ELEC 3105 Basic EM and Power Engineering Boundary Conditions Energy in Electric Field Corona Discharge PN Junction in Electrostatics

Boundary conditions Tangential Component of Around closed path (a, b, c, d, a) ELECTROSTATICS Boundary a b c d Potential around closed path

Boundary conditions Tangential Component of Boundary a b c d ELECTROSTATICS

Boundary conditions Tangential Component of a b c d The tangential components of the electric field across a boundary separating two media are continuous. ELECTROSTATICS

Boundary conditions Tangential Component of At the surface of a metal the electric field can have only a normal component since the tangential component is zero through the boundary condition. ELECTROSTATICS a b c d metal

Boundary conditions Normal Component of Gauss’s law over pill box surface ELECTROSTATICS Boundary

ELECTROSTATICS Boundary Boundary conditions Normal Component of

8 The normal components of the electric flux density are discontinuous by the surface charge density. Boundary conditions Normal Component of ELECTROSTATICS

9 Boundary conditions Normal Component of ELECTROSTATICS metal At the surface of a metal the electric field magnitude is given by E n1 and is directly related to the surface charge density.

Boundary conditions Normal Component of ELECTROSTATICS Gaussian Surface Air Dielectric Gaussian surface on metal interface encloses a real net charge  s. Gaussian surface on dielectric interface encloses a bound surface charge  sp, but also encloses the other half of the dipole as well. As a result Gaussian surface encloses no net surface charge.

Energy Stored in an Electric Field q1q1 ∞ q1q1 q2q2 ∞ q1q1 q2q2 r 12 q3q3 ∞ q1q1 q2q2 q3q3 r 13 r 23

Energy Stored in an Electric Field q1q1 ∞ q1q1 q2q2 ∞ q2q2 q3q3 ∞ q2q2 r 12 q3q3 r 13 r 23 q3q3 q3q3

Energy Stored in an Electric Field q1q1 ∞ q1q1 q2q2 r 12 r 13 q3q3 In both cases the same charge distribution is produced Total work can be considered as the sum of the two expression divided by 2

Energy Stored in an Electric Field q1q1 ∞ q1q1 q2q2 r 12 r 13 q3q3 In both cases the same charge distribution is produced In general for N charges Point charges

Energy Stored in an Electric Field Variations for other charge distributions

Corona Discharge If the electric field is very high in a medium, any ions present gain enough kinetic energy between collisions that they in turn ionize the molecules they run into. This is referred to as avalanche breakdown. Ions and free electrons recombine as well, releasing a photon in the process. High electric fields close to conductors can thus induce avalanche breakdown and a glow will be present around the conductor. This glow and process is the Corona discharge.

Corona Discharge

ARC Discharge

Corona Discharge Power line Corona discharge on power lines causes power to be dissipated giving power loss. An undesirable effect.

PN Junction in Electrostatics In crystalline silicon, each silicon atom shares its 4 valence electrons with neighbors in covalent bonds. Si

PN Junction in Electrostatics If a small number of phosphorus atoms are added, a free electron results as the covalent bonds with nearby silicon atoms form, leaving an extra electron weakly attached to the phosphorus. This electron breaks free by thermal action, leaving a positively charged ion behind. SiP+P+ - Free electron Impurities like this are called donors

PN Junction in Electrostatics If boron atoms are added, only 3 valence electrons are available, but an extra electron will usually be stolen from a nearby silicon atom, with thermal kinetic energy providing the small amount of energy required for this capture. In this case, the missing electron in the lattice behaves like a positively charged particle, which we refer to as a hole. SiB-B- + Missing electron “HOLE” Impurities like this are called acceptors

PN Junction in Electrostatics n-type silicon bonded to p-type silicon SiP+P+ P+P+ P+P+ P+P+ - B-B- B-B- B-B- B-B n-type with donor density N D p-type with acceptor density N A

PN Junction in Electrostatics n-type silicon bonded to p-type silicon SiP+P+ P+P+ P+P+ P+P+ - B-B- B-B- B-B- B-B Free electrons from n-type will diffuse into p-type region Holes from p-type will diffuse into n-type region

