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Let us find the current density and magnetic field intensity distribution in an infinitely long straight cylindrical conductor of radius R as shown in figure (3.1). In a system of cylindrical coordinates (r, q, z) where the z-axis is the cylinder axis, the transient electric field intensity vector H has circumferential direction as shown in figure (3.1). The current density vector J and the electric field intensity vector E are parallel to the conductor axis.
Heller[3.1] outlines two types of problems that are considered in the literature: those driven by an ideal current source and those driven by an ideal voltage source. When the conductor is in series with a homopolar generator, then the machine parameters determine that this problem should be modeled with a current source, see Perry [3.2]. The implication of this is that the driving current passing through the conductor is a controlled function of time.
We seek the solution to the problem where a step (Heaviside function in time) current excitation is applied to the cylindrical conductor (see figure 3.2) and observe that the current starts flowing entirely on the surface and then slowly diffuses into the conductor. We will not discuss the wave (high-frequency) problem, namely with how the fields establish themselves since this effect occurs much faster than the diffusion problem. The ratio of the time scales is: Lcn/s »107 (where L is the length of the conductor and c is
Figure 3.1 Cylindrical time-varying current carrying conductor, where the current density vector, J(r, t), and electric field intensity vector E(r, t) are parallel to the conductor axis. The magnetic field intensity vector H circulates about the axis.
Figure 3.2 The step current function I(t) = I0 for t > 0.
the velocity of an electromagnetic wave in the media, i.e. in copper). This high-frequency problem is outside our immediate interest, and the given approach has been justified by Perry [3.2] and Drake and Rathmann [3.3].
If we want to find the relationship between the current density vector J and the magnetic field intensity vector H, we shall write Maxwell's equations in the following form [3.4, 3.5]:
(3.1) Ń “ H = J
(3.2) Ń “ E = - m ¶H/¶t
where the displacement current has been neglected. Furthermore, we shall assume that Ohm's law applies to the above conducting cylinder. Then, at any given point at constant temperature, the electric field intensity E is proportional to the current density J. Thus
(3.3) E = 1/s J.
It is important to note that the expression for the curl in equation (3.1) and (3.2) cannot be obtained from the corresponding expression in rectangular coordinates by simply replacing x, y, z, by r, q, z and therefore the following procedure will be taken. The magnetomotive force along the curve G bounding the elementary shaded area in figure 3.1
(3.4) da = r dq dr
is
(3.5) - H r dq + (H +¶H/¶rdr) (r + dr)dq = ¶H/dr r dr dq + H dr dq
where on the right side of equation (3.5) the third order term is neglected. In this problem, where the vector H = (0, H(r, t), 0) has only the q component, we can write
(3.6) curlz H = (0, 0, ¶H/¶r + H/r )
and Maxwell's first equation (for the z-component) becomes
(3.7) ¶H/¶r + H/r = J
where Jz is understood to be the z-component of J.
The electromotive force along the circumference L, which is a boundary of the shaded area in figure 3.3 is
(3.8) E dz - (E + ¶E/¶r dr) dz = - ¶E/¶r dr dz.
Figure 3.3 A long cylindrical conductor of circular cross-section carrying a current I(t) with time-varying magnetic and electric field.
The division by an area da = dr dz, in the case when the vector E = (0, 0, E(r, t)) has only the z-component gives
(3.9) curlq E = (0, - ¶E/¶r, 0).
Then Maxwell's second equation can be written (for the q-component) in the following form:
(3.10) - ¶E/¶r = - m ¶H/¶t
or
(3.11) - ¶J/¶r = - ms ¶H/¶t
where the displacement current has been neglected. The derivative of equation (3.7) with respect to t and the derivative of equation (3.11) with respect to r give
(3.12) 1/r ¶H/¶t + ¶2H /¶r ¶t = ¶J/¶t
(3.13) ¶2J /¶r2 = ms ¶2H /¶r ¶t
with the initial condition J(r, t) = 0 for t<0, r²R and boundary conditios as in equation (3.22) below.
From these relationships taking into account equation (3.11) we get the classical diffusion equation for the current density in cylindrical coordinates
In a similar manner, the derivative of equation (3.7) with respect to r and taking into consideration equation (3.11) gives the diffusion equation for the magnetic field intensity
The above equations (3.14) and (3.15) can be solved with the aid of Laplace transform see references [3.6, 3.7, 3.8, 3.9]
Denoting by J(r, p) = L {J(r, t) }, the Laplace transform of J(r, t), equation (3.14) becomes under the zero initial conditions J(r, 0) = 0, an equation of the Helmholtz type
where
(3.17) g2 = m s p.
