GROUNDHOG DAY!

1. GROUNDHOG DAY! Alan Murray

2. Agenda • RC circuit, AC signal • using trigonometry (J?) • using phasors (K?) • using complex numbers (L?) • worked examples complex numbers Alan Murray – University of Edinburgh

3. T=1/f VS0 VS Phase Notation Alan Murray – University of Edinburgh

4. This is all we need ... Alan Murray – University of Edinburgh

5. I VR VS VC RC Phasor (K?) • Choose I =IC =IR horizontal • VR = RI (Ohm’s Law) VRalso horizontal i.e. VR and I are in phase • CIVIL → I leads VC by π/2 • Or VC is -π/2 behind I • VC points ↓ … rotated by -π/2 • VS = VC + VR Sonny & Cher sing…"I got you babe" ... Alan Murray – University of Edinburgh

6. I VR VS VC RC Phasor (K?) Alan Murray – University of Edinburgh

7. I VR VS VC RC Phasor (K?) Alan Murray – University of Edinburgh

8. I VS RC Phasor (K?) I and VR are in phase VC lags I and VR by π/2 VS is at an angle -Φ in between Plug in numbers forR,C and ω =2πf to get values for VR, VC, I and Φ (Leave this as a worked example, once we have the same result from the complex number method and from trigonometry) VR = RI Φ VC = I/ωC Alan Murray – University of Edinburgh

9. I VR Φ VS VC RC Complex numbers (L?)Here’s the idea ... • Write all currents/voltages as Cej(ωt+phase) • I = I0ejωt • VR = VR0ejωt = ZRI0ejωt • VC = VC0ejωt = ZCI0ejωt • VS = VS0ejωt= VR + VC= (ZR + ZC)I0ejωt • ωt spins the complex “phasors” • I0 and VR0are real • VC0is imaginary because of an e-jπ/2 term • -jπ/2 puts the CIV in CIVIL • VS0 is complex and will includean e-jΦ term • -Φ is the phase of VS • To get real voltages and currents - take real parts • I = Re(I0ejωt) • VR = Re(VR0ejωt) • VC = Re(VC0ejωt) • VS = Re(VS0ejωt) • The ejωt terms will cancel. We will leave them in for now J Alan Murray – University of Edinburgh

10. RC Complex numbers (L?) • And impedances, ZR, ZC? • VR = RI, VR and I are in phase “j x” = “I leads VC by 90°” The 90° phase shift is dealt with by the mathsautomatically.We no longer have to think about it explicitly. J Alan Murray – University of Edinburgh

11. RC Complex numbers (L?) • This chooses I0 to be real, = “horizontal in the phasor diagram” Sonny & Cher sing…"I got you babe" ... X X Alan Murray – University of Edinburgh

12. And if we had anticipated the cancellation of ejωt … • This chooses I0 to be real, = “horizontal in the phasor diagram” Alan Murray – University of Edinburgh

13. RC Complex numbers (L?) Insert numbers for |VS0|, R, C, f and thus ω Then … Alan Murray – University of Edinburgh

14. IS VS C R IC IR Fill in the blanks Draw a phasor diagram for VS, VR, VC, IS, IR and IC here Similar expressions forVR and VC VS= VR= VC Now use Ohm’s Law to Write an expression for IR, complete with its ejωt+phase Then do the same for IC And tidy it up a little. Now write this expression IC as IC0ejωt+phase, with allthe phase information inthe exponential part. Alan Murray – University of Edinburgh

15. From the phasor diagram From ejωt analysis Comparison Note – in the phasor equations, the sinusoidal nature of the voltages andcurrents and the phase differences between them are not spelt out. In the complex-number version these are explicitas the ejωt and e-jπ/2 terms respectively

16. (Notation) (Impedances) (Ohm’s Law on the total impedance) (Ohm, continued … ejωt cancels out) (More Ohm’s Law on R and C individually,then add VR0 and VC0 to get VS0) Summary : How to use ejωt Plug in numbers, take real parts, calculate relative phases NB – We can choose to leave out the ejωtthroughout the calculation, although it must go back in atthe end to retrieve the sinusoidal voltages and currents explicitly. Alan Murray – University of Edinburgh

17. Worked examples lecture Clicker exercise

18. RC Trigonometry! (J?)Should be easy, shouldn't it? Sonny & Cher sing…"I got you babe" ... • I = I0cos(ωt) • Equivalent to choosing I horizontal • VR = RI0cos(ωt) • VR is in phase with I • VC = I/(ωC) and lags I by π/2 (CIVIL) • VC = I0cos(ωt-π/2)/(ωC) … which is the same as … • VC = I0sin(ωt)/(ωC) • VS = VR + VC • VS = I0Rcos(ωt) + I0sin(ωt)/(ωC) • VS = I0[Rcos(ωt) + sin(ωt)/(ωC)] • VS = I0R[cos(ωt) + sin(ωt)/(RωC)] Alan Murray – University of Edinburgh

19. RC Trigonometry! (J?)Should be easy, shouldn't it? • VS = I0R[cos(ωt) + sin(ωt)(RωC)-1] • Now for a very non-intuitive step … • Set Ф = arctan[(RωC)-1], • tan(Ф) = (RωC)-1 = sin(Ф) cos(Ф) • VS = I0R[cos(ωt) + sin(ωt)tan(Ф)] • VS = I0R[cos(ωt)cos(Ф) + sin(ωt)sin(Ф)] cos(Ф) • VS = I0R cos(ωt+Ф) cos(Ф) • fatigue setting in? Alan Murray – University of Edinburgh

20. RC Trigonometry! (J?)Should be easy, shouldn't it? X X • VS = I0R cos(ωt+Ф) = VS0 cos(ωt+Ф) cos(Ф) • ... and after several lines of VERY tedious trigonometry(!) ... • cos(Ф) = [1+(ωRC)-2]-½ • VS is at an angle of Ф = tan-1(ωRC-1) Alan Murray – University of Edinburgh

21. Message? • Trigonometry is familiar from school, but really messy - even for only two components (R and C) • it gets MUCH worse VERY rapidly for more • Phasors are excellent for seeing what is happening, but make for messy algebra. • again - MUCH worse for 3, 4 or more components • because phasors take you straight back to trigonometry • Complex numbers are initially threatening, but make the actual maths MUCH easier, ONCE YOU HAVEACCEPTED THE IDEA. • Groundhog Day finally ends ... • Goodnight Alan Murray – University of Edinburgh