytically, hence it has to be determined experimentally for each type of material.
Magnetisation curve
When an initially non-magnetised piece of ferromagnetic material is magnetised by increasing
the external field slowly, then the flux density will also increase. The curve B(H) showing this
vary of flux density B vs. change of magnetic field intensity H called primary, original, dc or
normal magnetisation curve.
B
d
Bmax
c
Br
b
-Hc
a
H
Hmax
Typical shape of magnetisation curve
In the primer magnetisation curve 4 sections distinguished:
a - starting zone,
b - linear zone,
c - knee of the curve (intermediate zone),
d - saturated zone.
VIVEM111 Alternating current systems
2014
As field intensity achieved saturated zone and H is decreased from its maximum value Hmax to
zero slowly, the flux density B while also decreases it will not return along the original curve,
but on a new (somewhat higher) one, and a Br remanent magnetisation (residual, remanent
magnetic flux density or remanence, retentivity) remains although H become zero. The
change of B delayed with respect to change of H (hysteresis = delay). To reduce remanent
magnetic flux density to zero needs a field intensity in the opposite direction, called coercitive
force -Hc. As the figure shows the value of permeability, the values of relation B/H, is not
single-valued, its change is non-linear, it depends of the previous value of field intensity H,
the rate of change. In the saturated zone µr~1.
If the magnetising force is alternating (AC) the flux density changes along a loop known as
hysteresis loop.
The largest hysteresis loop belongs to the peak values Bmax and Hmax, determined by the flux
density of saturation. The hysteresis loops of less peak values located inside the largest loop.
If the magnetising force changes slowly the loop is called static hysteresis loop.
B
Bmax
H
Hmax
Hysteresis curves of different maximum value of Hmax
Dynamic hysteresis curves
In case of alternating magnetic field generated by alternating current of network frequency (or
higher) the points of the B(H) plane form a full hysteresis loop during each period. In addition, due to the alternating flux an induced voltage appears which produces eddy currents inside the ferromagnetic material. According to Lenz's law the magnetic field of these eddy
currents makes the change of the flux density more delayed, therefore increasing the frequency the dynamic hysteresis loops become spacious compared to static ones.
2
Properties of ferromagnetic materials, magnetic circuits – principle of calculation
B
 static
- - - dinamic
H
Static and dynamic hysteresis curves
Relative permeability
In each point of magnetisation curve the absolute permeability µ =
B
and the relative perH
B
may be determined. Because of the non-linearity of the curve in engiµ0H
neering practice several simplifications applied.
meability µ r =
B
αt
P
αd
αs
µ0H
Explanation of the full, the differential and the initial permeability
- full (ordinary) permeability: the slope of the line from origin to the points of primary magB
netisation curve (eg. to point P) µ r =
= tgα t ,
µ0H
- initial permeability: the relative permeability at low excitation level, the slope of the starting
zone of primary magnetisation curve µrs=tgαs,
- differential permeability: the slope of primary magnetisation curve at a given point (eg. at
dB
point P) µ rdiff =
= tgα d ,
µ 0 dH
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VIVEM111 Alternating current systems
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∆B
incremental
∆B
µ0∆H
reversible
µ0H
µ0∆H
Illustration of incremental and reversible permeabilities
- incremental permeability: specific for the narrow loop due to small cyclic changes in a point
of the magnetisation curve when a small alternating field is superimposed on a dc or slowly
∆B
,
changing field µ rinc =
µ 0∆ H
- reversible permeability: the same as incremental one, but the cyclic change so small, that the
elementary loop forms a single line.
In electrical power engineering practice mostly the full (ordinary) permeability is used.
Calculation of magnetic circuits
The magnetic circuit is a closed segment of the magnetic field in which the flux considered
constant, flux lines do not leave such a segment. Actually all closed flux line form a magnetic
circuit. In magnetic circuits usually ferromagnetic components determine the path of the flux
lines. The calculation is simple in cases the path of flux (the arrangement) is known.
Magnetic circuits examples
The required excitation for a desired field of a complex magnetic circuit composed from different segments may be easily calculated whereas to answer the inverse problem, calculation
the result of known excitation, is difficult because of the non-linearity of magnetisation curve.
