R, L and C components in AC circuits

Transcription

1 , L and C componnts in AC circuits sistor Gnral I-V dpndncis v i I-V dpndncis for sin-wavforms v i Capacitor i( t) C d v( t) dt? Inductor v( t) L di( t) dt?

2 Objctiv: to find th mathmatical functions or transformations rplacing th tim drivativs with algbraic actions, lik multiplication, division, tc. Exampl: if y(t) ωt, thn: dy dt ωt ω ω y Issu: th actual lctrical signals ar sinusoidal, not ωt typs; thr is no simpl transformation from cos(ωt) or sin(ωt) typ wavforms into ωt function and back

3 Eulr s formula Th fundamntal rlationship bridging complx numbrs and AC signals is providd by th Eulr s formula: φ j cos( φ) + jsin( φ)

4 Eulr s formula Gomtrical intrprtation φ j cos( φ) + jsin( φ) j*sin(φ) jy jφ φ Th modulus of a complx numbr N (th lngth of th arrow) cos(φ) x N + y x If N jφ, thn using Eulr s formula N ( sinφ) + ( cos ) φ

5 Eulr s formula Gomtrical intrprtation φ j cos( φ) + jsin( φ) jy jφ j*sin(φ) φ cos(φ) N jφ N ( sinφ) + ( cos ) φ

6 Any complx numbr can b asily dfind using Eulr s formula: jφ cos( φ) + jsin( φ) jy jφ Th modulus of jφ is j*sin(φ) φ cos(φ) x is th modulus; φ is th argumnt

7 Th Eulr s form dscribs a polar form of a complx numbr and hnc a phasor: Phasor dscribing AC currnt or voltag: N L φ Complx numbr in th Eulr s form: j*sin(φ) jy φ jφ Dscribs th complx numbr with th modulus and th angl φ cos(φ) x ( N L φ ) N jφ Phasor of an AC currnt or voltag Complx numbr in Eulr s form

8 An important cas of th Eulr s formula: φ π/ ( π / ) cos( / ) + jsin( / ) 0 + j j j π π j j ( π / ) Exampl: find j [ ] j( π / ) ( / ) j( / ) / π 4 j j

9 ultiplication and division of complx numbrs using Eulr s formula: ; ; jϕ jϕ N N Following rgular algbraic ruls, ( ) ) ( ϕ ϕ ϕ ϕ + j j j N N Th modulus of th product th product of th moduli; Th argumnt th sum of th argumnts ( ) ) ( / ϕ ϕ ϕ ϕ j j j N N Th modulus of th quotint th quotint of th moduli; Th argumnt th diffrnc of th argumnts

10 Using Eulr s formula to find tim drivativs of complx numbrs: Suppos th argumnt φ of a complx numbr N is a linar function of tim: φ ω t: th tim drivativ bcoms N j t ω dn j jω t ω dt or: d d d ( x ) x ( k x ) k x d x x k dn j ω N dt

11 AC circuit analysis using complx numbrs Th simplicity of tim drivativs using complx numbrs in th Eulr s form opns up a simpl way to analyz AC circuits. Th approach:. Bridg th actual wavform to th complx variabl in th Eulr s form; i.. crat a complx imag of a ral wavform.. Apply th KVL, KCL and th I-V rlationships to th complx imags of voltags and currnts in th AC circuit. (Th math involvd is much simplr than that rquird to solv th actual circuit). 3. Find th actual currnt or voltag wavforms by taking th ral part of th rsulting complx variabl.

12 Complx imags of rsistor voltag and currnt Th voltag across th rsistor: v( t) V cos( ωt ). Th phasor corrsponding to th rsistor voltag v(t) V jω t Phasor dsign ruls : Th phasor modulus th ral voltag amplitud. Th phasor argumnt ωt

13 . Find th complx imags corrsponding to th currnt through rsistor. Us th sam ruls that apply to th actual voltag currnt rlationship: i v(t) / V t j ω 3. Find th actual rsistor currnt by taking th ral part of th complx currnt: Phasors of rsistor voltag and currnt () V jω t V V cos( ωt) + j sin( ωt) Th rsistor currnt V i ( t) cos( ωt )

14 Complx imags of capacitor voltag and currnt Th voltag across th capacitor: v( t) V cos( ωt ) Not that th actual capacitor currnt can b found as: d v ic C C ω V sin( ωt) dt C ω V cos( ωt + π / ) W will now find th currnt using th complx imag tchniqu.

15 Complx imags of capacitor voltag and currnt (). Givn th actual voltag v( t) V cos( ωt ) Th complx voltag corrsponding to th actual voltag v(t) (shown as bold v): v(t) V jω t. Find th complx capacitor currnt using th sam ruls that apply to th actual voltag and currnt Using v(t ) ic C C jωv t j jπ/ jω t jω t jπ / jω t C ω ω j( ω t + π / ) ωc V i j C V C V

16 Complx imags of capacitor voltag and currnt (3) 3. Tak th ral part of th complx currnt : j( t / ) ic C V ω + π ω ωc V cos( ωt + π / ) Compar to th currnt found by taking th tim drivativ of th capacitor voltag: i C ω V cos( ωt + π / ) C

17 Complx imags of inductor voltag and currnt Th currnt across th inductor: i( t) I cos( ωt ) Not that th actual inductor voltag can b found as: d i vl L L ω I sin( ωt) dt L ω I cos( ωt + π / ) W will now find th voltag using th complx imag tchniqu.

18 Complx imags of inductor voltag and currnt (). Givn th actual currnt i( t) I cos( ωt ) Th complx currnt corrsponding to th actual currnt i(t) (shown as bold v): i(t) I jω t. Find th complx inductor voltag using th sam ruls that apply to th actual voltag and currnt Using i(t ) vl L L jω I t j jπ/ jω t jω t jπ / jω t L ω ω j( ω t + π / ) ω L I v j L I L I

19 Complx imags of inductor voltag and currnt (3) 3. Tak th ral part of th complx voltag: j( t / ) vl L I ω + π ω ω L I cos( ωt + π / ) Compar to th currnt found by taking th tim drivativ of th capacitor voltag: v L ω I cos( ωt + π / ) L

20 sistor I-Vs in tim-domain and on th complx plan Assuming v(t) and i(t) ar th sinusoidal signals with th angular frquncy ω: Tim domain (ral variabls) v ( t) i( t) Complx plan ( rotating phasors) ( ) i t Vˆ Iˆ ω Capacitor i( t) C d v( t) dt ˆ j t C jω C V ˆ ω I Inductor di( t) v( t) L dt Diffrntial quations ˆ j t j L ˆ ω L ω I V Linar V-I dpndncis similar to th Ohm s law Not that th trm jωt can b omittd: thy simply rmind you what th angular frquncy is; th rmaning phasors ar calld th complx amplituds.

21 ,C and L I-Vs on th complx plan sistor Complx amplituds Vˆ Iˆ Complx plan quivalnt circuit sistor with th rsistanc Capacitor Vˆ C ˆ jω C Vˆ I C ( / jω C) ˆ I Quazi-Ohmic (i.. linar) componnt with th complx impdanc Z C /(jωc) Inductor ˆ jω L V L Iˆ Quazi-Ohmic (i.. linar) componnt with th complx impdanc Z L jωl