The solution of equation (3.16) is
(3.18) J(r, p) = A(p) I0( g r ) + B(p) K0( g r )
where A(p) and B(p) are integration constants, I0( g r ) is the modified Bessel function of the first kind and zeroth order, and K0( g r ) is the modified Bessel function of the second kind and zeroth order. Using the boundary condition that there is no singularity at the origin (since we have a cylinder), we get that
B(p) = 0
The Laplace transform of the magnetic field intensity
(3.19) H(r, p) = L { H(r, t) }
is computed with the help of the equation
(3.20) curl E(r, p) = - p H(r, p)
and is expressed by
(3.21) H(r, p) = A(p) g/ mp I1 ( g r )
where I1( g r ) is the modified Bessel function of the first kind and first order.
From the operational form of Ampere's law, see Kovacs [3.10] applied to the surface, we obtain the current
(3.22) I(p) = L {i(t)} = 2pR H(R, p) = 2 p A(p) g R / p m I1( g R )
and hence:
If we consider a step current excitation I(t) suddenly applied at time t=0 (see figure 3.2), then replacing I(p) = I0/p (see reference [3.11]) in equation (3.10) we obtain the following value for the constant A
and substituting into equation (3.18) gives, See reference [3.12]
Finally, taking the inverse transform of equation (3.25) we obtain
where xn is the nth positive root of the Bessel function of first kind and first order
(3.27) J1(x) = 0
and t0 = s/n = 1/K2, where K is the diffusion coefficient. The diffusion coefficient relates the time it takes for the concentration gradient to travel per unit area. For the magnetic field H(r, t), we deduce the expression
and from Ampere's law
(3.29) I(t) = 2 p R H(R, t) = I0.
Formulas (3.25) and (3.28) have been established by Miller [3.13] and have been used for the computation of the transient resistance Rt(t) and the transient inductance Lt(t).
In our case, we are also interested in the analytical solution to a linear ramp current input. The ramp current is turned on at time t0= 0 and increases linearly to a value of 0.5 megaamps at t1=0.5 milliseconds and then maintaines this steady value. This ramp waveform can be represented by
t t
K(t) = I0 ŗ h(t) dt - I0 ŗ h(t-t1) dt,
0 0
where the Heaviside function h(t) = 0 for t<0 and h(t) = 1 for t³0. Since our differential equation is linear, the responce to this wave form is obtained by
performing the same operations on the responce to the step (Heaviside) function (eq. 3.26).
where Jr and Jc denote the same solution but at different times.
From a computational standpoint, we must look at the number of terms that we should take in order for our series to converge with sufficient accuracy. The question is to determine how many terms are necessary near t=0. The roots xn tend to space themselves out asympoticaly at intervals of p for n large, i.e. xn » pn + c, where c is an arbitrary constant (see reference [3.14], p. 371). Hence xn2 » p2n2. The term exp (-xn2 t/tn) tends to zero rapidly as n increases except near t=0. Moreover, since |J0(xn)| ² c/Ć(pn), this previous expression is not slowed down. Our overall convergence of the given series (3.30) is O(n3/2 e-Ktn2). The integrated form (3.30) is used in computations rather than (3.26) since as time tends to zero, Jr(r, t) also tends to zero. In our program, we have a routine which keeps summing terms until the next added term is less than a specified tolerence. This tolerence is choosen to give an accuracy of one more decimal place than we require in expression (3.30).
The radial distribution of the current density is plotted for several different values of time in figures 3.4 and 3.5. It can be seen that as time progresses, the current diffuses into the cylindrical conductor. From equation (3.26) and from figures 3.4 and 3.5, it is seen that the greatest current
Figure 3.4 The analytical solution of the current density [Amperes/meter2] plotted vs. the radial distance [meters] for a ramp input (risetime = 0.5 msec) at time = 0.1 msec, 0.2 msec, 0.3 msec, ... 1.0 msec.
