AEEL 1102 - Fundamentals of Electricity II

Topic 1 - Sinusoidal Alternating Waveforms

Course Learning Outcomes

• Understand the response of resistors, inductors, and capacitors to the application of a sinusoidal voltage or current.

Introduction to Alternating Waveforms

• Alternating Waveforms: The term 'alternating' indicates that the waveform alternates between two prescribed levels in a time sequence.

• Alternating Current (ac): Denoted as 'ac', refers to the alternating current itself.

Characteristics and Definitions of Sinusoidal ac Voltage

• Focus: The primary focus is on ac voltage.

  • Generated by power utilities.

  • Requires a §higher level of mathematical understanding.

• Sources of ac Power:

  • Generating plants

  • Portable ac generators

  • Wind-power stations

  • Solar panels

  • Function generators (generate ac voltage with controlled characteristics).

Definitions Related to Sinusoidal ac Voltage

• Instantaneous Value (e): The value of the waveform at any specific instant of time (e.g., e1 at time t1).

• Peak Amplitude (Em): The maximum value of the waveform.

• Peak-to-Peak Value (Ep-p): The total distance from the highest peak to the lowest peak in the waveform.

• Periodic Waveform: A waveform that repeats after a specific interval (T).

• Period (T): The duration of one complete cycle of the waveform.

• Cycle: Represents one complete waveform contained within one period.

• Frequency (f): Measures the number of cycles per second (cps); defined as:
  $1 ext{ Hz} = 1 ext{ cycle per second (cps)}$

• Important Parameters for Sinusoidal Voltage:

  • Vertical scale shows volts or amperes.

  • Horizontal scale is in units of time.

Example 13.1 - Sinusoidal Waveform Parameters

• For a given sinusoidal waveform:

  • a. Peak value: 8 V

  • b. Instantaneous values:

  • At 0.3 s: 8 V

  • At 0.6 s: 0 V

  • c. Peak-to-peak value: 16 V

  • d. Period: 0.4 s

  • e. Number of cycles: 3.5 cycles

  • f. Frequency: 2.5 cps (or 2.5 Hz).

Defined Polarities and Direction in Waveforms

• Notations:

  • e, i: Used for quantities that change over time.

  • E, I: Used for quantities that do not change over time.

• If voltage is above the axis, it is positive; similarly for current.

The Sinusoidal Waveform

• Radian Definition: A radian is defined such that the length of the circle portion equals the radius.

  • Conversion:

  • $2 ext{π rad} = 360⁰$

  • $1 ext{ rad} ext{ is approximately } 57.3⁰$

• Equation Conversions:

  • Radians to Degrees:
    $ext{Degrees} = ( ext{Radians} imes rac{180}{ ext{π}} )$

  • Example calculations for specific angles (30°, 90°, 270°) expressed in radians.

Plotting the Sinusoidal Waveform

• Comparisons shown in figures plotting the sine wave in degrees vs. radians.

Angular Velocity and Frequency Relationships

• The time required to complete one revolution is equal to the period (T) of the sinusoidal waveform. It is defined by the equation:
  $ext{Angular Velocity} (ω) = rac{2 ext{π}}{T}$

• Higher frequency of the sinusoidal waveform results in higher angular velocity.

Example 13.4 - Angular Velocity Calculation

• Given: Frequency of 60 Hz

• Find: Angular Velocity

  • Solution:
    $ω = 2 ext{πf} = (2 ext{π})(60 ext{ Hz}) = 377 ext{ rad/s}$

• This is related to the frequency predominance of 60 Hz.

General Format for Sinusoidal Voltage or Current

• Basic Mathematical Format:

  • Voltage: $e(t) = E_m ext{sin}(ωt + φ)$

  • Current: $i(t) = I_m ext{sin}(ωt + φ)$

• Definitions:

  • $E<em>m$ or $I</em>m$: Peak value.

  • $φ$: Phase angle.

• The angle corresponding to a particular voltage can be determined by rearranging the equations.

Example 13.11 - Current Calculation

• Given: ( i = 6 imes 10^{-3} ext{sin}(1000t) )

• Determine current at: $t = 2 ext{ ms}$

  • Solution:
    $i = 6 imes 10^{-3} ext{sin}(1000 imes 0.002) = 5.46 ext{ mA}$

Phase Relations in Sinusoidal Waveforms

• Phase Shift: If waveforms shift right or left from 0°

• Phase angle ($q$) is used to indicate the shift.

• Cosine Wave: Crosses the horizontal axis with a positive-going slope 90° prior to the sine wave.

Leading and Lagging Relationships

• Concept of Leading and Lagging:

  • Two sinusoidal waveforms of the same frequency can illustrate leading and lagging behaviors.

  • Example: A cosine curve leads a sine curve by 90°, while the sine lags the cosine by 90°.

Phase Relationship Examples

• Example 13.12:

  • Analyze the phase relationship between given sets of waveforms, leading to various outcomes based on shifts in angle and slope characteristics.

Average Value of Sinusoidal Waveforms

• Average value of a sine wave can be calculated over its positive or negative region.

  • The average over one complete cycle is zero for a pure sinusoidal waveform.

Effective (RMS) Values of Sinusoidal Waveforms

• rms (Root Mean Square) Value: Represented as the ac value of the current that delivers the same average power to a resistor as a dc current does.

  • For sinusoidal quantities:

  • $V<em>{rms} = rac{E</em>m}{ ext{√2}} = 0.707 E_m$

  • The rms value of a sinusoidal current or voltage is equivalent to 0.707 of its peak value.

• True rms Meters: Essential for measuring effective values of any waveform accurately.

Practice Questions and Examples

• Include various examples concerning finding peak values, periodic waveforms, and determining average values based on specified conditions (with answers for practice).