DIFFUSION OF CURRENT INTO CONDUCTORS

DIFFUSION OF CURRENT INTO CONDUCTORS J.Edwards* and T.K Saha** * Research Concentration in Electrical Energy Queensland University of Technology ** Department of Computer Science and Electrical Engineering University of Queensland. Currents are established on the surface of conductors by the propagation of electromagnetic waves in the insulating material between them. If the load is less than the characteristic impedance of the insulating material of the line, multiple reflections and retransmissions eventually build up the line current to that required by the load. The currents are initially established on the surface of the conductors before diffusing relatively slowly into the interior and gives rise to the skin effect. The diffusion velocity depends the conductivity, permeability, thickness of the conductor, and the frequency of the excitation, and such effects of the diffusion process are difficult to conceptually appreciate. Fortunately, the diffusion of heat into solids is very similar, and will be used as an analogy to aid understanding. This diffusion is the means whereby current moves into conductors and flux into of magnetic cores. (transmission line) to the load as displacement currents 1. INTRODUCTION in the insulation, at velocities approaching c. The displacement current builds up the line current on the Current flow in conductors is often associated with surface of the conducting cables by multiple liquid flow in pipes, at least conceptually. The ‘liquid’ reflections, and this current diffuses into the interior of being the sea of free electrons, in the conduction band the conductor [1]. Thus current changes (electron of the material, that drift along the conductor with a accelerations) actually move into the conductors from velocity that is proportional to the electric field and the outside surfaces and only have to diffuse through gives rise to the current, J = σE. This liquid flow the thickness of the conductors (half the diameter) analogy is quite reasonable for steady dc currents, rather than along the whole length of the cable from giving a conceptual understanding of Ohms law. the power source to the load. If it were not for the However, while good conductors such as copper displacement current setting up the surface currents in present little opposition to electrons moving at the first instance, energy transmission, (other than via constant velocity, they strongly oppose electron relatively steady dc currents) via copper conductors accelerations, because the relatively large currents would be virtually impossible because of the long (J=σE) produce magnetic fields that are much higher diffusion times and attenuations. than those produced by displacement currents in free space. These high magnetic fields move at relatively The skin effect results from the fact that at the low velocities since the back emfs induced by their particular frequency of operation the surface currents motion cannot be any larger than the driving E field. only have time to diffuse into the conductor to the skin These induced emfs generate eddy currents within the depth in the ¼ period of the supply frequency. Surface conductor, which also oppose the diffusion due to currents move into the conductor on the rising part of associated energy losses. These limited back emfs and the current waveform. Once the surface currents peak eddy currents cause any changes in the surface and start to fall, the interior currents move back out electric field (current) to move very slowly into the towards the surface of the conductor. At 50Hz the skin interior of conductors and suffer significant. depth in copper is approx 10mm. The relatively slow velocity of penetration depends on the conductivity, and permeability of the material. The higher the conductivity, permeability, and size of the conductor, the slower is diffusion velocity. For a very large copper conductor the penetration velocity at a frequency 50Hz is approximately 8 m/s. This relatively low velocity is not very apparent in every day applications because the currents needed to energise electrical loads initially propagate along the cable The same diffusion process also applies to surface magnetic flux moving into the interior of magnetic cores, since most cores are electrically conductive. Surface flux changes produce a driving