3. Alternating Current

Transcription

1 3. Alernaing Curren TOPCS Definiion and nroducion AC Generaor Componens of AC Circuis Series LRC Circuis Power in AC Circuis Transformers & AC Transmission

2 nroducion o AC The elecric power ou of a home or office power socke is in he form of alernaing curren (AC), as opposed o he direc curren (DC) of a baery. Alernaing curren is used because i is easier o ranspor, and easier o ransform from one volage o anoher using a ransformer. n Nigeria and UK, he frequency of oscillaion of AC is 50 Hz. n he USA i is 60 Hz. November 7, 007

3 The AC Generaor November 7, 007

4 A coil of area A and N urns roaing wih consan angular velociy in a uniform magneic field produces a sinusoidal emf. The slip rings and brushes allow he coil o roae wihou wising he connecing wires. Such a device is called a generaor.

5 Alernaing Curren Generaor Con d ε ε sin m d i sin( φ) December 5, 007 d

6 akes power o roae he coil, bu ha power can come from: moving waer (a waer urbine) air (windmill) gasoline moor (as in a car) seam (as in a nuclear power plan). November 7, 007

7 NBA d NBA and NBA m m m δ φ ε δ φ δ θ θ φ ) sin( ) ( November 7, 007 f f frequency NBA d m π δ ε ε δ ε ; ) sin( ) sin( + +

8 RLC Circuis wih AC Power When an RLC circui is driven wih an AC power source, he driving frequency d is he frequency of he power source, while he circui can have a differen resonan frequency. 1/ LC ( R / L) Le s look a hree differen circuis driven by an AC EMF. The device conneced o he EMF is called he load. Wha we are ineresed in is how he volage oscillaions across he load relae o he curren oscillaions. We will find ha he phase relaionships change, depending on he ype of load (resisive, capaciive, or inducive). December 5, 007

9 Leading & Lagging in Phasors The figure shows, in (a), a sine curve S() sin(d) and hree oher sinusoidal curves A(), B(), and C(), each of he form sin(d φ). (a) Rank he hree curves according o he value of φ, mos posiive firs and mos negaive las. Answer (a) C, B, A November 7, 007

10 (b) Which curve corresponds o which phasor in par (b) of he figure? (c) Which curve leads he ohers? (b) (c) 1 >A A >B 3 >S 4 >C November 7, 007

11 A Resisive Load Phasor Diagram: shows he insananeous phase of eiher volage or curren. For a resisor, he curren follows he volage, so he volage and curren are in phase (φ 0). φ f v R R sin d Then i R R sin d R R sin d December 5, 007

12 Resisive Load Con d November 7, 007

13 Resisive Load CURRENT is in phase wih OLTAGE November 7, 007

14 AC Power in a Resisor Poenial drop across he resisor, R Curren in he resisor Power dissipaed in he resisor, P Average power dissipaed in he resisor P average ε ε R max R, R, R R, R P P av R ( ) av R ( ( ) R ( )) av R

15 Roo-Mean-Square alues

16 A Capaciive Load For a capaciive load, he volage across he capacior is proporional o he charge q Q vc sind C C Bu he curren is he ime derivaive of he charge i C dq d C d C d n analogy o he resisance, which is he proporionaliy consan beween curren and volage, we define he capaciive reacance as 1 X C C So ha i. dc C X C The phase relaionship is ha φ 90º, and curren leads volage. d December 5, 007

17 Capaciive Load Con d November 7, 007

18 Capaciors in Alernaing Curren Circuis The poenial drop lags he curren by 90º Q C C Q C dq d ε ε C, max C C, sin CAPACTE REACTANCE C, C, 1 C C, 1 C ( + ; rms π ) C, rms 1 C Power delivered by he emf in he capacior: nsananeous and average

