11. Properties of alternating currents of LCRelectric circuits


 Lenard Singleton
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1 WS. Properies of alernaing currens of Lelecric circuis. Inroducion Socalled passive elecric componens, such as ohmic resisors (), capaciors () and inducors (L), are widely used in various areas of science and elecrical engineering. Loscillaory circuis deliver he ime base for many elecrically generaed oscillaion processes and elemens are uilised as frequency filers for signal forming. The resonance behaviour of Loscillaory circuis feaures pleny of similariies compared o oscillaioncapable sysems in oher physical areas. In he firs par of his lab course, he behaviour of a serial circui compared o a squarewave volage shall be examined. In he second par we ake a look a he properies of conducors and inducors (coils) compared o harmonic alernaing volages (A), where in he hird par we invesigae on he resonance behaviour of a L parallel oscillaory circui.. Theoreical par a) Squarewave volage on he serial circui A capacior and a resisor are conneced o a direc curren volage (D) source, which creaes a consan volage, via a swich S as i is shown in Fig... S Figure.: Generaion of a squarewave volage wih a serial circui. A ime = 0, he capacior shall be compleely discharged and he volage is being plugged by urning he swich. Afer he powerup of he volage i follows by he nd Kirchhoff s circui law = V + V = I + Q = dq d + Q (.)
2 . Properies of alernaing currens of Lelecric circuis This is a firsorder differenial equaion for he charge Q which is sored on he capacior. onsidering he iniial condiions V = V = Q = 0 for = 0 he soluion of he differenial equaion becomes Q() = Q 0 ( e / ) mi Q 0 = (.) I() = dq d = I 0 e / mi I 0 = Q 0 = (.3) V () = Q() = V 0 ( e / ) mi V 0 = Q 0 = (.4) The characerisic ime consan which defines he behaviour of he circui is τ =. A ime T τ he capacior shall be compleely charged and he volage is being broken by urning he swich again. The iniial condiions a ime T become Q = Q 0, V = and I = 0. Saring wih he differenial equaion for > T i follows: 0 = V + V = I + Q = dq d + Q (.5) Q() = Q 0 e ( T 0)/ mi Q 0 = (.6) I() = dq d = I 0 e ( T 0)/ mi I 0 = Q 0 = (.7) V () = Q() = V 0 e ( T 0)/ mi V 0 = Q 0 = (.8) The emporal progress of he volage for periodic powerup and shuoff is shown in Fig.., in paricular for he case T τ. If his condiion is no fulfilled hen he capacior is no compleely charged and he curren hasn dropped down o zero ye when urning he swich. In his case he emporal progress for he volage is shown in Fig..3. b) Harmonic alernaing volage on he circui and he Lcircui When plugging a harmonic alernaing curren V () = cos (ω) (.9) of he frequency ω o an ordinary elecrical circui, hen a harmonic alernaing curren of he same frequency sreams which, in he general case, is shifed by a phase ϕ agains he volage V (): I() = I 0 cos (ω ϕ) (.0) Laboraory manuals for Physics Majors  PHY/
3 .. THEOETIAL PAT 3 a) V = Q b) T V = I c) τ = T Figure.: Temporal progress of he volage for periodic powerup and shuoff of he inpu volage for T 0. The dashed line shows he volage characerisic for a squarewave volage wih mean zero, as i is being delivered from a squarewave generaor. a) V b) V c) Figure.3: Temporal progress of he volage for periodic powerup and shuoff of he inpu volage when he condiion T 0 is no fulfilled. Laboraory manuals for Physics Majors  PHY/
4 4. Properies of alernaing currens of Lelecric circuis The behaviour of he circui is compleely described by his phase and he absolue value of he impedance Z = I 0 (.) If he circui conains only one single elemen, one receives he impedance wih he help of Kirchhoff s nd law: Ohmic resisor : ~ V () = cos (ω) = V () = I() I() = cos (ω) Z =, ϕ = 0 apacior of capaciy : ~ oil of inducance L: ~ L V () = cos (ω) = V () = Q() I() = dq = ω sin (ω) = ω cos (ω + π d ) Z = ω, ϕ = π V () = cos (ω) = V L () = L di d I() = ω L sin (ω) = ω L cos (ω π ) Z L = ω L, ϕ L = + π The phase and he absolue value of he impedance can be displayed as a vecor in he complex plane. For he hree described special cases his is shown in Fig..4 Im Im Im Z ϕ = 0 e Z _ π e Z L + π e Figure.4: Depicion of he impedance in he complex plane. In case he circui conains muliple impedances serially, he oal impedance is given by vecor sum of he single impedances. For he example of he serial circui one receives: Laboraory manuals for Physics Majors  PHY/
