Chapter 4 AC Network Analysis

Transcription

1 haper 4 A Nework Analysis Jaesung Jang apaciance Inducance and Inducion Time-Varying Signals Sinusoidal Signals Reference: David K. heng, Field and Wave Elecromagneics.

2 Energy Sorage ircui Elemens Energy loss elemen: resisors Energy sorage elemen: capaciors and inducors (in he form of elecromagneic field Ideal capacior Ideal inducor In pracice, any componen of an elecric circui will exhibi phenomena of some resisance, some inducance, and some capaciance.

3 The Ideal apacior A physical capacior is a device ha can sore energy in he form of a charge separaion when appropriaely polarized by an elecric field, or volage. Tha is, he ideal capaciors sore energy (elecric charges on he conducing plaes in he form of elecric field. apaciance is he measure of how much elecric charge can be sored in a capacior. -> I depends on maerial properies only. The simples capacior consiss of wo parallel conducors separaed by a dielecric (insulaor, which has very large resisances. The insulaing maerial does no allow for he flow of D curren: hus, a capacior acs as an open circui for D curren. harging: Applying a volage o a (discharged capacior causes a curren o charge he capacior. Tha is, elecric charges move o he capacior, bu hey can go hrough he capacior. elecric fields Discharging: onnecing a pah across he erminals of a charged capacior causes curren o flow (because i has energy. capacior 3

4 harging & Discharging harging (lef swich closed, righ swich open The elecric charges from he volage source move o he capacior, so capacior volage and energy increases up o V B. Discharging (lef swich open, righ swich closed The elecric charges from he capacior move o he resisor, so he energy accumulaed on he capacior dissipaes in he resisor. 4

5 The Uni of apaciance and Energy The farad (F is he uni of capaciance. One farad of capaciance equals one coulomb of charge sored in he dielecric wih one vol applied. Mos capaciors have values less han F: µf (microfarad -6 F, nf (nanofarad -9 F, pf (picofarad - F harge on a capacior is generaed due o volage applied across he capacior: q V i υ W ( dq ( υ ( ( d[ υ ( ] dυ ( dυ ( i ( ( ( p( d υ ( i ( d υ ( d + d i d W ( υ ( Energy sored in a capacior (J : capaciance in farads υ : volage in vols d d dυ d ( d [ υ ( ] i ( d 5

6 i Series and Parallel apaciances onnecing capaciances in series is equivalen o increasing he disance beween he conducing plaes. Toal is less han he smalles individual value. / T / + / +... ec. v v + v + + v i i ( d + i ( d + i ( ( d i ( onnecing capaciances in parallel is equivalen o increasing plae area where can sore charge. Toal is he sum of individual s: T ec. Volage is he same across parallel capaciors. ( dv ( dv ( dv 3 i + i + i d d d EQ d 3 3 d ( + + dv d ( dv( EQ d 6

7 Magneic Field around an Elecric urren A circular magneic field is produced by he flow of curren hrough a sraigh conducor in he cener. The direcion of he magneic field inside a coil is perpendicular o he curren flowing hrough he coil. The polariy of he magneic field is based on he righ-hand rule. Righ-hand rule: The humb: B -> he oher fingers: i The humb: i -> he oher fingers: B 7

8 Induced urren When a moving conducor cus across magneic flux lines, curren is induced. The polariy of induced volage is deermined by enz s law. enz s law saes ha he direcion of an induced curren mus be such ha is own magneic field will oppose he change ha produced he induced curren. -> Wha if he permanen magne does no move? The direcion of he induced curren is deermined by righ-hand rule for curren flow. If he fingers coil around he direcion of curren shown, he humb will poin o he lef for he norh pole. change Example N S permanen magne Induced curren Induced curren produced by magneic flux cuing across urns of wire in a coil. 8

9 Induced Volage Faraday s aw of Induced Volage The amoun of volage induced is deermined by he following formula. v ind N dφ (webers d (seconds N number of urns dφ/d how fas he magneic flux cus across he conducor Eiher he flux or he conducor should move o induce volages. B Posiive charges flow Excess posiive charges A Volage induced across coil cu by magneic flux. (a Moion of flux generaing volage across coil. (b Induced volage acs in series wih coil. (c Induced volage is a kind of volage source ha can produce curren in an exernal load resisor R conneced across he coil. 9

10 Self-Induced Volage enz s law saes ha he direcion of an induced curren mus be such ha is own magneic field will oppose he change ha produced he induced curren. When i increases, v has polariy ha opposes he increase in curren. When i decreases, v has polariy o oppose he decrease in curren. In boh cases, he change in curren is opposed by he induced volage. Wha if he magniude of curren is consan? (D case Increasing curren Volage source + _ v Decreasing curren _ Volage source + v Volage source Increasing curren _ + v Decreasing curren Volage source + _ v

11 The Ideal Inducor The ideal inducors sore energy (elecric charges on he conducing plaes in he form of magneic field. A inducor is ypically made by winding a coil of wire around a core (an insulaor or a ferromagneic maerial. air Ferromagneic maerials include iron, seel, nickel, cobal, and cerain alloys (usually conducors. They can become srongly magneized in he same direcion as he exernal magneizing field. Inducance is he measure of he abiliy of a conducor o induce volage when he curren changes or abiliy o sore energy in a magneic field. I depends on maerial properies only.

