J.L. Kirtley Jr. September 4, 2010

Transcription

1 Massachusetts Institute of Technoloy Department of Electrical Enineerin and Computer Science Electromanetic Enery: From Motors to Lasers Supplemental Class Notes Manetic Circuit Analo to Electric Circuits J.L. Kirtley Jr. September 4, Introduction In this chapter we describe an equivalence between electric and manetic circuits and in turn a method of describin and analyzin manetic field systems which can be described in manetic circuit fashion. As it turns out, the equivalence is a fair approimation to reality and may be used with some confidence. Manetic circuits are those parts of devices that employ manetic flu to either induce voltae or produce force. Such devices include transformers, motors, enerators and other actuators (includin thins such as solenoid actuators and loudspeakers). In such devices it is necessary to produce and uide manetic flu. This is usually done with pieces of ferromanetic material (which has permeability very much larer than free space). In this sense, manetic circuits are like electric circuits in which conductive material such as aluminum or copper has hih electric conductivity and are used to uide electric current. The analoies between electric and manetic circuits are two: the electric circuit quantity of current is analoous to manetic circuit quantity flu. (Both of these quantities are solenoidal in the sense that they have no diverence). The electric circuit quantity of voltae, or electomotive force (EMF) is analoous to the manetic circuit quantity of manetomotive force (MMF). EMF is the interal of electric field E, MMF is the interal of manetic field H. 2 Electric Circuits and Kirchoff s Laws 2.1 Conservation of Chare and KCL To bein with, consider the law of Conservation of Chare: d J d a = qdv = 0 dt vol c 2003 James L. Kirtley Jr. 1

2 This assumes, of course, that there is no accumulation of chare anywhere in the system. This is not a wonderful assumption for any systems with capacitor plates, but if one considers capacitors to be circuit elements so that both plates of a capacitor are part of any iven element the riht hand side of this epression really is zero. Thsn, if we note current to be the interal of current density: Over some area, a fraction of the whole area around a node: i k = A k J d a then we have: ik = Faraday s Law k : d E dl = dt B d a The left-hand interal may be taken to be a number of sub-interals, each denoted by a discrete fractional interal: bk v k = E dl If we assume that there are no substantive flu linkaes amon the circuit elements: B = 0 a k then we have KVL: 2.3 Ohm s Law vk = 0 k At this point it is probably appropriate to note that Ohm s Law can be used to derive the constitutive relationship for a resistor. Suppose we have a conductive element similar to the rectanular solid shown in Fiure 1. Assume current is confined in this element and flowin perpendicular to the flat end shown in the fiure. Current density is I J = hw where I is to total current and h and w are heiht and width of the conductor, respectively. Electric field alon the lenth of the element is: J E = σ 2

3 where σ is the electrical conductivity of the material. Voltae developed is: J v k = E dl = l σ Which leads us to an epression for element resistance: v l R = = I hwσ h + V l I w Fiure 1: Simple circuit element 3 Manetic Circuits As it turns out, manetic circuits are very similar and are overned by laws that are not at all different from those of electric circuits, with only one minor difference. 3.1 Conservation of Flu: Gauss Law To start, Gauss law is: B d a = 0 This reflects that notion that there are no sources of flu: this is a truely sinusoidal quantity. It neither beins nor ends but just oes in circles. If we note a fraction of the surface around a node and call it surface k, the flu throuh that surface is: Φ k = A k B d a If we take the sum of all partial flues throuh a surface surroundin a note we come to the analo of KCL: Φk = 0 k 3

4 3.2 MMF: Ampere s Law Ampere s Law is simply stated as: H dl = J d a The interal of current density J is current, quantified in the SI system as Amperes. As it is enerally carried in wires which miht number, say, N, it is often quantified as: J d a = NI For this we will often use the term ManetoMotive Force or MMF, which ets the symbol F. If we use that symbol to denote the interal of manetic field over a manetic circuit element: F k = b k a k H dl Then, if we take enouh of these subinterals to cover the loop around a roup of elements, we have Fk = NI k Note that this is not eactly the same as KVL, as it has a source term on the riht. 3.3 Manetic Circuit Element: Analoy to Ohm s Law Manetic circuits have an equivalent to resistance. It is the ratio of MMF to flu and has the symbol R. I direct comparison with the derivation of resistance, a manetic circuit element is shown in Fiure 2. h + F l Φ w Fiure 2: Manetic circuit element Assume that the material of this element has permeability µ > µ 0, so that it has a constitutive relationship: 4

