એક પ્રથમ ક્રમની સિસ્ટમનો સમય અચળાંક 2 સેકન્ડ છે અને તેના પર યુનિટ સ્ટેપ ઇનપુટ લગાડેલ છે. સિસ્ટમ ગેઇન એક છે, તો
1. 2 સેકન્ડે આઉટપુટ તેના અંતિમ મૂલ્યના આશરે 63 ટકા સુધી પહોંચે છે.
2. આઉટપુટનું સ્થાયી-સ્થિતિ મૂલ્ય 1 બરાબર છે.
3. પ્રતિભાવ ઘાતાંકીય (exponential) છે અને દોલન દર્શાવતો નથી.
4. આઉટપુટ આશરે 2 સેકન્ડમાં અંતિમ મૂલ્યના 95 ટકા સુધી પહોંચે છે.
ઉપરોક્તમાંથી કયું/કયા વિધાન સાચું/સાચા છે?
જવાબ અને પગલાવાર ઉકેલ
τ=2 s માટે y(t)=1−e^(−t/2). t=2 s પર y≈0.632, સ્થાયી મૂલ્ય 1 છે અને પ્રતિભાવ દોલન-રહિત ઘાતાંકીય છે. 95% મૂલ્ય આશરે 3τ=6 s પર આવે છે. મુખ્ય સંબંધ: G(s)=1/(τs+1).
પરીક્ષા મુદ્દો: τ પર 63%, 3τ પર ~95%, 5τ પર ~99%. સામાન્ય ભૂલ: એક સમય અચળાંકને 95% સેટલિંગ માની લેવો.
વિગતવાર ઉકેલ (મૂળ ભાષામાં)
Statement-wise Verification
- Statement 1 - Correct.
The output reaches approximately 63 percent of its final value at 2 seconds.
Starting from zero with unit DC gain, the output reaches 63.2% at one time constant, about 95% near 3τ and over 99% near 5τ. - Statement 2 - Correct.
The steady-state value of the output is equal to
Starting from zero with unit DC gain, the output reaches 63.2% at one time constant, about 95% near 3τ and over 99% near 5τ. - Statement 1 - Correct.
- The response is exponential in nature and does not exhibit oscillations.
The output reaches approximately 63 percent of its final value at 2 seconds.
- Statement 4 - Incorrect.
The output reaches 95 percent of final value in about 2 seconds.
A first-order response reaches about 95% at roughly 3τ; for τ=2 s this is about 6 s
Core Concept
A standard first-order system has one real pole and an exponential step response controlled by a time constant τ. Starting from zero with unit DC gain, the output reaches 63.2% at one time constant, about 95% near 3τ and over 99% near 5τ.
Formula / Key Relationship
G(s)=1/(τs+1); y(t)=1−e^(−t/τ).
Step-by-Step Check
For τ=2 s, y(t)=1−e^(−t/2). At t=2 s, y≈0.632. A first-order response reaches about 95% at ≈3τ≈6 s, not 2 s.
Why the Other Options Are Wrong
Option A is incorrect because it omits correct statement(s) 2. Option C is incorrect because it omits correct statement(s) 1. Option D is incorrect because it includes incorrect statement(s) 4.
Exam Shortcut / Approach
63% at τ, ~95% at 3τ, ~99% at 5τ.
Common Mistake
Do not confuse one time constant with 95% settling.
Quick Revision
Time Constant and Step Response is linked with First-Order System, Time Response and Control Systems. 63% at τ, ~95% at 3τ, ~99% at 5τ.
A first-order system has a time constant of 2 seconds and is subjected to a unit step input. The system gain is unity, then
1. The output reaches approximately 63 percent of its final value at 2 seconds.
2. The steady-state value of the output is equal to 1.
3. The response is exponential in nature and does not exhibit oscillations.
4. The output reaches 95 percent of final value in about 2 seconds.
Which of the above statements is/are correct?
Answer & step-by-step solution
A standard first-order system has one real pole and an exponential step response controlled by a time constant τ. For τ=2 s, y(t)=1−e^(−t/2). At t=2 s, y≈0.632. Key relation: G(s)=1/(τs+1); y(t)=1−e^(−t/τ).
Exam focus: 63% at τ, ~95% at 3τ, ~99% at 5τ. Common trap: Do not confuse one time constant with 95% settling.
Statement-wise Verification
- Statement 1 - Correct.
The output reaches approximately 63 percent of its final value at 2 seconds.
Starting from zero with unit DC gain, the output reaches 63.2% at one time constant, about 95% near 3τ and over 99% near 5τ. - Statement 2 - Correct.
