The potential distribution in a constricted cylinder: an exact solution

Abstract

Abstract. An exact solution to the Laplace equation is derived for the distribution of electric potential within a long cylinder carrying a circular constriction along its axis. The expression obtained for the potential distribution is reduced to a form which may be readily evaluated and is highly accurate for a ratio of constriction radius to cylinder radius approaching unity. Exact expressions both for the electric current density within the constriction and for the spreading resistance (i.e., the increase in resistance of the cylinder due to constriction of current flow lines) are also obtained.

High speed test interface module using MEMS technology

“…In the proposed model, a single circular a-spot is considered to derive the electrical contact resistance. Using Laplace's equation [21,22], the constriction resistance for a circular a-spot in a cylindrical conductor with radius R is given by…”

Constriction Resistance of Thin-Film Contacts

“…(6) in the left-hand-side of Eq. (12) immediately leads to [4,5] a B n = (Iρ / J 1 2 (λ n )λ n πa 2 b) ∫ r J 0 (λ n r/b) / [1 -r 2 /a 2 ] 1/2 dr 0 Since it is known that [8,9] a ∫ r J 0 (λ n r/b) / [1 -r 2 /a 2 ] 1/2 dr = (ab / λ n ) sin(λ n a/b) 0 It follows that B n = Iρ / J 1 2 (λ n )λ n 2 πa The total resistance R T between the center of the constriction and the film surface at r = b is thus given as ∫ R T = (V 0 -V(0,0))/ I n=∞ = (ρ/πa)Σcoth(λ n L/b)sin(λ n a /b) / (J 1 (λ n )λ n ) 2 (14) n=1…”

“…since electrical current does not flow across the radial surfaces except in the region 0 < r < a at z = 0, then ∂V/∂z = 0 for 0 < r < b at z = L (3) and ∂V/∂z = 0 for a < r < b at z = 0 (4) ∂V/∂z = ρ j(r) for 0 < r < a at z = 0 (5) where ρ is the electrical resistivity of the conducting film and j(r) is the cylindrically symmetric current density distribution in the constriction, 2. similarly to the known distribution of electrical current in a cylindrical constriction in a semi-infinite solid [1,2,4], the current density distribution j(r) within the circular constriction in the film is assumed as…”

“…(6) is only approximate, the approximation will be shown to be reliable and representative of the true current distribution as long as the ratio a / b does not exceed about 0.5 [4,5], 3. finally, the potential distribution is assumed constant over the outer cylindrical surface of the film such that…”

Lumped circuit elements, statistical analysis, and radio frequency properties of electrical contact

“…Equation ͑2͒ was a useful, accurate formula, synthesized from the numerical solution of the Laplace equation 20,21. It gives Holm's celebrated formula for the a spot, R c = / 2a, as b → ϱ.…”