Why is rms value I0/√2 and what is "average value" if the full cycle average is zero?

7 answers & replies

• [+][–]Arjun Nair12thSample· 18d ago

  Your derivation for rms is exactly right, no need to say sorry. 👍

  Half-cycle average, take the positive half where $i = I_0\sin\omega t$ from 0 to T/2:

  $\langle i\rangle_{1/2} = \frac{1}{T/2}\int_0^{T/2} I_0\sin\omega t\,dt = \frac{2I_0}{\pi} \approx 0.637\,I_0$

  For the full cycle it is zero because positive half cancels the negative half. The half-cycle value is just the "mean of |i|", which is what rectifier circuits care about.

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• [+][–]Neel Chatterjee12thSample· 18d ago

  Your guess for 2 is correct. Combine the two terms into one sine wave:

  $3\sin\omega t + 4\cos\omega t = 5\sin(\omega t + \phi)$

  because amplitude is $\sqrt{3^2 + 4^2} = 5$. Then $I_{rms} = 5/\sqrt2 \approx 3.5$ A.

  If you want to avoid the phase trick, square it: the cross term $2\cdot3\cdot4\sin\omega t\cos\omega t = 12\sin\omega t\cos\omega t$ averages to zero over a cycle. So $\langle i^2\rangle = (9+16)/2 = 12.5$, and the root is the same $\sqrt{12.5}$.

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• [+][–]Tushar PawarDropperSample· 18d ago

  Why rms at all: heating. Heat produced is $i^2R$, so the average power in a resistor is $\langle i^2\rangle R = I_{rms}^2 R$. So $I_{rms}$ is the DC value that would heat the resistor equally. That is why 220 V is called "220 V" even though the peak is $220\sqrt2 \approx 311$ V.

  If rms was average of i, it would be zero and every bulb would be off. 😄

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• [+][–]Farhan ShaikhDropperSample· 18d ago

  For 3: normal AC ammeters and voltmeters (hot-wire type) read rms because they work on the heating effect. A moving coil meter would just show zero or shake, because the average over a cycle is zero.

  For exams, remember the ratio: form factor = rms/average = 0.707/0.637 ≈ 1.11. Sometimes asked directly.

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• [+][–]Pooja Iyer12thSample· 18d ago

  Mnemonic: rms 0.707, average 0.637, and both are fractions of peak. For a half cycle the average is also 0.707 I0, that's the one I remember as "seven-oh-seven" for both. 🙈

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  • [+][–]Vivek Mishra12thSample· 18d agoAuthor

    Pooja didi, Arjun bhaiya wrote $2I_0/\pi$ for the half cycle, which is 0.637. Which one is correct for half-cycle average?

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    • [+][–]Pooja Iyer12thSample· 18d ago

      omg you're right, I mixed them up. rms = 0.707 $I_0$, half-cycle average = 0.637 $I_0 = 2I_0/\pi$. Never say "both 0.707", that was my mistake. Thanks Arjun 🙏

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