Skin effect in the time domain via vector potentials

Acta Technica 59 (2014), 49–62

c 2014 Institute of Thermomechanics AS CR, v.v.i.

Skin effect in the time domain via vector potentials Malcolm S. Raven 1 Abstract. The vector potential method is used to solve the problem of transient conduction and skin effect in cylindrical conductors. Several time dependent current sources are analyzed including linear, quadratic and exponential. The stability factor is determined and the results for each case are presented. A simple method of measuring the skin effect using a digital multimeter is suggested. Key words. Vector potential, skin effect, transient currents, RMS, power series inversion convergence/divergence.

1. Introduction The problem of computing the skin effect for transient currents has received more attention recently as broadband applications increase and new energy sources are developed. There are numerous approaches to solving the transient current problem, both analytical [1], [2], empirical [3], [4] and digital simulation and modelling including the ElectroMagnetic Transient Program Theory Book (EMTPTB) [5] and Alternative Transient Program (ATP) [6]. Complex techniques have also been developed to tackle practical conductor and transmission line problems which include proximity effects, surface roughness, reflections, radiation, and other transmission line losses [7], [8]. In a previous paper [9], an alternative vector potential approach was examined for solving the skin effect in a solid cylindrical conductor which may contribute towards the more sophisticated and complex approaches. In this method, originally due to Maxwell [10], the emf acting round the circuit is found to be represented by infinite series of increasing derivatives of current with respect to time. This approach has recently been proved rigorously correct at low frequencies [2], [9], [10]. However, at higher frequencies or for ratios of conductor radius to skin depth a/δ ≥ 2.7, the series diverges leading to 1 28

The Crayke, Bridlington, YO16 6YP, UK; e-mail: [email protected]

http://journal.it.cas.cz

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M. S. RAVEN

diverging impedance with increasing frequency. For sinsusoidal currents a stability factor k was found which predicts the divergence/convergence conditions in terms of the conductor parameters and frequency. In this paper we consider transient current sources. Unlike the sinusoidal case, a unique value of k is only obtainable for specific source current functions. Hence, several different time dependent current sources are applied to a conductor including linear, quadratic and exponential, values of k calculated and the conductor transient emf determined as a function of time both with and without the skin effect.

2. Vector potential method For quasi-static fields and a good conductor with negligible displacement current, the fundamental equations relating the vector potential A, current density J and electric potential V are [11] ∇2 A = −µ0 J , ∂A −E = ∇V + ∂t

(1) (2)

where the total electric field E includes the gradient of the electric potential V . Maxwell identified the emf due to other causes than the current induction on itself as Vemf = −l ∇V (3) where l is the conductor length. Hence, Vemf = −

ρl 2 ∂A ∇ A+l . µ0 ∂t

(4)

For current flowing in the z-direction only in a solid cylindrical conductor Vemf

ρl =− µ0



∂ 2Az 1 ∂Az + 2 ∂r r ∂r

 +l

∂Az . ∂t

(5)

Maxwell [10] solved this equation by assuming Az to be represented by the series Az = S + T0 + T1 r2 + T2 r4 + T3 r6 + · · · + Tn r2n + . . . (6) where S, T0 , T1 , . . . are functions of time. After differentiating (6) and after some considerable algebra [9] or using matrix algebra [2], [12], the emf becomes Vemf = R0 I + L0 I (1) −

µ2 l2 (2) µ3 l3 (3) µ4 l4 (4) I + I − I + ... 12R0 48R02 180R03

(7)

SKIN EFFECT IN TIME DOMAIN

51

where µ = µ0 µr /(4π) and I (m) is the m-th derivative of the current. The symbol I denotes the current, R0 and L0 are the low frequency resistance and inductance, respectively. This inductance is the sum of the internal inductance of the conductor plus its external inductance, i.e. L0 = Lint + Lext where it is derived in the vector potential approach rather than assumed [9]. Equation (7) may be re-written as Vemf = R0 I + L0

∞  a 2m X dI − R0 (−1)m fm I (m) dt d m=2

(8)

where r d=

dm I 1 1 1 4ρ , I (m) = m , f2 = , f3 = , f4 = , ... µ0 dt 12 48 180

(9)

Previously, the f -coefficients were found to fit a power law relationship [9] fm = ar 10 br m ,

m = 1, 2, 3, . . .

