Magnetic electro-mechanical machinesLorentz ForceD'Arsonval GalvanometerDirect Current Permanent Magnet Electric Motor“Parasitic” DynamicsElectrical sideMechanical sideConnections:Electrical side:Mechanical side:Motor vs. GeneratorTachometerMagnetic electro-mechanical machines Neville Hogan This is a brief outline of the physics underlying simple electro-magnetic machines, especially the ubiquitous direct-current permanent-magnet motor. Lorentz Force A magnetic field exerts force on a moving charge. The Lorentz equation: f = q(E + v × B) where f: force exerted on charge q E: electric field strength v: velocity of the moving charge B: magnetic flux density Consider a stationary straight conductor perpendicular to a vertically-oriented magnetic field. Magnetic fluxBFiCurrentForceStationary conductorMotion of charges Figure 1: Forces on a current-carrying conductor in a magnetic field. An electric field is oriented parallel to the wire. As charges move along the wire, the magnetic field makes them try to move sideways, exerting a force on the wire. The lateral force due to all the charge in the wire is: f = ρAl (v × B) where ρ: density of charge in the wire (charge per unit volume) l: length of the wire in the magnetic field A: its cross-sectional area The moving charges constitute a current, i i = ρAv The lateral force on the wire is proportional to the current flowing in it. f = l (i × B) © Neville Hogan page 1For the orthogonal orientations shown in the figure, the vectors may be represented by their magnitudes. f = l B i This is one of a pair of equations that describe how electromagnetic phenomena can transfer power between mechanical and electrical systems. The same physical phenomenon also relates velocity and voltage. Consider the same wire perpendicular to the same magnetic field, but moving as shown Magnetic fluxBMoving conductorElectro-motive forceMotion of chargesVelocityv Figure 2: Voltage across a conductor moving in a magnetic field. A component of charge motion is the same as the wire motion. The magnetic field makes charges try to move along the length of the wire from left to right. The resulting electromotive force (emf) opposes the current and is known as back-emf. The size of the back-emf may be deduced as follows. Voltage between two points is the work required to move a unit charge from one to the other. If a unit charge moves along the wire from right to left the work done against the electromagnetic force is e = v B l This is the other of the pair of equations that describe how electromagnetic phenomena can transfer power between mechanical and electrical systems. Two important points: 1. The interaction is bi-lateral (i.e., two-way). If an electrical current generates a mechanical force mechanical velocity generates a back-emf. 2. The interaction is power-continuous. Power is transferred from one domain to the other; no power is dissipated; no energy is stored; electrical power in equals mechanical power out (and v.v.). Pelectrical = e i = (v B l) i = v (B l i) = v f = Pmechanical Power continuity is not a modeling approximation. It arises from the underlying physics. The same physical quantity (magnetic flux density times wire length) is the parameter of the force-current relation and the voltage-velocity relation. © Neville Hogan page 2D'Arsonval Galvanometer Many electrical instruments (ammeters, voltmeters, etc.) are variants of the D'Arsonval galvanometer. A rectangular coil of wire pivots in a magnetic field as shown in figure 3. Restraining springMagnetic fieldSquare coil+- S Nθ Figure 3: Conceptual sketch of a D'Arsonval galvanometer. Current flowing parallel to the axis of rotation generates a torque to rotate the coil. Current flowing in the ends of the coil generates a force along the axis. Assuming the magnetic flux is vertical across the length and width of the coil, the total torque about the axis is: τ = 2NBlh cos(θ) i where τ: clockwise torque about the axis N: number of turns of wire B: magnetic flux density l: length of the coil h: half its height. This torque is counteracted by a rotational spring. As the coils rotate, a back-emf is generated. e = 2NBlh cos(θ) ω where ω: angular speed of the coil. Note that the same parameter, 2NBlh cos(θ), shows up in both equations. © Neville Hogan page 3Direct Current Permanent Magnet Electric Motor With a different geometry the dependence on angle can be substantially reduced or eliminated. Radial flux linesStationary permeable coreRotating cylindrical coil(much of the magnetic field is not shown)Stationary permanent magnetsNS Figure 4: Schematic end-view of a D'Arsonval galvanometer modified to reduce the angle-dependence of the transduction equations. Features: • rotating cylindrical coil • stationary permeable core • shaped permanent magnets • constant radial gap between magnets and core • the magnetic field in the gap is oriented radially If all turns of the coil are in the radial field the torque due to a current in the coil is independent of angle. 0° 30° 60°-60° 90°-30°-90°Angle Figure 5: Sketch of half of the constant-current torque/angle relation resulting from the design of figure 4. From symmetry the torque/angle relation for the other half of the circle is the negative of that shown above. The reversal of torque can be eliminated by reversing the current when the angle passes through ±90°. A mechanical commutator is sketched in figure 6. © Neville Hogan page 4NSSplit-ring commutator rotates with coilBrushes are stationary electrical contactsElectrical connection to the coil is through the commutator+- Figure 6: Schematic of the mechanical commutation system used in a direct-current permanent magnet motor. Electrical connection is through a set of stationary conductors called brushes1. They contact a split ring called a commutator that rotates with the coil. This commutator design is used in a direct-current permanent-magnet motor (DCPMM). The same effect may be achieved electronically. That approach is used in a brushless DCPMM. Assuming perfect commutation the relation between torque and current for a DCPMM is τ = Kt i Kt: torque constant, a parameter determined by the mechanical, magnetic and electrical configuration of the device. There is also a corresponding relation between voltage and rotational speed. e = Ke ω Ke: back-emf or voltage constant. The following is excerpted from a manufacturer's