Unit-1 Chapter 1.1 Magnetic Fields and Magnetic Circuits LECTURE TOPICS: 1.1.1 Review of magnetic circuits - MMF, flux, reluctance, inductance 1.1.2 Review of Ampere Law and Biot Savart Law 1.1.3 Visualization of magnetic fields produced by a bar magnet and a current carrying coil - through air and through a combination of iron and air 1.1.4 Influence of highly permeable materials on the magnetic flux lines 1.1 .1 Lecture topic Review of magnetic circuits - MMF, flux, reluctance, inductance: Web link: http://ocw.nthu.edu.tw/ocw/upload/124/news/Saadany_Magnetic%20Circuits.pdf Video link: https://www.youtube.com/watch?v=q-nTad8SYOw Electric machines and transformers have electric circuits and magnetic circuits interlinked through the medium of magnetic flux. Electric current flow through the electric circuits, which are made up of windings. On the other hand, magnetic fluxes flow through the magnetic circuits, which are made up of iron cores. The interaction between the currents and the fluxes is the basic of the electromechanical energy conversion process that takes place in generators and motors. However, in transformers it is more feasible to think about the process in terms of an energy transfer. In transformers, the energy transfer is normally associated with change in voltage and current levels. Thus, magnetic circuits play an essential role. The magnetic flux is produced due to the flow of a current in a wire (electric magnet). The direction of the produced magnetic flux is determined by “the right-hand rule” as shown in Fig. 1. Fig. 1 The Right-hand rule for magnetic flux The unit for the flux Φ is (weber) and the magnetic flux density B is given as: Φ 𝐴 B= Wb/m2 (Tesla). The magneto-motive force mmf is the ability of a coil to produce magnetic flux. The mmf unit is Amp-turn and is given by: mmf = NI (AT). The magnetic flux intensity H is the mmf per unit length along the path of the flux and is given by: 𝐻 = 𝑚𝑚𝑓 𝑙 AT/m, where l is the mean or average path length of the magnetic flux in meters. The relation between the mmf and the flux is governed by the system reluctance ℜ, such that mmf = ℜ φ, where the reluctance is given by ℜ = 𝜌𝑙 𝐴 , where l = The average length of the magnetic core (m) A = The cross section area (m2 ) μ = The permeability of the material (AT/m2 ) The permeability of the material is given by μ = μ0μr where: μ0 is the permeability of air and μr is the relative permeability. From the above relationships, we can conclude that: 𝑙 Φ 𝑚𝑚𝑓/𝑅 (𝐻𝑙)/(𝜇𝐴) 𝐵= = = = 𝜇𝐻 𝐴 𝐴 𝐴 The relation B-H is known as the magnetization characteristics of the material and is broken to three different regions: Linear, knee and saturation as shown in Fig. 2. Fig. 2 Magnetization Curve 2. Analogy between magnetic and electric circuits 3. Magnetic Circuit Analysis: In order to analyze any magnetic circuit, two steps are mandatory as illustrated by Fig. 3: • Step #1: Find the electric equivalent circuit that represents the magnetic circuit. • Step #2: Analyze the electric circuit to solve for the magnetic circuit quantities. Fig. 3 Magnetic circuit analysis Inductances: The inductance (in Henry) is given by: 𝐿= 𝜆 𝑁∅ = 𝐼 𝐼 Since: 𝐹 = 𝑁𝐼 = ∅𝑅, and ℜ = 𝜌𝑙 𝐴 Therefore, 𝜆 𝑁∅ 𝑁 2 𝑙𝜇𝐴 𝑁 2 𝐿= = = = 𝐼 𝐼 𝐼𝑙 ℜ 𝐿11 = 𝜆11 = 𝑠𝑒𝑙𝑓 𝑖𝑛𝑑𝑢𝑐𝑡𝑎𝑛𝑐𝑒 𝐼1 𝐿21 = 𝜆21 = 𝑚𝑢𝑡𝑢𝑎𝑙 𝑖𝑛𝑑𝑢𝑐𝑡𝑎𝑛𝑐𝑒 𝐼1 where: 𝜆11 is the total flux linking coil 1 due to 𝐼1 when 𝐼2 and 𝜆21 is the total flux linking coil 2 due to 𝐼1 when 𝐼2 =0 𝐿11 𝑁1 2 = , 𝑅 𝐿22 𝑁2 2 = , 𝑅 𝐿21 = 𝑁1 𝑁2 = 𝐿12 𝑅 1.1.2 Lecture Topic Review of Ampere Law and Biot Savart Law: Web link: https://ciet.nic.in/moocspdf/Physics03/leph_10403_eContent2020.pdf https://ciet.nic.in/moocspdf/Physics03/leph_10402_eContent2020.pdf Video link: https://www.youtube.com/watch?v=JHNloU9Rfow https://www.youtube.com/watch?v=1kydon2HxQA You will recall that electric fields and magnetic fields might seem different, but they're actually part of one larger force called the electromagnetic force. Charges that aren't moving produce electric fields. But when those charges do move, they instead create magnetic fields. Charges moving in an electric wire also produce magnetic fields. If we move a compass near to an electric wire, the compass needle changes direction or