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Lecture 16: Sinusoidal Sources and PhasorsNilsson 9.1-9.6, App. B

ENG17 : Circuits I

Spring 2015

May 21, 2015

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Overview

• Complex Numbers• Sinusoidal Source• Sinusoidal Response• Phasors• Passive Circuit

Elements• KVL / KCL

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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)

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Graphical Representation

z = a + jb =

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Graphical Representation: example

or

Conjugate: z = a + jb = & z* = a – jb =

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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)

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Integer Power and RootsEasier to write the complex number in polar form

Examples:

(1) z = 3 + j4, z4 = ?

(2) , k-th root of z?

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Overview

• Complex Numbers• Sinusoidal Source• Sinusoidal Response• Phasors• Passive Circuit

Elements• KVL / KCL

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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

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Sinusoidal Source: Units

• ω – rad/s, ωt – rad

• φ – °

• The unit for ωt and φ should be consistent

o Convert ωt from rad to °

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Root Mean Square (rms)

• rms: root of the mean value of the squared function

• For sinusoidal voltage source,

General Expression

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Example

• A sinusoidal voltage

(1) Period?

(2) Frequency in Hz?

(3) Magnitude at t = 2.778ms?

(4) rms value?

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Useful Identities

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Overview

• Complex Numbers• Sinusoidal Source• Sinusoidal Response• Phasors• Passive Circuit

Elements• KVL / KCL

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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

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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.

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Overview

• Complex Numbers• Sinusoidal Source• Sinusoidal Response• Phasors• Passive Circuit

Elements• KVL / KCL

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Phasors

• Defined: complex number that represents the amplitude and phase of a sinusoid.

• Euler’s Identity

𝑒± 𝑗𝜃=cos𝜃± 𝑗 sin 𝜃

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Overview

• Complex Numbers• Sinusoidal Source• Sinusoidal Response• Phasors• Passive Circuit

Elements• KVL / KCL

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V-I Relationship for Resistor

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V-I Relationship for Inductor

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V-I Relationship for Capacitor

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Impedance and Reactance

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Overview

• Complex Numbers• Sinusoidal Source• Sinusoidal Response• Phasors• Passive Circuit

Elements• KVL / KCL

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KVL & KCL

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Recap

• Complex Numbers• Sinusoidal Source• Sinusoidal Response• Phasors• Passive Circuit

Elements• KVL / KCL