Mean (or Average) Value of Alternating Current (or Voltage)

The mean (or average) value of alternating current is the arithmetic average of all its instantaneous values over a given time interval.

Analogy: Imagine walking 5 steps forward and then 5 steps back. Your net displacement is zero — similar to AC over a full cycle. But if you only consider the forward walk (half-cycle), your average step count is meaningful.

In AC circuits:

• Current changes direction periodically.
• Over a full cycle, positive and negative halves cancel → mean = 0.
• Over a half-cycle, we get a finite, useful average → mean = 0.637 × peak value.

Given:

Mean value over time T:

Solving:

• \[\frac{I_0}{T}\left[-\frac{\cos\omega t}{\omega}\right]_0^T=\frac{I_0}{\omega T}(-\cos\omega T+\cos0)\]

Since ωT = 2π:

• \[\frac{I_0}{2\pi}(-\cos2\pi+\cos0)=\frac{I_0}{2\pi}(-1+1)=0\]

Conclusion: Mean value over full cycle = 0

Substitute T = \[\frac {2π}{ω}\]:

Conclusion: Mean value over half-cycle = 63.7% of peak value

If peak current I₀ = 10A, then:
→ Mean current over half-cycle = 0.637 × 10 = 6.37 A

This is the DC-equivalent average you’d measure if you “smoothed” only the positive half of the AC wave (as in half-wave rectifiers).

Real-Life Application:

• Half-Wave Rectifier: Converts AC to pulsating DC. The output’s average voltage is V_m = 0.637 V₀.
• Galvanometer in AC Circuit: Shows no deflection because the needle can’t follow 50Hz reversals → net average torque = 0.