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Passive components and circuits - CCP Lecture 10 1/36 Content Coils Short history Electrical properties Constructive elements of a coil Parameters Categories Transformers 2/36 The coil – history 1821 – Michael Faraday highlights the magnetic field lines that appear around a conductor through electric current flows. 1825 – William Sturgeon builds the first electromagnet 1831 – independently, Michael Faraday and Joseph Henry discover the law of the magnetic induction Faraday is the one who built the first electric engine, the first electric generator and the first transformer. Henry is the one who built the first telegraph then improved by Morse 1876 – Bell invents the first telephone and electromagnetic phonograf 3/36 Electrical properties The inductance depends on the geometry of the coil and on the magnetic properties of the media where the coil is placed. Formula (1) is valid for a length l of the coil greater than its diameter 2rc. Formula (2) is valid for a coil of length l smaller than its diameter 2rc. rw represents the diameter of the reeling string. ( 0 N 2 rc ) L Henries l 2 L 0 N 2 rc {ln( 8rc ) 2} rw (1) (2) 0 4 10 7 [Wb A1 m 1 ] 4/36 Electrical properties The inductance depends on the geometry of the coil (l, d=2r, h in mm). The formulas are valid for air. L 0,001 N 2d l 0,44 d [μH] N 2d 2 L 0,008 [μH] 3d 9l 10h 5/36 Electrical properties The inductance is dependent on the distance between the whirls (turns). The inductance is dependent on the magnetic property of the media in which the coil is placed, property characterized by the magnetic permeability, . air 1.257x10-6 H/m ferrite U M33 9.42x10-4 H/m nickel 7.54x10-4 H/m iron 6.28x10-3 H/m ferrite T38 1.26x10-2 H/m steell 5.03x10-2 H/m supermalloy 1.26 H/m 6/36 The coil – equivalent circuit ZL R p jL 1 2LC p jR pCp 7/36 The coil – the frequency characteristic ZL Rp Q L Inductive area 10% 1 C Rp 10% 2 ,2 R p L 0 ,3 0 0 1 LC p 8/36 Dimensioning the inductance when the wires are remoted km l 8 6 D 4 2 p/d 0 p d (ND) 2 L0 107 [H] l 0,45 D 1 2 3 4 7 L k m N D 10 [H] L1 L0 L 9/36 Calculus of the parasitic capacitance Cp[pF] 30 20 10 7 5 D=10cm 3 D=8cm 2 D=6cm D=4cm 1 D=2cm 0,7 0,5 p/d 1 1,1 1,3 1,5 1,7 2 2,5 3 3,5 4 10/36 Designing steps for a coil It starts from the value of the needed inductance L, its diameter D and from the domain in which it is going to be used. From these it can be deduced the maximum value for Cp. Then, calculate the number of wires depending on the geometric dimensions of the coil by solving the equation on the right. L, 0 , D C p max 1 L02 p / d km N 2D L[μH] 0,1k m ND dp 0,44 N 1 Dd Dimension the length of a coil with diameter of 2cm and the inductivity of 50 H that is executed in one layer and for which it is desired a parasite capacitance lower that 2 pF. 11/36 Constructive elements of a coil The coil turns (whirls) The body The impregnating material The core No core Iron core Ferrite core 12/36 Coil turns The most frequently used material for reeling conductors is cooper (due to its electrical and mechanical properties) but the aluminum is also used. The used conductors are isolated in order to avoid shortcircuits between the adjoined turns. The materials used for isolation are enamels (lacquers of different compositions), fabrics (silk, cotton) or inorganic fibers (glass fiber). The type of isolating material is chosen depending on the reached estimated temperature of the conductor. The materials with the lower thermal resistance are the textiles and the materials with the higher thermal resistance are the glass fibers. 13/36 Coil turns The diameter of the conductor is chosen depending on two criteria: The intensity of the current that flows through the conductor, gives the inferior limit of the diameter in order to avoid overheating. The maximum value accepted for the resistance of the coil (parasite parameter) can furthermore limit the dimension of the diameter. At high frequencies, due to the pelicular effect, stranded wired conductors or silvered cooper conductors are used. The conductors for reeling are delivered by producers with diameters of standardized dimensions: 0,05mm, 0,07mm, 0,1mm, ... 