1. Lecture 13: Complex Numbers and Sinusoidal Analysis Nilsson & Riedel Appendix B, 9.1-9.2 ENG17 (Sec. 2): Circuits I Spring 2014 1 May 13, 2014

2. Overview Complex Numbers Sinusoidal Source Sinusoidal Response 2

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

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 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 [°]

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. Overview Complex Numbers Sinusoidal Source Sinusoidal Response 13

14. Sinusoidal Response 14 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

15. Steady State Component 15 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.

16. Overview Complex Numbers Sinusoidal Source Sinusoidal Response 16