The flux leakage may be considered as a result of calculation or estimation but may be neglected as well.
4
Properties of ferromagnetic materials, magnetic circuits – principle of calculation
Along a magnetic circuit usually there are segments of different properties (geometry, permeability) and may occur ramifications.
The Ampère's law of excitation is valid for slow changes only: for direct current and time
instants of alternating current. In case of fast changes the effects of eddy currents have to be
taken into account too.
Series magnetic circuits
The series magnetic circuits usually consist of segments of different cross-sections and materials.
Calculation the excitation necessary to generate specified magnetic flux
Suppose the Φ flux is given, specified along the magnetic circuit investigated and the leakage
neglected Φl=0, according to the figure.
B2, H2, µ2
A1
B1, H1, µ1
l1
l2/2
µ2
l2/2
Bδ, Hδ, µ0 Aδ
A2
δ
Outline of a series magnetic circuit
In this case the flux density in the air gap is Bδ =
ments B1 =
Φ
, and in the further ferromagnetic segAδ
Φ
Φ
, B2 =
and so on.
A1
A2
Bδ
, while in the ferromagnetic
µ0
segments the field intensities H1, H2 etc. or the relative permeabilities µr1, µr2 etc. are usually
determined from the magnetisation curve.
B1
B2
Φ
Φ
H1 =
=
and H 2 =
=
.
µ 0 µ r1 µ 0 µ r1 A1
µ 0 µ r 2 µ 0 µ r 2 A2
Applying the law of excitation the resultant excitation for the whole circuit using terms
µi=µ0µri:
B
l
Φ
li = Φ ∑ i ,
Θ = ∑ Θ i = ∑ H i l i = δ δ + H1l1 + H 2 l 2 +K = ∑
µ0
i
i
i µ i Ai
i µ i Ai
since Φ provided to be constant. The resultant excitation can be defined as sum of partial
excitations for the single segments of circuit.
In such cases when the most of resultant excitation belongs to the air gap, the ferromagnetic
(iron) parts of magnetic circuit may often be neglected (µiron»µ0, thus Hδ »Hiron).
The field intensity in the air gap is simply calculated: Hδ =
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VIVEM111 Alternating current systems
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Example
Given Bδ=Biron=1T, δ=1 mm, liron=1 m, from the magnetisation curve µriron=106.
B
Bδ
A
The field intensity inside the air gap: Hδ = δ =
= 0 ,8 ⋅ 10 6 Bδ = 0,8 ⋅ 10 6 ,
−6
µ 0 1,256 ⋅ 10
m
Bδ
Bvas
A
=
= 0 ,8 Bδ = 0,8 .
inside the iron segments: H iron =
m
µ 0 µ riron µ 0 µ riron
The resultant excitation is the sum of the required excitation for iron and air gap: Θ=Θiron+Θδ.
The excitation for iron Θiron=Hironliron=0,8 A, while that for air gap Θδ=Hδlδ=800 A.
Θ 800,8
=
The current needed when the number of turns in the coil N: I =
(A ) .
N
N
The excitation needs for iron segment is 0.1% of the resultant excitation.
Using iron of less permeability the necessary excitation for iron increases and is not negligiA
ble. For example if µriron=103, H iron = 800 , Θiron=Hironliron=800 A, that is 50% of the full
m
excitation (Θ=1600 A).
The inverse problem, when the flux or flux density have to be answered for known current I,
is difficult to solve because the distribution of full excitation depends on the relation among
permeabilities of each segments, however to determine the permeabilities the field intensity
needed because of the non-linearity of magnetisation curve.
A practical solution is the calculation of excitations or currents which generate different
fluxes and then the drawn Φ(Θ) or Φ(I) relationship may give the answer of problem.
Parallel magnetic circuits
Since the flux lines are closed, the total flux at input and output are identical:
Φ=Φ1+Φ2.