The steady-state value of the output is equal to
Starting from zero with unit DC gain, the output reaches 63.2% at one time constant, about 95% near 3τ and over 99% near 5τ. - Statement 1 - Correct.
- The response is exponential in nature and does not exhibit oscillations.
The output reaches approximately 63 percent of its final value at 2 seconds.
- Statement 4 - Incorrect.
The output reaches 95 percent of final value in about 2 seconds.
A first-order response reaches about 95% at roughly 3τ; for τ=2 s this is about 6 s
Core Concept
A standard first-order system has one real pole and an exponential step response controlled by a time constant τ. Starting from zero with unit DC gain, the output reaches 63.2% at one time constant, about 95% near 3τ and over 99% near 5τ.
Formula / Key Relationship
G(s)=1/(τs+1); y(t)=1−e^(−t/τ).
Step-by-Step Check
For τ=2 s, y(t)=1−e^(−t/2). At t=2 s, y≈0.632. A first-order response reaches about 95% at ≈3τ≈6 s, not 2 s.
Why the Other Options Are Wrong
Option A is incorrect because it omits correct statement(s) 2. Option C is incorrect because it omits correct statement(s) 1. Option D is incorrect because it includes incorrect statement(s) 4.
Exam Shortcut / Approach
63% at τ, ~95% at 3τ, ~99% at 5τ.
Common Mistake
Do not confuse one time constant with 95% settling.
Quick Revision
Time Constant and Step Response is linked with First-Order System, Time Response and Control Systems. 63% at τ, ~95% at 3τ, ~99% at 5τ.
एक प्रथम कोटि सिस्टम का समय स्थिरांक 2 सेकंड है और उस पर यूनिट स्टेप इनपुट लगाया गया है। सिस्टम गेन एक है, तब
1. 2 सेकंड पर आउटपुट अपने अंतिम मान के लगभग 63 प्रतिशत तक पहुँचता है।
2. आउटपुट का स्थायी-अवस्था मान 1 के बराबर है।
3. प्रतिक्रिया घातांकी (exponential) है और दोलन नहीं दिखाती।
4. आउटपुट लगभग 2 सेकंड में अंतिम मान के 95 प्रतिशत तक पहुँचता है।
उपरोक्त में से कौन-सा/से कथन सही है/हैं?
उत्तर और चरणबद्ध हल
τ=2 s के लिए y(t)=1−e^(−t/2)। t=2 s पर y≈0.632, स्थायी मान 1 है और प्रतिक्रिया दोलन-रहित घातांकी है। 95% मान लगभग 3τ=6 s पर आता है। मुख्य संबंध: G(s)=1/(τs+1)।
परीक्षा फोकस: τ पर 63%, 3τ पर ~95%, 5τ पर ~99%। सामान्य गलती: एक समय स्थिरांक को 95% सेटलिंग मान लेना।
विस्तृत हल (मूल भाषा में)
Statement-wise Verification
- Statement 1 - Correct.
The output reaches approximately 63 percent of its final value at 2 seconds.
Starting from zero with unit DC gain, the output reaches 63.2% at one time constant, about 95% near 3τ and over 99% near 5τ. - Statement 2 - Correct.
The steady-state value of the output is equal to
Starting from zero with unit DC gain, the output reaches 63.2% at one time constant, about 95% near 3τ and over 99% near 5τ. - Statement 1 - Correct.
- The response is exponential in nature and does not exhibit oscillations.
The output reaches approximately 63 percent of its final value at 2 seconds.
- Statement 4 - Incorrect.
The output reaches 95 percent of final value in about 2 seconds.
A first-order response reaches about 95% at roughly 3τ; for τ=2 s this is about 6 s
Core Concept
A standard first-order system has one real pole and an exponential step response controlled by a time constant τ. Starting from zero with unit DC gain, the output reaches 63.2% at one time constant, about 95% near 3τ and over 99% near 5τ.
Formula / Key Relationship
G(s)=1/(τs+1); y(t)=1−e^(−t/τ).
Step-by-Step Check
For τ=2 s, y(t)=1−e^(−t/2). At t=2 s, y≈0.632. A first-order response reaches about 95% at ≈3τ≈6 s, not 2 s.
Why the Other Options Are Wrong
Option A is incorrect because it omits correct statement(s) 2. Option C is incorrect because it omits correct statement(s) 1. Option D is incorrect because it includes incorrect statement(s) 4.
Exam Shortcut / Approach
63% at τ, ~95% at 3τ, ~99% at 5τ.
Common Mistake
Do not confuse one time constant with 95% settling.
Quick Revision
Time Constant and Step Response is linked with First-Order System, Time Response and Control Systems. 63% at τ, ~95% at 3τ, ~99% at 5τ.
More Time Response questions