deflects. The Biot-Savart Law is a mathematical expression that describes the magnetic field created by a current carrying wire, and allows you to calculate its strength at various points. To derive this law, we first take this equation for electric field. This is the full version, where we use µ0/4π instead of the electrostatic constant k. Since we're looking at a wire, we replace the charge q with Idl, which is the current in the wire, multiplied by a length element in the wire. Basically, it's treating this little chunk of the wire as our charge. And we also replace the electric field E with a magnetic field element dB because a moving charge produces a magnetic field, not an electric field. Last of all, we have to realize that a current has a direction (unlike a charge). So, we need to make sure the direction of the current affects our result. We do that by adding sine of the angle between the current and the radius. That way, if the wire is curvy, we'll take that into account. And that's it - that's the Biot-Savart law. Fig. 3 Biot-Savart law The magnetic field dB due to this element is to be determined at a point P which is at a distance r from it. Let θ be the angle between dl and the displacement vector r. According to Biot-Savart’s law, the magnitude of the magnetic field dB is proportional to the current I, the element length |dl|, and inversely proportional to the square of the distance r. Its direction is perpendicular to the plane containing dl and r. Thus, in vector notation: 𝑑𝐵 = 𝑑𝐵 = 𝐼𝑑𝑙 × 𝑟 𝑟3 𝜇0 𝐼𝑑𝑙 × 𝑟 4𝜋 𝑟 3 Direction of the field is given by Right hand grip rule. AMPERE'S CIRCUITAL LAW: This law, given by Ampere, provides us with an alternative way of calculating the magnetic field due to a given current distribution. This law, is, in a way, similar to the Gauss’s law in electrostatics, which again provides us with an alternative way of calculating the electric field due to a given charge distribution. Ampere’s circuital law can be written as: The line integral of the magnetic field around some closed loop is equal to μ0 times the algebraic sum of the currents which pass through the loop. So, let us attempt to understand what is meant by: line integral closed loop algebraic sum of currents Fig. 4 Magnetic field around a current carrying conductor The figure shows the magnetic field around a current carrying conductor. From our previous knowledge we take the conventional direction of current (from +ve to –ve), the red concentric circles represent the magnetic field in a plane perpendicular to the wire. From Biot Savart’s Law, B is inversely proportional to the square of the distance from the source to the point of interest. The source of the field is a vector given by Idl or the magnetic field is produced by a vector source Idl. The magnetic field is perpendicular to the plane containing the displacement vector r and the current element Idl. Amperes’ circuital law makes the calculation of B easier in many cases: Consider a long straight current carrying wire encircled by magnetic field lines and imagine travelling around a closed path that also encircles the wire. Ampere’s law relates the magnetic field along the path to the electric current enclosed by this path. Let us travel along the path taking steps of length ∆l and let B be the component of the magnetic field parallel to these steps. According to ampere’s law over the entire closed loop, which we have taken as a circular loop. ∑ 𝐵. Δ𝐼 = 𝜇𝑜 𝐼 1.1.3 Lecture Topic Visualization of magnetic fields produced by a bar magnet and a current carrying coil - through air and through a combination of iron and air Web link: http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/elemag.html#c1 Video link: https://www.youtube.com/watch?v=nfSJ62mzKyY Bar Magnet The lines of magnetic field from a bar magnet form closed lines. By convention, the field direction is taken to be outward from the North pole and in to the South pole of the magnet. Permanent magnets can be made from ferromagnetic materials. As can be visualized with the magnetic field lines, the magnetic field is strongest inside the magnetic material. The strongest external magnetic fields are near the poles. A magnetic north pole