1. Introduction to Electrical & Computer Engineering Sine Waves & Simple Complex Numbers 1 Dr. Cynthia Furse University of Utah Dr. Cynthia Furse University of Utah

2. Sine Waves & Simple Complex Numbers Sine Waves in the… –TIME domain –FREQUENCY (phasor) domain Impedance (Z) –Resistance R –Capacitance 1/jωC –Inductance jωL 2

3. 3 http://mynasadata.larc.nasa.gov/science-processes/electromagnetic- diagram/

4. Sinusoidal Voltage AC = Alternating Current 4 v(t) = Vm cos(2πf t + φ) f(Hz) = 1/ T(sec)

5. Sinusoidal Voltage AC = Alternating Current 5 v(t) = Vm cos(2πt/T + φ) T(sec)=1 / f(Hz)

6. Capacitor V and I 6 i(t) = ?

7. Capacitor V and I 7 i(t) = 1 cos (2π t/T +0) (amps) T = 6.2 seconds A=1V

8. Capacitor V and I 8 v(t) = ?

9. Capacitor V and I Is φ + or ? 9 + time = - φ Voltage phase LAGS current

10. Capacitor V and I Finding φ using time 10 φ= -2π (1.55 sec / T ) = -π/2 φ T = 6.2 seconds

11. Capacitor V and I Finding φ using degrees or radians 11 φ=-1/4 cycle of the wave Full cycle = 360° so… φ=-360°/4= -90°=-π/2 φ

12. Capacitor V and I 12 v(t) = 1 cos (2π t/T - π/2) (volts) T = 6.2 seconds A=1V φ=-90°=-π/2

13. 13

14. Inductor V and I 14 i(t) = 1 cos (2π t/T +0) (amps)

15. Inductor V and I 15 v(t) = 1 cos (2π t/T + π/2) (volts) -time = + φ Voltage phase LEADS current

16. 16 http://mynasadata.larc.nasa.gov/science-processes/electromagnetic- diagram/ But really, ECE thinks in terms of frequency (f Hz), not time period (T)

17. 17 Inductor v(t) = cos (2π f t + π/2) Capacitor v(t) = cos (2π f t - π/2) i(t) = cos (2πf t)

18. 18 Inductor v(t) = cos (ω t + π/2) Capacitor v(t) = cos (ω t - π/2) ω(rad/s)=2πf (Hz) i(t) = cos (ωt)

19. 19 Inductor v(t) = cos (ω t + π/2) Capacitor v(t) = cos (ω t - π/2) ω(rad/s)=2πf (Hz) i(t) = cos (ωt)

20. Time Domain (cosine) math is not always convenient…. Quick! In your head! Add two voltages … cos (ω t –π/2) + cos (ω t +π/2) Multiply two voltages … cos (ω t –π/2) x cos (ω t +π/2) 20

21. Phasor Domain is easy to + and x sine waves… Complex numbers make ECE easier 21

22. Capacitor voltage… 22

23. 23 Inductor v(t) = cos (ω t + π/2) Capacitor v(t) = cos (ω t - π/2) ω(rad/s)=2πf (Hz)

24. Phasor Math 24

25. Phasor Math 25

26. Phasor Math – Adding (+/-) Convert to Rectangular Form V1 = xV1 + j yV1 V2 = xV2 + j yV2 V1 + V2 = (xV1 + xV2) + j (yV1+yV2) V1 - V2 = (xV1 - xV2) + j (yV1-yV2) 26

27. Phasor Math – Multiplication (x/divide) 27

28. Examples 28

29. Impedance Ohm’s Law v(t) = i(t) R Ohm’s Law (impedance) V = I Z Impedance Z (ohms) = V / I = Zx + j Zy ohms Resistance Z = R Capacitance Zc = 1/(jωC) Inductance ZL = jωL 29

30. Introduction to Electrical & Computer Engineering Antelope Island, Great Salt Lake, Utah 30 Dr. Cynthia Furse University of Utah