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7 February 2020 CIRCUITS 2

Sinusoids

Adonis S. Santos, ECE

Introduction Definition of Terms Sinusoidal Waveform Phase Relations

Sinusoidal Addition OUTLINE

Introduction

• Sinusoid
  • Signal that has the form of the sine or cosine function
• AC circuits
  • Circuits driven by sinusoidal current or voltage sources

• Periodic Waveform
  • A waveform that continually repeats itself after the same time interval
• Instantaneous Value
  • The magnitude of a waveform at any instant of time
  • Ex. e 1 , e SINUSOIDAL^2 ALTERNATING WAVEFORM

• Peak-to-peak Value
  • full voltage between positive and negative peaks of the waveform
  • Ex. E p-p
• Period (T)
  • time of one complete cycle or the number of seconds per cycle.
  • Seconds (s)
  • Ex. T 1 , T 2 , T 3 SINUSOIDAL ALTERNATING WAVEFORM

• Cycle
  • The portion of a waveform contained in one period of time
• Frequency (f)
  • The number of cycles that occur in 1 s
  • Unit is Hertz (Hz)
    • Heinrich Rudolph Hertz SINUSOIDAL ALTERNATING WAVEFORM

• The sinusoidal waveform is the only alternating waveform whose shape is unaffected by the response characteristics of R, L, and C elements. SINUSOIDAL ALTERNATING WAVEFORM

SINUSOIDAL ALTERNATING WAVEFORM

• For the sinusoidal waveform a) What is the peak value? b) What is the instantaneous value at 0.3 s and 0.6 s? c) What is the peak-to-peak value of the waveform? d) What is the period of the waveform? e) How many cycles are shown? Example 1^ f)^ What^ is the frequency of the waveform?

SINUSOIDAL ALTERNATING WAVEFORM

• Determine the frequency of the waveform Example 3

• Angular Velocity (ω),
  • velocity with which the radius vector rotates about the center
  • rad/s or degree/s
• Angular Velocity (ω) and Period (T) 𝑇 = 2𝜋 𝜔
• Angular Velocity (ω) and Frequency (f) 𝜔 = 2𝜋𝑓 v ( t ) = V m sin ωt SINUSOIDAL ALTERNATING WAVEFORM Vm = the amplitude of the sinusoid ω = the angular frequency in radians/s ωt = the argument of the sinusoid

SINUSOIDAL ALTERNATING WAVEFORM

• Determine the frequency and period of the sine wave Example 5

SINUSOIDAL ALTERNATING WAVEFORM

• Given ω = 200 rad/s, determine how long it will take the sinusoidal waveform to pass through an angle of 90°. Example 6

A

m

sin α

A

m

GENERAL FORMAT sin^ ωt

SINUSOIDAL

VOLTAGE AND

CURRENT

SINUSOIDAL ALTERNATING WAVEFORM

• Given e = 5 sin α, determine e at α = 40°

and α = 0.

• Determine the peak voltage and peak-to-

peak voltage.

Example 8