5 .. THEOETIAL PAT 5 ~ Z = ω Z = ϕ Z o V = V + V = I (Z + Z ) = I Z o (.) Z o = Z + Z = + /(ω ) (.3) an ϕ = ω and for he ampliude of he curren follows: I 0 = Z o = + /(ω ) (.4) (.5) The volage ampliudes a he capaciy and he resisor are given by V 0 = I 0 Z = V 0 = I 0 Z = + ω (.6) ω + ω (.7) Their dependence on he frequency ω of he inpu volage is shown in Fig..5. A he capaciy high frequencies and a he resisor low frequencies of he inpu signal are being suppressed. Therefore his circui can be used eiher as a highpass or as a lowpass filer. The frequency response of he oupu volage V a can be adjused in boh cases by varying he resisor. A pracical usage of his circui are one conrols in audio amplifiers. Va = V V a = V Low pass High pass V 0 V0 V 0 V0 ω ω Figure.5: serial circui as highpass or lowpass filer. In a similar manner one can combine a resisor and an inducance. In his case i holds ha: L Z o = Z ~ + Z L = w L + (.8) an ϕ = ωl (.9) Z o Z L = ω L ϕ Z = and he oupu ampliude a he resisor is: V 0 = I 0 = Z o = + ω L (.0) Laboraory manuals for Physics Majors  PHY/
6 6. Properies of alernaing currens of Lelecric circuis c) Elecrical parallel oscillaory circui (Lcircui) ~ m I I I L I L p V p Figure.6: parallel oscillaory circui. In his experimen we use he circui which is shown in Fig..6. The oscillaory circui iself is made up of a coil of inducance L which is parallelconneced wih a capaciy and an ohmic resisor p. Due o merological reasons a furher resisor m is conneced in series wih he oscillaory circui. When a harmonic inpu volage V () = cos ω is applied o he circui wih Kirchhoff s second law follows V () = m I() + V p () and herefore (.) I() = m ( cos ω V p ()) (.) and wih Kirchhoff s firs law I() = I () + I () + I L () = dv p d + V p() + L V p d (.3) By equalling Equ.. and.3 and differeniaing one ime wih respec o ime he characerisic differenial equaion for his oscillaory circui follows: ω ( sin ω = d V p m d + + ) dv p p m d + V p() L (.4) I is of he same form as he one for he forced oscillaion of he harmonic oscillaor in classical mechanics (see lab course from he winer semeser). Is general soluion is a superposiion of a freely damped oscillaion and a forced oscillaion wih exciaion frequency ω. The free oscillaion decays wih he ime consan τ = L/ and for imes τ he sysem oscillaes only wih he frequency ω of he exciaion volage and is shifed by a phase ϕ o his exciaion. V p () = V p0 cos (ω + ϕ) (.5) The ampliude V p0 and he phase ϕ follow by plugging ino he differenial equaion: Laboraory manuals for Physics Majors  PHY/
7 .. THEOETIAL PAT 7 V p0 = an ϕ = / m ( ) ( ) (.6) p + m + ωl ω ( ) / ( ωl ω + ) (.7) p m Hence, he ampliude and he phase of he volage ha decays a he oscillaory circui are dependen on he exciaion frequency ω and show he ypical resonance behaviour of oscillaioncapable sysems. The ampliude of he volage drop is a he maximum for he resonance frequency ω = ω 0 = L (.8) Here, he impedances of he capaciy and he coil cancel each oher ou and he phase is ϕ = 0. The expression on he righ side of Equ..6 can be simplified for frequencies in he close region of he resonance frequency as follows. Firs of all wih ω0 = L one has V p0 (ω) = / m ( ) p + m + (ω (.9) ω 0 ω) Near by he resonance frequency we have ω ω 0 and wih ω = ω ω 0 i is approximaely rue ha Plugged ino Equ..6 his approximaion leads o ω 0 ω = (ω 0 + ω) (ω 0 ω) ω ω (.30) V p0 ( ω) / m ( ) = p + m + (ω ω) ω / m ( ) (.3) p + m + 4 ω The resonance curve is shown in Fig..7. Is widh is usually specified by he socalled halfpower widh ω /, where ω / denoes he frequency difference for which he square of he ampliude of he volage drop i.e. he power in he parallel oscillaory circui has fallen off o half of he value a he resonance frequency ω 0. For ω / he ampliude of he volage drop iself has fallen off o / imes he value a he resonance frequency. From Equ..3 one ges ω / = ( + ) p m (.3) Laboraory manuals for Physics Majors  PHY/