will attract the south pole of another magnet, and repel a north pole. As can be visualized with the magnetic field lines, the magnetic field is strongest inside the magnetic material. The strongest external magnetic fields are near the poles. A magnetic north pole will attract the south pole of another magnet, and repel a north pole. Fig. 6 Magnetic field of a bar magnet The magnetic field lines of a bar magnet can be traced out with the use of a compass. The needle of a compass is itself a permanent magnet and the north indicator of the compass is a magnetic north pole. The north pole of a magnet will tend to line up with the magnetic field, so a suspended compass needle will rotate until it lines up with the magnetic field. Unlike magnetic poles attract, so the north indicator of the compass will point toward the south pole of a magnet. In response to the Earth's magnetic field, the compass will point toward the geographic North Pole of the Earth because it is in fact a magnetic south pole. The magnetic field lines of the Earth enter the Earth near the geographic North Pole. Fig. 7 Magnetic field using compass Electric and Magnetic Sources The electric field of a point charge is radially outward from a positive charge. Electric sources are inherently "monopole" or point charge sources. The magnetic field of a bar magnet. Magnetic sources are inherently dipole sources - you can't isolate North or South "monopoles". Bar Magnet and Solenoid The magnetic field produced by electric current in a solenoid coil is similar to that of a bar magnet. The magnetic field lines can be thought of as a map representing the magnetic influence of the source object in the space surrounding it. The properties of the magnetic field lines are can be summarized by: 1. The direction of the magnetic field is tangent to the magnetic field line at any point in space. 2. The strength of the magnetic field is visualized by the closeness of the lines to each other. It is proportional to the number of lines per unit area perpendicular to the lines. a commonly used phrase is "magnetic flux density". 3. Magnetic field lines never cross. The magnetic field at any point is unique. 4. Magnetic field lines are continuous, forming closed loops without beginning or end. Iron Core Solenoid An iron core has the effect of multiplying greatly the magnetic field of a solenoid compared to the air core solenoid on the left. Electromagnet: Electromagnets are usually in the form of iron core solenoids. The ferromagnetic property of the iron core causes the internal magnetic domains of the iron to line up with the smaller driving magnetic field produced by the current in the solenoid. The effect is the multiplication of the magnetic field by factors of tens to even thousands. The solenoid field relationship is 𝐵 = 𝑘𝜇0 𝑛𝐼 𝑤ℎ𝑒𝑟𝑒 𝜇 = 𝑘𝜇0 and k is the relative permeability of the iron, shows the magnifying effect of the iron core. 1.1.4 Lecture Topic Influence of highly permeable materials on the magnetic flux lines Web link: http://info.ee.surrey.ac.uk/Workshop/advice/coils/mu/ Video link: https://study.com/academy/lesson/what-is-magnetic-permeability-definitionexamples.html Permeability Quantity name permeability alias absolute permeability Quantity symbol μ Unit name henrys per metre Unit symbols H m-1 Base units kg m s-2 A-2 Duality with the Electric World Quantity Unit Formula Permeability henrys per metre μ = L/d Permittivity farads per metre ε = C/d Although, magnetic permeability is related in physical terms most closely to electric permittivity, it is probably easier to think of permeability as representing 'conductivity for magnetic flux'; just as those materials with high electrical conductivity let electric current through easily so materials with high permeabilities allow magnetic flux through more easily than others. Materials with high permeabilities include iron and the other ferromagnetic materials. Most plastics, wood, non-ferrous metals, air and other fluids have permeabilities very much lower: μ0. Just as electrical conductivity is defined as the ratio of the current density to the electric field strength, so the magnetic permeability, μ, of a particular material is defined as the