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## Lecture 10

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**Lecture 10**Coils (Inductors) Passive Electronic Components and Circuits (PECC)**Coils**Coils Short history Electrical Properties Constructive elements for coils Parameters Categories Transformers**Short history**• 1821 – Michael Faraday reveals the magnetic field lines which occur around a conductive material through which an electric current flows. • 1825 – William Sturgeon builds the very first electromagnet. • 1831 – independently, Michael Faraday and Joseph Henry discover the magnetic induction law. • Afterwards, Faraday builds the first electrical engine, the first electrical generator and the first transformer. • Henry builds the first telegraph improved later by Morse. • 1876 – Bell invents the first telephone and the first electromagnetic phonograph.**Coils**Coils Short history Electrical Properties Constructive elements for coils Parameters Categories Transformers**Electrical properties**• The inductance of a coil is strongly dependent of the coil’s geometry and the magnetic properties of the environment in which it is placed. • Equation 1 is suitable for the case in which the length “l” of the coil is higher then its diameter “2rc”. • Equation 2 is suitable for the case in which the length “l” of the coil is lower then its diameter “2rc”. The “rw” quantity represents the winding wire’s diameter.**Electrical properties**• The inductance is dependent of the coil’s geometry (l, d=2r, h in mm). All the above formulas are available when the environment in which the coil is placed is the air.**Electrical properties**• The inductance is dependent of the distance between its turns. • The inductance is dependent of the magnetic properties of the environment in which the coil is placed – magnetic permeability, μ. • Air: 1.257x10-6H/m. • Ferrite U M33: 9.42x10-4H/m. • Nickel: 7.54x10-4H/m. • Iron: 6.28x10-3H/m. • Ferrite T38: 1.26x10-2H/m. • Orr: 5.03x10-2H/m. • Super Malloy: 1.26H/m.**Electrical properties**• Equivalent circuit**Electrical properties**• Frequency characteristic**Electrical properties**• Dimensioning the inductance:**Electrical properties**• The parasitic capacitance value calculus:**Electrical properties**• Steps in designing a coil: • The procedure starts from the desired value for the inductance – “L”, its diameter “D” and the frequency domain in which it will work – “ω0”. • From the last slide characteristic the maximum value of the parasitic capacitance is chosen. • The turns number can be calculated in respect with the coil’s geometrical dimensions by resolving the adjacent equation. Calculate the length of a coil with a 2 cm diameter and an inductance of 50 H which is executed in a single layer and will have a desirable parasitic capacitance lower then 2pF.**Coils**Coils Short history Electrical Properties Constructive elements for coils Parameters Categories Transformers**Constructive elements for coils**• The winding (the turns). • The casing. • The impregnation (soak) material. • The core. Iron core Ferrite core without a core**Constructive elements for coils**• The coil’s winding • The most common used material for the conductive winding wire is the copper (due to its electrical and mechanical properties) and rarely aluminum. • The conductive wires are being insulated to avoid short-circuits between two adjacent turns. The materials used for the insulation are enamels (ro: emailuri) – different composition varnishes, textile fibers (silk, cotton) or mineralfibers (glass fiber). The insulating material type is generally chosen in strong dependency with the conductive wire’s estimated maximum temperature. The most thermal resistant materials are the glass fibers, the most susceptible ones are the textile fibers.**Constructive elements for coils**• The coil’s winding • The coil’s winding diameter can be estimated following 2 criteria: • The maximum estimated value of the current that passes through the conductive materials is inferiorly limiting this parameter due to excessive heating possibility. • The parasitic resistance maximum value introduces a supplementary limitation for the windings diameter. • At high frequencies, due to the pellicle effect, stranded (ro: litat) wires are used (thin bundles of wires – ro: manunchiuri de fire) or silvered copper wires. • The conductive winding wires are being delivered by the producers in a standardized fashion: 0.05 mm, 0.07 mm, 0.1 mm, … , 2 mm. The thickness of the insulating material is not included in the above values.