ENSC2003 - Lecture 12

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ENSC2003 - Lecture 11
Sinusoidal Sources - Slides

Review: AC & DC

• DC $-$ direction current → direction of current doesn’t change
  • We have currently been working in DC circuits
• AC $-$ alternating current → direction of current changes regularly

Sinusoidal Sources

• Trigonometric functions are used to model waveforms

  • Sine and cosine functions used widely for modelling

• Example: mains AC power (single phase)

  • 240 V RMS, at 50 Hz → sinusoidal waveform
    • Frequency $f=50$ Hz
    • Period $T=20$ ms
    • RMS amplitude of 240 V

  

• Root mean square (RMS) $-$ effective power of an AC voltage that would be delivered by equivalent DC voltage

Sine waveform: $f(t)=A\cdot \sin (\omega t+\varphi )$
Cosine waveform: $f(t)=A\cdot \cos (\omega t+\varphi )$

• $A$ $-$ amplitude
• $\omega$ $-$ angular frequency in radians/seconds
  • $\omega =2\pi f$ where $f$ is frequency in Hertz (Hz)
• $\varphi$ $-$ phase angle in degrees or radians

Review: Calculus Y12 > Derivative of Trigonometric Functions

• You should understand the sine-cosine relationship and be able to differentiate and integrate trigonometric functions

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Complex Numbers - Slides

Complex Numbers

• Review: Complex Number Y12
  $i^2=-1$
• $i$ is an imaginary number
  • Needed to solve equations with roots of negative numbers
    $z=Re\{z\}+i\cdot Im\{z\}$
• Complex numbers are a tool to move from DC → AC analysis

Why is $j$ symbol used instead of $i$?

To avoid confusion, the symbol $j$ is used in electrical engineering to represent complex number component ($i$): $z=x+jy$

$\angle \theta \to \text{cis}(\theta )=\cos (\theta )+j\sin (\theta )$

List of Key Complex Number Equations:

• Cartesian Form: $z=x+jy$
• Modulus: $|z|=\sqrt{x^2+y^2}$
• Argument: $\arg (z)=\theta$
• Polar Form: $z=r\angle \theta =|z|{\tan }^{-1}(\frac{y}{x})$
• Conjugate: $z^{\ast }=x-jy,z^{\ast }=r\angle (-\theta ),z^{\ast }=r{e}^{j\theta }$

Euler’s formula: $e^{i\theta }=\cos \theta +j\sin \theta$

• This gives us exponential formula:
  $z=r{e}^{j\theta }$

Review: Polar Form Equations

Polar form is used for easier use of multiplication and division

• Multiplication:
  $r_1\angle ({\theta }_1)\cdot r_2\angle ({\theta }_2)=r_1{r}_2\angle ({\theta }_1+{\theta }_2)$
• Division:
  $\frac{r_1\angle ({\theta }_1)}{r_2\angle ({\theta }_2)}=\frac{r_1}{r_2}\angle ({\theta }_1-{\theta }_2)$

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ENSC2003 - Lecture 13