(10)

where ar = 1.13721 and br = −0.5675. 2.1. Stability and normal emf The stability of Eq. (8) may be determined using the d’Alembert’s ratio test for the absolute convergence of a series [13], [14] am+1 =k. lim (11) m→∞ am If k < 1 the series converges. If k > 1 the series diverges. If k = 1 the series either converges or diverges. Substituting values in (8) we can show that fm+1  a  2 I (m+1) . (12) k= fm d I (m) This is dimensionless because (a/d)2 has dimension of seconds and the derivative ratio is s−1 . Hence, the stability factor k depends on the ratio of the current derivatives and may be determined when the current function is known since a and d are known and fm+1 /fm < 1. The first two terms on the right of (8) correspond to the normal equation for the total emf in a series RL circuit when the effect of restricted current penetration into the conductor is neglected, that is the skin effect is neglected. The summation term corrects for the skin effect. If the effective penetration depth is much greater than the wire radius, i.e. d  a, then this correction is insignificant and (8) becomes dI , (13) dt which is the usual expression for the total voltage in a series RL circuit. Vemf = R0 I + L0

52

M. S. RAVEN

2.2. Current impulse Maxwell completed Article 690 in [10] by considering a current impulse starting at a constant value I0 and rising to a steady value I1 . The result applied to (7) is Z Z I dt + L0 (I1 − I0 ) ,

Vemf dt = R

(14)

‘which is the same value of electromotive impulse as
if the current had been uniform throughout the wire’, where L0 = l A + 21 µ in Maxwell’s original equation. This is proved as follows. Integrating (7) with respect to time gives Z

Z Vemf dt = R

I dt + L0 [I] −

µ2 l2 dI µ3 l3 d2 I + + ... 12 R dt 48 R2 dt2

(15)

This is the indefinite integral which shows that the skin effect is still present. Now consider the definite integral where the integration is between t = 0 and t = t1 . Then t1

Z

t1

Z Vemf dt = R

0

I(t) dt +

L0 [I(t)]t01

0

so that Z t1

Z

 t  t µ2 l2 dI 1 µ3 l3 d2 I 1 − + + ... 12R dt 0 48R2 dt2 0 (16)

t1

  I(t) dt + L0 I(t1 ) − I(0) − 0    µ3 l3 d2 I(t1 ) d2 I(0) µ2 l2 dI(t1 ) dI(0) − + − + ... − 12R dt dt 48R2 dt2 dt2

Vemf dt = R 0

(17)

At t = 0 let I(0) = I0 = const and at t = t1 let I(t1 ) = I1 = const. Then Z

t1

Z

0

I1

I(t) dt + L0 [I1 − I0 ] −     µ2 l2 dI1 dI0 µ3 l3 d2 I1 d2 I0 − − + − + ... 12R dt dt 48R2 dt2 dt2

Vemf dt = R

I0

(18)

But since I0 and I1 are constants then the differentials are zero. Hence we obtain Z t1 Z I1 Vemf dt = R I(t) dt + L0 [I1 − I0 ] , (19) 0

I0

which is the same as (14). It could be useful in proving (7) by experimental measurements since the infinite series is removed and I(t) can be any analytic function of time. If (19) is divided by the period of the current function, the

53

SKIN EFFECT IN TIME DOMAIN

average emf is obtained. This is apparently independent of the skin effect. If the current is a sine wave and the integration is taken over one complete period then the average is zero as expected. A more common measurement is the Root Mean Square (RMS), Vrms = p 2 i, which avoids the zero average value of a sine wave. This is provided = hVemf on many Digital Multimeters (DMM’s) and specialized integrated circuits are available which provide RMS values for a wide range of signal inputs. However, squaring the emf retains the series terms in (7). Hence, RMS values include the skin effect. This is discussed further in the following RMS subsection. 2.3. RMS values In order to obtain the Root Mean Square value of a time dependent function we first square the function, then find its average value and finally calculate its square root. Thus for a time dependent emf Vrms =

q

#1/2 " Z T 1 2 2 i= V dt . hVemf T 0 emf

(20)

In (13) let V1 = RI + L0 and V2 = R

∞ X

(−1)m fm

m=2

dI dt

 a 2m d

(21)

I (m) .