Φ1, H1
A1
l1
µ1
Φ
Φ
Φ2, H2
A2
l2
µ2
Outline of a parallel magnetic circuit
According to excitation law for the l1-l2 closed loop:
H1l1 - H2l2 = 0, hereby H1l1 = H2l2 = Θp,
that is the excitations of parallel segments Θp are equal. Substituting field intensities H1 and
H2:
Φ
Φ2
µ A
µ A
Θ p = 1 l1 =
l 2 , from which follow Φ 1 = Θ p 1 1 , and Φ 2 = Θ p 2 2 .
µ 1 A1
µ 2 A2
l1
l2
The total flux Φ with excitation Θp:
µ A µ A 
µ A
Φ = Φ1 + Φ 2 = Θ p  1 1 + 2 2  = Θ p ∑ i i .
l2 
li
 l1
i
6
Properties of ferromagnetic materials, magnetic circuits – principle of calculation
The „magnetic Ohm’s law”
When modifying the formula of excitation law Θ = ∫ Hdl – because of formal analogy – the
equations for complex magnetic circuits often mentioned as „magnetic Ohm’s law”.
B
Φ
into the line integral of the magnetic field intensity the
With substitutions H = and B =
A
µ
formula for the excitation of series magnetic circuit
l
Φ
Θ=∑
li = Φ ∑ i
i µ i Ai
i µ i Ai
is similar to the formula for the resultant voltage of series conductor segments with finite resistances:
l
I
U =∑
l i = I ∑ i = I ∑ Ri ,
i γ i Ai
i γ i Ai
i
1
where γ = – is the specific electric conductivity, the reciprocal of the specific resistivity ρ.
ρ
The total excitation of series magnetic circuit may be written as
U m = Φ ∑ Rm i ,
i
where Um=Θ – the total magnetic voltage (excitation),
l
Rm i = i – the magnetic resistance (magnetic reluctance) of ith segment. The total magµ i Ai
netic resistance of the series segments: Rm = ∑ Rm i , herewith Um=ΦRm.
i
A
A
=
.
Vs Wb
The more the permeability the less the magnetic resistance and the excitation (magnetic voltage) of a segment in a magnetic circuit.
The excitation of a segment in a magnetic circuit may be termed as magnetic voltage of that
segment, for the ith segment:
l
Umi = Φ i .
µ i Ai
The excitation law may be formulated as follows: the line integral of the magnetic voltage
around a closed path is equal to the excitation Θ of the area enclosed by that path
Θ = ∑ U mi .
Physical units of variables above: [Um]= A, [Φ]= Vs=Wb, [ Rm ] =
i
The formula derived for the total flux of parallel circuit
µ A
Φ =Θ p∑ i i
li
i
may be written using the above relations as Φ = U m p ∑ Λ i ,
i
µ i Ai
1
– the magnetic conductivity (magnetic permeance) of the i-th seg=
Rm i
li
ment, the reciprocal of magnetic resistance. Physical unit of magnetic conductivity is
Vs Wb
[Λ m ] = A = A .
where Λ i =
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VIVEM111 Alternating current systems
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The total magnetic conductivity of the parallel segments:
Λ m = ∑ Λ m i , whereby
i
Φ=UmΛm=ΘΛm.
From the analogy above may be build up substituting electric circuits of magnetic circuits.
Such substitution have to be used carefully because the analogy is only formal, the physical
phenomena are essentially different.
a) The electric current I is a real flow of the charges (charge-carrier particles) while the magnetic flux Φ describes the state of the space without any movement of particles.
b) Keeping on the electric current (dc current as well) produces losses, while keeping on the
magnetic flux does not require energy (only the building up or the changing of magnetic
field).
c) The line integral of the electric voltage around a closed path is equal to zero if the path
dΦ
= 0 , while the line integral of the magnetic voltage
does not encircle changing flux
dt
around a closed path equal to zero if the path does not encircle current ΣI=0.
d) The electric conductivity γ is usually constant (at constant temperature) and currentindependent, while the permeability of ferromagnetic materials µ vary with flux (flux density)
significantly.
e) The relation between the conductivities of electric conductors and insulators is about 1020,
thus current leakage is usually negligible. This relation for magnetic conductors and insulators is about 103-106, thus flux leakage and its influence is often has to be taken into account.
f) The method of superposition unusable in circuits containing ferromagnetic components, in
most cases the excitations may be summarised only, summarising is not allowed for the magnetic flux densities of single exciting components.
