Sinusoids and Phasors

General Expression of a Sinusoid

• A sinusoid can be expressed mathematically as:
    $v(t) = V_m ext{sin}( ext{wt} + heta)$
      - Where:
        - $V_m$ = Amplitude of the sinusoid
        - $w$ = Angular frequency in radians/second
        - $ext{wt} + heta$ = Argument of the sinusoid
        - $heta$ = Phase angle

Historical Context

• Key figures in the establishment of alternating current (AC) as the primary mode of electricity transmission and distribution include Nikola Tesla and George Westinghouse.

• The debate between direct current (DC) and AC involved prominent figures, including Thomas Edison who advocated for DC. AC systems became commercially successful due to Tesla’s polyphase motor patents.

Key Terms

1. Sinusoid: A signal that follows the form of a sine or cosine function, often referred to as alternating current (AC). These currents have alternating positive and negative values over regular intervals.

2. AC Circuits: Circuits driven by sinusoidal current or voltage sources;

3. DC Circuits: Direct current circuits that operate with constant or time-invariant sources.

Sinusoidal Relationships

1. Sine to Cosine Conversions:
      - $ext{sin}(x) = ext{cos}(x - 90^ ext{°})$
      - $ext{cos}(x + 180^ ext{°}) = - ext{cos}(x)$

Graphical Representation of Sinusoids

• To add two sinusoids visually, transform them both to cosine or sine forms with positive amplitudes. The resultant amplitude can be found using:
    $C = ext{√}(A^2 + B^2)$
    - Example: If given 3 cos(wt) and 4 sin(wt), it can be expressed as:
    $5 ext{cos}(wt + 53.1^ ext{°})$

Sinusoidal Parameters

• Parameters of a sinusoid:
    - Amplitude ($V_m$): Maximum value of the sinusoidal function.
    - Period ($T$): Time for one complete cycle given by:
      $T = rac{2 ext{π}}{w}$
    - Frequency ($f$): Number of cycles per second calculated as:
      $f = rac{1}{T} = rac{w}{2 ext{π}}$

Example 1

• Given:
    $v(t) = 12 ext{cos}(50t + 10^ ext{°})$

• Determine:
    - Amplitude: 12 V
    - Phase: 10°
    - Angular Frequency: 50 rad/s
    - Period: $T = rac{2 ext{π}}{50} = 0.1257 ext{s}$
    - Frequency: $f rac{1}{T} = 7.958 ext{Hz}$

Example 2: Phase Angle Calculation

• Given: $v_1 = 10 ext{cos}(wt + 50^ ext{°})$ and $v_2 = 12 ext{sin}(wt - 10^ ext{°})$

• Convert to sinusoidal forms:
    - Method 1: Convert all to cosine phase.
    - Method 2: Convert all to sine phase.
    - Method 3: Graphical approach.

Historical Context on Electromagnetic Waves

• Heinrich Hertz (1857-1894): Demonstrated that electromagnetic waves obey the same laws as light, confirming Maxwell's theory. Noted for discovering the photoelectric effect in 1887.

Phasors

Definition

• Phasor: A complex number that denotes the amplitude and phase of a sinusoid. Can be represented as:
    $z = r_z e^{j heta}$
    - In rectangular form:
    $z = x + j y$
    - Where:
      - $j = ext{√}(-1)$
      - $j^{2} = -1$

Complex Number Properties

1. Rectangular Form: $z = a + jb$
      - Example: $z = 1 + j2$
      - Another example: $z = -3 - j8$

2. Polar Form: $z = r_z ext{∠} heta$

3. Exponential Form: $z = r_z e^{j heta}$

Operations of Complex Numbers

Addition and Subtraction

1. Addition:
      - Given two complex numbers, $z_1 = x_1 + jy_1$ and $z_2 = x_2 + jy_2$:
      $z_1 + z_2 = (x_1 + x_2) + j(y_1 + y_2)$

2. Subtraction:
      - $z_1 - z_2 = (x_1 - x_2) + j(y_1 - y_2)$

3. Tip: Prefer rectangular form for addition/subtraction process.

Multiplication and Division

1. Multiplication involves using polar coordinates:
      - $z_1 z_2 = |z_1| |z_2| e^{j( heta_1 + heta_2)}$

2. Division also utilizes polar representations:
      - $rac{z_1}{z_2} = rac{|z_1|}{|z_2|} e^{j( heta_1 - heta_2)}$

Imaginary Numbers Calculation

• An imaginary number is defined as numbers that yield negative results when squared.

• Example equation to solve: $x^2 + 10 = 0$

• Result: Imaginary roots present due to negative square root.

Example Calculations

Example #3

Evaluate:

• Calculation of $(40e^{j50^ ext{°}} + 20e^{-j30^ ext{°}})^{1/2}$

Example #4

Transform:
(a) To phasors: $i(t) = 6 ext{cos}(50r - 40^ ext{°}) A$
(b) $v = -4 ext{sin}(30t + 50^ ext{°}) V$

Voltage-Current Relationships

The relationships are summarized as follows:

• For resistors, capacitors, and inductors
     - In the frequency domain:
     - $V = I Z$
     - In the time domain:
     - $v = Ri$; $v = L rac{di}{dt}$; $i = C rac{dv}{dt}$

Example #8

• Total impedance for circuit combines resistors and capacitors, where:
     - $Z = Z_1 + Z_2$

Conclusion

• Understanding sinusoids and phasors is essential for analyzing AC circuits, allowing for simplification in electrical engineering analysis.

• The historical contributions of significant figures in this space paved the way for current electrical engineering applications.