Consider the following statements:
1. RMS value of sinusoidal current is its peak value divided by √2.
2. Average value of sinusoidal current over one cycle is zero.
3. Form factor of sine wave is 1.11.
Which of the above statements is/are correct?
Answer & step-by-step solution
For a sine wave x(t)=Xm sinωt, the RMS value is Xm/√2. The average of a rectified sine over a half cycle is 2Xm/π, giving sine-wave form factor RMS/average-rectified ≈1.11. Key relation: Xrms=Xm/√2; Xavg(rectified)=2Xm/π; form factor≈1.11.
Exam focus: Check whether the question asks full-cycle average or rectified average. Common trap: Using the half-cycle average as the full-cycle average is a common error.
Statement-wise Verification
- Statement 1 - Correct.
RMS value of sinusoidal current is its peak value divided by √2.
RMS value of sinusoidal current is its peak value divided by √. - Statement 2 - Correct.
Average value of sinusoidal current over one cycle is zero.
- Average value of sinusoidal current over one cycle is zero.
- Statement 3 - Correct.
Form factor of sine wave is 1.11.
The average of a rectified sine over a half cycle is 2Xm/π, giving sine-wave form factor RMS/average-rectified ≈1.11.
Core Concept
For a sine wave x(t)=Xm sinωt, the RMS value is Xm/√2. The average over a complete cycle is zero because positive and negative halves cancel. The average of a rectified sine over a half cycle is 2Xm/π, giving sine-wave form factor RMS/average-rectified ≈1.11.
Formula / Key Relationship
Xrms=Xm/√2; Xavg(rectified)=2Xm/π; form factor≈1.11.
Why the Other Options Are Wrong
Option A is incorrect because it omits correct statement(s) 3. Option B is incorrect because it omits correct statement(s) 1. Option C is incorrect because it omits correct statement(s) 2.
Exam Shortcut / Approach
Check whether the question asks full-cycle average or rectified average.
Common Mistake
Using the half-cycle average as the full-cycle average is a common error.
Quick Revision
RMS Average and Form Factor is linked with Sinusoidal Waveforms, AC Fundamentals and Basic Electrical Engineering. Check whether the question asks full-cycle average or rectified average.
निम्नलिखित कथनों पर विचार कीजिए:
1. साइनसॉइडल करंट का RMS मान उसके पीक मान को √2 से विभाजित करने पर मिलता है।
2. एक चक्र में साइनसॉइडल करंट का औसत मान शून्य होता है।
3. साइन तरंग का फॉर्म फैक्टर 1.11 है।
उपरोक्त में से कौन-सा/से कथन सही है/हैं?
उत्तर और चरणबद्ध हल
साइन तरंग x(t)=Xm sinωt के लिए RMS मान Xm/√2 है। अर्ध-चक्र पर रेक्टिफाइड औसत 2Xm/π है, जिससे फॉर्म फैक्टर ≈1.11 आता है। पूरे चक्र पर औसत शून्य है।
परीक्षा फोकस: पूरे चक्र का औसत और रेक्टिफाइड औसत अलग हैं। सामान्य गलती: अर्ध-चक्र औसत को पूरे चक्र का औसत मान लेना।
विस्तृत हल (मूल भाषा में)
Statement-wise Verification
- Statement 1 - Correct.
RMS value of sinusoidal current is its peak value divided by √2.
RMS value of sinusoidal current is its peak value divided by √. - Statement 2 - Correct.
Average value of sinusoidal current over one cycle is zero.
- Average value of sinusoidal current over one cycle is zero.
- Statement 3 - Correct.
Form factor of sine wave is 1.11.
The average of a rectified sine over a half cycle is 2Xm/π, giving sine-wave form factor RMS/average-rectified ≈1.11.
Core Concept
For a sine wave x(t)=Xm sinωt, the RMS value is Xm/√2. The average over a complete cycle is zero because positive and negative halves cancel. The average of a rectified sine over a half cycle is 2Xm/π, giving sine-wave form factor RMS/average-rectified ≈1.11.
Formula / Key Relationship
Xrms=Xm/√2; Xavg(rectified)=2Xm/π; form factor≈1.11.
Why the Other Options Are Wrong
Option A is incorrect because it omits correct statement(s) 3. Option B is incorrect because it omits correct statement(s) 1. Option C is incorrect because it omits correct statement(s) 2.
Exam Shortcut / Approach
Check whether the question asks full-cycle average or rectified average.
Common Mistake
Using the half-cycle average as the full-cycle average is a common error.
Quick Revision
RMS Average and Form Factor is linked with Sinusoidal Waveforms, AC Fundamentals and Basic Electrical Engineering. Check whether the question asks full-cycle average or rectified average.
નીચેના વિધાનો ધ્યાનમાં લો:
1. સાઇનુસોઇડલ કરંટનું RMS મૂલ્ય તેના પીક મૂલ્યને √2 વડે ભાગતાં મળે છે.
2. એક ચક્રમાં સાઇનુસોઇડલ કરંટનું સરેરાશ મૂલ્ય શૂન્ય હોય છે.
3. સાઇન તરંગનો ફોર્મ ફેક્ટર 1.11 છે.
ઉપરોક્તમાંથી કયું/કયા વિધાન સાચું/સાચા છે?
જવાબ અને પગલાવાર ઉકેલ
સાઇન તરંગ x(t)=Xm sinωt માટે RMS મૂલ્ય Xm/√2 છે. અર્ધ-ચક્ર પર રેક્ટિફાઇડ સરેરાશ 2Xm/π છે, જેથી ફોર્મ ફેક્ટર ≈1.11 મળે છે. સંપૂર્ણ ચક્ર પર સરેરાશ શૂન્ય છે.
પરીક્ષા મુદ્દો: સંપૂર્ણ ચક્રનું સરેરાશ અને રેક્ટિફાઇડ સરેરાશ અલગ છે. સામાન્ય ભૂલ: અર્ધ-ચક્ર સરેરાશને સંપૂર્ણ ચક્રનું સરેરાશ માની લેવું.
વિગતવાર ઉકેલ (મૂળ ભાષામાં)
Statement-wise Verification
- Statement 1 - Correct.
RMS value of sinusoidal current is its peak value divided by √2.
RMS value of sinusoidal current is its peak value divided by √. - Statement 2 - Correct.
Average value of sinusoidal current over one cycle is zero.
- Average value of sinusoidal current over one cycle is zero.
- Statement 3 - Correct.
Form factor of sine wave is 1.11.
The average of a rectified sine over a half cycle is 2Xm/π, giving sine-wave form factor RMS/average-rectified ≈1.11.
Core Concept
For a sine wave x(t)=Xm sinωt, the RMS value is Xm/√2. The average over a complete cycle is zero because positive and negative halves cancel. The average of a rectified sine over a half cycle is 2Xm/π, giving sine-wave form factor RMS/average-rectified ≈1.11.
Formula / Key Relationship
Xrms=Xm/√2; Xavg(rectified)=2Xm/π; form factor≈1.11.
Why the Other Options Are Wrong
Option A is incorrect because it omits correct statement(s) 3. Option B is incorrect because it omits correct statement(s) 1. Option C is incorrect because it omits correct statement(s) 2.
Exam Shortcut / Approach
Check whether the question asks full-cycle average or rectified average.
Common Mistake
Using the half-cycle average as the full-cycle average is a common error.
Quick Revision
RMS Average and Form Factor is linked with Sinusoidal Waveforms, AC Fundamentals and Basic Electrical Engineering. Check whether the question asks full-cycle average or rectified average.
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