ELECTRONIC DESIGN NOTES #5
Inductors, Transformers
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| In this page are presented inductors
and transformers together, because they are closely related. The structure employed for this task is: 1. Electromagnetic Field 2. Electromagnetic Induction 3. Equivalent series and parallel inductors 4. Current growth/decay in inductive circuits 5. Transformers 6. Power Factor in AC circuits NOTE The basic notions highlighted in this page are related to electronic design topics presented in the first part Hardware Design of Learn Hardware Firmware and Software Design. |
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In addition to creating an Electrostatic Field, electric current flowing through conductors creates an Electromagnetic Field, perfectly similar to the (mechanical) magnetic one plus few electrical characteristics. Specific to both the magnetic and electromagnetic fields is, the magnetic flux lines are distinct (discrete): they go out from the conventional North Pole, then they enter the conventional South Pole. In one particular point, we can measure the number of magnetic field lines (Φ) per unit area (A), and that results in Magnetic Field Density (also known as the Electromagnetic Induction): Magnetic Field Density: B [tesla] = Φ [weber] / A [m2] 1 T = 1 Wb/m2 = 104 G [gauss--this is an older notation] Each conductor develops an electromagnetic field when the current passes through it. However, in order to amplify the electromagnetic field we need to coil our conductor few times: the result is a solenoid having a number of turns (N), a length (L), and a coil area (A). Note that inside our solenoid (the core) there is air, but we could easily introduce various metallic alloys instead. Now, different core materials have the quality of increasing or decreasing magnetic field density based on their characteristic Magnetic Permeability (μ): Magnetic Permeability in vacuum ( and also in air, because it is almost the same) is: μo = 4*PI*10-7 [Wb/A*m] or μo = 4*PI*10-7 [T*m/A] Few formulas are of particular interest, therefore they are listed in the following table:
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Suppose we have a coil and it is wired in series with a amp-meter. That is all: there is no power source and nothing else. Next, we bring a magnet close to our coil, and we start moving it in and out the coil: we will immediately notice on our amp-meter that current is flowing in one sense or the other in our basic, rudimentary circuit! The voltage that appears inside the coil when it is subjected to crossing (moving) magnetic field lines is named Electromotive Force (Uemf), and it is produced by (Electro) Magnetic Induction. Faraday's Law of Induction The electromotive force induced in a conductor is equal to the rate of change of the magnetic flux through that conductor. Electromotive Force: Uemf = dΦ/dt (with Φ = B * A) dΦ/dt = magnetic flux lines variation in time For a closed loop circuit having only wires (one closed coil, or a ring) the above equation may be also written as: Electromotive Force for one coil: Uemf = -B * L * v B = Magnetic Flux Density L = length of the conductor v = perpendicular velocity across magnetic lines. If v is not perpendicular, and it comes at an (x) angle to perpendicular then: Electromotive Force for one coil: Uemf = -B*L*v*sin x If we have a coil with N turns, then the relation becomes: Electromotive Force for multiple turns: Uemf = -B * N * L * v Now, we have used a permanent magnet to induce power in our (one turn) coil, but we get exactly the same effect if we use another coil, generating an alternative electromagnetic field. Suppose we have two separate circuits: the first one is described above: it is just a ring or a one-turn closed coil; the second one is a coil powered by alternative current. The new Induced Electromotive Force is named Mutual Inductance in this case, and it is calculated with: Electromotive Force for Mutual Inductance: Uemf = -M * (di/dt) M is Mutual Inductance, a constant specific to the system of coils used. It is the only unknown in the above relation; we calculate it with: Mutual Inductance: M = -Uemf / (di/dt) M is expressed in [H] Henry; Uemf is [V]; di/dt is [A/s]. The tricky part is, if we have only the second coil (described above in the second case) it is capable to self-induce Uemf! In this case Self-Induction is: Self Induction: L = -Uemf / (di/dt) L is expressed in [H] Henry; Uemf is [V]; di/dt is [A/s]. The above relation defines the Inductance (L) of a coil as being an AC circuit element due to current variation in time. Practically, in continuous DC both the capacitor and the inductor do exactly nothing. Sorry, we need to clarify this: in continuous DC a capacitor creates an electrostatic field and an inductor creates an electromagnetic field, but they do not interfere with the DC circuit. However, when the DC current varies, or if we bring close another moving magnetic or electrostatic field, our DC circuit becomes an active/transitive AC one. NOTE Please discover the Impedance and Inductive Reactance formulas in previous Design Notes. |
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The equivalent of series inductors is calculated with: LT = Σ Li The equivalent of parallel inductors is calculated with: 1/LT = Σ 1/Li Calculation examples for three inductors:
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A inductor in series with a resistor forms a timing circuit, just as the capacitor does. Time constant T: T [s] = L [H] / R [Ω] In order to reach 100% current, it takes 5 time constants (L/R) calculated with the above formula. The decay curve behaves perfectly similar to the growth one, having only an inverse second derivate (the curve holds water). Fig 3: Rising current curve |
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The beauty with AC is, we can easily step up or down the voltage, and in order to do that we use transformers. A transformer is an electric device having two coils placed on a closed loop metallic core. Suppose we have a transformer with a primary coil having Np number of turns, and a secondary coil with Ns number of turns. Our transformer uses Up as primary voltage and it outputs Us in the secondary coil. We work with the following set of equation: Transformer's Formulas: Us / Up = Ns / Np which is exactly the same as: Us * Np = Up * Ns In the same time we have: Us / Up = Ip / Is which is, again, exactly the same as: Us * Is = Up * Ip Now, because Up = -Np * dΦ/dt then Us = -Ns * dΦ/dt With the above set of formulas we can calculate the dimensions of the metallic core. This should be all you need to work successfully with transformers. However, the manufacturers of metallic cores have Data Sheets specific to their products, therefore you do need to consult them. |
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| Power in AC circuits is: P =
I2 * Z which may be: Apparent Power: PA [VA] = √[PR2 + (PXL - PXC)2] True Power: PR [W] = PA * cos(x) Reactive Power: PX [VAR] = PA * sin(x) = PXL - PXC PA is Apparent Power in [V*A] and you can see in Fig 4 where it comes from: it is the red vector PR is True Power in [W] given by the (pure) resistance of the AC circuit PXC is Reactive Capacitive Power in [V*Ar], and it is lagging PR by (-PI/2) PXL is Reactive Inductive Power in [V*Ar]; it is ahead of PR by (+PI/2) PX is the difference PXL-PXC. It is mandatory that Reactive Inductive Power is greater than Reactive Capacitive Power, due to the issues I presented in Design Notes 2 cos(x) is Power Factor (Pf) VAR is read as "Volt Ampere Reactive" and it is used to mark Inductive and Capacitive reactive currents (only the current is considered as being reactive) Power factor: Pf = cos(x) = PR/ PA = True Power / Apparent Power NOTE Because they are opposed by PI (180 degrees), Inductive and Capacitive reactances compensate each other. However, it is better to have slightly inductive AC circuits rather than capacitive ones, because capacitive reactance is very dangerous. Of course, too much inductive reactance is not good, therefore we employ capacitors to control it. |
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