VI International Workshop “Computational Problems of Electrical Engineering”
Zakopane 2004
Power Emitted Inside a Conducting Cylinder
Placed in Longitudinal Uniform Magnetic Field
of a Character an Attenuated Sinusoid
Zygmunt Piątek, Bernard Baron, Marian Pasko, Borys Borowik
Abstract —In the paper we determine transient
electromagnetic field in a conducting cylinder placed in
external magnetic field having the character of an
attenuated sinusoid through the solution of Bessel
equation in cylindrical co-ordinates using Laplace
integral transformation. Then, we use Poynting theorem
to determine the superficial density of the instantaneous
power flux diffused into the cylindrical charge and the
volume density of the instantaneous power converted
into heat inside this charge.
M
I. INTRODUCTION
agnetic field of a character of an attenuated
sinusoid is used in metal forming and consists of
applying energy of impulse magnetic field to the
process. The impulse of the field is obtained due to the
flow of impulse current, generated by a high-current
impulse generator, through an operating head (a
solenoid or flat bobbin) [1, 2]. In the conducting
cylinder forming the impulse magnetic field is external
in relation to the cylinder and has one component
along the Oz axis (Fig.1) and it is determined by the
following formula
H zew (t ) = 1 z H zzew (t ) ,
the magnetic field strength in rad, 1(t ) - Headviside
unit step.
Typical average values of the above quantities
appearing in the impulse magnetic field metal forming
are: ξ = 0 ,
where:
In the impulse magnetic field metal forming besides
the basic problem consisting of determining the spacetime distribution of the pressure in the element being
formed (having previously determined the volume
density of electrodynamic forces) it is also important
to determine the power emitted inside the metal. We
will deal with this question after the determination of
the space-time distribution of current density J ( r , t )
I
and of the magnetic field strength H (r , t ) in the
element being formed, i.e. of the electromagnetic field.
z
Hzew(t)
(1)
J(r,t)
HI(r,t)
(1a)
HII(r,t)
H 0 - magnetic field amplitude when,
EI(r,t)
z
-1
attenuation in A·m is non-existent, ω - pulsation of
proper oscillation of the system element being formedoperating head- capacitor bank in rad·s-1 , η - magnetic
field attenuation coefficient in s-1 , ξ initial phase of
Zygmunt PIĄTEK is with Czestochowa University of Technology,
Faculty of Electrical Engineering, Electrotechnics Department, 42200 Częstochowa, Aleja Armii Krajowej 17, Poland, E-mail:
zpiatek@el.pcz.czest.pl Bernard BARON and Marian Pasko are
with the Institute of Theoretical and Industrial Electrotechnics,
Faculty of
Electrical Engineering, Silesian University of
Technology, ul. Akademicka 10, 44-100 Gliwice, Poland. E-mail:
berbar@zeus.polsl.gliwice.pl, mpasko@zeus.polsl.gliwice.pl. Borys
Borowik is with the Czestochowa University of Technology, Faculty
of Mechanical Engineering and Computer Science, Institute of
Machines Technology and Production Automation, 42-200
Częstochowa, Aleja Armii Krajowej 21, Poland, E-mail:
borygo@zim.pcz.czest.pl
108
η = 5 ⋅ 10 3 s -1 ,
ϖ = π ⋅ 10 4 rad ⋅ s -1 .
where the component of the magnetic field strength
along the Oz axis
H zzew (t ) = H 0 e −η t sin(ϖ t + ξ ) 1(t ) ,
H 0 = 10 7 A ⋅ m -1 ,
θ
1z
r
y
1θ
1r
I
II
x
R
Fig. 1 Conducting cylinder in external longitudinal uniform
magnetic field of a character of an attenuated sinusoid
II. ELECTROMAGNETIC FIELD
In the case of an infinitely long conducting cylinder
placed in external longitudinal magnetic field (Fig.1)
the values describing the electromagnetic field as for
the symmetry of the system depend only on the r coordinate of the cylindrical co-ordinate system. Then,
we deal here with a one-dimensional question with a
constant magnetic permittivity of the cylinder
VI International Workshop “Computational Problems of Electrical Engineering”
µ = µ0
and its constant conductivity γ . As the field
H zew (t ) has got only one component along the Oz
axis,
form
the
second
rotE zew (r , t ) = − µ
Maxwell
∂H zew (t )
, the electric field
∂t
, i.e.