Self induction, self inductance
When a current I flows in a coil it builds up a magnetic flux Φ inside the coil, changing current causes changing the magnetic flux. According to Faraday's induction law the changing
dψ (t )
magnetic flux produces an induced voltage ui (t ) =
. In this process the coil induces a
dt
voltage in itself, and therefore this voltage is called self induced voltage. As follows from
Lenz's law the effect of induced voltage opposes the changing what has produced that voltage
(back emf.).
In general, taking into account that the flux linkage is a function of current ψ = ψ ( i(t )) , the
induced voltage:
ui (t ) =
dψ ( i(t ))
dt
=
dψ (t ) di(t )
.
di(t ) dt
The relation between the flux linkage and the current is represented by the self inductance
dψ (t )
L=
. The SI unit of inductance in honour of Henry's1 scientific activity
di(t )
Vs
= Ωs .
A
di(t )
. By means of the changing in
The formula for induced voltage become as ui (t ) = L
dt
magnetic field may converted to changing in electric circuit.
[L] = H = henry =
1
Henry, Joseph (1797-1878) American physicist
8
Properties of ferromagnetic materials, magnetic circuits – principle of calculation
Ψ
µ = const.
Ψ
Ψ
µ ≠ const.
Ψ
L
L
I
I
Current-dependence of the inductance – the magnetisation curve is
linear
non-linear
In non-ferromagnetic substance the relation ψ(i) is linear thus L =
dψ (t ) Ψ
=
= const., in
di(t )
I
ferromagnetic substance L≠const.
Since the field outside the coil is negligible, by the law of excitation for the homogeneous
field inside solenoid:
NΦ
Φ
Ψ
A
Ψ
NI = Hl =
l=
l=
l , and L =
= N 2µ 0 = N 2Λ .
Nµ0 A
Nµ0 A
µ 0A
I
l
i
H, φ
N
l
A
Approximate calculation of inductance in a solenoidal coil
The inductance of solenoidal coil depends on the number of turns, the geometric dimensions
and the material. In ferromagnetic substance the inductance is current-dependent.
Explanation the square of N: on the one hand the magnetic field is generated by current I in N
turns of coil, on the other hand the magnetic field induces voltage also in N turns of coil.
To fabricate a circuit component (eg. wound resistor) with low inductance (pure-resistive,
inductance-free component) the double-wound (or bifilar) technology may be used. There are
two coils, a right-thread and a left-thread, accordingly the generated fluxes are in opposite
direction, they destroy each other and the result is a very low (ideally zero) flux, thus dΨ is
small (the voltage of self-induction ui is small) therefore the inductance L is also small.
Scheme of inductance-free (bifilar) coil
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VIVEM111 Alternating current systems
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Mutual induction, mutual inductance
Consider two coils placed near each to other. A portion of the magnetic flux generated by the
current of the first coil affects the second coil or part of it. Changing the current i1(t) of the
first coil (primer) the change of the flux φ21(t) linked with the turns of the second coil
(secondary) induces voltage in the second coil:
dψ 21 (t ) dψ 21 (t ) di(t )1
u21 (t ) =
=
,
dt
di1 (t ) dt
where ψ21(t)=N2φ21(t).
The first index shows the coil what is affected by the magnetic field of the coil with the second index: φ21 - the flux, linked with the second coil, produced by the current in the first coil.
φ21
i1
N1
N2
µ1
µ2
u21
A2
H21
l21
Illustration of coupled coils
dψ 21
is termed mutual inductance, usually marked as M21 or L21, the SI unit is
di1
the same as that of self inductance: [M]=H (henry). The mutual inductance depends on the
numbers of turns, the dimensions and the material. In ferromagnetic substance the mutual
inductance is current-dependent. If the permeability is constant (eg. in iron-less coil) in steady
Ψ
state the mutual inductance is constant M 21 = 21 . Applying the law of excitation to the
I1
closed path of the magnetic field determined by flux φ21 results (simplified):
φ
θ1 = N1i1 = H 21l 21 = 21 l 21 = φ 21 Rm21
A2 µ 2
φ 21 = N1i1Λ 21
Nφ
ψ
M 21 = 21 = 2 21 = N1 N 2 Λ 21 .
i1
i1
In the last formula includes the product of the numbers of turns N1N2, because N1 turns are
magnetising and the voltage induced in N2 turns. The link exists in the opposite direction too,
when exciting the second coil, the voltage induced in the first coil.