8 8. Properies of alernaing currens of Lelecric circuis Vp0( ω) Vp0( ω) Vp0,max Vp0,max Vp0,max Vp0,max ω ω / ω/ ω.3 Experimenal par Figure.7: esonance curve. In his experimen we observe he measured quaniies wih a cahoderay oscillograph (O). The funcional principle of he O is explained in he appendix of he lab course manual for EM. The O can only display volages. urrens can be displayed indirecly via he volage drop V on a known ohmic es resisor, due o V = I. Therefore he resisor has o be included ino he circui, whereby is value has o be small enough such ha i s no affecing he measuremen. In his experimen a es resisor of Ω is being used. For he abiliy o display wo measured quaniies a he same ime (e.g. inpu volage and curren in he circui) o read off heir phase relaion we use a woray O. Le he assisan give you an inroducion o he working principles of he O Take ino accoun ha he ouer cable (shielding) of he inpus of he O are conneced inernally wih he ground! Therefore, always link he inpus such ha boh ouer cable lead o he same poin in he circui. For his experimen no error calculaion has o be done. a) Squarewave volage n he serial circui For an applied squarewave volage he oupu volages a he capacior and he resisor of a serial circui shall be observed. Assemble he circui o arge he arrangemen of Fig..8. Use a resisor of = 470 Ω and a capacior wih a capaciy of = µf. Wih he volage generaor se up a squarewave volage wih an ampliude of 5 V and a frequency of 00 Hz. On he O, simulaneously sudy he inpu volage V () and he volage V () a he capacior. Plo he volage characerisic as a funcion agains he ime Laboraory manuals for Physics Majors  PHY/
9 .3. EXPEIMENTAL PAT 9 ~ V () ~ V () Figure.8: Measuremens on he serial circui. graphically and from he curves deermine he periods T and τ =. ompare your resuls wih he expeced values. epea he measuremen for a squarewave volage wih a frequency of 5000 Hz. ese he squarewave volage o a frequency of 00 Hz and simulaneously sudy he inpu volage V () and he volage V () a he resisor. Plo he volage characerisic as a funcion agains he ime graphically and from he curves deermine he periods T and τ =. ompare your resuls wih he expeced values. epea he measuremen for a squarewave volage wih a frequency of 5000 Hz. b) Harmonic inpu volage a conducor and coil For a harmonic inpu volage you shall explore he impedances of a single capacior and a single coil as well as he oupu volages of a serial circui. I() I() ~ L ~ Impedance of a capaciy: Figure.9: Measuremen of he impedance of a capacior and a coil. Assemble he circui as shown in Fig..9. Use a capacior wih a capaciy of = µf. Wih he volage generaor se up a squarewave volage wih an ampliude of 5 V and a frequency of 0 Hz. Simulaneously sudy he inpu volage V () and he curren I() (via he volage drop over he es resisor ) wih he aid of he O. Deermine Z and ϕ and compare your resuls wih he expeced values. epea he measuremens for frequencies of 50, 50, 500 and 000 Hz and plo Z as a funcion of he frequency. Impedance of a coil: Laboraory manuals for Physics Majors  PHY/
10 0. Properies of alernaing currens of Lelecric circuis Exchange he capaciy by a coil of inducance L = 35 mh. Wih he volage generaor se up a squarewave volage wih an ampliude of 5 V and a frequency of 00 Hz. Simulaneously sudy he inpu volage V () and he curren I() on he O. Deermine Z L and ϕ L and compare your resuls wih he expeced values. epea he measuremens for frequencies of 500, 000 and 000 Hz and plo Z L as a funcion of he frequency. c) esonance curve of he L parallel oscillaory circui The resonance curve of a L parallel oscillaory circui shall be measured for differen values of he resisor p. Assemble he circui afer Fig..0. Firs, use a resisor of p = 9. kω. Wih he O measure he oscillaion ampliude of he oscillaory circui as a funcion of he inpu (exciaion) frequency. Iniially, raise he inpu frequency in big seps of 00 khz o 600 khz and measure he range around he resonance frequency in more delicae seps. hose an appropriae sep size o do ha. Plo he ampliude as a funcion of he inpu frequency and deermine he halfpower widh. epea he measuremen for a resisor of p = 57.6 kω. Plo he halfpower widh agains / p and lay a sraigh line hrough boh measuremen poins. Deermine he capaciy from he slope of he line and he resisor m from he abscissa segmen. Finally, deermine he inducance L from he measured resonance frequency. Oscilloscope H H Ex. Z Ex. poeniomeer VG IN Signal generaor SWEEP OUT HI Frequency couner ~ ouner Generaor KO y m 50Ω L p KO Probe KO y Figure.0: Seup for he measuremen of he resonance curve of a L parallel oscillaory circui. Laboraory manuals for Physics Majors  PHY/
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