E
(r , t ) = −1Θ E
zew
Θ
(r , t ) . So we have to
deal with a question of the cylindrical wave cast on the
lateral surface of the conducting cylinder.
In the genral case of a conductor of a chosen kind
placed in alternating electromagnetic field some
currents are bound to appear, as the total electric field
cannot equal zero everywhere in the whole conductor.
Those currents are called Foucault currents [5 ,6] and
are determined by the current density vector J ( r , t ) Fig.1. These currents generate the so-called return
oz
interaction magnetic field H ( r , t ) , which, in the
system we are considering, has got one component
along the z axis, thus
papers [5, 6, 7, 8] it
equals zero. The zero
magnetic field in r >
H oz (r , t ) = 1 z H zoz (r , t ) . In
has been shown that this field
value of the return interaction
R2 area results form the fact
that the lines of the density of current J (r ) induced
in the tubular charge are concentric circles of Oz
axis– Fig.1. Then they do not generate any magnetic
field outside the tube, as it is also the case with the
current in the infinitely long solenoid. Then the total
magnetic field in the considered area
zew
z
H (t) = 1z H (t) = 1z H (t) = 1z Im{H (t)}, (2)
II
II
z
zew
z
where
H z (t ) = H 0 e -η t e jϖ t 1(t ) ,
zew
t=0
I
H z (r ,0) = 0 .
(3b)
We solve equation (3) with the boundary condition
(3a) applying Laplace integral transform. In order to
do this let us denote by
I
H z (r , s ) the Laplace
I
transform complex function H z ( R, t ) in relation to
variable t , and then we perform the Laplace
transformation of the following terms of the
differential equation (3), taking into account the zero
I
initial condition H z ( r ,0) = 0 . Thus, we obtain the
following equation [5, 6, 8]:
I
I
I
∂ 2 H z (r, s) 1 ∂ H z (r, s)
+
− sµ γ H z (r, s) = 0 (4)
2
r
∂r
∂r
with the boundary condition
1
,
s − s0
I
H z ( R, s ) = H 0
where
(4a)
s 0 = −η + j ϖ .
(4b)
Equation (4) is the Bessel equation of zero order of
variable r, whose solution is the function [5, 6, 7, 8]
I 0 ( s µγ r )
I
H z (r , s) = H 0
where I 0 ( s
( s − s 0 ) I 0 ( s µγ R)
µγ r )
is the modified
function of the complex variable
(2a)
where the complex amplitude of the external magnetic
field
H 0 = H 0 e jξ .
Moreover we assume a zero initial condition, i.e. for
equation
strength has also got one component along the axis Θ
zew
Zakopane 2004
,
(5)
Bessela
s µγ r of first
kind and zero order. The function zeros of the
denominator in formula (5) are s = s 0 and
x k2
s = s k = −σ k = −
< 0,
µ γ R2
(2b)
(5a)
where
The required magnetic field strength
the area of
H zI (r , t ) in
( 0 ≤ r ≤ R ) is written as
I
I
z
I
z
H (r , t ) = Im{H (r , t )}, where H (r , t ) is the
I
z
complex magnetic field function of real variables r
and t . This function fulfils the scalar wave equation
in cylindrical co-ordinates
I
I
I
∂ H z (r, t )
∂ 2 H z (r, t ) 1 ∂ H z (r, t )
= 0 . (3)
+
−µγ
2
r
∂r
∂t
∂r
For r = R it has to be the case of the continuity of
the magnetic field strength, i.e. we have the following
boundary condition for the complex values:
I
zew
H z ( R, t ) = H z (t ) .
(3a)
xk ≅ ϕ k +
1
8ϕ k
−
124
120928
+
− ... , (5b)
3
3(8ϕ k ) 158(8ϕ k ) 5
where
1
4
ϕ k = (k − )π ,
(k = 1, 2, 3,...) .