In isotropic substance M12=M21, since Λ12=Λ21.
The mutual inductance often has to be determined by measurement data. The voltage induced
in the secondary coil ui21(t):
dψ 21
di
ui21 =
= M 21 1 ,
dt
dt
when the current i1(t) is sinusoidal with time then Ui21rms= X21I1rms (using linear approxima1 U i 21rms
tion) and the mutual inductance: M 21 =
.
2 fπ I1rms
The derivative
10
Properties of ferromagnetic materials, magnetic circuits – principle of calculation
Magnetic field of coupled coils, decomposition
The coils are coupled (in most simple case 2 coils) when they are placed into the magnetic
field each others and the mutual influence of these fields is not negligible. Depending on the
application the aim may be the strong linkage (e.g. for energy conversion) or the weak linkage
(eg. to reduce the electromagnetic noise).
φ11
φ21
φl 1
φm
i1
φ12
i2
φl 2
φ22
Decomposition of magnetic field
Depending on the arrangement and geometric dimensions the only existing complete magnetic field is linked with the individual coils not equally. To make the illustration and explanation more simple the flux of the magnetic field usually decomposed into four parts:
- the part φ21 of flux φ11 generated by the current i1 of the coil 1 is linked with the coil 2, the
rest is the flux leakage φl1 of the coil 1 (which linked only with the coil 1),
φ11=φ21+φl1.
- the part φ12 of flux φ22 generated by the current i2 of the coil 2 is linked with the coil 1, the
rest is the flux leakage φl2 of the coil 2 (which linked only with the coil 2),
φ22=φ12+φl2.
The complete flux becomes: φ=φ11+φ22=φ21+φl1+φ12+φl2.
These components usually divided according to its „origin” or „function”.
If the components are selected by „origin” (coupled-circuit model), the resultant flux of each
coil is the sum of the whole „own” flux and the additional part from the other coil:
the resultant flux linked with coil 1 is
φ1=φ11+φ12=φ21+φl1+φ12,
the resultant flux linked with coil 2 is
φ2=φ22+φ21=φ12+φl2+φ21.
If the components are selected by „function” (magnetic field theory), the resultant flux of
each coil is the sum of the mutual part of the complete flux φm and the „own” flux leakage φl:
the resultant flux linked with coil 1 is
φ1=φm+φl1=φ21+φ12+φl1,
the resultant flux linked with coil 2 is
φ2=φm+φl2=φ12+φ21+φl2.
The mutual flux φm has two components: φm1=φ21 and φm2=φ12, thus φm=φm1+φm2=φ21+φ12.
The complete flux of the system by both interpretations is certainly equal.
For description of electrical machines (e.g. transformers, induction motors) the magnetic field
is usually approached in accordance with field theory and the flux components are considered
as those created in suitable inductances: flux leakages are created in leakage inductances,
magnetising (or mutual) flux is created in magnetising (or mutual) inductance.
ψm1=i1Lm
ψm2=i2Lm
ψm=ψm1+ψm2=imLm=( i1+ i2)Lm.
ψl1=i1Ll1
ψl2= i2Ll2
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VIVEM111 Alternating current systems
φl 1
φ1
Ll1
2014
i1
i2
im
φm
φl 2
Ll2
φ2
Lm
Equivalent circuit for decomposed magnetic field
Coupling and leakage coefficients
Generally, the quality of inductive coupling that exists between the two coils is expressed as a
fractional number between 0 and 1, where 0 indicates zero or no inductive coupling, and 1
indicating full or maximum inductive coupling. The coupling coefficient k1 expresses that the
φ
portion of flux generated in coil 1 by current i1 which linked with coil 2: k1 = 21 . Similarly,
φ 11
coupling coefficient k2 expresses the portion of flux generated in coil 2 by current i2 linked
φ
with coil 1: k 2 = 12 .
