💡 What is RMS Value?
RMS (Root Mean Square) value of AC current wo steady current (DC) hota hai jo kisi resistance me utna hi heat produce karta hai jitna alternating current same time me karta hai.
Iska matlab:
RMS value = Equivalent DC current that produces the same heating effect in a resistor as AC does.
🔄 Example to Understand:
Agar ek alternating current I = I₀ sinωt kisi resistor me flow karta hai, to usse jo heat produce hoti hai, wahi heat agar kisi DC current Irms se hoti ho to us Current ko hi RMS value kehte hain.
🔧 Mathematical Derivation:
Let:
- Alternating current ho:
I = I₀ sin(ωt) - Resistance = R
- Time = T (1 complete cycle)
✅ Step 1: Heat produced in small time dt
H = I²R dt
✅ Step 2: Total Heat in one cycle (0 to T)
H = ∫₀ᵗ I²R dt
= ∫₀ᵗ (I₀² sin²(ωt)) R dt
Use identity:sin²(ωt) = (1 - cos(2ωt))/2
H = I₀²R ∫₀ᵗ (1 - cos(2ωt))/2 dt
= (I₀²R/2) × [T - (sin(2ωT)/2ω)]
Since ωT = 2π, sin(2ωT) = 0, so:
H = (I₀²RT)/2 ...(Eq. 3)
✅ Step 3: If DC (Irms) current is used, heat will be:
H = Irms² × R × T ...(Eq. 4)
Now, equating Eq. 3 and Eq. 4:
Irms² × RT = (I₀²RT)/2
=> Irms = I₀ / √2
=> Irms = 0.707 I₀
🔍 Final Formula:
🔹 Irms = 0.707 × I₀
🔹 Vrms = 0.707 × V₀
✅ Conclusion:
- RMS value is 0.707 times the peak value of AC.
- This means RMS value = 70.7% of the maximum (peak) value.
⚠️ Important Note:
- AC current or voltage is measured by AC ammeter or voltmeter.
- These instruments show RMS value, not peak value.
- They are called hot wire instruments and are designed to measure effective (RMS) values only.
📌 Summary Chart:
| Quantity | Peak Value | RMS Value |
|---|---|---|
| Current | I₀ | 0.707 I₀ |
| Voltage | V₀ | 0.707 V₀ |
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