I
H z (r , t )
Then to calculate the original
I
z
operational function H ( r , s )
theorem, obtaining [5, 6, 7, 8]
(5c)
of the
we use the distribution
∞
⎡ I
⎤
I
I
H z (r , t ) = ⎢ H z ,0 (r , t ) + ∑ H z ,k (r , t )⎥ 1(t ) , (6)
k =1
⎣
⎦
I
where H z , 0 ( r , t ) is the original of function (5) in
the pole s = s 0 ( k = 0 ), while
I
H z ,k (r , t ) is the
109
VI International Workshop “Computational Problems of Electrical Engineering”
original of this function in the pole s = s k ( k = 1, 2,
3,... ). These originals are given by the following
formulas:
H
I
z ,0
I ( Γ r ) −ηt jϖt
(r , t ) = H 0 0
e e ,
I 0 ( Γ R)
(6a)
Zakopane 2004
where its module
Ak ( xk ) =
1
η
(2k 2 R2 ) 2 + (2k 2 R 2 − xk2 ) 2
2xk
ϖ
and its argument
and
r
I0 (−j xk )
2
I
R exp[- xk t] ,
Hz,k (r,t) = H0
Ak (xk ) I1(−j xk )
µγ R2
α k ( x k ) = arc tg
first kind and first order.
The complex propagation constant
(7)
whose module
η
Γ = ϖµ γ 1 + ( ) 2 = κ k
ω
(7a)
and the argument
1
π 1
ϖ
η
ϕ = arc tg(- ) = + arc tg
η
ϖ
2
4 2
The exponential form of the
appearing in formulas (6a) and (6b)
ϖ µγ
2
,
1
2
=
.
k
ϖ µγ
⋅ sin[ϖ t + β 0 ( Γ r ) − β 0 ( Γ R) + ξ ]
(7d)
r
⋅ sin[β0,k (-jxk ) − β1,k (-jxk ) −αk (xk ) +ξ]
R
(7e)
η
) =κ k ,
ϖ
(7f)
η
) = κ e jϕ .
ϖ
(7f)
Then the complex constant
η
1
[2k 2 R 2 + j (2k 2 R 2 -xk2 )] =
2 xk
ϖ
, (8)
= Ak ( xk ) exp[jα k ( xk )]
Ak ( xk ) =
110
(10)
r
M0 (-jxk )
2
R exp[- xk t]⋅
HzI,k (r,t) = H0
Ak (xk ) M1(-jxk )
µ γ R2 (10a)
where
κ = 2 (j−
,
and
The complex propagation constant can also be
written as
Γ = k 2 (j−
M 0 ( Γ r ) −ηt
e ⋅
M 0 ( Γ R)
(7c)
whose inverse is the depth of the diffusion of the wave
inside the well conducting medium and it is
δ =
Bessel function
lets us to write the functions (6a) and (6b) in real
forms, i.e. as real functions of variable r of the
cylindrical co-ordinate system and of time t . We
obtain respectively
(7b)
and the coefficient
k=
(8b)
⎫
⎪
I 0 ( Γ R) = M 0 ( Γ R) exp[j β 0 ( Γ R)],
⎪⎪ (9)
r
r ⎬
r
I 0 (-j x k ) = M 0,k (-j x k ) exp[j β 0,k (-j x k )],⎪
R
R
R
⎪
I 1 (-j x k ) = M 1,k (-j x k ) exp[j β 1,k (-j x k )].
⎪⎭
where
η
κ = 2 1 + ( )2
ω
.
I 0 ( Γ r ) = M 0 ( Γ r ) exp[j β 0 ( Γ r )],
H zI, 0 (r , t ) = H 0
,
η
− x k2
ϖ
2k 2 R 2
(6b)
where I 1 ( − j x k ) is the modified Bessel function of
Γ = − η µ γ + jϖ µ γ = Γ e j ϕ ,
2k 2 R 2
(8a)
Finally the magnetic field strength in a conducting
cylinder placed in external longitudinal magnetic field
of a character of an attenuated sinusoid has the
following form
∞
⎤
⎡
H zI (r, t ) = ⎢ H zI,0 (r, t ) + ∑ H zI,k (r, t )⎥ 1(t ) . (10b)
k =1
⎦
⎣
It is convenient to perform the analysis of the
electromagnetic field in relative units. That is why we
introduce the variable x corresponding to the variable
r of the cylindrical co-ordinate system, as
x=
r
,
R
0 ≤ x ≤ 1.
