Influence of the Corona Discharge on DC Current2pdf

Influence

of the corona discharge

on DC current

Influence of the corona discharge

on DC current

B.Tsoniff

Engineering Department, Trans – Africa Projects

PO Box 6583, Halfway House, Midrand, 1685, JHB, South Africa

Phone: +27-011-2077710, E-mail: [email protected]

A.A. Burger

Eskom Technology division

Eskom MWP, Maxwell Drive, Sunninghill, JHB, RSA

Phone: +27-011-8003973, E-mail: [email protected]

Chapter 1

Physical model of electric current in the frame of the Maxwellian

electromagnetic theory.

Electric charge on the surface of a current carrying conductor

An DC electric current in any cross-section of a conductor is related to the field by the local version of Ohm's law (G.L. Pollack, D.

R. Stump, ELECTROMAGNETISM, 2002)

The source of can only be a charge distribution (no time dependence). Since the inner volume charge density of the conductor must

be zero, and the environment around the conductor is a dielectric, this charge distribution should only exist on the surface of the

conductor. This surface charge determines both outside and inside the conductor, and hence the current in the conductor.

To provide with the longitudinal (axial) component of electric field the surface charge distribution should be non-uniform. The

analytical solutions exist for the distributions with high degree of symmetry:

for the single conductor (A. Sommerfeld) and for the two long parallel conductors (A. Assis). In both cases there is the surface charge

distribution with linear gradient in the direction of electric current.

j Eσ=

E

Experimental mapping of the external electric field of

a straight conductor carrying a steady current (O. Jefimenko)

The picture clearly demonstrates the structure of external electric field created by the surface electric charge on the

current-carrying conductor. The electric lines of force start and finish on the conductors surface which means that there is

a longitudinal component of the electric field.

This aspect differentiates it from the electrostatic field outside of the conductors (in which case the external electric field

in steady state is normal to the conductor at every point of its surface).

Calculation mapping of the external electric field of

a straight conductor carrying a steady current (Peeter Saari)

Electric field of a ring of charge

The axial electrostatic force is a resultant force of a charged ring. If a ring-shaped conductor in YZ plane with radius R

carries a total charge Q uniformly distributed around it,

then the electric field at a point P that lies on the X axis of the ring at a distance x from its centre is

Gradient of surface electric charge

Any conductor may be considered as a set of rings with uniformly distributed charges around them. Rings of equal

charge density correspond to the electrostatic case. Linear gradient of the surface charge creates an internal constant axial

electrostatic force in each point parallel to the conductor.

Under the influence of this external force, the free electrons of a conductor move longitudinally with a constant speed (in

accordance with Aristotle's mechanics for the motion with friction where the speed is proportional to the applied force).

The surface charge approach describes the real physics of the electric current generation in a conductor.

Relation between the surface charge gradient and the current magnitude

The excessive electrical charges are accumulated on the surface of a conductor as a result of the

charge separation by a source of electricity. The gradient of charge is reciprocal to the charge: if

excessive charge tends to infinity its gradient tends to zero which means that the longitudinal

component of electric field also tends to zero as well as the current density.

Poynting vector for DC current conductor

Poynting Vector defines the magnitude and direction of the electromagnetic energy flux density. In the simplest form it is written as a vector product

Where is electric field intensity and is magnetic field intensity.

The electric field on the surface of a straight cylindrical conductor of radius r and resistance R may be expanded

into the sum of two orthogonal components (I.Sefton)

where the radial component represents the electrostatic field and the axial component is proportional to the

gradient of surface charge . Magnetic induction provided by the current I is orthogonal to the plane

Poynting Vector also may be expanded into the sum of two orthogonal components: axial and radial

E

B

0

BS Eµ

= ×

r aE E E= +

rE

aE

B

{ },r aE E

Flux of the Poynting Vector’s radial component through the lateral surface

The flux of the Poynting Vector’s radial component through the lateral surface of the cylindrical conductor of length L is

where

Potential energy cannot pass to heat directly. Firstly it has to be converted into kinetic energy. So there should be the two-step process of converting

the potential energy of the surface charge into kinetic energy of electrons with the following dissipation on the ionic lattice.

( )S ⊥

Φ 2A rLπ=

( ) ( ) ( )2

sin 90

2

Sa

A A

A

S dA E B dA

IR I dA I RL rπ

⊥ ⊥Φ = = ⋅ ⋅ =

⋅ = ⋅

∫∫ ∫∫

∫∫

( )

0

a

BS E

µ⊥ ×=

aI RE

L⋅

= 0

2B I

rµπ

=

Corona discharge internal losses

As a result of a corona discharge the conductor loses excessive charges which means that surface charge density gradient locally increases. This increase

lasts for a time which is needed to compensate the charge and to restore the quasi-static state. During this time interval the current should also

experience the local increase.

Formally, in the frame of the Poynting Vector approach the corona effect results in redistribution of the power density flux between the axial and radial

components in favor of the latter. The redistributed part of the radial component may be attributed to the corona internal power losses.

Corona discharge on the conductor’s surface generates the local pulse currents in a conductor. Those pulse currents are

responsible for the heat losses and generate the wide band electromagnetic radiation.

Chapter 2

Experimental estimation of corona losses for HVDC lines.

Based on

Estimation of ionic current

Js – ionic current density

Estimation of ionic current

The shape OMN may be approximated by the polynomial of the second order . The definite integral ( area under the curve) yields Corona power losses due to the ionic current for the 800m length line are:

20.3 10y x x= − +

( )40

2 3

0

0.3 10 1.6 10 / 1.6 /x x dx nA m A mµ− + = × =∫

corP = 1.6 A / m × 800 m × 800 kV 1kWµ ≈

Direct measure of corona losses

Chapter 2: Conclusions

There is at least one more physical source of corona losses.

This source is negligible for low voltages but making a substantial contribution

to the losses for high voltages.

It is logical to assume that:

- the new source of losses is placed inside of a current-carrying conductor;

- the additional losses is a result of influence of the corona effect on a conductor’s current

A.Sommerfeld interpretation

This rather surprising result was obtained by A.Sommerfeld. In his book ‘Electrodynamics, § 17, pg. 130) he put it the following way:

“Accordingly we obtain the following total picture of the behaviour of the energy: Outside of the wire the energy flows from the electrodes from all sides towards the surface of the wire. After entering it flows radially toward the axis of the wire, being converted at the same time into heat. There is no energy flux parallel to the wire axis within the wire. This picture is materially different from the popular concept of the energy transfer in a wire

carrying current. From the Maxwellian standpoint there is no doubt, however, about the inner consistency and unique validity of our picture. It

indicates the fundamental change which Maxwell's theory has brought in the concepts conductor and nonconductor: The conductors are nonconductors

of energy. Electromagnetic energy is transported without loss only in nonconductors; in conductors it is destroyed, or rather transformed. The notation

‘conductor’ and ‘nonconductor’ refers only to the behaviour, with respect to charge; it is misleading if applied to behaviour with regard to energy.”