A common-emitter amplifier circuit is shown in Figure 3(a). Neglect r_x and r_o and assume the current source to be ideal.

(i) Derive an expression for the midband gain.

(ii) Derive expressions for the break frequencies caused by C_E and C_C.

(iii

Question

A common-emitter amplifier circuit is shown in Figure 3(a). Neglect r_x and r_o and assume the current source to be ideal.

(i) Derive an expression for the midband gain.

(ii) Derive expressions for the break frequencies caused by C_E and C_C.

(iii) Give an expression for the amplifier voltage gain A(s).

(iv) For R_sig = R_C = R_L = 10 kΩ, β = 100 and I = 1 mA, find the value of the midband gain.

(v) Select values for C_E and C_C to place the two break frequencies a decade apart and to obtain a lower 3 dB frequency of 100 Hz while minimizing the total capacitance.

(vi) Sketch a Bode plot for the gain magnitude and estimate the frequency at which the gain becomes unity.

(b)(i) Apply the Routh-Hurwitz (R-H) criterion to the polynomial P(s) = s⁴ + 4s³ + 8s² + 12s + 15 in order to determine the number of roots, with positive real parts, with zero real parts and with negative real parts. Also, state about the stability of the system represented by P(s).

(b)(ii) For the network shown in the Figure 3(b)(ii), find the impulse response.

(c) A 4-pole single phase 50 Hz induction motor is having values of R₂ and X₂ equal to 0·02 ohm and 0·5 ohm respectively. Calculate the slip for maximum torque and the speed corresponding to maximum torque. Stator resistance and leakage reactance are to be neglected.

UPSC Answer Check · Accepted Answer

Derive expressions systematically across all sub-parts, allocating approximately 35% time to part (a) covering amplifier analysis, 25% to Routh-Hurwitz and impulse response in part (b), and 15% to induction motor calculations in part (c). Begin with small-signal equivalent circuit construction for the common-emitter amplifier, proceed through frequency response analysis with proper pole-zero identification, then apply R-H criterion with complete array construction, solve for network impulse response using Laplace transforms, and conclude with torque-slip characteristics for the induction motor. Ensure all sketches include labeled axes and critical frequency points.
- Small-signal equivalent circuit of common-emitter amplifier with r_π, g_m, and neglecting r_x and r_o as specified; correct identification of input and output resistances
- Midband gain expression: A_M = -g_m(R_C||R_L) · [r_π/(R_sig+r_π)] with proper sign convention for inverting amplifier
- Break frequency expressions: ω_L1 = 1/[C_E(R_E||(r_π+R_sig)/(1+β))] for emitter bypass and ω_L2 = 1/[C_C(R_C+R_L)] for coupling capacitor
- Complete transfer function A(s) = A_M · s²/[(s+ω_L1)(s+ω_L2)] showing second-order high-pass characteristic
- Numerical calculation: g_m = I_C/V_T = 40 mA/V, r_π = β/g_m = 2.5 kΩ, yielding A_M ≈ -160 V/V or 44 dB
- Capacitor selection: C_E and C_C values satisfying ω_L2 = 10ω_L1 with f_L = 100 Hz, minimizing C_total = C_E + C_C
- Routh-Hurwitz array construction showing no sign changes in first column, two pairs of complex conjugate roots with negative real parts, indicating marginal stability with oscillatory response
- Induction motor: s_maxT = R₂/X₂ = 0.04, n_maxT = n_s(1-s_maxT) = 1440 rpm for 4-pole 50 Hz machine

Dimension	Weight	Max marks	Excellent	Average	Poor
Concept correctness	20%	12	Correctly identifies hybrid-π model parameters; properly applies Miller approximation limitations; recognizes that r_o neglect simplifies analysis; correctly interprets R-H criterion showing marginal stability with complex roots on stability boundary; understands that ideal current source implies infinite output resistance in small-signal model	Uses correct basic formulas but confuses r_π with βr_e or misapplies current source conditions; R-H array constructed with minor arithmetic errors leading to incorrect root classification; understands s_maxT formula but applies to wrong motor type	Uses wrong small-signal model (e.g., T-model incorrectly); fails to recognize inverting nature of CE amplifier; R-H array incomplete or with sign errors; calculates synchronous speed incorrectly for 4-pole machine
Numerical accuracy	20%	12	Precise calculations: g_m = 40 mS, r_π = 2.5 kΩ, A_M = -160 V/V (or 44 dB); C_E ≈ 12.7 μF, C_C ≈ 0.8 μF for decade separation with f_L = 100 Hz; R-H array with exact values (7.5, 4.29, 15 in auxiliary rows); s_maxT = 0.04 exactly, n_maxT = 1440 rpm	Correct formulas but arithmetic errors (e.g., off by factor of 10 in g_m due to mA/A confusion); C values approximately correct but decade separation not exact; R-H array with calculation errors in auxiliary rows; slip calculation correct but speed calculation uses wrong synchronous speed	Major numerical errors: wrong V_T value (e.g., 26 mV not used), incorrect parallel combinations, or completely wrong capacitor sizing; R-H array with sign errors leading to wrong stability conclusion; treats single-phase motor as three-phase for speed calculation
Diagram quality	20%	12	Clear small-signal equivalent circuit with all nodes labeled, ground reference shown, and controlled source properly oriented; Bode plot with accurate slope changes (+40 dB/decade below break frequencies, flat midband, -20 dB/decade at high frequencies), labeled break frequencies, and unity-gain frequency marked; impulse response sketch showing decaying oscillatory behavior for marginally stable system	Diagrams present but missing labels or incorrect controlled source direction; Bode plot shows correct general shape but incorrect slope values or missing decade markings; impulse response shows decay but not oscillatory nature	Missing diagrams or unrecognizable circuits; Bode plot with completely wrong slopes (e.g., -20 dB/decade in low frequency); no sketch for impulse response or incorrect stable exponential shown
Step-by-step derivation	20%	12	Complete derivation from first principles: KCL at emitter node for C_E effect, Thevenin equivalent for C_C circuit; explicit Laplace transform for impulse response with partial fraction expansion or residue method; full R-H array with all rows calculated showing systematic reduction; torque equation derivation leading to s_maxT condition	Correct final expressions but skips key intermediate steps (e.g., jumps to time constant formula without showing RC equivalent); impulse response stated without inverse Laplace working; R-H array shown but missing calculation steps for auxiliary polynomials	No derivations shown, only final formulas stated; incorrect application of formulas (e.g., uses high-frequency hybrid-π for low-frequency analysis); R-H criterion stated without array construction; s_maxT quoted without any derivation from torque equation
Practical interpretation	20%	12	Explains why C_E >> C_C typically (bypass vs coupling requirements); discusses trade-off between low-frequency response and capacitor size/cost; interprets marginal stability from R-H as indicating oscillatory system needing compensation; relates unity-gain frequency to gain-bandwidth product limitations; connects single-phase induction motor torque-slip to practical starting torque issues	Mentions practical aspects superficially (e.g., 'larger capacitors cost more') without linking to design constraints; recognizes stability conclusion but doesn't discuss implications for control system design; understands motor operates below synchronous speed but doesn't explain why maximum torque slip matters for starting	No practical discussion; treats all calculations as purely mathematical exercises; fails to recognize that unity-gain frequency estimation requires extrapolation beyond given data; no interpretation of what marginal stability means for system response

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