**Constructive elements for coils**• The coil’s casing • The coil’s casing fulfill the role of keeping the stiffness of the winding. • The materials used must have adequate electrical (dielectric rigidity, low dielectric losses) and mechanical properties (thermal and humidity stability). • Examples – ascending order of theirs performances: electro-insulating carton, pertinax, textolit, thermo-rigid materials (Bakelite), thermo-plastic materials (polystyrene, polyethylene, Teflon) or ceramic materials. • From the geometrically point of view the material can be of different cross-sections: circular, square, rectangular. • At very high frequencies, the coils can be made without a casing.**Constructive elements for coils**• The coil’s impregnation (soak) material • The impregnation material has the role to protect the coil against humidity and also realizes a supplementary stiffness (especially for the cases when the coils are not using casings). • The impregnation advantages: • Winding stiffness. • Improves heat dissipation. • Improves dielectric properties of the turns insulation. • Avoids humidity penetration between turns. • The impregnation disadvantages: can concur to a higher parasitic capacitance (by growing the electrical permittivity of the insulating material between the turns).**Constructive elements for coils**• The coil’s core • To increase the obtained inductance, magnetic cores are being displaced inside coil’s winding. So a magnetic circuit is created which has the major contribution in concentrating the magnetic field lines. In this way the magnetic flux increases, almost all the magnetic lines intersect the turns, in conclusion the coil’s inductance increases. • The magnetic materials have a non-linear behavior when are being placed in an exterior magnetic field. This non-linearity is related with the dependency of the magnetic induction “b” and the magnetic field intensity “H”. The ratio between the above two quantities is the environment's magnetic permeability:**Constructive elements for coils**• The magnetic materials properties – Hysteresis phenomena • Hc – coercive field – cancels the magnetic induction. • Br – residual magnetic induction. • Hs – the magnetic field intensity at which saturation phenomena occurs. • Bs – the magnetic induction at saturation point.**Constructive elements for coils**• The magnetic materials properties – Hysteresis phenomena • The magnetic materials have atoms with an own magnetic moment, and the neighbor atomic moments are being orientated identically, so the material will have a residual magnetization. • When applying an exterior magnetic field, a reorientation of the magnetic domains occurs. The exterior field intensity at which the magnetic induction is canceling is called coercive field. When “H” increases, the saturation phenomena occurs (“B” remains constant). • The phenomena are dependent with the direction on which the magnetic field changes (hysteresis). • The residual magnetization exhibits until a certain temperature (Curie temperature) at which the thermal agitation destroys the well-ordered orientation domains.**Constructive elements for coils**• Magnetic materials applications • Soft magnetic materials – Hc<80A/m (narrow hysteresis) • Rough magnetic materials – Hc>80A/m (wide hysteresis) • Soft magnetic materials with the ratio Br/Bm (ratio which characterizes the hysteresis inclination) lower then 0.5 are used for constant inductances, the ones with 0.5< Br/Bm<0.8 for common used cores, the ones with Br/Bm>0.8 (rectangular hysteresis) are used in switching and memorizing applications. • Rough magnetic materials with the ratio Br/Bm<0.4 are used for magnetically recording the information and the ones with Br/Bm>0.4 for permanent magnets fabrication.**Constructive elements for coils**• Core constructive types • Core-trays(ro: tole),bands, columns, coatings for transformers magnetic circuits. • Cylindrical bars for high frequencies inductances (also for adjustable inductances). • Torus (ro: miezuritoroidale) and pots (miezuri tip “oala”) for high frequency and pulses applications. • Different forms of yokes (ro: miezuri tip “jug”) • For high frequency applications, the cores are being obtained by compressing magnetic powders. That results in obtaining magneto-electric cores (the magnetic powder is a ferromagnetic material) or magneto-ceramic cores (ferrites).