H field (∆Hs = ∆Bs/µ) that drives any change in surface flux into interior of the core. The changing flux levels induce back emfs as they move into the core, with resulting eddy currents, whose H fields oppose the driving H field. These induced H fields, due to changing surface flux levels moving into the interior of the core cannot be any greater than the driving H field at any point in the conductor and this is the fundamental reason for the low diffusion velocity. Eddy current losses also attenuate the fields and further reduce the diffusion velocity. The phase velocity for transformer steels at a 50Hz ≈ 110mm/sec resulting in a skin depth of ≈ 0.5mm, and is the reason for laminating the core. The velocity at which a wave penetrates into a conductive material also depends upon the thickness of the conductor and frequency of the excitation. These are difficult to appreciate but fortunately the diffusion of heat into solids is similar, and will be used to conceptually understand this diffusion. 2.0 CURRENT DIFFUSION The high conductivity of good conductors, such as copper, provide a good medium for electrons to move at constant velocity (steady dc currents) but opposes electron acceleration. Accelerating electrons produce changing magnetic fields with resulting back emfs that oppose the acceleration. The currents in conductors (per unit of E) are very much larger than displacement currents in free space (377 ohms), as also are the resulting changes in internal magnetic flux levels as any surface changes in E move into the interior. Changes in surface current produce the surface electric field (∆Es= ∆Js/σ) that drives the current changes into the interior. As these surface current changes move into the interior, back emfs are induced that can be no bigger than the driving E field. The high magnetic fields associated with the high current levels (due to low conductivity) is the fundamental reason why electric fields move very slowly into the interior of a conductor compared with free space. The velocities of these fields in free space is also limited by back emfs but in this case the much smaller magnetic fields have to move at velocity c. These induced emfs, due to the accelerating electrons, also create eddy currents in the material that absorb energy and further reduce the diffusion velocity due to associated losses that reduce the effective driving E field. These eddy current losses cause any changing electro-magnetic fields to be greatly attenuated as they go into the interior of a conductors. 2.1 Propagation in Conducting Materials At any point inside the conductor electromagnetic equations are relevant. Conduction Current, J = σE Flux Density, B = µH following .......(1a) ...... (1b) Current Density, i = J + Displacement Current ∂D ∂E = σE + ε o ε r ∂t ∂t ∂D ∂E Curl H = i = σE + = σE + ε o ε r ....... (1c) ∂t ∂t ∂E For sinusoidal excitation, = jω E , so ∂t Displacement Current = jωεoεrE & Curl H = ( σ + jωεo εr ) E The displacement current (jωεoεrE) rises with frequency and for copper equals the conduction current (σE) when ω = σ/ε ≈ 1019. Thus displacement currents can be neglected in comparison to the conduction current at normal frequencies so: Curl H = σ E . If they could occur, any net like charges inside a conductor would repel each other, and quickly move to the surface with a time constant, τ = εo/σ ≈ 10-19, for copper. Thus at normal frequencies there is virtually no charge build up inside the conductor, and the electric field inside a conductor can be considered to be divergence free (Div E = 0). i = σE + Thus at any point inside a conductor: Div E = ∇.E = 0 & Curl H ≈ J = σE ...... (2a) ...... (2b) ∂B ∂H = −µ .…. (2c) ∂t ∂t = − jωB = − jωµH , for sinusoids & ∇. H = ∇ .B = 0 ...... (2d) Now, Curl E = − 2.2 Electromagnetic Diffusion in Conductors Taking the Curl of both sides of (2b) we get Curl Curl H = σ Curl E Substituting (2c) we get: Curl Curl H = −σµ Now, B = µ H so: ∂H ∂t ….. (3a) ∂B ...... (3b) ∂t Similarly, taking the Curl of (2c) gives ∂Curl B ∂Curl H Curl Curl E = − = −µ ∂t ∂t Substituting Curl H =σ E gives: ∂J ∂E Curl Curl E = − µ = −σµ ….. (3c) ∂t ∂t Now J = σ E so: ∂J Curl Curl J = −σµ ...... (3d) ∂t These 4 equations are basically the same, so (3d) can represent them all on the understanding that J can be replaced by E, H, or B. Curl Curl B = −σµ Magnetic Diffusion Equation Applying the vector identity Curl Curl J = Grad Div J − ∇2 J = − ∇2 J , since Div J = 0 to (4d) gives the Magnetic Diffusion Equation ∂J ∂J 1 2 ∇ 2 J = σµ , or = ∇ J ...... (4a) ∂t ∂t σµ where, J can be replaced by E,H, or B. 1 Magnetic Diffusivity, α = ….. (4b) σµ The diffusivity α, and hence the penetration speed, decreases with increase in the conductivity and permeability of the material. For copper, σ = 5.8x107, µ = 4π x10-7, α = 13.7x10-3 m2/sec For transformer steels, σ = 2x106, µ = 104 x 4π x10-7, α = 0.04x10-3 m2/sec 2. HEAT CONDUCTIVITY Heat flow in solids will now be considered to gain a conceptual understanding of this diffusion. 