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 B through any loop is proportional to the current I making the flux All our formulas for B field show B I K K K K 0 I d A r Biot Savart dB Ampere B d A 0 I thru v 2 4 r L B I I So the ratio I is independent of I Definition Self inductance L of a coil of wire B B L I The inductance L B I is independent of I I Current I makes B which makes units of inductance L 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 H 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 L I E L dI dt d dI L dt dt E Faraday 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 Last update 10 30 2009 Dubson Phys1120 Notes University of Colorado AC 2 Computing the inductance of a single turn coil or a few turn coil is quite messy because the Bfield in a loop of wire is non uniform The non uniform B makes computing the magnetic flux K K B B da quite difficult In practice one determines the inductance of a coil by measuring S it using E L dI put in a known dI dt measure emf compute L dt Computing the inductance of a long solenoid is easy because the B field is uniform Self inductance L of a solenoid length z If the coil is very long B o n I inside area A so total flux is N B A N 0 n I A inductance L 0 N n A 0 n 2 A z I I We use z for length here to avoid confusion with I n N z N turns L for inductance Magnetic Energy Density Recall that for a capacitor the stored electrostatic potential energy is U 1 2 is in the electric field and the energy density energy per volume is u E U vol For an inductor the stored energy is U 1 2 1 2 1 2 0 E 2 L I 2 This energy is stored in the magnetic field so we call it magnetostatic potential energy and the energy density is u B Proof of U C V 2 This energy U vol 1 B2 2 0 L I 2 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 P so U dU ILdI Last update 10 30 2009 L I d I 1 2 dU dI so dU P dt I L d I IV IL dt dt L I2 Dubson Phys1120 Notes University of Colorado AC 3 Exercise for the motivated student Show that u B Start with U 1 2 1 B2 for the case of a long solenoid 2 0 L I 2 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 E L dI The direction dt 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 A switch Switch at position A for a long time E0 R B L I constant so EL 0 I E0 R EL At t 0 switch B and the circuit becomes R The emf in the inductor keeps the current going Apply Loop Law EL I R L L I dI dt Note on signs dI dt 0 so EL 0 dI R I dt L This is a differential equation with an expontial solution R t L I t I0 e I0 e t L R I0 e t L time constant of LR circuit time for anything in circuit to change by factor of e R Last update 10 30 2009 Dubson Phys1120 Notes University of Colorado AC 4 I I0 I0 e t Another LR circuit Close switch at t 0 switch At t 0 I 0 since I cannot change instantly R E0 Apply Loop Law inductor acts like second battery L E0 L dI IR dt Initially I 0 so E0 L dI dI 0 dt dt t 0 E0 L As t I VR I R EL L dI dt As t EL 0 E0 IR I I t E0 R E0 1 e t L R R t 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 Last update 10 30 2009 Dubson Phys1120 Notes University of Colorado AC 5 V V wall socket AC Vo V constant time battery DC time Vo period T 1 60 s t Wall socket voltage V V t Vo sin 2 Vo sin 2 f t Vo sin t T 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 120 VAC AC voltage causes AC current in resistor Current actually flows back and forth 60 times a second I symbol for AC voltage I V V 0 sin t Io sin t R R The instantaneous voltage is as often as so V Vavg 0 but V avg 0 Electrical engineers always report AC voltage using a kind of average called root mean square or rms average VAC volts AC Vrms V 2 120 V in US The average voltage Vrms is less than the peak voltage Vo by a factor of 2 Vrms Vo 2 V Vo Vrms t Vo Why 2 V sin t V 2 sin 2 t sin varies from 1 to 1 sin 1 0 1 0 1 0 1 0 …
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