(11)
The frequency of the sinusoidal external magnetic
field and the conductivity of the charge with regard to
its external radius are taken into account through the
coefficient
α=
R
=k R.
δ
Γ r =κ k r =κ α x,
Thus,
we
have:
Γ R =κ k R =κ α .
VI International Workshop “Computational Problems of Electrical Engineering”
The magnetic field is then described by the
following formulas:
∞
⎡
⎤
H zI ( x, t ) = ⎢ H zI, 0 ( x, t ) + ∑ H zI,k ( x, t )⎥ 1(t ) . (12)
k =1
⎣
⎦
where
H zI, 0 ( x, t ) = H 0
M 0 (κ α x) −ηt
e ⋅
M 0 (κ α )
,
(12a)
⋅ sin[ϖ t + β 0 (κ α x) − β 0 (κ α ) + ξ ]
and
HzI,k (x, t) = H0
M0 (-j xk x)
ϖ x2
exp[- k2 t] ⋅
Ak (xk ) M1 (-j xk )
2α
(12b)
⋅ sin[β0,k (-j xk x) − β1,k (-j xk ) −αk (xk ) + ξ ]
In the conducting cylinder the complex magnetic
I
I
field strength H (r , t ) = 1 z H z ( r , t ) is connected
with
the
complex
current
density
J (r , t ) = 1Θ J Θ (r , t ) by the first Maxwell equation
Zakopane 2004
Pr ( x, t ) = −
∞
1⎡
⎤
J
(
x
,
t
)
J Θ , l ( x, t ) ⎥
+
∑
Θ,0
⎢
γ ⎣
l =1
⎦
∞
⎤
⎡ I
H
(
x
,
t
)
H zI,k ( x, t )⎥ 1(t )
+
∑
⎢ z ,0
k =1
⎦
⎣
, (14a)
where the fixed component and the transient
component of the current density are given
respectively by the formulas (13a) and (13b), while
for the magnetic field by the formulas (12a) and (12b).
The influence of the parameter α on the distribution
of the superficial density of the power flux inside the
circular charge is shown in Fig. 2 at t = T/4, i.e. for
the instantaneous value of the external magnetic field
equal to its amplitude. This graph has been worked out
for relative values, that is to say in relation to the
value of the power flux on the surface of the
cylindrical charge, i.e. as the coefficient
P1 =
Pr ( x, t = T / 4) .
Pr ( x = 1, t = T / 4)
Fig.2. Diffusion and attenuation of the superficial density of the
power flux inside a circular charge placed in external uniform
I
rot H (r , t ) = J (r , t ) , from which, taking into
account the fact that the above vectors have got only
one component, we obtain
∞
⎡
⎤
J Θ ( x, t ) = ⎢ J Θ,0 ( x, t ) + ∑ J Θ,k ( x, t )⎥ 1(t ) , (13)
k =1
⎣
⎦
where
J Θ , 0 ( x, t ) = − H 0
M 1 (κ α x)
Γ e −ηt ⋅
M 0 (κ α )
(13a)
⋅ sin[ϖ t + β 1 (κ α x) − β 0 (κ α ) + ϕ + ξ ]
where
M 1 (κ α x) and β 1 (κ α x)
are
respectively the module and the argument of the
modified Bessel function of first kind and first order
I 1 (κ α x) = M 1 (κ α x) e j β1 (κ α x ) , and
M1(-jxk x) k
ϖ x2
xkexp[- 2k t]⋅
Ak (xk ) M1(-jxk ) α
2α
(13b)
⋅ cos[β1,k (-jxk x) − β1,k (-jxk ) −αk (xk ) +ξ]
JΘ,k (x,t) = H0
magnetic field of a character of an attenuated sinusoid at t = T/4, ω
= π·104 rad·s-1, η = 5·103 s-1, γ = 58·106 S·m-1, ξ = 0: 1 – α = 3,
2 – α = 6, 3 – α = 12
The time-space distribution is shown in Fig.3. This
graph has been worked out for relative values, that is
2
to say in relation to H 0 Γ , i.e. as the coefficient
γ
P2 =
γ
H 02 Γ
Pr ( x, t ) .
