Upload
harriet-howard
View
230
Download
2
Tags:
Embed Size (px)
Citation preview
1
Lecture 16: Sinusoidal Sources and PhasorsNilsson 9.1-9.6, App. B
ENG17 : Circuits I
Spring 2015
May 21, 2015
2
Overview
• Complex Numbers• Sinusoidal Source• Sinusoidal Response• Phasors• Passive Circuit
Elements• KVL / KCL
3
Complex Numbers: notation
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 = cejθ j = sqrt (-1)
6
Arithmetic OperationsSimilar to the arithmetic operations of vectors
Examples:
(1) z1 = 8 + j16, z2 = 12 – j3, z1+z2 = ?
(2) z1 = , z2 = , z1+z2 = ?
(3) z1 = 8 + j10, z2 = 5 – j4, z1z2 = ? (rectangular form & polar form)
(4) z1 = 6 + j3, z2 = 3 – j, z1/z2 = ? (rectangular form & polar form)
7
Integer Power and RootsEasier to write the complex number in polar form
Examples:
(1) z = 3 + j4, z4 = ?
(2) , k-th root of z?
8
Overview
• Complex Numbers• Sinusoidal Source• Sinusoidal Response• Phasors• Passive Circuit
Elements• KVL / KCL
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 [°]
Periodicity
10
Sinusoidal Source: Units
• 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)
• rms: root of the mean value of the squared function
• For sinusoidal voltage source,
General Expression
12
Example
• A sinusoidal voltage
(1) Period?
(2) Frequency in Hz?
(3) Magnitude at t = 2.778ms?
(4) rms value?
14
Overview
• Complex Numbers• Sinusoidal Source• Sinusoidal Response• Phasors• Passive Circuit
Elements• KVL / KCL
15
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
16
Steady State Component
1. The steady state response is also sinusoidal.2. Frequency of the response = Frequency of the source3. Max amplitude of the response ≠ Max amplitude of the
source in general4. 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
18
Phasors
• Defined: complex number that represents the amplitude and phase of a sinusoid.
• Euler’s Identity
𝑒± 𝑗𝜃=cos𝜃± 𝑗 sin 𝜃
19
Overview
• Complex Numbers• Sinusoidal Source• Sinusoidal Response• Phasors• Passive Circuit
Elements• KVL / KCL
24
Overview
• Complex Numbers• Sinusoidal Source• Sinusoidal Response• Phasors• Passive Circuit
Elements• KVL / KCL