2mm. These diameters do not include the thickness of the isolator layer. 14/36 The carcass of the coil Its role is to insure the stiffening of the reeling (and through this keep the electric properties of the coil ). The materials used must have adequate properties both electrical (dielectric stiffness, small dielectric loses ) and mechanical (thermal stability and resistance to the action of humidity). Examples in increasing order of performances: cardboard, electro isolating cardboard, pertinax, textolite, thermorigide materials (bakelite), thermoplastic materials ( polystyrene, polyethylene, teflon), ceramic materials. Constructively, they can have different sections: circular, square, rectangular, with or without flanges. At very high frequencies, the coils can be manufactured without a body (carcass). 15/36 The impregnating material It has the role of increasing protection against humidity and a role in extra stiffening (especially when the coils are not placed in carcasses). Advantages of impregnating: Stiffening of the wires; Improves the heat dissipation; Improves the electrical properties of the isolation between the wires; Avoids the humidity access between the wires; Disadvantages of impregnating: can lead to the increase of the parasite capacities (by increasing the relative permeability of the dielectric between the wires). 16/36 The core of the coil For increasing the usual inductance, magnetic cores are introduced in the coil. They make up a magnetic circuit (sometimes with interruptions) that has the purpose of concentrating the magnetic field lines. In this way, the magnetic flux increases, most of the lines intersecting the surface of the wires, and so the inductance of the coil increases too. Magnetic materials have a nonlinear behavior when placed into an exterior magnetic field. This nonlinearity refers to the dependence of the magnetic induction B on the intensity of the magnetic field H. The ratio between the two represents the magnetic permeability of that media: B 1 B ; r H 0 H 17/36 The properties of the magnetic materials – the histerezis phenomena Hc – coercive field, anules the magnetic induction; Br – remanent magnetic induction; Hs – the intensity of the magnetic field to which the saturation phenomena appear; Bs – the magnetic induction at saturation . B Bs Bm Br -Hs -Hm 0 -Hc H c Hm H s H -Br -Bm -Bs 18/36 The properties of the magnetic materials – the histerezis phenomena The magnetic materials have atoms with their own magnetic moments, and neighboring atomic moments are oriented identically, the material presenting a remanent magnetization. When applying an exterior field, the magnetic domains are reoriented. The intensity of the exterior field at which the magnetic inductance is zero is called coercive field. When H increases at a certain moment, there appears a saturation phenomena (B no longer modifies). The phenomena are dependent on the sense in which the magnetic field modifies (histerezis). The remanent magnetization manifests itself up to a certain temperature (Curie temperature) at which the thermal agitation destroys the domains of ordered orientation. 19/36 The use of magnetic materials Clasification: Soft magnetic materials – Hc<80 A/m (narrow histeresis ) Rough magnetic naterials – Hc>80 A/m (broad histeresis) The soft magnetic materials with the ratio: Br/Bm (ratio that characterizes the histeresis inclination) <0,5 for inductances approximately constant ; 0,5<Br/Bm<0,8 for common cores; Br/Bm>0,8 (rectangular histeresis ) in memorization and commutation circuits. The rough magnetic materials with the ratio: Br/Bm<0,4 are used for the magnetic recording of the information; Br/Bm>0,4 for manufacturing permanent magnets. 20/36 Constructive shapes for cores Sheet plates, bands, cloaks for manufacturing the magnetic circuit for transformers; Cylindrical bars for inductances used in high frequencies (sometimes they are adjustable); Toroidal coils and pot cores used in high frequency and pulses; Yokes of different shapes in circuits of magnetic deflection; The cores for high frequencies are obtained by pressing magnetic powders. This is how are obtained magnetoelectric cores (powder is a ferromagnetic material), but also the magneto-ceramic ones (also called ferrites). 