III. SUPERFICIAL DENSITY OF THE INSTANTANEOUS
POWER FLUX
According to the Poynting theorem [3, 9] the
superficial density of the instantaneous power flux is
defined by the following formula:
1
P( x, t ) = J ( x, t ) × H I ( x, t ) = Pr ( x, t ) (−1r ) . (14)
γ
Substitution of (15) and (16) into this formula yields
111
VI International Workshop “Computational Problems of Electrical Engineering”
Zakopane 2004
p cal (t ) =
1
l 2π R
J
γ ∫ ∫∫
2
Θ
(r , t ) r dr dΘ dz =
0 0 0
=
2π l
γ
∫J
2
Θ
(r , t ) r dr
0
Substitution
x=
(19)
R
of
the
variable
r
→ r = x R → dr = R dx, where 0 ≤ x ≤ 1
R
yields respectively the instantaneous volume density
of the power
Fig.3. Time-space distribution of the power flux inside a circular
charge placed in external uniform magnetic field of a character of an
attenuated sinusoid; α = 5 , ω = π·104 rad·s-1, η = 5·103 s-1, , ξ = 0,
γ = 58·106 S·m-1
The determined superficial density of the power flux
inside the cylindrical charge allows us to determine the
energy supplied to the charge in the time interval of
< 0, t 0 > . It is determined from the instantaneous
power per unit length of the conductor, diffused
through its lateral surface.
p (t ) = 2 π R Pr ( x = 1, t ) 1(t ) =
α
= −2 π
kγ
∞
⎡
⎤
J
(
x
=
1
,
t
)
+
J Θ ,l ( x = 1, t )⎥ ⋅ (15)
∑
⎢ Θ,0
l =1
⎣
⎦
⎡ I
⎤
I
⎢ H z , 0 ( x = 1, t ) + ∑ H z , k ( x = 1, t )⎥ 1(t )
k =1
⎣
⎦
∞
p cal ( x, t ) =
1
γ
J Θ2 ( x, t )
(20)
as well as the instantaneous power converted into heat
in area V
pcal(t) = ∫ pcal(x, t) dV =
V
2π l R2
γ
1
∫J
2
Θ
(x, t) x dx =
0
. (21)
2π l α 2 2
=
JΘ (x, t) x dx
γ k 2 ∫0
1
The time-space distribution of the volume density of
the loses of calorific power is shown in Fig. 4. The
graph is worked out for relative values, i.e. in relation
to
H 02 k 2
γ
p3 ( x, t ) =
integration of this power in relation to the time from
0 to t 0 , i.e.
,
i.e.
γ
H 02 k 2
in
the
form
of
function
pcal ( x, t ) .
t0
W = ∫ p (t )dt .
(16)
0
IV.
VOLUME DENSITY OF THE INSTANTANEOUS POWER
In order to determine the temperature distribution
inside the charge one has to define the so called
external heat sources, which are defined by the volume
density of the power converted into heat. According to
Poynting theorem the instantaneous volume density of
the power converted into heat in area V in [ W ⋅ m ]
-3
p cal (r , Θ, z, t ) =
1
γ
J 2 ( r , Θ, z , t )
(17)
and then the instantaneous power in [W]
pcal(t) = ∫ pcal(r,Θ, z,t) dV =
V
1
J (r,Θ, z,t) dV .
γ∫
2
(18)
V
In the case of the current density depending only on
variable r of cylindrical co-ordinate system, i.e. for
J = J Θ (r , t ) this power is given by the formula:
112
Fig.4. Time-space distribution of the volume density of heat loses
inside a cylindrical charge placed in external uniform magnetic field
of a character of an attenuated sinusoid j; α = 5 , ω = π·104 rad·s-1,
η = 5·103 s-1, γ = 58·106 S·m-1, ξ = 0
The curve of the instantaneous power converted into
heat inside a cylindrical charge is shown in Fig..5. The
graph is worked out for relative values, that is to say in
2 π l α H 02 , i.e. in the form of function
relation to
γ
p4 (t ) =
γ
2 π l α H 02
pcal (t ) .
