AC Circuits AC Circuit with only R AC circuit with only L AC circuit with only C AC circuit with LRC phasors Resonance Transformers

Transcription

1 A ircuis A ircui wih only A circui wih only A circui wih only A circui wih phasors esonance Transformers Phys 435: hap 31, Pg 1

2 A ircuis New Topic Phys : hap. 6, Pg

3 Physics Moivaion as ime we discovered ha a circui was a naural oscillaor. However, any real aemp o consruc a circui mus accoun for he resisance of he inducor. This resisance will cause oscillaions o damp ou Quesion: Is here any way o modify he circui above o susain he oscillaions wihou damping? Answer: Yes, if we can supply energy a he rae he resisor dissipaes i! How? A sinusoidally varying emf (A generaor) will susain sinusoidal curren oscillaions! i()icos(ω) or v()vcos(ω) Bu firs, le us look a,, separaely before puing hem ogeher. Phys 435: hap 31, Pg 3

4 Phasor Diagram How o graphically represen an A source? i I cos! A phasor is a vecor in he x-y plane, roaing counerclockwise wih angular velociy ω. Magniude is he peak value Projecion on he x-axis is he value a a given ime Same for emf: v V cos! Phys : hap. 31, Pg 4

5 MS curren and volage i I cos! roo-mean-square: I rms i i T T I 1 I! i d! (1 + cos " ) d T T 0 0 I rms I V rms V In he US, he 10-V A volage supplied o household refers o he rms value. Wha is he peak volage? 170 V Phys : hap. 31, Pg 5

6 A ircui onaining only esisance The A source is given as So he volage across he resisor is i I cos! v i I cos! V cos! The resisor volage is in phase wih he curren. The power dissipaed in he resisor is piv, or a an average rae P I rms V rms Phys 435: hap. 31, Pg 6

7 A ircui onaining only Inducor The A source is given i I cos! So he volage across he inducor is v di! I" sin" V cos( d " The inducor volage leads he curren by 90 degrees ) Define reacance of inducor ω, hen V I. Is uni is Ohm. For example, a 0.3-H inducor conneced o a 60-Hz ac-source has a reacance of πx60x Ω. Phys 435: hap. 31, Pg 7

8 A ircui onaining only apacior i I cos! The volage across he capacior is I I q " id sin# cos( # # # 0! 90 0 ) v q / Define reacance of capacior 1/ω, hen V I. Is uni is Ohm. For example, a 1-µF capacior conneced o a 60-Hz ac-source has a reacance of 1/(πx60x10-6 ).7 kω. I v cos( " "! 90 The capacior volage lags he curren by 90 degrees. 0 ) Phys 435: hap. 31, Pg 8

9 Summary: A source Define V I i I cos! v I cos! resisance. v I! cos(! + 90 inducive reacance ω. 0 ) I v cos( " "! 90 capaciive reacance 1/(ω). 0 ) Phys 435: hap. 31, Pg 9

10 Frequency Dependence resisance. inducive reacance ω. capaciive reacance 1/(ω). Phys 435: hap. 31, Pg 10

11 A-driven ircui in Series Given ac-source iicos(ω) e v, v, v represen he volages a a given ime. Physics quesions: Wha is he impedance Z of he enire circui as defined by he peak values VIZ? Does he source volage lead or lag he curren? By how much? In oher words: If one can wrie v v +v +v Vcos(ω+ϕ), hen wha is V and ϕ? Or: How o add he volages v v +v +v, noing ha hey are varying wih ime and in general ou of phase wih each oher? Phys 435: hap. 31, Pg 11

12 A-driven ircui in Series (Graphical Mehod) The goal is o cas he oal volage a any ime ino he form I v v + v + v V cos( " +!) I where φ is he phase angle wih which v leads i. From he riangle: V V ( I) + ( V + ( I + (! V! )! I ) ) i cos! The impedance The phase difference V IZ wih Z + (! an" or cos"! Z ) Phys 435: hap. 31, Pg 1

13 Given: v i I cos! v v A-driven ircui in Series (Trigonomery Mehod) I cos(! + " / ) wih! I Soluion: i I cos! cos(! #" / ) wih V cos" I V sin" and I(! ) 1! V Find he sum: v V cos(! IZ v + v wih Z + v I cos(! ) # + ") + (! an" or cos" I( V cos" cos(! ) # V Used: (desired form) # cos( A + B) cos Acos B! sin sin" sin(! )! )sin(! ) Z Asin B ) Phys 435: hap. 31, Pg 13

14 The sign of he phase difference φ The oal volage a ime is v V cos( " +!) i I cos! where φ is he phase angle by which volage leads he curren. The figure shows he case > which means φ is posiive. Or he volage leads he curren. If <, hen φ is negaive. Or he volage lags he curren. The figure is I can be seen more easily from anφ.! an" Phys 435: hap. 31, Pg 14

