ECE 2100 Circuit Analysis Lessn 25 Chapter 9 & App B: Passive circuit elements in the phasr representatin Daniel M. Litynski, Ph.D. http://hmepages.wmich.edu/~dlitynsk/
ECE 2100 Circuit Analysis Lessn 23 Chapter 9 & App B: Intrductin t Sinusids & Phasrs
Sinusids and Phasr Chapter 9 9.1 Mtivatin 9.2 Sinusids features 9.3 Phasrs 9.4 Phasr relatinships fr circuit elements 9.5 Impedance and admittance 9.6 Kirchhff s laws in the frequency dmain 9.7 Impedance cmbinatins 3
9.1 Mtivatin (1) Hw t determine v(t) and i(t)? v s (t) = 10V Hw can we apply what we have learned befre t determine i(t) and v(t)? 4
9.2 Sinusids (1) A sinusid is a signal that has the frm f the sine r csine functin. A general expressin fr the sinusid, v( t) V sin( t ) m where V m = the amplitude f the sinusid ω = the angular frequency in radians/s Ф = the phase 5
9.2 Sinusids (2) A peridic functin is ne that satisfies v(t) = v(t + nt), fr all t and fr all integers n. T 2 f 1 T Hz 2f Only tw sinusidal values with the same frequency can be cmpared by their amplitude and phase difference. If phase difference is zer, they are in phase; if phase difference is nt 6 zer, they are ut f phase.
9.2 Sinusids (3) Example 1 Given a sinusid, 5sin(4t 60 ), calculate its amplitude, phase, angular frequency, perid, and frequency. Slutin: Amplitude = 5, phase = 60, angular frequency = 4 rad/s, Perid = 0.5 s, frequency = 2 Hz. 14
9.2 Sinusids (4) Example 2 Find the phase angle between i1 4sin(377t and i 5cs(377t 40 ), des i 1 lead r lag i 2? 2 25 ) Slutin: Since sin(ωt+90 ) = cs ωt i 1 i 2 4sin(377t 5sin(377t 25 40 90 ) 4sin(377t ) 5sin(377t 180 25 50 ) ) 4sin(377t 205 ) therefre, i 1 leads i 2 155. 15
9.3 Phasr (1) A phasr is a cmplex number that represents the amplitude and phase f a sinusid. It can be represented in ne f the fllwing three frms: Rectangular z x jy r(cs j sin ) Plar z r Expnential j z re where r x 2 tan 1 y y x 2 16
9.3 Phasr (2) Example 3 Evaluate the fllwing cmplex numbers: a. [(5 j2)( 1 j4) 560 ] b. 10 j5 340 3 j4 10 30 Slutin: a. 15.5 + j13.67 b. 8.293 + j2.2 17
9.3 Phasr (3) Mathematic peratin f cmplex number: 1. Additin 2. Subtractin 3. Multiplicatin 4. Divisin z 1 z2 y z z ( x1 x2 ) j( y1 2 ) 1 z2 y ( x1 x2) j( y1 2) 1z2 r1 r2 1 2 z z 1 1 1 2 2 2 r r 5. Reciprcal 6. Square rt 1 1 z r z r 2 7. Cmplex cnjugate z x jy r re j 8. Euler s identity e j cs j sin 18
9.3 Phasr (4) Transfrm a sinusid t and frm the time dmain t the phasr dmain: v( t) V cs( t m ) V V m (time dmain) (phasr dmain) Amplitude and phase difference are tw principal cncerns in the study f vltage and current sinusids. Phasr will be defined frm the csine functin in all ur prceeding study. If a vltage r current expressin is in the frm f a sine, it will be changed t a csine by subtracting frm the phase. 19
9.3 Phasr (5) Example 4 Transfrm the fllwing sinusids t phasrs: i = 6cs(50t 40 ) A v = 4sin(30t + 50 ) V Slutin: 6 40 a. I A b. Since sin(a) = cs(a+90 ); v(t) = 4cs (30t+50 +90 ) = 4cs(30t+140 ) V Transfrm t phasr => V 4140 V 20
9.3 Phasr (6) Example 5: Transfrm t the sinusids crrespnding t phasrs: a. b. V 1030 I j(5 j12) V A Slutin: a) v(t) = 10cs(t + 210 ) V b) Since I 12 j5 5 tan i(t) = 13cs(t + 22.62 ) A 12 2 2 1 ( 5 12 ) 13 22.62 21
9.3 Phasr (7) The differences between v(t) and V: v(t) is instantaneus r time-dmain representatin V is the frequency r phasr-dmain representatin. v(t) is time dependent, V is nt. v(t) is always real with n cmplex term, V is generally cmplex. Nte: Phasr analysis applies nly when frequency is cnstant; when it is applied t tw r mre sinusid signals nly if they have the same frequency. 22
9.3 Phasr (8) Relatinship between differential, integral peratin in phasr listed as fllw: v(t) V V dv dt vdt jv V j 23
9.3 Phasr (9) Example 6 Use phasr apprach, determine the current i(t) in a circuit described by the integr-differential equatin. di 4i 8idt 3 50cs(2t 75) dt Answer: i(t) = 4.642cs(2t + 143.2 ) A 24
9.3 Phasr (10) In-class exercise fr Unit 6a, we can derive the differential equatins fr the fllwing circuit in rder t slve fr v (t) in phase dmain V. d v dt 2 2 5 3 dv dt 0 20v 0 400 3 sin(4t 15 Hwever, the derivatin may smetimes be very tedius. Is there any quicker and mre systematic methds t d it? ) 25
9.3 Phasr (11) The answer is YES! Instead f first deriving the differential equatin and then transfrming it int phasr t slve fr V, we can transfrm all the RLC cmpnents int phasr first, then apply the KCL laws and ther therems t set up a phasr equatin invlving V directly. 26
9.4 Phasr Relatinships fr Circuit Elements (1) Resistr: Inductr: Capacitr: 27
9.4 Phasr Relatinships fr Circuit Elements (2) Summary f vltage-current relatinship Element Time dmain Frequency dmain R v Ri V RI L C v L i C di dt dv dt V jli I V jc 28
Example 7 9.4 Phasr Relatinships fr Circuit Elements (3) If vltage v(t) = 6cs(100t 30 ) is applied t a 50 μf capacitr, calculate the current, i(t), thrugh the capacitr. Answer: i(t) = 30 cs(100t + 60 ) ma 29