Inductancia inductor inductancia

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Julián Ávila Rivas
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1 INDUCTANCIA

2 Inductancia El inductor es un elemento de un circuito que guarda energía en el campo magnético que rodea a sus alambres portadores de corriente. Del mismo modo que un capacitor guarda dicha energía en el campo eléctrico formado entre sus placas cargadas. El inductor se caracteriza por su inductancia, la cual depende de la forma de dicho inductor.

3 Inductancia Inductancia L: ε L = L di dt

4 Inductancia We used a coil and the solenoid assumption to introduce the inductance. But the definition L L E di dt holds for all types of inductance, including a straight wire. Any conductor has capacitance and inductance. An inductor is usually made of a coil to make a large inductance (more loops = more flux). The circuit symbol is The self-induced emf through this inductor under a changing current I is given by: di EL = L dt

5 Unidades de la inductancia The SI unit for inductance is the henry (H) V s 1H = 1 A Named for Joseph Henry: American physicist First director of the Smithsonian Improved design of electromagnet Constructed one of the first motors Discovered self-inductance

6 Inductancia V b > V a V b < V a V b V a = L di dt

7 Cálculo de la inductancia L = NΦ i B NΦ B conexiones de flujo Por ser Φ B proporcional a la corriente i, la razón de dicha ecuación no depende de i y, por consiguiente, la inductancia (como la capacitancia) depende sólo de la forma del dispositivo.

8 Cálculo de la inductancia de un solenoide When a current flows through a coil, there is magnetic field established. If we take the solenoid assumption for the coil: B = µ ni When this magnetic field flux changes, it induces an emf, E L, called self-induction: Φ d ( NAB) d ( NAµ ni ) E 0 d di di µ L dt dt dt dt dt B 0 L = = = = 0n V or: E L di L dt For a solenoid: This defines the inductance L, which is a constant related only to the coil. The self-induced emf ε L is generated by (changing) current in the coil. According to Lenz s Law, the emf generated inside this coil is always opposing the change of the current which is delivered by the original emfε. L = µ E 0n V I + E L Where n: # of turns per unit length. N: # of turns in length l. A: cross section area V: Volume for length l.

9 La inductancia de un toroide Recordemos

10 Magnetic Field of a Toroid The toroid has N turns of wire Find the field at a point at distance r from the center of the toroid (loop 1) B r d s r = B( πr ) = µ on I µ on I B = πr There is no field outside the coil (see loop )

11 La inductancia de un toroide

12 Inductores con materiales magnéticos Recordemos

13 Magnetización La permeabilidad de la mayor parte de los materiales comunes (excepto los ferromagnéticos) tiene valores cercanos a 1. Con respecto a otros materiales que no son ferromagnéticos, la permeabilidad puede depender de propiedades como la temperatura y la densidad del material, pero no del campo B 0. Para los ferromagnéticosκ m depende del campo aplicado B 0. r B = κ m r B 0

14 Put inductor L to use: the RL Circuit An RL circuit contains a resistor R and an inductor L. There are two cases as in a RC circuit (charging and discharging) but in an RL circuit one changes current, not electric charge. Current increases: When S is connected to position a and when switch S 1 is closed (at time t = 0), the current through R and L begins to increase Current decreases: When S is connected to position b.

15 RL Circuit Applying Kirchhoff s loop rule to the circuit in the clockwise direction gives d I ε I R L = dt Here because the current is increasing, the induced emf has a direction that should oppose this increase. Solve for the current I, with initial condition that I(t=0) = 0, we find ε ε I = e e R R ( ) ( τ 1 Rt L t 1 ) Where the time constant is defined as: 0 τ L R Constante de tiempo inductiva

17 RL Circuit When switch S is moved to position b, the original current disappears. The self-induced emf will try to prevent that change, and this determines the emf direction (Lenz Law). IR di + L dt = 0 Solve for the current I, with initial condition that I t = = E we find ε R ( ) 0 R ε R Rt L t I = e e τ

19 Energy stored in an inductor The increasing current I from the battery supplies power not only to the resistor, but also to the inductor. From Kirchhoff s loop rule, we have ε = I R + L d I dt Multiply both sides with I: d I εi = I R + LI dt This equation reads: power battery =power R +power L So we have the rate of energy increase in the inductor as: dul d I = LI dt dt 1 Solve for U L : UL = LId I = LI I 0

20 Stored energy type and the Energy Density of a Magnetic Field Given U L = ½ L I and assume (for simplicity) a solenoid with L = µ o n V 1 B B U L = µ on V = V µ on µ o Since V is the volume of the solenoid, the magnetic energy density, u B is u B UL B = V µ o So the energy stored in the solenoid volume V is magnetic (B) energy. And the energy density is proportional to B. This applies to any region in which a magnetic field exists (not just the solenoid)

21 RL and RC circuits comparison RL RC Charging Discharging I I ε = R = ε e R Rt L ( 1 e ) Rt L ( ) I t ( ) I t = = ε e R Q e RC t RC t RC Energy UL = 1 LI Q 1 ( ) UC = = C V C Magnetic field Electric field Energy density u B = B µ o u E = 1 εoe

22 Energy Storage Summary Inductor and capacitor store energy through different mechanisms Charged capacitor Stores energy in the electric field When current flows through an inductor Stores energy in the magnetic field A resistor does not store energy Energy delivered is transformed into thermo energy

24 Oscilaciones electromagnéticas: cualitativas

25 Oscilaciones electromagnéticas: cualitativas

26 Oscilaciones electromagnéticas: cualitativas Analogía con el MAS q x i v 1/C k L m ω = πf = k m ω = πf = 1 LC

27 Oscilaciones electromagnéticas: cualitativas

28 Oscilaciones electromagnéticas: cuantitativas U = U B + U E U = 1 Li + 1 q C du U = cte = 0 dt

29 Oscilaciones electromagnéticas: cuantitativas d dt q 1 + q LC = 0 d dt x k + x m = 0 x = x cos( ω t + φ) m q = q cos( ω t + φ) m

30 Oscilaciones electromagnéticas: cuantitativas U E 1 q qm = = cos ( ω t + φ ) C C U B Sustituyendo ω: 1 1 = Li = Lω qmsen ( ωt + φ) U B qm = sen ( ω t + φ) C

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