1. Lecture 16: Sinusoidal Sources and Phasors Nilsson 9.1-9.6, App. B ENG17 : Circuits I Spring 2015 1 May 21, 2015

2. Overview Complex Numbers Sinusoidal Source Sinusoidal Response Phasors Passive Circuit Elements KVL / KCL 2

3. Complex Numbers: notation 3 Rectangular form: z = a + jb a = Real, b = Imaginary and j = sqrt (-1) Polar form: z = c = amplitude, θ = angle or argument Relationship: Exponential form: z = ce jθ j = sqrt (-1)

4. Graphical Representation 4 z = a + jb =

5. Graphical Representation: example 5 or Conjugate: z = a + jb = & z* = a – jb =

6. Arithmetic Operations 6 Similar to the arithmetic operations of vectors Examples: (1) z 1 = 8 + j16, z 2 = 12 – j3, z 1 +z 2 = ? (2) z 1 =, z 2 =, z 1 +z 2 = ? (3) z 1 = 8 + j10, z 2 = 5 – j4, z 1 z 2 = ? (rectangular form & polar form) (4) z 1 = 6 + j3, z 2 = 3 – j, z 1 /z 2 = ? (rectangular form & polar form)

7. Integer Power and Roots 7 Easier to write the complex number in polar form Examples: (1) z = 3 + j4, z 4 = ? (2), k-th root of z?

8. Overview Complex Numbers Sinusoidal Source Sinusoidal Response Phasors Passive Circuit Elements KVL / KCL 8

9. Sinusoidal Source Basics 9 A source that produces signal that varies sinusoidally with t A sinusoidal voltage source: V m – max amplitude [V] T – period [s] f – frequency [Hz] ω – angular frequency [rad/s] φ – phase angle [°] Periodicity

10. Sinusoidal Source: Units 10 ω – rad/s, ωt – rad φ – ° The unit for ωt and φ should be consistent o Convert ωt from rad to °

11. Root Mean Square (rms) 11 rms: root of the mean value of the squared function For sinusoidal voltage source, General Expression

12. Example 12 A sinusoidal voltage (1)Period? (2)Frequency in Hz? (3)Magnitude at t = 2.778ms? (4)rms value?

13. Useful Identities 13

14. Overview Complex Numbers Sinusoidal Source Sinusoidal Response Phasors Passive Circuit Elements KVL / KCL 14

15. Sinusoidal Response 15 General response of the circuit with a sinusoidal source (1) (2) i(t) = 0 for t < 0 What is i(t) for t ≥ 0? Transient componentSteady-state component

16. Steady State Component 16 1.The steady state response is also sinusoidal. 2.Frequency of the response = Frequency of the source 3.Max amplitude of the response ≠ Max amplitude of the source in general 4.Phase angle of the response ≠ Phase angle of source in general Will use phasor representation to solve for the steady state component in the future.

17. Overview Complex Numbers Sinusoidal Source Sinusoidal Response Phasors Passive Circuit Elements KVL / KCL 17

18. Phasors Defined: complex number that represents the amplitude and phase of a sinusoid. Euler’s Identity 18

19. Overview Complex Numbers Sinusoidal Source Sinusoidal Response Phasors Passive Circuit Elements KVL / KCL 19

20. V-I Relationship for Resistor 20

21. V-I Relationship for Inductor 21

22. V-I Relationship for Capacitor 22

23. Impedance and Reactance 23

24. Overview Complex Numbers Sinusoidal Source Sinusoidal Response Phasors Passive Circuit Elements KVL / KCL 24

25. KVL & KCL 25

26. Recap Complex Numbers Sinusoidal Source Sinusoidal Response Phasors Passive Circuit Elements KVL / KCL 26