(22)

Then, Vemf = V1 − V2 ,

(23)

2 Vemf = V12 − 2V1 V2 + V22 ,

Vrms =

1 T 1/2

"Z

T

V12

Z dt − 2

0

T

(24) Z

V1 V2 dt + 0

#1/2

T

V22 dt

.

(25)

0

The RMS voltage therefore contains the skin effect term V2 given by (22).

3. Examples of current source functions 3.1. Linear current source Consider a current source I = I0 t where I0 is a constant current. Then d2 I d3 I dn I dI = I0 , = = · · · = =0. dt dt2 dt3 dtn

(26)

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M. S. RAVEN

Thus ki = 0 and the series converges. Equation (8) then becomes Vemf = R0 I0 t + L0 I0

(27)

where L0 is the total inductance of the wire. Hence, the voltage step at t = 0 is Vemf (0) = L0 I0 . If L0 is 1 nH and I0 = 1 As−1 then the voltage step at t = 0 is 1 nV. If I0 = 106 As−1 , the step is 1 mV which is a more detectable voltage. Equation (27) could have been obtained by assuming a simple series RL circuit with linear current source I = I0 t and both R and L constants. In this case summing the voltages gives dI = R0 I0 t + L0 I0 , (28) dt which is the same as (27). This suggests that a linear time dependent current flowing in a solid mass of conductor is spread uniformly across the conductor section for any rate of current flow. Hence, there is no skin effect for linear time dependent current sources. This seems rather surprising particularly if the current changes in nanoseconds or less corresponding to GHz frequencies. However, linear current changes at such high rates may not be achievable in practice due to external inductance and capacitance. Vemf = IR0 + L0

3.2. Quadratic current source A current source is defined by I = I0 t2 . Then dI d2 I d3 I dn I = 2I0 t , = 2I , = · · · = =0. (29) 0 dt dt2 dt3 dtn Since the derivatives above second are zero, the stability factor is from (12)  a 2 . (30) k = 10 br d For copper wire this becomes k = 5a2 . (31) For k = 1 it holds that a = 0.447 m. Thus Vemf diverges with t if a ≥ 0.447 m and converges if a ≤ 0.447 m. Equation (8) becomes  a 4 Vemf = R0 I0 t2 + L0 2I0 t − R0 f2 2I0 . (32) d At t = 0 and since f2 = 1/12 then Vemf (0) = −

la2 µ20 I0 96ρ

(33)

where l is the wire length. A negative step voltage with this value is therefore expected at t = 0. As an example, consider copper wire length 1 m, a = 1 mm, I0 = 1 A. This gives a voltage step at t = 0 of Vemf (0) = −0.9676 pV.

SKIN EFFECT IN TIME DOMAIN

55

3.3. Exponential current source Consider a current source defined by   I = I0 1 − e−t/τ

(34)

driving a time dependent current I with time constant τ through a conducting wire length l, radius a. The quantity I0 is the steady state current when t  τ . The wire emf is given by (8). Differentiating (34) gives I (1) =

dI I0 e−t/τ dI 2 I0 e−t/τ = , I (2) = 2 = − , ... , dt τ dt τ2 I0 e−t/τ dI m . I (m) = m = (−1)m+1 dt τm

(35)

Hence, Eq. (7) becomes   τ1 1  a 4 −2 1  a 6 −3 1  a 8 −4 Vemf = I+I0 e−t/τ + τ + τ + τ + ... R0 τ 12 d 48 d 180 d (36) where 4π  a 2 πa2 L0 L0 τ1 L0 τ −1 = = = (37) µ0 d ρτ R0 τ τ and

L0 τ1 = , d= R0

r 4

ρ = 0.2326 for copper , µ0

(38)