12 Example of Inducance Inducance is a funcion of he number of urns (N, a cross secional area (A, permeabiliy of core(µ r, and he lengh of a core (l. alculaing he Inducance of a ong oil air-core symbol (µ r µ r N A Where: is he inducance in henrys. µ r is he relaive permeabiliy of he core N is he number of urns A is he cross secional area in square meers l is he lengh in meers l 4 π 7 H iron-core symbol (µ r >>

13 The Uni of Inducance and Energy The henrys (H is he uni of inducance. One henrys of inducance means ha one vol of volage is induced due o a rae of change of one A/sec. v i W ( ( i ( di d ( di ( v ( ( ( p( d v ( i ( + v d d [ i ( ] ( W ( i ( Energy sored in an inducor (J : inducance in henrys i : curren in amperes d di d Read he able 4. (Analogy beween elecric and fluid circuis!! i ( v d ( d 3

14 Energy Accumulaion & Dissipaion Energy accumulaion (lef swich closed, righ swich open The curren flows hrough he inducor increasing up o I B and energy is sored. Energy dissipaion (lef swich open, righ swich closed he energy accumulaed on he inducor dissipaes in he resisor. 4

15 Series and Parallel Inducances Series: Toal is he sum of individual s: T ec. urren is he same hrough he series inducors. di ( di ( di3( di ( ( di( v v + v + v d d Parallel: Toal is less han he smalles individual value. / T / + / +... ec. Volage is he same across parallel inducors. i i + i + i 3 v d ( d + v ( d + v ( d + + v ( d v ( 3 3 d EQ d EQ d 5

16 Time-Dependen Signal Sources onsider sources ha generae ime-varying volages and currens and, in paricular, sinusoidal sources. One of he mos imporan ime-dependen signals is periodic signal. x ( x( + nt n,, 3, Kand T is he period of x( Several ypes of waveforms are provided by commercially available funcion (signal generaors. periodic signals 6

17 Time-Dependen Signal Sources (con. A generalized sinusoid is defined as x Acos ω + φ where A is he ampliude, ω ( ( and φ he phase (angle. x ( Acos( ω and x( Acos( ω + φ where f is frequency (cycles or Hz T dθ ω he radian frequency πf (rad/sec d φ he phase (angle π (rad or 36 (deg T T he radian frequency 7

18 Time-Dependen Signal Sources (con. The following specific values are used o compare one wave o anoher: Peak value: Maximum value for currens or volages. This applies o he posiive or negaive peak. I I av I Peak-o-peak: Usually, bu no always, double he peak value, as i measures disance beween wo ampliudes. Average value: Arihmeic average of all values in one half-cycle (he full cycle average. Roo-Mean-Square (RMS or Effecive Value: The amoun of a sine wave of volage or curren ha will produce he same power compared o he D values. ω rms I m sin dθ d π π Idθ I π π ( ω θ θ θ π m sinθdθ I π m ~ I m m av π π π θ for consan angular ve π ( I sinθ dθ I P I R dθ R I dθ I R lociy moion The average value is.637 peak value. The rms value is.77 peak value. π P I rms R for D cases 8

19 RMS vs. D V rms V Ω V rms is he effecive value. The heaing effec of hese wo sources is idenical. + V Ω Same power Dissipaion wih rms values in A 9

20 Phase Angle Phase angle (Θ is he angular difference beween he same poins on wo differen waveforms of he same frequency. Two waveforms ha have peaks and zeros a he same ime are in phase and have a phase angle of. When one sine wave is a is peak while anoher is a zero, he wo are 9 ou of phase. When one sine wave has jus he opposie phase of anoher, hey are 8 ou of phase. Two sine-wave volages are 9 ou of phase.

21 The 6-Hz A Power ine Almos all homes in he US are supplied alernaing volage beween 5 and 5 V rms, a a frequency of 6 Hz. Alhough he frequency of house wiring in Norh America is 6 Hz, many places ouside N. America use a 5 Hz sandard for house wiring. Residenial wiring uses ac power insead of dc, because ac is more efficien in disribuion from he generaing saion. House wiring in he US uses 3-wire, single-phase power. A value higher han V would creae more danger of faal elecric shock, bu lower volages would be less efficien in supplying power. Higher volage can supply elecric power wih less I R loss.