φ 22
The unity coupling coefficient means that all the flux lines of one coil cuts all the turns of the
other coil.
The leakage coefficient σ1 expresses that the portion of flux generated in coil 1 by current i1
does not linked with coil 2, thus the complement of coupling coefficient k1. Similarly defined
σ2 for the coil 2.
φ
φ
φ − φ 21
φ
φ − φ 12
σ 1 = l1 = 11
= 1 − 21 = 1 − k1 and σ 2 = l 2 = 22
= 1 − k2 .
φ 11
φ 11
φ 11
φ 22
φ 22
Coupled coils in series
Due to series connection a common current flows through both coils. Say the resistances are
negligible compared to inductances. If the fluxes of the coils tend towards the same direction
(aiding or cumulative coupling), then the resultant voltage ui equal to the sum of self induced
voltages and mutual induced voltages:
di
di
di
di
di
di
ui = L1 + M12
+ L2 + M 21
= ( L1 + L2 + M12 + M 21 ) = Le ,
dt
dt
dt
dt
dt
dt
Le - the equivalent inductance.
M12
i
Le
M21
i
ui
ui
Cumulative coupled series coils
In case the numbers of turns in the coils N1 and N2 are equal and M12=M21=M, then
Le=L1+L2+2M, while without coupling M12=M21=0 and Le=L1+L2.
12
Properties of ferromagnetic materials, magnetic circuits – principle of calculation
With perfect coupling σ=0, k=1, φ21=φ11,
ψ
φ

ψ 11 = N1φ 11 → L1 = 11 = N1 11 
i1
i1 
ψ 21
φ 11 

ψ 2 1 = N2 φ 11 → M 21 =
= N2
i1
i1 
N
M12 = 1 L2
N2
M 21 =
N2
L1
N1


N 
N 
Le = L1 1 + 2  + L2 1 + 1  .
N1 
N2 


N 22
N2
N1
Provided M12=M21, then L1
= L2
or L1 = L2 2 ,
N1
N2
N1
substituting into the formula of the equivalent inductance

N
N2
Le = L1 1 + 2 2 + 22  .
N1 N1 

ψ
φ
If N1=N2, then Le=4 L1. while ψ11=N1φ11, hereby L1 = 11 = N1 11 .
i1
i1
For close coupled identical coils L1=L2=M12=M21=L, thus Le=4L, while without coupling
M12=M21=0 and Le=2L.
If the fluxes of the coils tend towards the opposite direction (opposing or differential coupling), then the mutual induced voltages have to be subtracted from the sum of self induced
voltages:
di
di
ui = ( L1 + L2 − M12 − M 21 ) = Le ,
dt
dt
If M12=M21=M, then Le=L1+L2-2M.
M12
Le
i
i
M21
i
ui
ui
Opposing coupled series coils
With perfect coupling the equivalent inductance Le=0 (equal to that of bifilar coil), while
without coupling, when the mutual inductances are negligible, the equivalent inductance is
also Le=L1+L2.
The mutual inductance may be determined experimentally by measurement data: the equivalent inductances measured when coupled in the same direction minus that in opposite direction: L1+L2+2M - (L1+L2-2M)= 4M.
Coupled coils in parallel
Due to parallel connection the voltage of both coils is common whereas the currents are ordinarily different. The voltage equation if the coupling is aiding:
13
VIVEM111 Alternating current systems
ui = L1
2014
di
di
di1
di
+ M12 2 = L2 2 + M 21 1 .
dt
dt
dt
dt
In case M12=M21=M, then
di1
di
di di L − M
di
di L − M
L1 − M ) = 2 ( L2 − M ) , consequently 1 = 2 2
and 2 = 1 1
,
(
dt
dt
dt
dt L1 − M
dt
dt L2 − M
with restriction L1≠M and L2≠M.
ui
i
ui
L1
i1
i
M21
M12
i2
Le
L2
Cumulative coupled parallel coils
Substituting the current derivatives into the voltage equation
di 
L − M  di1 L1 L2 − M 2 di2 
L − M  di2 L1 L2 − M 2
ui = 1  L1 + M 1
=
.
