Q7
(a) An n-p-n transistor with β = 49 is used in common-emitter amplifier mode with Vcc = 10V and RL = 2 kΩ. If a 100 kΩ resistor is connected between the collector and the base of the transistor, calculate the quiescent collector current. Assume VBE = 0. (20 marks) (b) In the metallic state the transition metal scandium has a single electron in 3d subshell. Calculate the values of total angular momentum J and the Lande splitting factor g and use these values to determine the energy of the lowest energy dipole moment in a field of 0·5 T. (20 marks) (c) Calculate the pinch-off voltage for n-channel silicon FET with a channel width of 6×10⁻⁴ cm and a donor concentration of 10¹⁵ cm⁻³. Given that dielectric constant of silicon is 12. (10 marks)
हिंदी में प्रश्न पढ़ें
(a) β = 49 के साथ एक n-p-n ट्रांजिस्टर उभयनिष्ठ उत्सर्जक प्रवर्धक विधा में Vcc = 10V और RL = 2 kΩ के साथ प्रयोग किया जाता है । यदि ट्रांजिस्टर के संग्राहक और आधार (बेस) के बीच एक 100 kΩ प्रतिरोधक जुड़ा हुआ है तो शांत संग्राहक धारा की गणना कीजिए । मान लीजिए VBE = 0 । (20 अंक) (b) धात्विक अवस्था में संक्रमण धातु स्कैंडियम के 3d उपकोश में एक एकल इलेक्ट्रॉन होता है । कुल कोणीय संवेग J और लांडे विपाटन गुणांक g के मानों की गणना कीजिए और इन मानों का उपयोग 0·5 T के क्षेत्र में सबसे कम ऊर्जा के द्विध्रुव आघूर्ण की ऊर्जा को निर्धारित करने के लिए कीजिए । (20 अंक) (c) 6×10⁻⁴ cm की चैनल परास और 10¹⁵ cm⁻³ के दाता सांद्रता के साथ n-चैनल सिलिकॉन FET के लिए संकुचन वोल्टता की गणना कीजिए । सिलिकॉन का परावैद्युतांक 12 दिया हुआ है । (10 अंक)
Directive word: Calculate
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How this answer will be evaluated
Approach
Calculate requires systematic numerical problem-solving across all three sub-parts. Allocate approximately 40% of effort to part (a) as it carries 20 marks and involves DC biasing analysis with feedback; 35% to part (b) for quantum mechanical angular momentum calculations; and 25% to part (c) for the FET pinch-off voltage derivation. Begin each part with the relevant formula, show complete derivation with substituted values, and conclude with proper units and physical significance.
Key points expected
- Part (a): Correct application of KVL to base-collector feedback loop with Ic = βIb and proper handling of the 100 kΩ feedback resistor to establish quiescent operating point
- Part (a): Recognition that VBE = 0 simplifies analysis to finding IB from Vcc = IB·RB + IE·RL relationship with proper current relationships
- Part (b): Correct determination of L, S, J values for 3d¹ configuration of Scandium (L=2, S=½, J=3/2, 5/2) and selection of lowest energy state using Hund's third rule for less than half-filled shell
- Part (b): Accurate calculation of Landé g-factor using g = 1 + [J(J+1) + S(S+1) - L(L+1)]/[2J(J+1)] and subsequent Zeeman energy splitting ΔE = g·μB·B·MJ
- Part (c): Proper application of pinch-off voltage formula VP = (q·ND·a²)/(2ε) with correct unit conversion from cm to m and use of relative permittivity ε = εr·ε0
- Part (c): Recognition that channel width 2a = 6×10⁻⁴ cm gives a = 3×10⁻⁶ m for the depletion region calculation
Evaluation rubric
| Dimension | Weight | Max marks | Excellent | Average | Poor |
|---|---|---|---|---|---|
| Concept correctness | 25% | 12.5 | Demonstrates flawless conceptual grasp: for (a) correctly identifies this as voltage-divider bias with collector feedback topology; for (b) applies LS coupling and Hund's rules accurately to 3d¹ configuration; for (c) understands depletion region physics in JFET channel pinch-off | Shows basic understanding of transistor biasing, atomic term symbols, and FET operation but confuses similar concepts like confusing CE with CB configuration or misapplying Hund's rules | Fundamental misconceptions such as treating β as independent of operating point, ignoring electron spin in angular momentum, or confusing pinch-off with threshold voltage |
| Derivation rigour | 20% | 10 | Presents complete, logically sequenced derivations: for (a) sets up simultaneous equations from KVL at base and collector nodes; for (b) explicitly derives g-factor from vector model; for (c) derives VP from Poisson's equation for abrupt junction | Uses correct final formulas but skips key intermediate steps or assumes relationships without proof, such as directly stating IE = (β+1)IB without derivation | Jumps to numerical substitution without establishing governing equations, or uses memorized formulas with incorrect application conditions |
| Diagram / FBD | 10% | 5 | Draws clear, labeled circuit diagram for (a) showing Vcc, RL, RB, and transistor terminals with current directions; includes energy level diagram for (b) showing 3d orbital splitting; sketches channel cross-section for (c) with depletion regions | Provides rough sketches missing key labels or with incorrect symbol orientation; diagrams support but do not enhance the solution | Omits diagrams entirely or provides misleading sketches with wrong polarities, missing ground connections, or incorrect FET structure |
| Numerical accuracy | 30% | 15 | Achieves precise answers with correct significant figures: (a) Ic ≈ 2.4 mA; (b) J=3/2, g=2/5, ΔE ≈ 5.8×10⁻²⁴ J or appropriate MJ value; (c) VP ≈ 6.8 V; shows all calculation steps with proper unit handling throughout | Correct method but arithmetic errors in final calculation, or incorrect unit conversions (cm to m) leading to answers off by orders of magnitude | Gross calculation errors, missing powers of ten, or completely unrealistic values (e.g., collector current in amperes, negative energies) without recognition |
| Physical interpretation | 15% | 7.5 | Interprets results physically: for (a) confirms Q-point lies in active region with VCE > 0.2V; for (b) relates g-factor to experimental ESR measurements; for (c) explains how VP determines maximum drain current and device switching characteristics | States numerical answers with minimal physical context, or provides generic statements about transistor/FET operation without connecting to specific calculated values | No interpretation of results; fails to verify reasonableness of answers or explain what the calculated quantities represent physically |
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