Q8
(a) A 415 V, 4-pole, 3-phase, 50 Hz, star-connected squirrel cage induction motor has per phase parameters of r_s = 1·1 Ω, r_r = 1·3 Ω, X_m = 167 Ω, and X_ls = X_lr = 3·5 Ω, all at rated frequency. The motor has a rated slip of 4%. Its speed is to be controlled by VVVF method using a VSI (Voltage Source Inverter). Find the voltage to be applied to the motor at 5 Hz operating frequency to maintain same peak torque as in 50 Hz. Also determine the speed at which rated torque appears at this frequency. (Neglect core losses in the motor and all motor parameters referred from stator side) (20 marks) (b) (i) A 300 W carrier is modulated to a depth of 70%. Calculate the total power transmitted in case of Vestigial Side Band (VSB) modulation. Assume 15% of the other side band is transmitted along with wanted side band. Also find the saving in power when compared to Double Side Band (DSB) transmission. (10 marks) (ii) Find the temperature of the attenuator in the system shown in the figure below so that the overall noise figure of the system does not exceed 3·5 dB. The attenuator introduces a loss of 3 dB. (10 marks) (c) A bank of three identical single-phase transformers having 11000 V/231 V voltage ratio are connected in delta-star combination with delta side connected to 11 kV, 3-phase balanced supply. The star side is supplying a balanced load of 120 kVA at 0·8 pf lag. A single-phase load of 40 kW, upf is now connected between one line and neutral of the secondary side. Calculate the input line currents at the delta side under this condition. (Neglect any magnetising currents of the transformers) (10 marks)
हिंदी में प्रश्न पढ़ें
(a) एक 415 V, 4-ध्रुव, त्रि-कला, 50 Hz, तारा-संयोजित गिलहरी पंजर प्रेरण मोटर के निर्धारित आवृत्ति पर प्रति कला प्राचल r_s = 1.1 Ω, r_r = 1.3 Ω, X_m = 167 Ω, तथा X_ls = X_lr = 3.5 Ω हैं । मोटर का निर्धारित सर्पण 4% है । मोटर की गति को एक VSI (वोल्टता स्रोत प्रतिलोमक) की सहायता से VVVF पद्धति द्वारा नियंत्रित किया जाना है । 5 Hz की संक्रियात्मक आवृत्ति पर मोटर में अनुप्रयुक्त की जाने वाली वह वोल्टता ज्ञात कीजिए जो उच्चतम बलाघूर्ण को 50 Hz की आवृत्ति पर उच्चतम बलाघूर्ण के बराबर बनाए रखे । वह गति भी ज्ञात कीजिए जो इस आवृत्ति पर निर्धारित बलाघूर्ण प्रदान करे । (मोटर के सभी प्राचल स्टेटर सिरे से संदर्भित हैं तथा मोटर के कोर क्षरण को अनदेखा कीजिए) (b) (i) एक 300 W के वाहक को 70% गहराई तक मॉडुलित (प्रमाणित) किया गया है । अवशेषी पार्श्व बैंड (VSB) मॉडुलन (तारतम्य) में कुल संचारित शक्ति परिकलित कीजिए । यह मानिए कि चाहे गए पार्श्व बैंड के साथ दूसरे पार्श्व बैंड के 15% हिस्से का संचारण किया जाना है । द्वि पार्श्व बैंड (DSB) के संचारण की तुलना में शक्ति में बचत भी ज्ञात कीजिए । (ii) चित्र में प्रदर्शित तंत्र में शीणक का वह तापमान ज्ञात कीजिए जिससे तंत्र का समग्र रवांक 3·5 dB से अधिक न हो । शीणक 3 dB का क्षरण उत्पन्न करता है । (c) तीन एकसमान एकल-कला परिणामित्रों, जिनका बोल्टता अनुपात 11000 V/231 V है, का समूह त्रिकोण-तारा संयोजन में जोड़ा गया है जहाँ त्रिकोण सिरा 11 kV के त्रि-कला संतुलित प्रदाय से जुड़ा है । तारा सिरा 0·8 शक्ति गुणांक पश्चता पर 120 kVA का संतुलित भार प्रदाय कर रहा है । अब द्वितीयक सिरे की एक लाइन तथा न्यूट्रल के मध्य एक 40 kW, इकाई शक्ति गुणांक, का एकल-कला भार जोड़ा जाता है । इस स्थिति में त्रिकोण सिरे पर निवेश लाइन धाराएँ परिकलित कीजिए । (परिणामित्रों की चुंबकन धाराओं को अनदेखा कीजिए)
Directive word: Calculate
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How this answer will be evaluated
Approach
Calculate the required quantities systematically across all four sub-parts. For part (a), spend approximately 50% of time (20 marks) on VVVF induction motor analysis including torque-slip characteristics and voltage-frequency relationships. For part (b)(i)-(ii), allocate 25% of time (10+10 marks) on VSB power calculations and noise figure analysis with attenuator temperature. For part (c), use remaining 25% of time (10 marks) on unbalanced transformer loading with delta-star connection. Present clear per-phase equivalent circuits, modulation spectra diagrams, and transformer connection diagrams where applicable.
