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Alternating Current (AC) and Electromotive Force (EMF)

• Definition of Alternating Current (AC): A current whose magnitude continuously varies with time and whose direction reverses periodically.
• Alternating EMF: Corresponding voltage that varies in a similar manner.

AC Waveform

• Complete Cycle: During one complete cycle, the current or EMF first rises from zero to maximum in one direction, falls to zero, becomes maximum in the reverse direction, and falls back to zero.
• Instantaneous AC: At any instant 't', it is represented mathematically as: $ext{I} = ext{I}_0 ext{sin}( ext{w}t)$
  • Where,
  • $ext{I}_0$ = maximum value of current (peak value of AC).
  • $ext{w} = 2 ext{π}f$ = angular frequency.
  • $T$ = time period.
  • $ext{w}t$ = phase of AC current or EMF.
• Instantaneous AC EMF: Given by: $E = E_0 ext{sin}( ext{w}t)$
  • Where,
  • $E_0$ = maximum value of EMF (peak value of AC EMF).
• Sinusoidal Waveform: The shape of the AC waveform is similar to a sine function, referred to as a sinusoidal waveform.

Peak Value of AC

• Peak Value of AC Current: Maximum value of AC current in either direction, denoted as $ext{I}_0$ (current amplitude).
• Peak Value of AC Voltage: Maximum value of AC voltage in either direction, denoted as $E_0$ (voltage amplitude).

Mean Value (Average Value) of AC

• Total Contribution: The value of AC over the positive half cycle equals the value over the negative half cycle, leading to a net mean value of zero over one complete cycle.
• Mean Value Calculation:
  • Average value of AC over a half cycle
  • Represents the value of steady current that transmits the same amount of charge as AC in the same time interval.
• Mathematical Calculation: $q = ext{T} imes ext{I}_0 ext{sin}( ext{w}t) ext{dt}$
  • Total charge in half cycle computed by integrating from t = 0 to t = T/2.

Root Mean Square (RMS) Value of AC (Effective Value)

• Definition: Represents a value of steady (DC) current that produces the same heating effect as AC under similar conditions.
• Mathematical Expression for Heating:
  • $dH = I^2R ext{dt}$
  • Total heat produced over one complete cycle using integration:
    $H = ext{I}_0^2 R ext{∫} ext{sin}^2( ext{θ}) ext{dt}$
• Final Calculated RMS Value:
  • $ext{H} = ext{I}_{rms}^2 RT$
  • Solving yields:
    $ext{I}<em>{rms} = 0.707 ext{I}</em>0$
  • Therefore, RMS value of AC is approximately 70.7% of the peak value.
  • Similarly, RMS value of alternating EMF is written as:
    $ext{E}<em>{rms} = 0.707 ext{E}</em>0$

AC in a Circuit

AC through Resistor

• Instantaneous EMF:
  $E = E_0 ext{sin}( ext{w}t)$
• Instantaneous Current: $I = \frac{E_0}{R} ext{sin}( ext{w}t)$
  • This shows that both voltage and current are in the same phase.

AC through Inductor

• Case: Pure inductor connected to an AC source.
• Instantaneous EMF across Inductor:
  $E = E_0 ext{sin}( ext{w}t)$
• Induced EMF Expression:
  $E + (-L \frac{dI}{dt}) = 0$
• Phase Relationship:
  • Current lags voltage by a phase angle of $\frac{ ext{π}}{2}$ in an inductor.
• Inductive Reactance (XL):
  • Definition: Resistance offered by the inductor against the flow of AC.
    $X_L = ext{w}L = 2 ext{π} f L$

AC through Capacitor

• Instantaneous EMF across Capacitor:
  $E = E_0 ext{sin}( ext{w}t)$
• Charge Stored in Capacitor:
  $V = \frac{q}{C}$
• Phase Relationship:
  • Current leads voltage by a phase angle of $\frac{ ext{π}}{2}$ in a capacitor.
• Capacitive Reactance (XC):
  • Resistance to the flow of AC by a capacitor is given by:
    $X_C = \frac{1}{ ext{w}C} = \frac{1}{2 ext{π} f C}$

LR Series Circuit

• Components: An inductor (L) and a resistor (R) connected in series with an AC source.
• Voltage Drops:
  • Across Inductor: $V<em>L = I X</em>L$
  • Across Resistor: $V_R = I R$
• Impedance (Z):
  • Definition: It is the total opposition offered by the LR circuit to the flow of AC current.
    $Z = \frac{E}{I} = \frac{ ext{√(V<em>R^2 + V</em>L^2)}}{I}$
• Phase Angle:
  $an( θ ) = \frac{V<em>L}{V</em>R} = \frac{X<em>L}{R}, ext{ where } θ = an^{-1}(</em> )$