Characteristics Of Alternating Currents (Copy)

Key Terminology

• Alternating Current (a.c.):
  • A current that reverses direction periodically.
  • In contrast to direct current (d.c.), which flows in only one direction.
• Period (T):
  • Time taken for one complete cycle of the waveform.
  • Unit: seconds (s)
• Frequency (f):
  • Number of complete cycles per second.
  • Unit: hertz (Hz)
  • Related to period by:
    f = 1 / T
• Peak (maximum) Value (I₀ or V₀):
  • The maximum value reached by current or voltage in either direction.

Written and Compiled By Sir Hunain Zia, World Record Holder With 154 Total A Grades, 7 Distinctions and 11 World Records For Educate A Change A2 Level Physics Full Scale Course

Sinusoidal Representation of A.C.

• For an alternating current or voltage that follows a sine wave, the variation over time is expressed as:
  • Current:
    I = I₀ sin(ωt)
  • Voltage:
    V = V₀ sin(ωt)

  Where:

  • I = instantaneous current (A)
  • V = instantaneous voltage (V)
  • I₀, V₀ = peak values
  • ω = angular frequency = 2πf (rad/s)
  • t = time (s)

Root-Mean-Square (r.m.s.) Values

• Definition:
  The r.m.s. value of a.c. is the value of direct current (or voltage) that would produce the same heating effect in a resistor as the a.c. does.
• Formulas for a sinusoidal waveform:
  • Iᵣₘₛ = I₀ / √2
  • Vᵣₘₛ = V₀ / √2
• These are often approximated as:
  • Iᵣₘₛ ≈ 0.707 I₀
  • Vᵣₘₛ ≈ 0.707 V₀
• Important Note:
  The r.m.s. value is not the average value of the waveform over a full cycle — that would be zero for a sine wave since the positive and negative halves cancel out.

Written and Compiled By Sir Hunain Zia, World Record Holder With 154 Total A Grades, 7 Distinctions and 11 World Records For Educate A Change A2 Level Physics Full Scale Course

Mean Power in a Resistor

• For a purely resistive load, the power at any instant is:

  P(t) = I(t)² × R = I₀² R sin²(ωt)

• Since the mean of sin²(ωt) over a full cycle is 0.5, the mean power is:

  P_mean = (1/2) × I₀² × R

• Alternatively:

  P_mean = Iᵣₘₛ² × R = Vᵣₘₛ² / R

• This shows that:
  • Mean power in an a.c. circuit is half the maximum instantaneous power.

Graphical Representation

• Voltage/Current vs Time Graph:
  • Smooth sine wave oscillating between +V₀ and −V₀ (or +I₀ and −I₀)
  • Period (T) is the horizontal length of one complete wave.
  • Amplitude is the vertical height from axis to peak (V₀ or I₀).

Comparison Table: Peak vs RMS

Written and Compiled By Sir Hunain Zia, World Record Holder With 154 Total A Grades, 7 Distinctions and 11 World Records For Educate A Change A2 Level Physics Full Scale Course

Example Question

A sinusoidal a.c. supply has a peak voltage of 340 V. Calculate its r.m.s. voltage and the mean power in a 20 Ω resistor.

• V₀ = 340 V
• Vᵣₘₛ = 340 / √2 = 240.4 V
• P_mean = Vᵣₘₛ² / R = (240.4)² / 20 = 2888 W