R0 is the steady state ohmic resistance of the wire and L0 the steady state inductance of the wire, τ is the rise time of the input pulse and τ1 the rise time of the circuit. Equation (39) is similar to the sinusoidal case with ω ≡ τ −1 except that the (1 − e−t/τ ) function ensures that all terms in the power series are positive. Substituting for I from (34) into (36), multiplying by R0 and noting that (−1) × (−1)2m+1 = +1, gives  2m ∞ X L0 I0 −t/τ a Vemf = R0 I + e (39) + R0 I0 e−t/τ fm τ δ 1 m=2 where δτ = dτ 1/2 = 2

r

ρτ , µ0

(40)

δτ is a penetration depth or skin depth in metres; τ is then equivalent to 1/(2ω) where ω is the angular frequency in the sinusoidal case. For copper δτ = 0.2326 τ 1/2 . Equation (39) may be written as   Vemf = I0 R0 1 + S e−t/τ (41)

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M. S. RAVEN

where  2m ∞ X τ1 a L0 S = −1 + + S∞ , S∞ = fm . , τ1 = τ δτ R0 m=2 Equation (39) may also be written as   L0 Vemf = R0 I + + R0 S∞ I0 e−t/τ . τ

(42)

(43)

where L0 = R0 τ1 . For small a/d or if the skin depth δτ  a then S∞ can be neglected and (43) reduces to the total low frequency voltage across a series RL circuit dI Vemf = R0 I + L0 . (44) dt For the general skin effect case, Eq. (43) can be written as Vemf = R0 I + L0

dI dI + L2 dt dt

(45)

where L2 = R0 S∞ τ ,

(46)

(L2 having dimension of Ωs = H). Generally we can put Vemf = VR0 + VL1 + VL2

(47)

VR0 = IR0 = I0 R0 e−t/τ ,

(48)

where dI L0 I0 −t/τ = e , dt τ dI L2 I0 −t/τ VL2 = L2 = e = I0 R0 S∞ e−t/τ . dt τ If the pulse is measured for a period T , Eqs. (20) and (41) give for the value   S2τ  1/2 2Sτ  −T /τ −2T /τ e e Vrms = I0 R 1 − −1 − −1 . T 2T VL1 = L0

(49) (50) RMS

(51)

3.3.1. Input rise-time equals circuit rise-time (τ = τ1 ). If we make the risetime of the input pulse equal to the rise-time of the circuit, i.e. τ = τ1 = L0 /R0 , then (41) becomes   Vemf = I0 R0 1 + S∞ e−t/τ .

(52)

SKIN EFFECT IN TIME DOMAIN

57

Thus if S∞ = 0 or S = 0 then Vemf = I0 R0 which is a constant. This result is confirmed by using the low frequency equation (44) which gives Vemf = R0 I + L0 and Vemf

 L I  dI 0 0 −t/τ1 = I0 1 − e−t/τ1 + e dt τ1

  L0 −t/τ1 −t/τ1 = I0 R0 1 − e + e = I0 R0 R0 τ 1

(53)

(54)

since τ1 = L0 /R0 . 3.3.2. Stability factor . The stability factor is am+1 fm+1 (a/δτ )2(m+1) = k = lim . m→∞ am fm (a/δτ )2m Thus fm+1 k= fm



a δτ

(55)

2 .

(56)

10 br a2 d2 τ

(57)

Substituting for fm from (10) then k = 10

br

and S = −1 +



a δτ

2 =

∞ X 4πL0 k + a km . r µ0 10 br m=2

(58)

Using ar and br values from (10) we find that the series converges if k < 1 or a/δτ < 1.922 and the or a/δτ > 1.922. p series diverges if k > 1 √ √ For copper, d = 4ρ/µ0 = 0.2326, δτ = d τ = 0.2326 τ and δτ2 = d2 τ = = 0.05411 τ . Thus 0.2707 a2 a2 = 5.0028 . (59) k= 0.05411 τ τ 3.3.3. Exponential decay. For a source current which decays exponentially with time according to I = I0 e−t/τ (60) then

−t/τ dm I m I0 e = (−1) , (61) dtm τm which has the opposite sign to the former case equation (35). This yields

I (m) =

Vemf = I0 R0 e−t/τ S2 ,

(62)

58

M. S. RAVEN

S2 = 1 −

τ1 L0 − S∞ = −S , τ1 = . τ R0

(63)

If τ1 = τ then Vemf = −I0 R0 e−t/τ S∞ .