2.1 Thermal Conductivity Heat flows in a material from regions of high temperature to ones at lower temperature, the driving force being the temperature gradient. T1 (hot) Q T2 (cold) Fig 1: Steady-State Heat Flow Fourier’s Law Heat Flux, Q = − k ∇T (W/m2 ) where, k = Thermal conductivity, ( W/m oC ) ≈ 388 W/m oC , for copper ≈ 62 W/m oC , for iron The thermal conductance of the material = kA/L, similar to G = σA/L for electrical conductance. In the steady state, if there are no heat sources or sinks inside the material, the temperature gradient throughout the material is constant. The heat leaving the cooler surface is equal to that flowing through the material from the hot surface and ∇ . Q = 0 . 2.2 Thermal Diffusion Considering a small internal elemental volume of the material, the heat flow rate equation is given by [2]: Heat Flow In – Heat Flow Out = Change in Internal Energy – Heat Generated ∂  ∂T  ∂  ∂T  ∂  ∂T  ∂T dq − k  + k  + k  = cρ ∂x  ∂x  ∂x  ∂x  ∂x  ∂x  ∂t dt where, c = specific Heat ( J/kg oC ) ρ = density ( kg/m3 ) ∂T dq Thus k ∇ 2 T = cρ − ∂t dt c ρ ∂ T 1 dq ∇2 T = − k ∂t k dt ∴ 2.2.1 Steady State Conditions In the steady state the rate of change of internal energy is zero. If internal energy sources exist then 1 dq ∇2 T = − ..... Poison’s Equation k dt If there are no internal sources then ∇2 T = 0 ..... Laplace’s Equation 2.2.2 Unsteady (Changing) Conditions If there are no internal sources and constant, steady state conditions have not yet been achieved then: Net Rate of Heat Flow into an element = Rate of increase in internal energy Thermal Diffusion (Fourier Equation) ∂T k∇ 2 T = cρ ∂t c ρ ∂ T ∂ T ∇2 T = , or = α∇ 2 T ….(5) k ∂t ∂t k Thermal Conductivity Therm Diffusivity, α = = cρ Thermal Capacitance ≈ 114 x 10-6 m2/sec, for copper ≈ 18 x 10-6 m2/sec, for iron The material absorbs energy as its temperature rises and so acts like a capacitance, C = cρ. Any heat stored in the material has to flow via the conductance k, so the thermal conductivity aids diffusion while c and ρ oppose it. Thermal diffusivity is a measure of the ability of a material to propagate energy compared with its energy storage requirements. Heat flow can be modelled with a distributed RC electrical circuit. The rate of heat flow is directly proportional to the temperature gradient, and this gradient is the driving force for the diffusive flow of heat. T1 T1 T1 B C A T2 (a) ∇ > 0 2 T2 (b) ∇ = 0 2 T2 (c) ∇ < 0 2 Fig2: Unsteady & Steady Heat Flows In Fig2(a) the heat flowing into point A exceeds that flowing out so the temperature of point A rises. In Fig2(b) the heat flowing into point B is equal to that flowing out so the temperature of B is constant. In Fig2(c) the heat flowing out of point C exceeds that flowing in, so the temperature of point C will fall. 3. COMPARISON OF DIFFUSIONS Thermal Diffusion At any point in the material ∂T k = α ∇ 2T , where α = ∂t cρ In this case T is a scalar so: ∇ 2 T = div of grad T = ∇. ∇T  ∂2 ∂2 ∂ 2  = + + T  ∂x 2 ∂y 2 ∂z 2    For one-dimensional diffusion in the x direction: ∂T ∂ 2T =a .… (6a) ∂t ∂x 2 Magnetic Diffusion (F = J, E, H, B, or T ) ∂F 1 = a∇2F , where α = ∂t σµ Since F is a vector ∇ 2 F = grad of div F - Curl of Curl F In rectangular co-ordinates this is equal to the vector sum of the Laplacian operation on the 3 scalar components of F.  ∂2 ∂2 ∂ 2  ∇2F =  + + (F a + F y a y + F z a z )  ∂x 2 ∂y 2 ∂z 2  x x   For one dimensional diffusion in the x direction let: ∂F y ∂Fy F = Fy a y , and = =0 ∂y ∂z ∴∇ F= 2 ∴ ∂ 2Fy ∂x 2 ∂F y a y , so =a ∂F y ∂t ay =a ∂ 2Fy ∂x 2 ay ∂ 2 Fy …. (6b) ∂t ∂x 2 Although the thermal (6a) and magnetic diffusion equations (6b), operate on scalar and vector fields respectively they are essentially the same. In each case the diffusive flow of the entity is due to the spatial concentration gradient of the entity itself. Heat diffusion is opposed by the heat capacitance (cρ) of the material, and aided by thermal conductivity. However, in the case of magnetic diffusion both the electrical conductivity and permeability oppose the diffusion. Material Diffusivity, α (m2/sec) Thermal Magnetic Copper 1.14 x 10-4 137 x 10-4 Transformer Steel 0.18 x 10-4 0.4 x 10-4 The thermal and electrical diffusivities of transformer steels are similar to each other and give an appreciation of the slow diffusion rates of surface flux into magnetic cores laminations. The electrical diffusivity of copper is approx 100 times its thermal. Since diffusion velocities are proportional to the √α, electric fields (and hence surface currents) will diffuse into copper approx 10 times faster than heat. These comparisons with heat give some appreciation of the impossible situation that would exist if currents had to diffuse longitudinally through the length of copper conductors instead of transversely across half their thickness. It is very fortunate that surface currents are initially established along the length of conductors at velocities around c, by means of the displacement currents that flow in the insulating medium. 4. EXAMPLES OF DIFFUSION Irrespective of whether we consider electromagnetic or thermal diffusion the situation is described by applying the relevant boundary conditions to the generalised diffusion equation: ∂F ∂2F =a ∂t ∂x 2 , where F = J, E, B, H or T. 4.1 Semi-Infinite Slab This is the simplest case for examining diffusion, since there is only one boundary. Although we could use J, E, B, H or T, we will use temperature (T) since it gives a conceptual understanding of this diffusion. Consider a semi-infinite block, initially at 0oC, whose exposed surface temperature is suddenly raised to a constant T s of 100oC, as indicated in Fig 3. Ts 0 x Initial Condition T (x,0) = 0 Boundary Conditions At surface T(0,t) = Ts Deep into Block T(∞ ,t) = 0 Fig 3: Step Temp at Surface of Semi-Infinite Block As the temperature of the exposed surface rises, a temperature gradient is produced which will drive heat into the material. At any point in the block the rate of temperature rise must satisfy the diffusion equation. ∂T ( x, t ) ∂ 2 T ( x, t ) = α ∇ 2 T ( x, t ) = α , for 0 < x < ∞ ∂t ∂x 2 Taking the Laplace Transform and inserting the boundary conditions gives. T ( x, s ) = Ts − x e s s  x ⇒ T ( x, t) = Ts erfc   4a t a where, erfc ( x) = 1 − erf ( x) = 1 − 2 ∞ Consider a plate of thickness L, whose exposed surface temperature is raised as shown below.     − y2 ∫ e .dy p x The temperature at an internal surface, distance x into the block at time t, is given by.  x   T ( x, t) = Ts erfc  …. (8a)   4a t  where, Ts = 100oC, α = 1.14x10-4 Temperatures in the copper block are as follows. TEMPERATURE DIFFUSION INTO COPPER BLOCK (Semi-Infinite) 100 Diffusivity = 0.000114 90 80 5 Secs Temperature 70 60 1 Sec 50 40 Ts 0 0 Fig 4: Step Temp at Surface of Insulated Plate In this case the temperature gradient at the insulated face is always zero since there is no heat flow out of this surface. Using (8a), and a series of images to satisfy the boundary conditions gives: ∞  x   2 nL + x   + T s ∑ (− 1) n erfc   T ( x, t ) = Ts erfc     n =1  4a t   4 at  ∞  2nL − x   − ∑ ( −1) n erfc  ….(8b)  n=1  4at  Heat diffusion into a copper plate 20mm thick is as shown below. 200 mSecs 30 TEMPERATURE DIFFUSION INTO 20mm COPPER PLATE (insulated at far face) 100 50 mSecs 20 80 Diffusivity = 0.000114 2 4 6 8 10 12 Distance (mm) 14 16 18 70 20 Temperature and electromagnetic diffusions into copper & steel blocks after 50msecs are shown below. Temperature 0 It can be seen that the temperature diffuses fairly slowly into the block. After 5 secs the temperature at 20mm is only 55% because material beyond 20mm still requires heat, and acts like a heat sink. 100 50 40 200 mSecs 30 50 mSecs 20 10 mSecs 10 0 2 4 6 8 10 12 Distance (cm) 14 16 18 20 The temperature diffuses faster into plates since there is no heat sinking beyond the plate thickness. The end surface of the plate at 20mm rises to 96% after 5 secs. This helps to appreciate how current & flux diffusion velocities reduce with material thickness, although in these cases it is due to reductions in eddy