Massachusetts nstitute of Technoloy Department of Electrical Enineerin and Computer Science 6.685 Electric Machines Class Notes 2 Manetic Circuit Basics September 5, 2005 c 2003 James L. Kirtley Jr. 1 ntroduction Manetic Circuits offer, as do electric circuits, a way of simplifyin the analysis of manetic field systems which can be represented as havin a collection of discrete elements. n electric circuits the elements are sources, resistors and so forth which are represented as havin discrete currents and voltaes. These elements are connected toether with wires and their behavior is described by network constraints (Kirkhoff s voltae and current laws) and by constitutive relationships such as Ohm s Law. n manetic circuits the lumped parameters are called Reluctances (the inverse of Reluctance is called Permeance ). The analo to a wire is referred to as a hih permeance manetic circuit element. Of course hih permeability is the analo of hih conductivity. By oranizin manetic field systems into lumped parameter elements and usin network constraints and constitutive relationships we can simplify the analysis of such systems. 2 Electric Circuits First, let us review how Electric Circuits are defined. We start with two conservation laws: conservation of chare and Faraday s Law. From these we can, with appropriate simplifyin assumptions, derive the two fundamental circiut constraints embodied in Kirkhoff s laws. 2.1 KCL Conservation of chare could be written in interal form as: J nda + volume dρ f dt dv = 0 (1) This simply states that the sum of current out of some volume of space and rate of chane of free chare in that space must be zero. Now, if we define a discrete current to be the interal of current density crossin throuh a part of the surface: i k = J nda (2) surface k and if we assume that there is no accumulation of chare within the volume (in ordinary circuit theory the nodes are small and do not accumulate chare), we have: i k = 0 (3) J nda = k 1
which holds if the sum over the index k includes all current paths into the node. This is, of course, KCL. 2.2 KVL Faraday s Law is, in interal form: E d l = d dt B nda (4) where the closed loop in the left hand side of the equation is the ede of the surface of the interal on the riht hand side. Now if we define voltae in the usual way, between points a and b for element k: v k = bk a k E d l (5) Then, if we assume that the riht-hand side of Faraday s Law (that is, manetic induction) is zero, the loop equation becomes: v k = 0 (6) k This works for circuit analysis because most circuits do not involve manetic induction in the loops. However, it does form the basis for much head scratchin over voltaes encountered by round loops. 2.3 Constitutive Relationship: Ohm s Law Many of the materials used in electric circuits carry current throuh a linear conduction mechanism. That is, the relationship between electric field and electric current density is J = σ E (7) Suppose, to start, we can identify a piece of stuff which has constant area and which is carryin current over some finite lenth, as shown in Fiure 1. Assume this rod is carryin current density J (We won t say anythin about how this current density manaed to et into the rod, but assume that it is connected to somethin that can carry current (perhaps a wire...). Total current carried by the rod is simply = J A and then voltae across the element is: v = from which we conclude the resistance is E dl = l σa R = V = l σa 2
J E l Fiure 1: Simple Rod Shaped Resistor Of course we can still employ the lumped parameter picture even with elements that are more complex. Consider the annular resistor shown in Fiure 2. This is an end-on view of somethin which is uniform in cross-section and has depth D in the direction you can t see. Assume that the inner and outer elements are very ood conductors, relative to the annular element in between. Assume further that this element has conductivity σ and inner and outer radii R i and R o, respectively. + v Electrodes Resistive Material Fiure 2: Annular Resistor Now, if the thin is carryin current from the inner to the outer electrode, current density would be: Electric field is Then voltae is v = J = i r J r (r) = E r = J r σ = Ro 2πDr 2πDrσ R i E r (r) = 2πσD lo R o R i so that we conclude the resistance of this element is R = lo Ro R i 2πσD 3
3 Manetic Circuit Analos n the electric circuit, elements for which voltae and current are defined are connected toether by elements thouht of as wires, or elements with zero or neliible voltae drop. The interconnection points are nodes. n manetic circuits the analoous thin occurs: elements for which manetomotive force and flux can be defined are connected toether by hih permeability manetic circuit elements (usually iron) which are the analo of wires in electric circuits. 3.1 Analoy to KCL Gauss Law is: B nda = 0 (8) which means that the total amount of flux comin out of a reion of space is always zero. Now, we will define a quantity which is sometimes called simply flux or a flux tube. This miht be thouht to be a collection of flux lines that can somehow be bundled toether. Generally it is the flux that is identified with a manetic circuit element. Mathematically it is: k = B nda (9) n most cases, flux as defined above is carried in manetic circuit elements which are made of hih permeability material, analoous to the wires of hih conductivity material which carry current in electric circuits. t is possible to show that flux is larely contained in such hih permeability materials. f all of the flux tubes out of some reion of space ( node ) are considered in the sum, they must add to zero: k = 0 (10) 3.2 Analoy to KVL: MMF k Ampere s Law is H d l = J nda (11) Where, as for Faraday s Law, the closed contour on the left is the periphery of the (open) surface on the riht. Now we define what we call Manetomotive Force, in direct analo to Electromotive Force, (voltae). bk F k = H d l (12) a k Further, define the current enclosed by a loop to be: F 0 = J nda (13) Then the analoy to KVL is: F k = F 0 k 4