(64)

Hence, the output voltage is only due to the skin effect. Generally, the output emf is then Vemf = VR0 + VL1 + VL2 (65) where VR0 = IR0 = I0 R0 e−t/τ , dI L0 I0 −t/τ =− e , dt τ

(67)

dI L2 I0 −t/τ =− e = −I0 R0 S∞ e−t/τ , dt τ

(68)

VL1 = L0 VL2 = L2

(66)

L2 = R0 S∞ τ .

(69)

The stability factor is the the same as in the preceding case, see Eq. (57). 3.3.4. Copper wire example. Here we consider a copper wire subjected to an exponential current source defined by   I = I0 1 − e−t/τ .

(70)

Figure 1 shows the results for the low frequency voltages VR0 , VL1 and the skin effect voltage VL2 obtained from (48–(50) for copper wire: τ = 1 µs, a = 0.1 mm, length = 1 m, k = 0.05. 3.3.5. Normalization. In the second example the source current is I0 = 1 A, time constant τ is made equal to the circuit time constant, i.e. τ = τ1 = = L0 /R0 = 92.3 µs. The wire has radius a = 0.675 mm, L0 = 1.2 µH/m, R0 = 0.013 Ω/m. The results are plotted in Fig. 2. A more sensitive approach to detecting the skin effect is to calculate the ratio of the output emf to the normal emf. We first consider the ideal case where there is no skin effect and then consider the skin effect. Ideal case: The emf across the wire is VL = IR0 + L0

  L I e−t/τ dI 0 0 = I0 R0 1 − e−t/τ + . dt τ

(71)

Thus, VN =

VL L0 −t/τ = 1 − e−t/τ + e . I0 R0 R0 τ

(72)

SKIN EFFECT IN TIME DOMAIN

59

Fig. 1. Exponential current pulse in copper wire, τ = 1 µs, a = 0.1 mm, length = 1 m, k = 0.05; voltages developed along the wire are VR —DC/LF ohmic, V L1 —DC/LF inductive and V L2 skin effect, total voltage is Vemf = VR + V L1 + V L2

If τ = L0 /R0 then for all t VN = 1 .

(73)

Skin Effect: In this case we have from (41) VN = 1 + S e−t/τ where S=

∞ X

ar 10

br m

m=2



k 5d2

(74) m .

(75)

p For copper d = 4ρ/µ0 = 0.2326, ar = 1.13721, and br = −0.5675. The stability factor is obtained from (59), k = 5.0028 a2 /τ = 5.0028 × 0.6752 × 10−6 /(92.3 × 10−6 ) = 0.02468 .

(76)

From the values of k, ar , br , and d it can easily be found that the series S in (75) rapidly converges. Let us denote sp = ar 10 br p



k 5d2

p .

(77)

60

M. S. RAVEN

Fig. 2. Exponential current pulse in copper wire, τ = 92.3 µs, a = 0.675 mm, length = 1 m, k = 0.02468

Fig. 3. As Fig. 2 but normalized skin effect emf and drive current

61

SKIN EFFECT IN TIME DOMAIN

The partial sums Sm =

m X

sp

(78)

p=2

of the series S practically do not vary when m > 5, attaining the value of 7.1178 × 10−4 , as seen from the table below: m

sm

Sm

2 3 4 5 6

6.9419 × 10−04 1.7151 × 10−05 4.2376 × 10−07 1.0470 × 10−08 2.5867 × 10−10

6.9419 × 10−04 7.1134 × 10−04 7.1176 × 10−04 7.1178 × 10−04 7.1178 × 10−04

The normalized emf is then VN − 1 = 7.1178 × 10−4 e−t/(9.23×10

−5

)

.