currents. DISTRIBUTIONS AFTER 50 mSECS 90 J,E (Copper), Diff = 0.0137 80 1 Sec 60 0 DIFFUSSION INTO LARGE BLOCKS (Semi-infinite) Entity, F = (E, J, H, B) or T 5 Secs 90 10 mSecs 10 0 Insulation Initial Condition T (x,0) = 0 Boundary Conditions At exposed surface T(0,t) = Ts At insulated surface x ∇ T(L,t) = 0 L 70 60 Temp (Copper), Diff = 0.000114 50 40 B,H (Trans Steel), Diff = 0.00004 30 20 Temp (Iron), Diff = 0.000018 10 0 0 2 4 6 8 10 12 Distance (mm) 14 16 18 20 It can be seen that that electrical diffusion rates for copper are approx 10 times the thermal ones. 4.2 Semi-Infinite Plate (insulated at far surface) 4.3 Surface Current Diffusion into Copper Plate In this case surface current changes diffuse into the copper from both sides of the plate. No diffusion flows cross the centre line, that in terms of the analogous heat flow can be considered a perfectly insulating surface. With this refinement Eq (8b) can be used to graph current diffusions into copper plate copper plate 20mm thick as follows. CURRENT DIFFUSION INTO 20mm COPPER PLATE (Diffusivity = 0.0137) 100 10 mSec 90 80 5 mSecs 70 Current 60 50 40 2 mSecs 30 20 1 mSec 10 0.01 mSec 0.05 mSec 0 0 2 4 6 8 10 12 Distance (mm) 14 16 18 20 Centre line currents (at 10mm depth) reach approx 75% of the surface currents after 5msecs. 4.3 Surface Flux Diffusion into Laminations The case for surface flux diffusion into a transformer steel lamination 1mm thick is indicated below. FLUX DIFFUSION INTO 1mm STEEL LAMINATION (Diffusivity = 0.00004) 10 mSec 80 Magnetic Flux 60 2 mSecs 40 30 1 mSec 10 0 0 0.01 mSec 0.1 0.2 0.3 0.05 mSec 0.4 0.5 0.6 Distance (mm) 0.7 0.8 0.9 1 In this case centre line flux levels (at 0.5mm depth) reach approx 85% of the surface flux after 5msecs. 4.5 Sinusoidal Excitations at Surface of Block Consider the flow of sinusoid surface temperatures into a semi-infinite block as indicated below. +Ts + + In Out 0 − x δ Sin (ωt − −x ) δ x δ Sin( ωt −x −π ) δ 4 …. (9a) …. (9b) 2α …. (9c) ω & Phase Velocity, u = ωδ …. (9d) The amplitude of these wave is attenuated by 1/e = 0.37 in one radian length, δ, of the propagating wave. The magnetic field H in conductors lags the electric field E by 450 due to the magnetic fields resulting from conduction rather than displacement currents losses. E & H fields in non-conductive materials are in phase . 70 20 E = Ese where, Skin Depth , δ = 5 mSecs 50 Magnetic Diffusion with Sinusoids. Whether it is surface current or flux that diffuses into a conductive material, both E & H fields are produced whose amplitudes decrease as they go further in [3]. H = H se 100 90 through the surface of the block for the first 90o of the waveform and back out for the next 90o. Similarly negative heat flows in and out of the block during the negative half cycle of the surface temperature. The lower the frequency the greater is the time for the heat to penetrate into the material before it moves back out. Thus the penetration depth reduces with increase in frequency. The rate of change of surface temperature increases with frequency and so do the spatial temperature gradients in the material. As gradients are the driving force for this diffusive flow, the diffusion velocity in the material increases with excitation frequency. - In Out x -Ts 0 Fig 5: Sinusoidal Temp at Surface of Large Block The diffusive flows depend upon the spatial gradient of the temperature. Heat flows into the block as the surface temperature rises and out of the block when the surface temperature falls. Positive heat flows in Diffusion Type Copper –Elect Copper -Therm TraSteel-Mag TraSteel -Therm Diffus α 137x10-4 1.14x10-4 0.40x10-4 0.18x10-4 SkDepth δ 9.34 mm 0.85mm 0.51mm 0.34mm Ph Vel u 2.93m/s 0.27m/s 0.16m/s 0.11m/s 5. CONCLUSIONS Liquid flow in pipes gives a good realistic analogy only for constant dc current flows in conductors since good conductors are a hostile medium for electrons to accelerate in. All changes have to diffuse in and out of conductors through their longitudinal surfaces, and a good analogy for this is heat flow in solids. References [1] J.Edwards,T.K.Saha,”Establishment of Current in Electrical Cables via Electromagnetic Energies & the Poyting Vector”, AUPEC’98 ,Vol2 385-388, Sept 98. [2] F.Kreith,”Principles of Heat Transfer”, 1973. [3] P.Lorrain, D.Corson,”Electromagnetic Fields and Waves”,2nd ed, pp 471-481, Freeman, 1970.