5 B = µh If flu density in the material is uniform, total flu throuh the element is: Φ = hwb = hwµh And the MMF is simply the interal of manetic field H from one end to the other: aain we assume uniformity so that: F = lh The reluctance of this element is then: F R = = Φ l hwµ 3.4 Manetic Gaps In reality, manetic circuits tend to be made up of very hihly permeable elements (pieces of iron) and relatively small air-aps. A sketch of such a ap is shown in Fiure 3. w µ µ h Fiure 3: Gap between manetic elements It is usually permissible to assume that iron elements have very hih permeability (µ ), so that there is neliible MMF drop. In this sense the iron elements serve in the same role as copper or aluminum wire in electric circuits. The ap, on the other hand, has reluctance: R = hwµ0 5

6 Bm B Ha Hm µ µ Frinin Field Fiure 4: Gap details 3.5 Boundary Conditions Shown in Fiure 4 is the cross-section of a ap. It is assumed that the elements shown have some depth into the paper which is reater than the ap width. If the permeability of the elements to the riht and the left is very hih, we say that manetic flu is larely confined to those elements. Note that the boundary condition associated with Ampere s Law dictates that the manetic field intensity H a in the air adjacent to the permeable material and parallel with the surface must be equal to the manetic field intensity H m just inside the manetic material and parallel with the surface. If the material is very hihly permeable (µ ), that manetic field must be nearly zero: H a = H m 0. This means that manetic field must be perpendicular to the surface of very hihly permeable material. This is the case in the ap itself, where: B = B m We should note, however, that there will be frinin fields in the reion near the ap, so that our epression for the reluctance of the ap will not be quite correct. The accuracy of the epression which inores frinin is best for really small aps and enerally over-estimates the reluctance. 4 Faraday s Law and Inductance Chanin manetic fields ive rise to electric fields and consequently produce voltae. This is how inductance works. Consider the situation shown rather abstractly in Fiure 5. Faraday s Law is, in interal form: d E dl = dt Area B nda If the contour shown is hihly conductin (say, if it is a wire), there is zero electric field over that part of the contour. Voltae across the terminals is: b V ab = E dl and that is the whole of the interal above. Thus we may conclude that: a 6

7 a + V b dl i n Fiure 5: Loop for Faraday s Law d V ab = dt Area B nda Now, if we define flu linked by this contour to be: λ = B Area nda then voltae is, as we epect: dλ V ab = dt As it turns out, current flowin in the wire with sense shown by i in Fiure 5 tends to produce flu with sense opposite to the normal vector shown in that fiure, and so produces positive flu. Generally, in calculatin inductance, one uses the riht hand rule in determin the direction of flu linkae: if the finers of your riht hand follow the direction of the windin, from the positive terminal, positive flu is in the direction of your thumb. 4.1 Eample: Solenoid Actuator Shown in Fiure 6 is a representation of a common solenoid actuator. When current is put throuh the coil a manetic flu appears in the aps and pulls the pluner to the left. We will eamine the force and how to calculate it in later chapters. For now, however, we are concerned with manetic fields in the device and with the calculation of inductance. Assume that the stator and pluner are both made of hihly permeable materials (µ ). If the coil carries current I in N turns in the sense shown, manetic flu will cross the narrow air-ap to the riht in the direction from the stator to the pluner and then return in the sense shown in the variable lenth ap of width. It is clear that this is also positive sense flu for the coil. The manetic circuit equivalent is shown in Fiure 7. All of the flu produced crosses the variable width ap which has reluctance: 7

8 Coil Depth D perpendicular to view NI Pluner h B Stator w µ Fiure 6: Cross-Section of Solenoid Actuator Φ F Fiure 7: Manetic Equivalent Circuit of Solenoid Actuator R = µ0 hd Half of the flu crosses each of the other two aps, which are in parallel and have reluctance: Total flu in the manetic circuit is R = µ0 wd F Φ = R R And since λ = NΦ and F = NI, the inductance of this structure is: N 2 N 2 L = = R R R 8

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