15 Z # 1 # an" The Impedance Triangle + (!! ) or cos" heck special cases: Z General resuls, valid for any combinaion of,, in series. For example, if circui, se 0. only : 0, 0, so Z,! 0 only : 0, 0, so Z,! only : 0, 0, so Z,"! 90 0 Phys : hap. 31, Pg 15

16 Who is driving? We derived he resuls wih a driving curren i I cos! v V cos( " +!) φ is he angle by which volage leads he curren. The same resuls for Z and φ apply if we have a driving emf v V cos! i I cos( " #!) φ is he same angle by which curren lags he volage. I s all relaive: he physics is he same. Phys 435: hap. 31, Pg 16

17 Phys 435: hap. 31, Pg 17 Frequency Dependence Frequency Dependence Z c Z IZ V V v I i!! + + " " # # $ # # os or an 1 ) ( ) cos( cos

18 The Power Facor The power is only dissipaed in he resisor, so he average power is P I rms Bu Z cos! P 1 I Z cos! I V cos! IV cos! rms rms rms The facor cosφ is called he power facor. For example: For a pure resisor (φ0), cosφ1. For a pure inducor (φ90 0 ) or capacior (φ-90 0 ), cosφ0. Phys 435: hap 31, Pg 18

19 onceptes 31.1 (Pre) Phasor onsider his phasor diagram. e us assume ha i describes a series circui conaining a resisor, a capacior, and an inducor. The curren in he circui has ampliude I, as indicaed in he figure. Which of he following choices gives he correc respecive labels of he volages across he resisor, he capacior, and he inducor? Phasor Diagram 1) V 1 ; V ; V 3 ) V 1 ; V ; V 4 3) V 1 ; V 4 ; V 4) V 3 ; V ; V 4 5) V 1 ; V ; V 3 6) V 3 ; V 4 ; V Phys 435: hap 31, Pg 19

20 onceptes 31. (Pre) A ircui An A-curren source (purple line) is conneced in series o a resisor, an inducor, and a capacior. The volage across each elemen (blue line) is given separaely in he picures shown below. Give he order in which he picures correspond o he inducor, resisor, and capacior, respecively. 1) AB ) BA 3) AB 4) AB 5) BA 6) BA (A) (B) () Phys 435: hap 31, Pg 0

21 onceptes 31.3 (Pre) ircui In an -- series circui as shown, he curren has a very small ampliude if he emf oscillaes a a very high frequency. Which circui elemen causes his behavior? 1) he resisor ) he inducor 3) he capacior 4) misleading quesion he curren acually has a very large ampliude if he frequency is very high Phys 435: hap 31, Pg 1

22 esonance in A-circui New Topic Phys 435: hap. 31, Pg

23 ecall: esonance in A ircui I V Z ( ) For fixed,, he curren I will be a maximum a he resonan frequency ω 0 which makes he impedance Z purely resisive. Z is minimum when: So he frequency a which his condiion is saisfied is given from: ωo 1 ω ω o 1 o Noe ha his resonan frequency is idenical o he naural frequency of he circui by iself! A his frequency, he curren and he driving volage are in phase! cos φ 1, soφ 0 Z + V Phys : hap. 31, Pg 3

24 esonance in A ircui A resonance, I and V are in phase. V, I and P are a heir maximum. (V100 V,.0 H, 0.5 µf) Phys 435: hap. 31, Pg 4

25 Applicaions of A ircuis apacior as filers Tweeer and woofer 1!f!f Tuning in a adio or TV saion by resonance. f 1! Phys 435: hap. 31, Pg 5

26 More abou A circuis Impedance maching: when (or Z ) 1 1 Z maximum power is delivered from circui 1 o circui. Three-phase A (4-wire ransmission lines) V V V 1 3 V 0 V V 0 0 sin" sin( " sin( " +! /3) + 4! /3) 1) More power delivered (3 imes) ) Smooher flow. Phys 435: hap. 31, Pg 6

27 Transformers New Topic Phys 435: hap. 31, Pg 7

28 Transformers Transformers change alernaing (A) volage o a bigger or smaller value Inpu A volage V p in he primary produces a flux hanging flux in secondary induces emf V s Same ΔΦ /Δ!! Transformer equaion: V s V p N N s p Phys 435: hap. 31, Pg 8

29 Transformers Nohing comes for free, however! volage increase comes a he cos of curren oupu power canno exceed inpu power power in power ou (assume no hea loss) I V p p I s V s If volage increases, hen curren decreases If volage decreases, hen curren increases Phys 435: hap. 31, Pg 9

30 Example: Transmission ines An average of 10 kw of elecric power is sen o a small own from a power plan 10 km away. The ransmission lines have a oal resisance of 0.4 Ω. alculae how much power is saved if he volage is sepped up from 40 V o 4,000 V and hen down o 40 V again, raher han simply ransmiing a 40 V. Phys 435: hap. 31, Pg 30

31 onceptes 31.4(Pre) Transformers Wha is he volage across he ligh bulb? (1) 30 V () 60 V (3) 10 V (4) 40 V (5) 480 V 10 V Phys 435: hap. 31, Pg 31