Figure 3.5 The analytical solution of the current density [Amperes/meter2] plotted vs. the radial distance [meters] for a ramp input (risetime = 0.5 msec) at time = 2.0 msec, 4.0 msec, 6.0 msec, 8.0 msec, 10.0 msec.
densities occur at the outer surface of the conductor. This phenomenon is the so-called the skin effect.
It is also desirable to obtain the analytical solution of the magnetic vector potential. A naive approach would be to obtain it from the current constraint
(3.31) ŗ J dS = ŗ Js dS + ŗ Je dS = I(t),
where we assume I(t)=0 for t<0 and I(t) = I for t³0. Integrating in time, we get
(3.32) ŗŗ Js dS dt - ŗŗ s ¶A/¶t dS dt =
ŗŗ Js dS dt - s [A(t) - A(0)] dS dt = ŹI*t.
However, one is puzzled by what values to assign to Js. One approach is to set it equal to zero by the argument that its magnitude is arbitrary to within a constant as was the definition of what to call the eddy current and what to call the source current. This approach is misleading because even though the definition is arbitrary, the mathematics is not. First, the value of Js is not necessarily constant but changes in time so that the current constraint equation is satisfied. More importantly, the change in the magnetic vector potential must be properly accounted for from the correct influences. A more convincing approach is to run the finite difference program and check the values of Js to convince oneself that the values are not constant.
The correct derivation of the magnetic vector potential is obtained by the substitution of Jr(r0, t) from equation (3.26) into the equation for the magnetic vector potential (2.50). The magnetic vector potential at a time t and radius r is given by
Conclusion:
A one-dimensional transient field and eddy-current problems in a cylindrical conductor is described by the diffusion equations governing the magnetic field intensity and current density vector. We assume that a ramp current excitation is applied to the above conductor. The field equations are solved with the aid of operational calculus. The field parameters, J(r, t) and H(r, t), are expressed in terms of modified Bessel functions.
Thus we have a theoretical formuation for the case of a ramp and step current in terms of the field parameters and a convergent series of modified Bessel functions which is reasonably computable for t > 0. The convergence is slow for t very close to zero.
References:
[3.1] Heller, B., Transient phenomena in solid iron. Electrotechnicky Obzor, Prague, Vol. 42, (1953), No. 7-8, pp 368-375
[3.2] Perry, M., "Electromagnetic Diffusion in Rails", General Electric IR&D Report, 1986
[3.3] Drake, P. A., and Rathmann, C. E. "Two Dimensional Current Diffusion in an EML Rail with Constant Properties," IEEE Transactions on Magnetics, Vol MAG-22, No. 6, Nov. 1986, pp.1448-1452.
[3.4] Silvester, P. Modern Electric Fields, Prentice-Hall, Inc., Englewood Cliffs, New Jersey, (1986)
[3.5] Wangsness, R. K., Electromagnetic Fields. John Wiley and Sons., New York (1979).
[3.6] Doetsch, G., An Introduction to the Theory and Application of the Laplace Transform. Springer-Verlag, New York, Berlin (1974).
[3.7] Papin, M. and Kaufmann, A., "Exercices de Calcul Operationnel avec Leurns Solutions (transformation de Carson-Laplace), 3rd edition, Editions Eyrolles, Paris (1966).
[3.8] Wagner, K. W. and Thoma, A. Operatoren rechnung und Laplacesche Transformation. 3rd Auflage, Johann Ambrosius Barth/Verlag/Leipzig (1982).
[3.9] Heger, W., "Diffusion of the Electromagnetic Field into a Conducting Semi-Infinite Solid," Research Report, General Electric Co., Advanced Electrical Systems Division. Ballston Spa, New York, August 1985.
[3.10] Kovacs, P. K., Transient Phenemena in Electrical Machines, Elsevier, Amsterdam - NY (1984).
[3.11] Whitesides, I., Operational Calculus, Springer Vergag. Berlin-New York.
[3.12] Mocanu, C. I., "Equivalent Schemes of the Circular Conductor at Transient State With Skin Effect," Rev. Roum. Sci. Techn.- Electrotechnique et Energetique, Bucarest, vol. 16 (1971), No. 2, pp. 235-254.
[3.13] Miller, K. W., "Diffusion of Electric Current into Rods, Tubes, and Flat Surfaces," AIEE Transactions (1947), 66, pp. 1496-1504.
[3.14] Mathematical Tables. National Bureau of Standards, Washington, D.C., (1975)
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