**Constructive elements for coils**• Designing a core coil • If a coil without a core has the “L0” inductance, the core displacements inside its windings changes its inductance: • The effective magnetic permeability, μeff, is dependent with the material’s relative permeability, with its geometry and with the relative position in respect with the winding. • The ferrite producers indicate in the datasheets a so called inductance factor, AL, having the following meaning: the inductance factor is the obtain inductance if on the ferrite core is made only 1 turn (nH/turn or μH/turn). Using this parameter , the total inductance can be obtained:**Coils**Coils Short history Electrical Properties Constructive elements for coils Parameters Categories Transformers**Parameters**• The inductance and its tolerance. • The coil’s own resistance. • The loss-angle tangent. • The quality factor. • The temperature coefficient.**Coils**Coils Short history Electrical Properties Constructive elements for coils Parameters Categories Transformers**Categories**• Constructive types • Toroidal (A). • Cylindrical (B). • Encapsulated (C). • Adjustable (D, E).**Categories**• Spiral circular plane coil**Categories**• Spiral square plane coil c – trace width**Coils**Coils Short history Electrical Properties Constructive elements for coils Parameters Categories Transformers**Transformers**• A transformer is an electrical component which consists on two coils mounted on the same magnetic core. • The magnetic core links the magnetic flux,ФB , from the two coils. • Taking into consideration the Faraday law: The transformer’s equation Voltage increasing transformers Voltage decreasing transformers**Transformers**• Ideal transformer • An ideal transformer doesn’t have losses, so: Input Power = Output Power Real transformers can reach an efficiency higher then 99% • A transformer makes its job only if the voltage/current varies through one of its windings. This variation will generate a variable flux which will lead to a variable voltage into the second winding.**Transformers**• Mutual inductance • The time variation of the current from the primary winding determines the occurrence of a induced voltage in the secondary winding. Obviously, the current through the second winding appears only if at the second windings terminals, a load is connected. • Let us consider the coil 1 with N1 – turns number and coil 2 with N2 – turns number and Ф21 – the magnetic flux in the second coil occurred due to the current that flows through the first coil: Unit – Henry 1H=Vs/A=Ωs Mutual inductance**Transformers**• Mutual inductance • The induced voltage in the second coil can be expressed:**Transformers**• Mutual inductance • Similar, it can be shown that the induced voltage into the first coil by the second coil’s current variation is: • But, generally M21=M12 and can be written as follows, where k – coupling factor: • In conclusion:**Transformers**• The transformer – circuit analysis • The transformer has two coils (as the symbol from the circuit above). The one from the circuit where the Vo source is applied is called primary coil and has the Lp inductance, and the one from the circuit where the Rl – load resistance is , is called secondary coil and has the Ls inductance. • Both inductances work on theirs designated circuit as was presented before, but supplementary the inductances are being magnetically coupled through the mutual inductance, M.**Transformers**• The transformer – circuit analysis • The voltage across the primary coil will be: • The voltage across the secondary coil will be:**Transformers**• The transformer – circuit analysis • Applying the KVL in the primary winding: • Applying the KVL in the secondary winding:**Transformers**• The transformer – circuit analysis • Extracting, from the second equation, the is current dependent of the ip current and introducing it in the first equation (KVL for the primary winding), and also taking into consideration that: • It can be obtained that:**Transformers**• The transformer – constructive types • Cylindrical (solenoidal) • Torus • Yoke**Important formulas**• Resistors • Coils • Capacitors Cu=5,344 x 10-7 -cm 0=8,8542·10-12 F/m 0=4·π·10-7 H/m**Problem**Using a copper wire (φCo = 5.344x10-7Ωm) with 1 mm diameter, a winding of 40 turns is being executed on insulated cylindrical support with the diameter of 10 mm. • Please determine the electrical parameters of the realized coil. • What is the modulus of the coil’s impedance at 50Hz? What is the modulus of the coil’s impedance at 500kHz?