AC-1 Inductors, Transformers, and AC circuits Inductors An inductor is simply a coil of wire. Inductors are used in circuits to store energy in the form of magnetic field energy. Important point: The magnetic flux ΦB through any loop is proportional to the current I making the flux. All our formulas for B-field show B ∝ I: Biot-Savart: 02ˆIdrdB4rµπ×=KKA Ampere: 0thruBd Iµ⋅ =∫KKAvL ⇒ Φ ∝ B ∝ I ⇒ Φ ∝ I . So the ratio Φ/I is independent of I. Definition: Self-inductance L of a coil of wire: The inductance is independent of I. BLIΦ≡BL/= Φ ICurrent I makes B, which makes Φ. units of inductance [L] = [Φ] / [I] = T⋅m2 / A = 1 henry (H). An inductor is a coil of wire. One or a few centimeter-sized loops of wire has L ≅ 1 µH (usually insignificant). A coil with many thousands of turns has L ≅ 1 H (big!). So, why do we care about inductors? An inductor acts like a "current regulator". An inductor helps to maintain constant current. How is that? FaradayddILI Ldt dtΦΦ = ⋅⇒ ==−E ⇒ Changing the current in an inductor creates an emf which opposes the change in I (by Lenz's Law). The induced emf is often called a "back emf". So, • It is difficult (requires a big external voltage) to change quickly the current in an inductor. • The current in an inductor cannot change instantly. If it did, there would be an infinite back emf, an infinite E-field to fight the change. I B dILdt= −E Last update: 10/30/2009 Dubson Phys1120 Notes, ©University of ColoradoAC-2 Computing the inductance of a single turn coil (or a few turn coil) is quite messy because the B-field in a loop of wire is non-uniform. The non-uniform B makes computing the magnetic flux quite difficult. In practice, one determines the inductance of a coil by measuring it, using BSBdaΦ = ⋅∫KKdILdt= −E : put in a known dI/dt, measure emf, compute L. Computing the inductance of a long solenoid is easy, because the B-field is uniform: Self-inductance L of a solenoid: If the coil is very long, B = µo n I inside , so total flux is 0NBA N nIAµΦ ==200inductance L Nn A n AzIµµΦ== = (We use z for length here, to avoid confusion with L for inductance). Magnetic Energy Density Recall that for a capacitor, the stored electrostatic potential energy is 212UC= V. This energy is in the electric field, and the energy density (energy per volume) is 21E02UuEvol.ε== For an inductor, the stored energy is 212UL= I . This energy is stored in the magnetic field (so we call it magnetostatic potential energy) and the energy density is 2B0U1uBvol. 2µ==. Proof of212UL= I : It takes work to get a current flowing in an inductor. The battery which make the current flow in an inductor must do work against the back emf, which opposes any change in current. Watch closely: power dU dIPIVIdt d t===L , so , sodU Pdt ILd I==212UdU ILdILIdI L== = =∫∫ ∫I length z area A I I N turns n = N / z Last update: 10/30/2009 Dubson Phys1120 Notes, ©University of ColoradoAC-3 Exercise for the motivated student: Show that 2B01u2 µ= B for the case of a long solenoid. Start with 212UL= I, and use the previously found expressions for L and B for a solenoid. LR circuits (circiuts with L's and R's) 3 things to remember about inductors in circuits: • An inductor acts like a battery when its current is changing: dILdt= −E. The direction of the battery voltage is such as to fight any change in the current. • The current through an inductor cannot change instantly (because that would cause an infinite E). • In the steady-state (after a long time), when the current is constant, I = const ⇒ EL = 0 ⇒ the inductor acts like a short (a zero-resistance wire). Example: Simple LR circuit. switch Switch at position A for a long time: A I = constant, so EL = 0, I = E0 / R . At t = 0, switch →B, and the circuit becomes: The emf in the inductor keeps the current going. Apply Loop Law: LdIIR Ldt==−E (Note on signs: dI/dt < 0 so EL > 0 . ) dI RIdt L= − This is a differential equation with an expontial solution. 00 0RtLt/(L/R)t/I(t) Ie Ie Ieτ⎛⎞⎟⎜⎟⎜⎟⎜⎟⎜⎟⎜⎟⎜⎝⎠−⋅−−== = , LRτ == time constant of LR circuit = time for anything in circuit to change by factor of e R LE0 E L B R LI Last update: 10/30/2009 Dubson Phys1120 Notes, ©University of ColoradoAC-4 I I0 I0/e t τ Another LR circuit: Close switch at t = 0. At t = 0+, I = 0 (since I cannot change instantly), switch R Apply Loop Law, inductor acts like second battery: 0dILIdt− =E R Initially, I = 0, so 00t0dI dIL0,dt dt L=− ==EE As t ↑ , I ↑ , VR = I R ↑ , | EL | = | L dI / dt | As t → ∞ , | EL | → 0 , E0 = IR 0t/(L/R)I(t) 1 eR−⎡⎤= −⎢⎥⎣⎦E AC Voltage and Current Batteries produce voltage that is constant in time, DC voltage. The wall socket produces sinusoidally-varying voltage, AC voltage. (DC originally stood for "direct current" but now it just means "constant in time". AC is short for "alternating current" but now means "sinusoidally-varying".) L E0 I E0 / R t Last update: 10/30/2009 Dubson Phys1120 Notes, ©University of ColoradoAC-5 V Vwall socket (AC) Vo Wall socket voltage: oootV V(t) V sin 2 V sin(2 f t) V sin( t)T⎛⎞== π= π= ω⎜⎟⎝⎠ In the US, the frequency of "line voltage" is f = 60 Hz = 60 cycles per second (Recall f = 1 / T, period T = 1/60 s) AC voltage causes AC current in resistor. Current actually flows back and forth, 60 times a second. 0oVVIsin(t)IsiRRn(t)==ω=ω The instantaneous voltage is (+) as often as (–), so avgVV 0== , but avgV0≠. Electrical engineers always report AC voltage using a kind of average called "root-mean-square" or rms average. 2rmsVAC "volts AC" = V V 120V (in US)=== The average voltage Vrms is less than the peak voltage Vo by a factor of √2 : ormsVV 2= Why √2 ? 22V sin( t) , V sin ( t)∝ω ∝ ωsin varies from +1 to –1 (sin = +1 → 0 → –1 → 0 → +1→ 0 → –1 → 0 → .. ) tV Vo–VoVrmstime time battery (DC) V = constant –Voperiod T = 1 / 60 s I 120 VAC symbol for AC voltage Last update: 10/30/2009 Dubson Phys1120 Notes, ©University of ColoradoAC-6 sin2 varies from 0 to +1 (sin2 = +1 → 0 → +1 → 0 → +1→ 0 → +1 → 0 → .. ) The average of sin2 is ½ . 222 2orms o o oV1VVVsin(t)Vsin(t)V22== ω= ω== Wall socket voltage or "line voltage" : Vrms = 120 V , Vpeak = V0 = rms2 V
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