Lecture 13: Complex Numbers and Sinusoidal Analysis Nilsson & Riedel Appendix B, 9.1-9.2

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

2. Overview • Complex Numbers • Sinusoidal Source • Sinusoidal Response

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

4. Graphical Representation z = a + jb =

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

6. Arithmetic Operations Similar to the arithmetic operations of vectors Examples: • z1 = 8 + j16, z2 = 12 – j3, z1+z2 = ? • z1 = , z2 = , z1+z2 = ? • z1 = 8 + j10, z2 = 5 – j4, z1z2 = ? (rectangular form & polar form) • z1 = 6 + j3, z2 = 3 – j, z1/z2 = ? (rectangular form & polar form)

7. Integer Power and Roots Easier to write the complex number in polar form Examples: • z = 3 + j4, z4 = ? • , k-th root of z?

8. Overview • Complex Numbers • Sinusoidal Source • Sinusoidal Response

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

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

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

12. Example • A sinusoidal voltage • Period? • Frequency in Hz? • Magnitude at t = 2.778ms? • rms value?

13. Overview • Complex Numbers • Sinusoidal Source • Sinusoidal Response

14. Sinusoidal Response • 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 component Steady-state component

15. Steady State Component The steady state response is also sinusoidal. Frequency of the response = Frequency of the source Max amplitude of the response ≠ Max amplitude of the source in general 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