ratio of flux density to magnetic field strength μ=B/H This information is most easily obtained from the magnetization curve. Figure shows the permeability (in black) derived from the magnetization curve (in color) using above equation. Note carefully that permeability so defined is not the same as the slope of a tangent to the B-H curve except at the peak (around 80 A m-1 in this case). The latter is called differential permeability, μ′ = dB/dH. In ferromagnetic materials the hysteresis phenomenon means that if the field strength is increasing then the flux density is less than when the field strength is decreasing. This means that the permeability must also be lower during 'charge up' than it is during 'relaxation', even for the same value of H. In the extreme case of a permanent magnet the permeability within it will be negative. There is an analogy here with electric cells, since they may be said to have 'negative resistance'. If you use a core with a high value of permeability then fewer turns will be required to produce a coil with a given value of inductance. For a given core B is proportional to flux and H is proportional to the current so that inductance is also proportional to μ: the ratio of B to H. Unlike electrical conductivity, permeability is often a highly non-linear quantity. Most coil design formula, however, pretend that μ is a linear quantity. If you were working at a peak value of H of 100 A m-1, for example, then you might take an average value for μ of about 0.006 H m-1. This is all very approximate, but you must accept inaccuracy if you insist on treating a non-linear quantity as though it was actually linear. This form of permeability, where μ is written without a subscript, is known in SI parlance as absolute permeability. It is seldom quoted in engineering texts. Instead a variant is used called relative permeability described next. μ = μ0 × μr Relative permeability Quantity name Relative permeability Quantity symbol μr Unit symbols dimensionless Relative permeability is a very frequently used parameter. It is a variation upon 'straight' or absolute permeability, μ, but is more useful to you because it makes clearer how the presence of a particular material affects the relationship between flux density and field strength. The term 'relative' arises because this permeability is defined in relation to the permeability of a vacuum, μ0 μr = μ / μ0 For example, if you use a material for which μr = 3 then you know that the flux density will be three times as great as it would be if we just applied the same field strength to a vacuum. This is simply a more user-friendly way of saying that μ -6 -1 = 3.77×10 H m . Note that because μr is a dimensionless ratio that there are no units associated with it. Many authors simply say "permeability" and leave you to infer that they mean relative permeability. In the CGS system of units these are one and the same thing really. If a figure greater than 1.0 is quoted then you can be almost certain it is μ r. Material μ/(H m-1) μr Application Ferrite U 60 1.00E-05 8 UHF chokes Ferrite M33 9.42E-04 750 Resonant circuit RM cores Nickel (99% pure) 7.54E-04 600 - Ferrite N41 3.77E-03 3000 Power circuits Iron (99.8% pure) 6.28E-03 5000 - Ferrite T38 1.26E-02 10000 Broadband transformers Silicon GO steel 5.03E-02 40000 Dynamos, mains transformers superalloy 1.26 1000000 Recording heads Approximate maximum permeabilities Note that, unlike μ0, μr is not constant and changes with flux density. Also, if the temperature is increased from, say, 20 to 80 centigrade then a typical ferrite can suffer a 25% drop in permeability. This is a big problem in high-Q tuned circuits. Another factor, with steel cores especially, is the microstructure, in particular grain orientation. Silicon steel sheet is often made by cold rolling to orient the grains along the laminations (rather than allowing them to lie randomly) giving increased μ. We call such material anisotropic. Before you pull any value of μ from a data sheet ask yourself if it is appropriate for your material under the actual conditions under which you use it. Finally, if you do not know the permeability of your core then build a simple circuit to measure it.