MC1013: AC Circuit Analysis Study Notes

Electrical and Electronic Circuits MC1013: Lecture 3 - AC Circuit Analysis

Introduction to Waveform Analysis

• Waveforms of voltage and current can take various forms:
    - Constant DC value (Figure a)
    - Step waveform (Figure b)
    - Exponentially decaying shape (Figure c)
    - Sinusoidal waveform (Figure d)
    - Rectangular waveform (Figure e)
    - Triangular waveform (Figure f)

• Characteristics of waveforms:
    - Waveforms a, b, and c are unidirectional.
    - Waveforms d, e, and f have positive and negative values.
    - Waveforms d, e, and f are repetitive (periodic).
    - Mean values of d and e are zero; f has a positive mean value.

Repetitive Waveforms

• Repetitive waveforms can be represented by a combination of waveforms:
    - Mean value zero: Alternating component.
    - Positive or negative mean value: Direct component.

DC and AC Waveforms

• Description of waveforms:
    - Direct waveform:
      $a(t) = A$
      (Figure 1(a))
    - Sinusoidal waveform:
      $a(t) = A ext{sin}( ext{ω}t + 0)$
      (Figure 1(b))
    - AC + DC waveform:
      $a(t) = A + A ext{sin}( ext{ω}t + 0)$
      (Figure 1(c))
    - Other waveform:
      (Figure 1(d))

Significant Magnitudes of Waveforms

• Instantaneous value:
    - Definition: Value of $a(t)$ at a given instant of time $t$.

• For a sinusoidal wave:
    - $v(t) = V_p ext{sin}( ext{ω}t + ø)$
    - Peak value: Maximum instantaneous value,
      $V_p$.

• Mean Value:
    - Definition: Mean value of waveform.
    - For a pure sinusoid,
      $ext{Mean value} = rac{1}{T} imes rac{1}{T_0} ext{A}_{ ext{msin}}( ext{ω}t + ø) ext{dt} = 0$

Average Value

• Definition: Average value of full-wave rectified waveform.
    - For sinusoidal AC:
      $A_{ ext{avg}} = rac{1}{T} ext{positive half cycle} + ext{negative half cycle}$
    - Example Calculation:
      - If $T_1 = 0$,
        $ext{Average value} = rac{1.2 A ωτ}{T}$

Effective Value of Waveform

• Definition: Defined based on power equivalence.

• RMS Value:
    - $V_{ ext{effective}}^2 = rac{1}{T} ext{v}^2(t) ext{dt}$
    - Root-mean-square or RMS defined as the square root of mean of squared waveform.
    - For sinusoidal AC waveform:
      - RMS value specified for AC voltage or current waveforms.

Dependence of Defined Values on Each Other

• Relationships between waveform values:
    - Form Factor:
      - For a sinusoid,
        $ext{Form Factor} = rac{V_{ ext{rms}}}{V_{ ext{avg}}} = rac{V_p}{2 imes ext{√}2} ext{(approximately } 1.111$
    - Peak Factor:
      - For a sinusoid,
        $ext{Peak Factor} = rac{V_{ ext{rms}}}{V_p} = rac{ ext{√}2}{2} ext{(approximately } 1.4142$.

Calculation of Relevant Magnitudes of Repetitive Waveforms

• Square Waveform:
    - Peak value = $E$
    - Period: $T$, Frequency: $f = rac{1}{T}$, Angular frequency: $ext{ω} = rac{2 ext{π}}{T}$
    - Mean value = 0
    - Average value (rectified) = $E$
    - RMS value:
      $ext{RMS} = ext{√} rac{E^2 imes rac{1}{2}T + (-E)^2 imes rac{1}{2}T}{T} = E$
    - Form Factor = $rac{E}{E} = 1$
    - Peak Factor = $rac{E}{E} = 1$
    - Conclusion: Square waveform with equal positive and negative half cycles has all values equal.

Mean Value of Rectangular Waveform

• Definition: Rectangular waveform with magnitudes and durations indicated.
    - Positive Peak = $E_1$, Negative Peak = $E_2$.

• Average value (rectified):
    $A_{ ext{avg}} = rac{E_1(T_1 + T_2) + E_2(T - T_1 - T_2)}{T}$.

• Example:
    - If $E_1 = 100V$, $E_2 = 40V$, $T = 10s$, $T_1 = 4s$, $T_2 = 2s$, calculate mean.
      - Mean:
        $rac{100 imes 6 - 40 imes 14}{20} = 2V$
      - Average:
        $rac{100^2 imes 6 + 40^2 imes 14}{20} = 64.19V$
      - RMS:
        $rac{100 imes 6 + 40 imes 14}{20} = 58V$
        Form factor:
        $rac{64.19}{58} = 1.1107$; Peak factor:
        $rac{100}{64.19} = 1.558$.

Symmetrical Triangular Waveform

• Peak value = $E$

• Mean value = 0

• Average value:
     $A_{ ext{avg}} = rac{1}{2} imes E$
     - RMS value:
      $ext{RMS} = ext{√} rac{4E}{3}$

• Relationships:
    - Form Factor = 0.5773E/0.5E = 1.155
    - Peak Factor = $rac{E}{0.5773E} = 1.732$.

Saw-tooth Waveform

• Peak value = $E_1$

• Mean value:
    $A_{ ext{mean}} = rac{1}{T} imes rac{(E_1 + E_2)}{2}$

Irregular Waveform

• Period = 20s, Peak value = 20V

• Mean value:
     $rac{1 imes 20 imes 10 + 20 imes 5 + 10 imes 5}{20} = -7.5V$

• Rectified Average Value:
    - RMS value = $ext{√} rac{1}{T^2} ext{integrated function}$

• Example Calculations: Mean = 12.5V, RMS = 13.844V, Form factor = 1.108, Peak factor = 1.445.