19 Capaciive Load CURRENT Leads OLTAGE By 90 degrees November 7, 007

20 An nducive Load For an inducive load, he volage across he inducor is proporional o he ime derivaive of he curren v L di L d Bu he curren is he ime derivaive of he charge i L L L sin d d d L d L L Again in analogy o he resisance, which is he proporionaliy consan beween curren and volage, we define he inducive reacance as X L L So ha il d. X L The phase relaionship is ha φ +90º, and curren lags volage. L d December 5, 007

21 nducive Load Con d November 7, 007

22 nducors in Alernaing Curren Circuis L d d d L ε ε max L, d L, L, L, sin L L L NDUCTE REACTANCE The poenial drop across he inducor led he curren 90º (ou of phase) L, L ( ; rms L, rms L π ) nsananeous power delivered by he emf o he inducor is no zero The average power delivered by he emf o he inducor is zero.

23 nducive Load CURRENT Lags OLTAGE By 90 degrees November 7, 007

24 Summary Table Circui Elemen Symbol Resisance or Reacance Phase of Curren Phase Consan Ampliude Relaion Resisor R R n phase wih v R 0º (0 rad) R R R Capacior C X C 1/ d C Leads v R by 90º 90º ( π/) C C X C nducor L X L d L Lags v R by 90º +90º (π/) L L X L December 5, 007

25 Driven RLC Series Circuis The Kirchhoff s rules govern he behavior of poenial drops and curren across he circui. (a) When any closed-loop is raversed, he algebraic sum of he changes of poenial mus equal zero (loops rule) (a) A any juncion (branch poin) in a circui where he curren can be divided, he sum of he currens ino he juncion mus equal he sum of he currens ou of he juncion (juncion rule)

26 ) ( ;,, δ C Q R d dq d Q d L d dq C Q R d d L app app

27 Phasors Poenial drop across a resisor can be represened by a vecor R, which is called a phasor. Then, he poenial drop across he resisor R, is he x componen of vecor R, Poenial drop across a series RLC circui d app L + R + d Q C

28 Power delivered o he series RLC circui δ ε δ ε π ε ε 1 ) (, rms app rms av rms rms av R R P resisor he in dissipaed R P P P Power facor: δ δ δ δ 1 / /,,,, rms rms app app app rms app rms av P Z and Z R as Z R P

29 The Transformer A ransformer is a device o raise or lower he volage in a circui wihou an appreciable loss of power. Power losses arise from Joule heaing in he small resisances in boh coils, or in currens loops (eddy currens) wihin he iron core. An ideal ransformer is ha in which hese losses do no occur, 100% efficiency. Acual ransformers reach 90-95% efficiency Because of he iron core, here is a large magneic flux hrough each coil, even when he magneizing curren m in he primary circui is very small. The primary circui consiss of an ac generaor and a pure inducance (we consider a negligible resisance for he coil). Then he average power dissipaed in he primary coil is zero. Why?: The magneizing curren in he primary coil and he volage drop across he primary coil are ou of phase by 90º Secondary coil open circui The poenial drop across he primary coil is 1 N N 1 dφ d urn f here is no flux leakage ou of he iron core, he flux hrough each urn is he same for boh coils, and hen dφ d urn N N 1 1

30 The Transformer A resisance R, load resisance, in he secondary circui A curren will be in he secondary coil, which is in phase wih he poenial drop across he resisance. This curren ses up and addiional flux Φ urn hrough each urn, which is proporional o N. This flux opposes he original flux ses up by he original magneizing curren m in he primary. However, he poenial drop in he primary is deermined by he generaor emf According o his, he oal flux in he iron core mus be he same as when here is no load in he secondary. The primary coil hus draws an addiional curren 1 o mainain he original flux Φurn. The flux hrough each urn produced by his addiional curren is proporional o N11. Since his flux equals Φ urn, he addiional curren 1 in he primary is relaed o he curren in he secondary by N 1, rms N 1, rms 1 1 These curens are 180 º ou of phase and produce couneracing fluxes. Since is in phase wih, he addiional curren 1 is in phase wih he poenial drop across he primary. Then, if here are no losses, rms, rms

31 THANK YOU November 7, 007