21/36 Dimensioning coils with core If a coil without a core has the inductance L0 by introducing a core it becomes: L ef L0 The actual permeability, ef is dependent on the relative permeability of the material, on its geometry and on its position relative to the reeling. The ferrites manufacturers indicate, in catalogs, an inductance factor for them, AL meaning the inductance that is obtained if a single wire is made on the ferrite (in nH/wire or H/wire). Using this parameter, the total inductance is obtained with the formula: L AL N 2 22/36 Parameters of coils Its inductance and tolerance The parasitic resistance The loss angle tangent The quality factor The temperature coefficient tg L QL vRL vL RL L L RL 1 dL L L dT 23/36 A few categories of coils Toroidal (A) Cylindrical (B) Incapsulated (C) Adjustable (D,E) Color Digit Multiplier Black Brown Red Orange Yellow Green Blue Purple Gray White None Silver Gold 0 1 2 3 4 5 6 7 8 9 1 10 100 1000 . Tolerance Code of colors for encapsulated coils ±20% ±10% ±5% 24/36 Transformers This component consists of two coils manufactured on the same magnetic support (core), like iron. The magnetic core couples the magnetic flux, B, between the two coils. In accordance with Faraday’s induction law: d B VP N P dt d B VS N S dt VS NS VP N P The ecuation of the transformer NS NP Step-up transformer NS NP Step-down transformer 25/36 The ideal transformer An ideal transformer has no loses, therefore: The input power = The output power VP I P VS I S I S VP N P I P VS NS The real transformers that are well manufactured can have the efficiency over 99%. A transformer achieves its function only if the voltage/current varies through one of the wires. This will generate a variable voltage in the second wire. 26/36 The mutual inductance The variation in time of the current in circuit 1 causes an induced voltage in circuit 2, v2. The curent through circuit 2 apares only if it is closed on a load. Let coil 1 have N1 wires and coil 2 with N2 wires. Φ21 = the magnetic flux in coil 2 due to the current i1 from coil 1 N221 I1 M 21 N 2 21 i1 N221 (constant) i1 M 21i1 Mutual inductance Unit = Henry 1 H = V.s/A = Ω.s 27/36 The mutual inductance The induced voltage in coil 2 can be expressed: d 21 v2 N 2 dt M 21 21 I1 d 21 M 21 dI1 N2 dt N 2 dt M 21 di1 di1 v2 N 2 M 21 N 2 dt dt 28/36 The mutual inductance Similary, it can be proved that the induced voltage in coil 1 by the variation of the current in coil 2 is: v1 M12 di2 dt It can be proved that: M21 = M12 = M=k2LPLS In an ideal case, k, the coupling factor, is 1. di2 v1 M dt di1 v2 M dt 29/36 The transformer – circuit functioning Ro vo ip vp vs Primar Secundar NP LP NS LS is Rl As the symbol shows, the transformer has two coils. The one in the circuit where the voltage source is, is called primary (Ls), and the one found in the other circuit where the load RL is called secondary (LL). Each inductance functions in the circuit where it is placed in accordance with the properties studied, and moreover they are coupled by the mutual inductance, M. 30/36 The transformer – circuit functioning Ro vo ip vp vs Primar NP LP is Rl Secundar NS LS The voltage on the primary coil will be: v p jLpi p jMis The voltage on the secondary coil will be: vs jLsis jMi p 31/36 The transformer – circuit functioning Ro vo ip vp vs Primar NP LP is Rl Secundar NS LS The sum of the voltages from the primary circuit obeys TKV: vo Roi p jLpi p jMis The sum of the voltages from the secondary circuit obeys TKV: 0 R is jLsis jMi p 32/36 The transformer – circuit functioning Ro vo ip vp vs Primar Secundar NP LP NS LS Taking into consideration that: It is obtained: LP N P LS N S is Rl 2 2 N vo Ro ip jLP || Rl S ip NP 33/36 Transformers – constructive variants Cylindrical (solenoidal) Toroidal Yoke 34/36 Basic formulas for dimensioning the studied components l l R Cu r S S Cu=5,344 x 10-7 -cm A C 0 r d 0=8,8542·10-12 F/m A L 0 r N 2 l 0=4·π·10-7 H/m 35/36 Problems Using a 1mm diameter Copper wire (=5,344 x 10-7 -cm) 40 turns are made on a cylindrical insulating substrate, with 10 mm diameter. Determine the electrical parameters of the coil. How is the value of modulus of coil impedance at 50 Hz and 500KHz frequencies? 36/36