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UNIVERSITY OF TECHNOLOGY, SYDNEY FACULTY OF ENGINEERING 48531 Electromechanical Systems Steady Electric Currents Topics to cover: 1. Electric Current and Current Density 2. Ohm’s Law 3. Electric Resistance 4. Electric Power Dissipation and Joule’s Law Ohm’s Law • Point Form J =σE where σ is the conductivity. The SI unit for conductivity is ampere per volt-meter (A/Vm) or siemens per meter (S/m). The reciprocal of conductivity is known as resistivity in ohm-meters (Ωm). • Circuital form I= V R where V is the voltage across the conductor and R is the resistance of the conductor. The unit of resistance is ohms (Ω). Electric Current and Current Density • Electric charges in motion constitutes an electric current. The unit of current is ampere or A, defined as the rate of movement of charge crossing a given plane in coulomb per second, or dQ I = dt • Electric current density is defined as the current per unit cross sectional area: ∆I ∴ I = J • ds J = lim n ∆s → 0 ∆ s S ∫ where n is the normal direction of the plane. The unit of J is A/m2. • Current density is a vector while current is not. Conductivity • Conduction currents in conductors and semiconductors are caused by drift motion of conduction electrons and/or holes. • Consider the steady motion of one kind of charge carriers, each of charge q (which is negative for electrons), across an element of surface ∆s with a velocity u. If N is the number of charge carriers per unit volume, then in time ∆t each charge carrier moves a distance u∆t, and the amount of charge carrier passing through the surface ∆s is ∆Q = Nqu • a n ∆s∆t Conductivity - cont. Electric Resistance Since current is the time rate of change of charge, we have ∆I = ∆Q = Nqu • an ∆s = J • ∆s ∆t where J=Nqu (A/m2) is the volume current density or simply current density. It can be justified analytically that for most conducting materials the average drift velocity is directly proportional to the applied external electric field strength, or u=−µeE, where µe is the mobility of electrons in a conductor. Therefore, we obtain the point form of Ohm's law: J = −NqµeE =σE where σ is conductivity. The SI unit for conductivity is ampere per voltmeter (A/Vm) or siemens per meter (S/m). The reciprocal of conductivity is known as resistivity in ohm-meters (Ωm). Electric Resistance - cont. • The reciprocal of resistance is defined as conductance or G=1/R. The unit for conductance is Siemens (S) or (Ω-1). • Resistance Calculation or R1 R1 R2 R2 R s = R1 + R2 Rs Rp • The voltage-current relationship Consider a piece of homogeneous material of conductivity σ, length l, and uniform cross section A, as shown below. Within the conductor, J=σE, where both J and E are in the direction of current flow. The potential difference or voltage between terminals 1 and 2 is V12 = El E = V12 l or and the total current is I = ∫ J • dA = JA = σEA = A or I= V12 R where or G p = G1 + G 2 l σA (Ω) is the electric resistance between the terminals. Electric Resistance - Example A metal hemisphere of radius Re, buried with its flat face lying in the surface of the ground, is used as an earthing electrode. It may be assumed that a current flowing to earth spreads out uniformly and radially from the electrode for a great distance. Show that, as the distance for which this is true tends to infinity, the resistance between the electrode and earth tends to the limiting value ρ/2πRe, where ρ is the resistivity of the earth. metallic hemisphere 1 1 1 = + R p R1 R2 R= σA V l 12 ground hemispherical cap with resistance dR, thickness dr and cross-sectional area A. Electric Resistance - Example Solution: To determine the total resistance between the metallic cap and earth (at ∞) we can sum the incremental resistances of the thin hemispherical caps (extending from Re to ∞). First, choose a hemispherical cap of thichness dr, and the incremental resistance of the cap is dR = ρ dr , A A = 2 πr 2 where Therefore, the total resistance ρ R= 2π ∞ ρ dr = ∫ r 2 2π Re − Power Dissipation and Joule's Law • Under the influence of an electric field, conduction electrons in a conductor undergo a drift motion macroscopically. Microscopically, these electrons collide with atoms on lattice sites. Energy is thus transmitted from the electric field to the atoms in thermal vibration. • The Joule's law states that for a given volume Vc the total electric power converted into heat is ∞ 1 ρ = r Re 2 πRe P = ∫ E • Jdv (W) Vc as required. Power Dissipation and Joule's Law - cont. • In a conductor of uniform cross section, dv=dAdl, with dl measured in the direction of J. The above equation becomes P = ∫ Edl ∫ JdA = VI l Summary I = ∫ J • ds or I= • Ohm’s Law J =σE • Resistance l R= σA 1 1 1 = + R p R1 R2 R s = R1 + R2 G p = G1 + G 2 P = I2R This is an expression for power dissipation in a resistor of resistance R. and I = A where I is the current in the conductor. Since V=RI, we have dQ dt • Electrical Current • Joule’s Law P = VI = I 2 R S V R