VI International Workshop “Computational Problems of Electrical Engineering”
Zakopane 2004
Pr ( x, t ) = −
=
1
γ
J Θ,0 ( x, t ) H zI,0 ( x, t ) 1(t ) =
ϖ µ H M 0 ( 2 j α x) M 1 ( 2 j α x)
⋅
γ 2
M 02 ( 2 j α )
, (25)
2
0
π
⎫
⎧
⎪cos[β1 ( 2 j α x) − β 0 ( 2 j α x) + 4 ] − ⎪
⎪⎪
⎪⎪
⋅ ⎨− cos[2 ϖ t + β1 ( 2 j α x) +
⎬ 1(t )
⎪
⎪
π
⎪+ β 0 ( 2 j α x) − 2 β 0 ( 2 j α ) + + 2 ξ ]⎪
4
⎭⎪
⎩⎪
Fig.5. Curve of the instantaneous power converted into heat inside a
cylindrical charge placed in external uniform magnetic field of a
character of an attenuated sinusoid;
α = 5 , ω = π·104 rad·s-1,
η = 5·103 s-1, γ = 58·106 S·m-1, ξ = 0
Energy converted into heat
during the time
0 ≤ t ≤ t 0 is given by the formula
Instantaneous power per unit length of the circular
charge diffused through its lateral surface
p (t ) = 2 π R Pr ( x = 1, t ) =
= −2 π
t0
Wcal = ∫ p cal (t ) dt .
(22)
0
For the practical purposes
η = 5 ⋅ 10 3 s -1
and
ϖ = π ⋅ 10 4 rad ⋅ s -1
which means that the relation
T
H (t = + 5 T )
does not surpass 1%. In order to
4
T
H zzew (t = )
4
determine the energy converted into heat inside a
cylindrical charge we can then (see also Fig. 5) quite
acurately assume the time t 0 = 5 T .
zew
z
V. CONCLUSION
For the case of the fixed sinusoidal electromagnetic
field ( η = 0 ) in the electric and magnetic field the
transient components disappear, i.e. the magnetic field
is given by the formula
M 0 (κ α x)
⋅
M 0 (κ α ) , (23)
⋅ sin[ϖ t + β 0 (κ α x) − β 0 (κ α ) + ξ ] 1(t )
H zI ( x, t ) = H zI,0 ( x, t ) 1(t ) = H 0
R
γ
J Θ,0 ( x = 1, t ) H zI,0 ( x = 1, t ) 1(t )
, (26)
Then the active power
P=
1T
ϖ µ H02 M1 ( 2 j α )
=
⋅
(
)
d
2
p
t
t
R
π
γ 2 M 0 ( 2 j α ) , (27)
T ∫0
π
⋅ cos[β1 ( 2 j α ) − β0 ( 2 j α ) + ]
4
which is, after the suitable modifications, the
equivalent of the solution obtained by M. Krakowski
in [3] – formula (9.111), p. 200.
The determined electromagnetic field and the
superficial and volume density of the power can be
used to define the substitute parameters of the system
working head-charge during the process of metal
forming with impulse magnetic field. The results
presented here, describing the volume density of the
power converted into heat inside a cylindrical charge
determine the so-called external heat sources and can
be used to describe the temperature field in the charge
being formed both in transient and fixed states of the
electromagnetic field.
REFERENCES
while the current density
M (κ α x)
J Θ ( x, t) = J Θ,0 ( x, t) 1(t) = −H0 1
Γ⋅
M 0 (κ α )
, (24)
⋅ sin[ϖ t + β1 (κ α x) − β 0 (κ α ) + ϕ + ξ ] 1(t)
in which κ = 2 j , Γ = ϖµ γ = 2 k and ϕ = π .
4
Then the superficial density of the instantaneous
power is given by the formula
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