An AC voltage is given by v(t) = 100 sin(ωt). Then
1. Peak voltage is 100 V.
2. RMS voltage is 100/√2.
3. RMS value ≈ 70.7 V.
4. Average value over full cycle is 100 V.
Which of the above statements is/are correct?
Answer & step-by-step solution
The average of a rectified sine over a half cycle is 2Xm/π, giving sine-wave form factor RMS/average-rectified ≈1.11. For v(t)=100 sinωt, Vm=100 V and Vrms=Vm/√2≈70.7 V. The full-cycle average of a sine wave is zero. 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.
Peak voltage is 100 V.
For v(t)=100 sinωt, Vm=100 V and Vrms=Vm/√2≈70.7 V. - Statement 2 - Correct.
RMS voltage is 100/√2.
For v(t)=100 sinωt, Vm=100 V and Vrms=Vm/√2≈70.7 V. - Statement 3 - Correct.
RMS value ≈ 70.7 V.
For v(t)=100 sinωt, Vm=100 V and Vrms=Vm/√2≈70.7 V. - Statement 4 - Incorrect.
Average value over full cycle is 100 V.
The correct principle is: The full-cycle average of a sine wave is zero.
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.
Step-by-Step Check
For v(t)=100 sinωt, Vm=100 V and Vrms=Vm/√2≈70.7 V. The full-cycle average of a sine wave is zero.
Why the Other Options Are Wrong
Option A is incorrect because it omits correct statement(s) 3. Option B is incorrect because it includes incorrect statement(s) 4 and omits correct statement(s) 1, 3. Option D is incorrect because it includes incorrect statement(s) 4.
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
Peak RMS and Average Values is linked with Sinusoidal Voltage, AC Fundamentals and Basic Electrical Engineering. Check whether the question asks full-cycle average or rectified average.
एक AC वोल्टेज v(t) = 100 sin(ωt) द्वारा दिया गया है। तब
1. पीक वोल्टेज 100 V है।
2. RMS वोल्टेज 100/√2 है।
3. RMS मान ≈ 70.7 V है।
4. पूरे चक्र पर औसत मान 100 V है।
उपरोक्त में से कौन-सा/से कथन सही है/हैं?
उत्तर और चरणबद्ध हल
v(t)=100 sinωt के लिए Vm=100 V और Vrms=Vm/√2≈70.7 V। साइन तरंग का पूरे चक्र पर औसत मान शून्य होता है। मुख्य संबंध: Xrms=Xm/√2; Xavg(रेक्टिफाइड)=2Xm/π; फॉर्म फैक्टर≈1.11।
परीक्षा फोकस: देखें कि प्रश्न पूरे चक्र का औसत पूछ रहा है या रेक्टिफाइड औसत। सामान्य गलती: अर्ध-चक्र औसत को पूरे चक्र का औसत मान लेना।
विस्तृत हल (मूल भाषा में)
Statement-wise Verification
- Statement 1 - Correct.
Peak voltage is 100 V.
For v(t)=100 sinωt, Vm=100 V and Vrms=Vm/√2≈70.7 V. - Statement 2 - Correct.
RMS voltage is 100/√2.
For v(t)=100 sinωt, Vm=100 V and Vrms=Vm/√2≈70.7 V. - Statement 3 - Correct.
RMS value ≈ 70.7 V.
For v(t)=100 sinωt, Vm=100 V and Vrms=Vm/√2≈70.7 V. - Statement 4 - Incorrect.
Average value over full cycle is 100 V.
The correct principle is: The full-cycle average of a sine wave is zero.
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.
Step-by-Step Check
For v(t)=100 sinωt, Vm=100 V and Vrms=Vm/√2≈70.7 V. The full-cycle average of a sine wave is zero.
Why the Other Options Are Wrong
Option A is incorrect because it omits correct statement(s) 3. Option B is incorrect because it includes incorrect statement(s) 4 and omits correct statement(s) 1, 3. Option D is incorrect because it includes incorrect statement(s) 4.
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
Peak RMS and Average Values is linked with Sinusoidal Voltage, AC Fundamentals and Basic Electrical Engineering. Check whether the question asks full-cycle average or rectified average.
એક AC વોલ્ટેજ v(t) = 100 sin(ωt) વડે આપેલ છે. તો
1. પીક વોલ્ટેજ 100 V છે.
2. RMS વોલ્ટેજ 100/√2 છે.
3. RMS મૂલ્ય ≈ 70.7 V છે.
4. સંપૂર્ણ ચક્ર પર સરેરાશ મૂલ્ય 100 V છે.
ઉપરોક્તમાંથી કયું/કયા વિધાન સાચું/સાચા છે?
જવાબ અને પગલાવાર ઉકેલ
v(t)=100 sinωt માટે Vm=100 V અને Vrms=Vm/√2≈70.7 V. સાઇન તરંગનું સંપૂર્ણ ચક્ર પર સરેરાશ મૂલ્ય શૂન્ય હોય છે. મુખ્ય સંબંધ: Xrms=Xm/√2; Xavg(રેક્ટિફાઇડ)=2Xm/π; ફોર્મ ફેક્ટર≈1.11.
પરીક્ષા મુદ્દો: પ્રશ્ન સંપૂર્ણ ચક્રનું સરેરાશ પૂછે છે કે રેક્ટિફાઇડ સરેરાશ તે જુઓ. સામાન્ય ભૂલ: અર્ધ-ચક્ર સરેરાશને સંપૂર્ણ ચક્રનું સરેરાશ માની લેવું.
વિગતવાર ઉકેલ (મૂળ ભાષામાં)
Statement-wise Verification
- Statement 1 - Correct.
Peak voltage is 100 V.
For v(t)=100 sinωt, Vm=100 V and Vrms=Vm/√2≈70.7 V. - Statement 2 - Correct.
RMS voltage is 100/√2.
For v(t)=100 sinωt, Vm=100 V and Vrms=Vm/√2≈70.7 V. - Statement 3 - Correct.
RMS value ≈ 70.7 V.
For v(t)=100 sinωt, Vm=100 V and Vrms=Vm/√2≈70.7 V. - Statement 4 - Incorrect.
Average value over full cycle is 100 V.
The correct principle is: The full-cycle average of a sine wave is zero.
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.
Step-by-Step Check
For v(t)=100 sinωt, Vm=100 V and Vrms=Vm/√2≈70.7 V. The full-cycle average of a sine wave is zero.
Why the Other Options Are Wrong
Option A is incorrect because it omits correct statement(s) 3. Option B is incorrect because it includes incorrect statement(s) 4 and omits correct statement(s) 1, 3. Option D is incorrect because it includes incorrect statement(s) 4.
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
Peak RMS and Average Values is linked with Sinusoidal Voltage, AC Fundamentals and Basic Electrical Engineering. Check whether the question asks full-cycle average or rectified average.
More AC Fundamentals questions