Note that the analo is not exact as there is a source term on the riht hand side whereas KVL has no source term. Note also that sin counts here. The closed interal is taken in such direction so that the positive sense of the surface enclosed is positive (upwards) when the surface is to the left of the contour. (This is another way of statin the celebrated riht hand rule : if you wrap your riht hand around the contour with your finers pointin in the direction of the closed contour interation, your thumb is pointin in the positive direction for the surface). 3.3 Analo to Ohm s Law: Reluctance Consider a ap between two hih permeability pieces as shown in Fiure 3. f we assume that their permeability is hih enouh, we can assume that there is no manetic field H in them and so the MMF or manetic potential is essentially constant, just like in a wire. For the moment, assume that the ap dimension is small and uniform over the ap area A. Now, assume that some flux is flowin from one of these to the other. That flux is = BA where B is the flux density crossin the ap and A is the ap area. Note that we are inorin frinin fields in this simplified analysis. This nelect often requires correction in practice. Since the permeability of free space is µ 0, (assumin the ap is indeed filled with free space ), manetic field intensity is H = B µ 0 and ap MMF is just manetic field intensity times ap dimension. This, of course, assumes that the ap is uniform and that so is the manetic field intensity: F = B µ 0 Which means that the reluctance of the ap is the ratio of MMF to flux: R = F = µ 0 A µ Area A y x Fiure 3: Air Gap 5
3.4 Simple Case Consider the manetic circuit situation shown in Fiure 4. Here there is a piece of hihly permeable material shaped to carry flux across a sinle air-ap. A coil is wound throuh the window in the manetic material (this shape is usually referred to as a core ). The equivalent circuit is shown in Fiure 5. Reion 1 Reion 2 Fiure 4: Sinle air-capped Core Note that in Fiure 4, if we take as the positive sense of the closed loop a direction which oes vertically upwards throuh the le of the core throuh the coil and then downwards throuh the ap, the current crosses the surface surrounded by the contour in the positive sense direction. + F = N Fiure 5: Equivalent Circuit 3.5 Flux Confinement The ap in this case has the same reluctance as computed earlier, so that the flux in the ap is simply = N R. Now, by focusin on the two reions indicated we miht make a few observations about manetic circuits. First, consider reion 1 as shown in Fiure 6. µ Fiure 6: Flux Confinement Boundary: This is Reion 1 6
n this picture, note that manetic field H parallel to the surface must be the same inside the material as it is outside. Consider Ampere s Law carried out about a very thin loop consistin of the two arrows drawn at the top boundary of the material in Fiure 6 with very short vertical paths joinin them. f there is no current sinularity inside that loop, the interal around it must be zero which means the manetic field just inside must be the same as the manetic field outside. Since the material is very hihly permeable and B = µ H, and hihly permeable means µ is very lare, unless B is really lare, H must be quite small. Thus the manetic circuit has small manetic field H and therfore flux densities parallel to and just outside its boundaries aer also small. µ B is perpendicular Fiure 7: Gap Boundary At the surface of the manetic material, since the manetic field parallel to the surface must be very small, any flux lines that emere from the core element must be perpendicular to the surface as shown for the ap reion in Fiure 7. This is true for reion 1 as well as for reion 2, but note that the total MMF available to drive fields across the ap is the same as would produce field lines from the area of reion 1. Since any lines emerin from the manetic material in reion 1 would have very lon manetic paths, they must be very weak. Thus the manetic circuit material larely confines flux, with only the relatively hih permeance (low reluctance) aps carryin any substantive amount of flux. 3.6 Example: C-Core Consider a apped c-core as shown in Fiure 8. This is two pieces of hihly permeable material shaped enerally like C s. They have uniform depth in the direction you cannot see. We will call that dimension D. Of course the area A = wd, where w is the width at the ap. We assume the two aps have the same area. Each of the aps will have a reluctance R = Suppose we wind a coil with N turns on this core as shown in Fiure 9. Then we put a current in that coil. The manetic circuit equivalent is shown in Fiure 10. The two aps are in series and, of course, in series with the MMF source. Since the two fluxes are the same and the MMF s add: µ 0 A F 0 = N = F 1 + F 2 = 2R and then = N 2R = µ 0AN 2 7
µ Area A Fiure 8: Gapped Core N Turns Fiure 9: Wound, Gapped Core and correspondin flux density in the aps would be: B y = µ 0N 2 3.7 Example: Core with Different Gaps As a second example, consider the perhaps oddly shaped core shown in Fiure 11. Suppose the ap on the riht has twice the area as the ap on the left. We would have two ap reluctances: R 1 = µ 0 A R 2 = 2µ 0 A Since the two aps are in series the flux is the same and the total reluctance is Flux in the manetic circuit loop is R = 3 2 µ 0 A = F R = 2 µ 0 AN 3 and the flux density across, say, the left hand ap would be: B y = A = 2 µ 0 N 3 8
F + Fiure 10: Equivalent Manetic Circuit N Turns Fiure 11: Wound, Gapped Core: Different Gaps 9