(79)

This is plotted in Fig. 3 along with some values obtained from a basic programme. When t = 0, Eq. (74) or (79) give VN − 1 = S as shown in Fig. 3. When t = τ , these equations give VN − 1 = Se−1 = 2.6185 × 10−4 as indicated in the figure. The normalized emf is shown to decrease to 37 % of its maximum value at a time of 92.3 µs. This corresponds to the time constant τ = L0 /R0 .

4. Conclusions In this paper we have shown that Maxwell’s vector potential method of determining time dependent current flow in a conductor leads to a relatively simple but important solution of the transient skin effect. Time average values of a current impulse are found to be independent of the skin effect but not RMS values. This suggests that simple measurements with a RMS voltmeter or DMM may in principal be used to detect the skin effect. Three types of current functions were examined including linear, quadratic and exponential. For the linear current source no skin effect is predicted although in practice at very high frequencies linear current functions are unlikely to be realised. For the non-linear current functions examined, stable power series were obtained but only for particular values of time constants and wire diameter. Outside these values the power series diverge. It is suggested that this method could find useful applications in checking more sophisticated methods and programmes. However, avoidance of the power series divergence problem for particular conductor parameters is still required to make the Maxwell approach more widely useful.

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M. S. RAVEN

Acknowledgements. The author would like to thank the following for interesting discussions and helpful comments: Professor J. A. Brandao Faria, Technical University of Lisbon, Portugal; Dr Oldřich Coufal, Brno University of Technology, Czech Republic; Dr David W. Knight, Technical Director Cameras Underwater, Ltd., Ottery Saint Mary Devon, U.K.; Mrs M. A. Raven, Bridlington, U.K.; Dr Malcolm Woolfson, University of Nottingham, U.K.

References [1]

[2]

[3]

[4] [5] [6] [7] [8]

[9] [10] [11] [12] [13] [14]

S. K. Mukerji, G. K. Singh, S. K. Goel, S. Manuja: A theoretical study of electromagnetic transients in a large conducting plate due to current impact excitation. Progr. Electromagn. Res. 76 (2007), 15–29. J. A. M. Brandao Faria, M. S. Raven: On the success of electromagnetic analytical approaches to full time-domain formulation of skin effect phenomena. Progr. Electromagn. Res. M 31 (2013), 29–43. S. Barmada, L. Rienzo, N. Ida, S. Yuferev: The use of surface impedance conditions in time domain problems: Numerical and experimental validation. Appl. Comput. Electromagn. Soc. J. 19 (2004), 76–83. C.-S. Yen, Z. Fazarinc, R. L. Wheeler: Time domain skin effect model for transient analysis of lossy transmission lines. Proc. IEEE 70 (1982) 750–759. H. W. Dommel: Electromagnetic transient program. Theory book. 2nd Ed., Microtran Power System Analysis Corporation, Vancouver 1992; www.dee.ufrj.br/ipst/EMTPTB.PDF. Alternative transient program (ATP). www.emtp.org, 2010. D. M. Pozer: Microwave engineering. Adison-Wesley, New York 1990, 26. T. Liang, S. Hall, H. Heck, G. Brist: A practical method for modeling PCB transmission lines with conductor surface roughness and wideband dielectric properties. IEEE MTT-S International Microwave Symposium, San Francisco, CA (USA), June 11–16, 2006, 1780–1783. M. S. Raven: Maxwell’s vector potential method, transient currents and the skin effect. Acta Techn. 58 (2013), 337–350. J. C. Maxwell: A treatise on electricity and magnetism. Vol. 2, Oxford University Press, Oxford 1892, Articles 689, 690. R. Plonsey, R. E. Collin: Principles and applications of electromagnetic fields. McGraw-Hill, New York 1961, 325, 328. O. Coufal: On inductance and resistance of solitary long solid conductor. Acta Techn. 57 (2012), 75–89. G. Stephenson: Mathematical methods for science students. Longmans, London 1966, 74. G. Arfken: Mathematical methods for physicists. Academic Press, New York 1970, 519.

Received January 6, 2014