PN Junction in Electrostatics n-type silicon bonded to p-type silicon Where electrons and holes meet they will cancel each other out as the electron fills the hole - +

PN Junction in Electrostatics n-type silicon bonded to p-type silicon - + n-type silicon with donor density N D p-type silicon with acceptor density N A electrons holes Junction

PN Junction in Electrostatics n-type silicon bonded to p-type silicon SiP+P+ P+P+ P+P+ P+P+ - B-B- B-B- B-B- B-B Positive net charge left behind negative net charge left behind

PN Junction in Electrostatics n-type silicon bonded to p-type silicon n-type polysilicon with donor density N D p-type polysilicon with acceptor density N A Positive net charge left behind Negative net charge left behind N D >> N A

PN Junction in Electrostatics n-type silicon bonded to p-type silicon Junction Charge separation produces an electric field starting on positive charges and ending on negative charges + - N D >> N A What are the details of this electric field across the junction and what is the potential across the junction.

PN Junction in Electrostatics n-type silicon bonded to p-type silicon N D >> N A Thus x p >> x n n p  Plot of net charge density versus x coordinate Gives

PN Junction in Electrostatics n-type silicon bonded to p-type silicon  Plot of net charge density versus x coordinate For a 1-D case as for the the direction transverse to the junction then: Overall charge neutrality requires that x n N D = x p N A

PN Junction in Electrostatics n-type silicon bonded to p-type silicon  Plot of net charge density versus x coordinate Slope of the curve of the x component of the electric field versus position x across the junction. Note on the n side and p side have different slope signs and magnitude.

PN Junction in Electrostatics n-type silicon bonded to p-type silicon  Plot of net charge density versus x coordinate On the n side of the junction Positive value n side p side ExEx

PN Junction in Electrostatics n-type silicon bonded to p-type silicon  Plot of net charge density versus x coordinate On the p side of the junction Negative value ExEx n side p side

PN Junction in Electrostatics n-type silicon bonded to p-type silicon  Plot of net charge density versus x coordinate ExEx n side p side Plot of x component of electric field versus x coordinate

PN Junction in Electrostatics n-type silicon bonded to p-type silicon ExEx Plot of x component of electric field versus x coordinate E x = 0 at x = x p At x = 0 gives With

PN Junction in Electrostatics n-type silicon bonded to p-type silicon ExEx Plot of x component of electric field versus x coordinate We can now obtain an expression for the potential

PN Junction in Electrostatics n-type silicon bonded to p-type silicon ExEx Plot of x component of electric field versus x coordinate We can now obtain an expression for the potential

PN Junction in Electrostatics n-type silicon bonded to p-type silicon ExEx Plot of x component of electric field versus x coordinate We can now obtain an expression for the potential

PN Junction in Electrostatics n-type silicon bonded to p-type silicon ExEx Plot of x component of electric field versus x coordinate At x = 0 gives At x = W gives

PN Junction in Electrostatics n-type silicon bonded to p-type silicon ExEx Plot of x component of electric field versus x coordinate V bi V bi = built in barrier potential at junction

PN junction in Electrostatics n-type silicon bonded to p-type silicon Plot of  versus x coordinate V bi V bi = built in barrier potential at junction Width of depletion region:

PN Junction in Electrostatics FORWARD BIAS - + Applied Potential V a between end of semiconductor

PN Junction in Electrostatics  Plot of net charge density versus x coordinate ExEx n side p side Plot of x component of electric field versus x coordinate FORWARD BIAS

PN Junction in Electrostatics FORWARD BIAS Plot of x component of electric field versus x coordinate V bi V bi -V a = reduced barrier potential at junction ExEx V bi -V a

PN Junction in Electrostatics REVERSE BIAS + - Applied Potential V a between end of semiconductor

PN Junction in Electrostatics  Plot of net charge density versus x coordinate n side p side Plot of x component of electric field versus x coordinate REVERSE BIAS ExEx

PN Junction in Electrostatics REVERSE BIAS Plot of x component of electric field versus x coordinate V bi V bi +V a = Increased barrier potential at junction V bi +V a ExEx

ELEC 3105 Basic EM and Power Engineering