Key points expected
- Part (a): Calculate Thevenin equivalent parameters (V_TH, R_TH, X_TH) for induction motor; determine breakdown torque at 50 Hz using torque-slip relation; apply VVVF control maintaining V/f constant with voltage boost for low frequency; find required voltage at 5 Hz and corresponding speed for rated torque
- Part (a): Recognize that for constant torque operation, slip frequency must remain constant; calculate synchronous speed at 5 Hz (150 rpm) and rotor speed at rated torque condition
- Part (b)(i): Calculate carrier power (300 W), sideband powers (m²P_c/4 each), total VSB power with 15% vestigial component; compare with DSB power (P_c + m²P_c/2) to find percentage saving
- Part (b)(ii): Apply Friis formula for cascaded systems; relate attenuator noise figure to physical temperature using F = 1 + (L-1)T/T_0; solve for attenuator temperature given overall NF ≤ 3.5 dB with 3 dB loss
- Part (c): Determine transformer turns ratio (11000/231); calculate secondary line-neutral voltage (231/√3 = 133.4 V) and line-line voltage (231 V); analyze unbalanced loading with 120 kVA balanced load plus 40 kW single-phase load; apply symmetrical components or direct phase analysis for delta primary currents
Evaluation rubric
| Dimension | Weight | Max marks | Excellent | Average | Poor |
|---|---|---|---|---|---|
| Concept correctness | 20% | 8 | Correctly applies Thevenin equivalent for induction motor torque calculation; understands VVVF control with constant flux principle; correctly interprets VSB modulation with vestigial sideband definition; applies proper noise figure cascade formula with temperature relation; correctly analyzes delta-star transformer with unbalanced loading using phase shift relationships | Minor errors in equivalent circuit simplification or torque formula; understands V/f control but misses low-frequency voltage boost; basic understanding of modulation but confuses VSB with SSB; applies Friis formula but makes temperature-noise figure relation errors; attempts transformer analysis but confuses line and phase quantities | Fundamental misunderstanding of induction motor equivalent circuit; treats VVVF as simple V/f without torque considerations; unable to distinguish modulation types; incorrect noise figure formula application; major errors in three-phase transformer connections and current relationships |
| Numerical accuracy | 20% | 8 | Accurate calculations: Thevenin voltage ~236.5 V, R_TH ~1.12 Ω, X_TH ~3.47 Ω; breakdown torque ~45 Nm; 5 Hz voltage ~41.5 V with proper boost; VSB power ~334.7 W with 35.3 W saving; attenuator temperature ~290 K; primary line currents correctly computed with magnitude and phase angles for unbalanced condition | Correct method but arithmetic errors in Thevenin parameters or torque calculation; approximate V/f ratio without boost; sideband power calculation errors; noise temperature calculation with wrong reference temperature; transformer currents with wrong scaling or phase sequence | Order of magnitude errors in calculations; completely wrong formulas leading to unrealistic values; inability to perform unit conversions (dB to ratio, kW to per-phase); ignores √3 factors in three-phase systems; nonsensical final answers without physical verification |
| Diagram quality | 20% | 8 | Clear per-phase equivalent circuit for induction motor with all parameters labeled; torque-slip characteristic sketch showing 50 Hz and 5 Hz curves; VSB spectrum diagram showing carrier, wanted sideband, and vestigial component; system block diagram for noise figure cascade with attenuator and amplifier; delta-star transformer connection diagram with current directions | Basic equivalent circuit without Thevenin indication; simple V/f plot without torque curves; generic modulation spectrum without vestigial labeling; incomplete cascade diagram; transformer connection without current labeling | Missing essential diagrams; incorrect circuit topology; wrong phasor relationships; no attempt to illustrate key concepts diagrammatically; messy unlabeled sketches |
| Step-by-step derivation | 20% | 8 | Systematic derivation: Thevenin equivalent step-by-step from T-equivalent; torque formula derivation from power transfer; clear VVVF voltage calculation with flux linkage preservation; explicit power formulas for AM showing carrier and sideband components; Friis formula derivation with temperature substitution; symmetrical component or direct phase analysis for unbalanced transformer loading | Skips some intermediate steps but maintains logical flow; states formulas without full derivation; some steps implied but not explicit; partial analysis of unbalanced conditions | Jumps directly to final formulas without justification; no logical sequence in problem solving; missing crucial steps like Thevenin conversion or power conservation; no derivation of noise temperature relationship |
| Practical interpretation | 20% | 8 | Interprets VVVF results for practical motor control: explains why voltage boost needed at low frequency due to stator resistance drop; discusses VSB bandwidth efficiency for TV broadcasting (Indian context: Doordarshan standards); explains attenuator temperature relevance for low-noise receiver design; discusses transformer unbalance implications for power quality and protection settings; relates to industrial drives and communication systems in Indian infrastructure | Brief mention of practical relevance without elaboration; generic statements about energy saving or efficiency; basic awareness of applications without specific context | Purely mathematical treatment with no physical interpretation; no awareness of why these calculations matter in practice; fails to connect to any real-world electrical engineering application |
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