9. Alternating currents

Transcription

1 WS 9. Alernaing currens 9.1 nroducion Besides ohmic resisors, capaciors and inducions play an imporan role in alernaing curren (AC circuis as well. n his experimen, one shall invesigae heir behaviour in AC circuis and heir influence on he curren- and volage characerisics as a funcion of he applied AC volages frequency. This will be done on he basis of simple inegraed circuis wih one or wo elemens. The sudies are confined on harmonic AC volages and currens: V ( = V 0 cos ω and ( = 0 cos(ω φ where V 0 = volage ampliude 0 = curren ampliude ω = 2πν = 2π/T = angular fre φ = phase shif beween applied volage and running curren The emporal progression of he curren and volage are being depiced wih he help of a cahoderay oscilloscope. You ge he opporuniy o make yourself familiar wih he working principles and he operaion of an oscilloscope. Oscilloscopes can be found quie exensively in research and indusry, e.g. for roubleshooing and adjusmens of elecrical devices of any kind (e.g. radios, elevisions, microwaves and radars. n medicine, hey re used for surveillance of biological funcions which manifes hemselves in form of elecrical signals. Devices like an EEG (elecroencephalography, procedure o measure and rack he elecric aciviy of he brain or an ECG (elecrocardiogram, procedure o record he elecric processes in he hear use oscilloscopes. Keywords for his lab course are: elecric circuis measuremen of volage and curren as a funcion of ime Kirchhoff s laws alernaing curren resisance or impedance impedance of an ohmic resisor, inducor and capacior 1

2 2 9. Alernaing currens 9.2 Theory mpedance of an inducor The alernaing curren properies of an inducor are characerised by an inducance L. The inducance is deermined by he lengh l of he coil, is cross secional area A and he winding number N. For a long inducor one has: L = µ 0 N 2 A l [L] = 1 V s A = 1 H (Henry (9.1 Where µ 0 = 4π 10 7 (Vs/Am is he so-called inducion consan. When insering a magneized core ino he coil, he equaion above becomes L = µµ 0 N 2 A l where µ is he magneic permeabiliy of he induced maerial. Since he magneic permeabiliy is µ 1 for iron, one can drasically increase he inducance of he coil in ha way. A simple circui, consising of an AC generaor, which produces a harmonic volage V ( = V 0 cos ω and an inducor of inducance L, is being depiced in Fig The inducor is assumed o be ideal, meaning is ohmic resisance R is being negleced. The applicaion of Kirchhoff s 2nd law (volage law o his circui yields he relaion beween he volage V ( and he curren (: ~ V = V 0 cos ω Figure 9.1: negraed circui consising of an inducor wih inducance L. L V 0 cos ω L d d = 0 negraing he equaion above once, yields he soluion of his differenial equaion: ( = V 0 ω L sin ω = V 0 ωl cos(ω φ mi φ = +π 2 (9.2 Hence, he curren is being shifed agains he volage by a phase φ = π/2 (see Fig The ampliude of he alernaing curren is 0 = V 0 /(ωl. A comparison wih he general definiion of an elecric resisor ( = V/R shows, ha he quaniy ωl plays he role of a resisor. One calls Z L = ωl (9.3 he alernaing curren resisance or he impedance of an ideal inducor of inducance L. decreases proporional o he frequency of he applied AC volage.

3 9.2. THEORY 3 V V 0 V 0 ω L ϕ ω V = V 0 cos ω = 0 cos (ω -ϕ Figure 9.2: Volage and curren as a funcion of ime for one inducor n general, he impedance for an elemen i is defined as Z i = V 0 0 = volage ampliude curren ampliude (9.4 Quesion 1: A 10 cm long inducor has a cross secional area A of 4 cm 2 and 2000 windings. Wha is is inducance? mpedance of a capacior Jus as he inducor, he capacior represens an AC resisor as well. For he inegraed circui, depiced in Fig. 9.3, consising of an AC generaor and capacior, Kirchhoff s 2nd law yields: V 0 cos ω = V C = Q C Differeniaing once wih respec o ime yields ωv 0 sin ω = C ( dq d = ~ V = V 0 cos ω C Figure 9.3: negraed circui wih capacior wih capaciy C. and herefore: ( = ωc V 0 sin ω = ωc V 0 cos (ω φ where φ = π 2 (9.5 So he curren is shifed agains he volage by a phase φ = π/2 (see Fig. 9.4 and he impedance Z C of a capacior wih capaciy C is: Z C = 1 (9.6 ωc is inversely proporional o he frequency ω of he AC volage.