=
 L2 + M 2
=
dt 
L2 − M  dt L2 − M
dt 
L1 − M  dt L1 − M
di
di
Integrating the current derivatives 1 and 2 the resultant current i become:
dt
dt
L2 − M
L1 − M
L + L2 − 2 M
1
i = i1 + i2 =
u dt +
u dt = 1
ui dt =
ui dt .
2 ∫ i
2 ∫ i
2
∫
L1 L2 − M
L1 L2 − M
L1 L2 − M
Le ∫
Thus the equivalent inductance Le:
L L − M2
Le = 1 2
.
L1 + L2 − 2 M
LL
In case the mutual inductance negligible M≈0, then Le = 1 2 = L1 × L2 ,
L1 + L2
L
If L1=L2=L then Le = .
2
ui
i
ui
L1
i1
i
M21
M12
i2
Le
L2
Opposing coupled parallel coils
14
Properties of ferromagnetic materials, magnetic circuits – principle of calculation
If the coupling is opposing then the induced voltage:
di
di
di
di
ui = L1 1 − M12 2 = L2 2 − M 21 1 .
dt
dt
dt
dt
In case M12=M21=M, then
di1
di
di
di L + M
di
di L + M
, and 2 = 1 1
.
(L1 + M ) = 2 (L2 + M ) , consequently 1 = 2 2
dt
dt
dt
dt L1 + M
dt
dt L2 + M
Substituting the current derivatives into the voltage equation
di 
L + M  di1 L1 L2 − M 2 di2 
L2 + M  di2 L1 L2 − M 2
ui = 1  L1 − M 1
L
M
=
=
+
.
=

 2
dt 
L2 + M  dt L2 + M
dt 
L1 + M  dt L1 + M
di
di
Integrating the current derivatives 1 and 2 the resultant current i become:
dt
dt
L2 + M
L1 + M
L + L2 + 2 M
1
i = i1 + i2 =
u dt +
u dt = 1
ui dt =
ui dt .
2
2 ∫ i
2 ∫ i
∫
L1 L2 − M
L1 L2 − M
L1 L2 − M
Le ∫
Thus the equivalent inductance Le:
L L − M2
Le = 1 2
.
L1 + L2 + 2 M
For close coupled identical coils L1=L2=M12=M21=L, the equivalent inductance – using the
l'Hospital's rule Le=L.
In case the mutual inductance negligible M≈0, then the result certainly not depends on the
direction of the produced fluxes:
LL
Le = 1 2 = L1 × L2 ,
L1 + L2
L
if L1=L2=L then Le = .
2
Composed by: István Kádár
March 2014.
15
VIVEM111 Alternating current systems
2014
Questions for self-test
1. Review the main properties of ferromagnetic materials.
2. Illustrate the representative sections of normal magnetisation curve
3. Illustrate and interpret the properties of hysteresis curve.
4. Interpret the static and dynamic hysteresis curves.
5. Represent some permeability definitions.
6. Define the absolute permeability.
7. Define the differential permeability.
8. Define the initial permeability.
9. Define the incremental and reversible permeabilities.
10. Define the magnetic circuit.
11. Review the concept for calculation of series magnetic circuit.
12. Review the concept for calculation of parallel magnetic circuit.
13. Review the principle of „magnetic Ohm’s law” and the limitations of analogy.
14. Review the phenomenon of self induction.
15. Explain the self inductance.
16. What approximation used in calculation the inductance of a solenoidal coil without ferromagnetic core.
17. Illustrate the current-dependence of inductance.
18. Illustrate the used components of coupled coils.
19. Explain the ways used to group the flux components of coupled coils.
20. Review the phenomenon of mutual induction.
21. Explain the mutual inductance.
22. How determined the equivalent inductance of cumulative coupled series coils.
23. How determined the equivalent inductance of opposing coupled series coils.
24. How determined the equivalent inductance of cumulative coupled parallel coils.
25. How determined the equivalent inductance of opposing coupled parallel coils.
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