4 4 9. Alernaing currens V V 0 V 0 ω C ϕ ω V = V 0 cos ω = 0 cos (ω -ϕ Figure 9.4: Volage and curren as funcions of ime for one capacior Cahode-ray oscilloscope (CO n his experimen, all currens and volages shall be measured wih he help of a CO. The fundamenal se-up of a CO is shown in Fig Cahode Anode Elecron beam Deflecion plaes y x Screen Heaing Ligh spo _ + Vy V x Figure 9.5: Schemaic depicion of a cahode-ray oscilloscope. nside of an evacuaed glas flask, elecrons, which are being emied from a heaed cahode (glowing wire, are being acceleraed owards an anode, which is equipped wih a lile hole. Elecrons, which pass hrough he hole, form a hin elecron beam. This beam successively passes wo perpendicular pairs of plaes before finally hiing a fluorescen screen, where i creaes a luminous spo. When applying a volage o one of he pair of plaes, he negaively charged plaes repels he beam while he posiively charged plae aracs i. As a consequence, he beam ges defleced in horizonal or verical direcion. The deflecion is always proporional o he applied volage: x V x and y V y. For he usual usage of he oscilloscope, a sawooh volage is being applied inernally o he x-pair of plaes. This paricular volage increases linearly wih ime unil i reaches a maximum value and hen quickly decreases o is saring value (see Fig.9.6. This causes he ligh poin o repeaedly move over he screen from lef o righ and jumping back o is saring posiion afer reaching he righ end side of he screen. When applying an arbirary volage o he y-pair of plaes (e.g. a harmonic AC volage, he curve on he screen becomes a graphic represenaion of his volage as a funcion of ime. For he abiliy o measure wo volages

5 9.2. THEORY 5 V x Figure 9.6: Sawooh volage for deflecion in x-direcion. a he same ime, we use a 2-ray oscilloscope. This enables us o easily represen phase shifs. Nowadays, one usually uses digial oscilloscopes. They ransform he applied volages ino digial values. Ou of his daa, he appropriae curves are being calculaed and depiced on a screen.

6 6 9. Alernaing currens 9.3 Experimenal par Definiion of ask Deerminaion of he inducance of an inducor Deerminaion of he capaciy of a capacior nvesigaion on he frequency response of he AC resisance for he capacior and he inducor Measuremen principles ~ V = V 0 cos ω Signal generaor KO (V Z Channel 1 R Z KO (V R Channel 2 Figure 9.7: negraed circui for he measuremen of he curren and he volage wih he CO. Wih he CO one can only measure volages in a direc way. A curren can be measured indirecly via he poenial drop over a known resisor R: = V R R, 0 = V R0 R (9.7 The resisor R has o be sufficienly smaller han he impedance Z one wans o measure, o make sure ha he volage measuremen on Z is no being falsified. For R Z one has V Z + V R V Z and he volage on Z can be measured coincidenally wih he curren, using a 2-ray CO. This is shown in Fig Le he eaching assisan explain o you and demonsrae he handling of he oscilloscope. Noe ha he ouer cable (shielding of he inpus of he CO are inernally wired and conneced o he ground! Therefore, always wire he inpus in such a way, ha he wo ouer cables lead o he same poin in he circui. AC generaor seings: ~ V = V 0 cos ω Signal generaor KO (V L Channel 1 L V 0 = 5V ν = 1000 Hz R = 22 Ω R KO (V R Channel 2 Figure 9.8: negraed circui for he measuremen of he inducance of a coil.

7 9.3. EXPERMENTAL PART 7 Pay aenion o he phase shif beween curren and volage on he CO screen by moving he wo curves one above he oher. Skech he resul. Using he CO, measure he ampliudes of V L and V R. Calculae he curren ampliude according o Eqn. (9.7. Using his resul, calculae Z L according o Eqn. (9.4 and L according o Eqn. (9.3. To do so, use ω = 2πν Deerminaion he capaciy of a capacior AC generaor seings: V = V 0 cos ω ~ Signal generaor KO (V L Channel 1 C V 0 = 5V ν = 500Hz R = 22Ω R KO (V R Channel 2 Figure 9.9: negraed circui for he measuremen of he capaciy. Pay aenion o he phase shif beween curren and volage on he CO screen by moving he wo curves one above he oher. Skech he resul. Using he CO, measure he ampliudes of V C and V R. Calculae he curren ampliude according o Eqn. (9.7. Using his resul, calculae Z C according o Eqn. (9.4 and C according o Eqn. (9.6. To do so, use ω = 2πν Frequency dependence of Z L and Z C Repea he above measuremens for he inducor and capacior as a funcion of he applied AC volage. Use he coil wih an insered iron core. Chose he following frequency values: for Z L : ν = 500, l000, 2000 and 3000 Hz for Z C : ν = 20, 50, 150, 250, 500 and 750 Hz Pu he measuremen values ogeher in an clearly arranged able. Calculae Z L and Z C for each frequency according o Eqn. (9.4 and express Z L and Z C as funcions of ω on ploing paper.

8 8 9. Alernaing currens Repor The repor should include he following: Calculaion of he inducance (Quesion l. Draf of he experimenal se-up. Draf of he observed phase shif for he inducor and capacior. Calculaion of he waned capaciy. Calculaion of he waned inducance. Frequency dependence of Z L and Z C : Table of he measuremen values. Graphic represenaion of Z L (ν and Z C (ν.