(a) What are the limitations of (i) Proportional (P), (ii) Integral (I), (iii) Derivative (D), and (iv) PID Controllers? What is…

(a) Limitations of P, I, D and PID controllers; applications of positive feedback

(i) Proportional (P) control, u = K_p e

• It leaves a steady-state error (offset) for a step input in a type-0 loop, e_ss = 1/(1 + K_pK), because some error is needed to hold the controller output. It cannot remove the offset caused by a constant load disturbance.
• Raising K_p to cut the offset reduces damping, increases overshoot and can make a third- or higher-order loop unstable.

(ii) Integral (I) control, u = K_i ∫e dt

• It adds a pole at the origin (90° phase lag), so the response becomes slower and more oscillatory and relative stability falls.
• If the actuator saturates while the error persists, the integrator keeps charging (integral wind-up). This produces a large overshoot when the loop recovers.
• It acts slowly on sudden changes, because its output depends on the accumulated error.

(iii) Derivative (D) control, u = K_d de/dt

• It gives no output for a constant error, so it cannot remove steady-state error and is never used alone.
• Its gain rises with frequency, so it amplifies measurement noise and needs a filter, K_d s/(1 + τs).
• A step change in set-point produces a derivative "kick" that can saturate the actuator.

(iv) PID control

• Three interacting gains must be tuned (Ziegler–Nichols, Cohen–Coon or trial), and they need retuning when plant parameters change.
• It inherits both noise sensitivity (D) and wind-up (I), so practical controllers need anti-windup and derivative filtering.
• As a fixed, linear, single-loop law it performs poorly on strongly nonlinear, time-varying or long dead-time processes.

Applications of positive feedback. Positive feedback raises loop gain and reduces stability, so it is used where regeneration or switching is wanted:

• Oscillators (RC phase-shift, Wien-bridge, LC): sustained oscillation when the Barkhausen condition holds, loop gain Aβ = 1 with 0° (360°) loop phase.
• Schmitt triggers and comparators with hysteresis, bistable and astable multivibrators, and latches (flip-flops, the two-transistor thyristor model).
• Regenerative amplifiers and receivers, which increase gain; voltage build-up of a self-excited DC generator.
• Positive current feedback (IR compensation) in DC drives, which offsets the speed drop caused by armature resistance.

(b) 8085 arithmetic instructions

(i) ADD M: (A) ← (A) + [(HL)] The byte at the memory address held in HL is added to the accumulator, and the result stays in A. It uses register-indirect addressing and is 1 byte long, taking 2 machine cycles (opcode fetch, memory read) and 7 T-states. All five flags (S, Z, AC, P, CY) are set by the result. Example: A = 35H, HL = 2050H, [2050H] = 49H gives A = 7EH, with CY = 0, Z = 0, S = 0, AC = 0 and P = 1 (six 1s).

(ii) ADC M: (A) ← (A) + [(HL)] + CY The same memory byte and the carry flag are added to A. It is 1 byte long and takes 7 T-states, and all flags are affected. It is used for multi-byte addition: add the low bytes with ADD, then the higher bytes with ADC so the carry propagates.

(iii) DAD rp: (HL) ← (HL) + (rp), with rp = BC, DE, HL or SP This is a 16-bit addition of the register pair to HL, with the result left in HL. It is 1 byte long, taking 3 machine cycles (opcode fetch plus two bus-idle cycles) and 10 T-states. Only CY is affected, set by a carry out of bit 15. DAD H doubles HL, which acts as a 16-bit left shift.

(iv) SBI d8: (A) ← (A) − d8 − CY The 8-bit immediate data and the borrow (CY) are subtracted from A, with the result left in A. It is 2 bytes long, taking 2 machine cycles (opcode fetch, memory read of d8) and 7 T-states. All flags are affected, and CY = 1 indicates a borrow. It is used for multi-byte subtraction.

(v) DCR reg: (reg) ← (reg) − 1 The selected 8-bit register (B, C, D, E, H, L or A) is decremented by one. It is 1 byte long, taking 1 machine cycle and 4 T-states. S, Z, AC and P are affected, but CY is not changed. A typical use is as a loop counter: DCR C followed by JNZ.

(c) Ohmmeter reading at 0.5 FSD

Circuit. Rx is in series with V_B and R₁. The meter (R_m = 50 Ω) is shunted by the zero-adjust resistor R₂ = 50 Ω.

At 0.5 FSD: I_m = 0.5 × 50 μA = 25 μA V_m = I_mR_m = 25×10⁻⁶ × 50 = 1.25 mV I₂ = V_m/R₂ = 1.25 mV/50 Ω = 25 μA I_b = I_m + I₂ = 50 μA

Total resistance: R_total = V_B/I_b = 1.5/(50×10⁻⁶) = 30 000 Ω

Internal resistance: R_in = R₁ + R_m‖R₂ = 15 000 + 25 = 15 025 Ω

Scale reading: R_x = R_total − R_in = 30 000 − 15 025 = 14 975 Ω ≈ 15 kΩ

Checks. With Rx = 0, I_b = 1.5/15 025 = 99.8 μA and I_m = 49.9 μA ≈ FSD, so the scale zero sits at full deflection. The half-scale reading always equals the internal resistance R₁ + R_m‖R₂. The scale is non-linear: 0.25 FSD reads 44.98 kΩ and 0.75 FSD reads 4.98 kΩ.

(d) Power loss in cable insulation

Equivalent circuit. A lossy dielectric behaves as an ideal capacitor C in parallel with a high resistance R. The capacitive current I_C leads V by 90°. The small resistive current I_R is in phase with V and produces the heat. The total current leads V by φ = 90° − δ, where δ is the loss angle.

Calculation. ω = 2π × 50 = 314.16 rad/s; tan δ = tan 0.05° = 8.727×10⁻⁴ I_C = ωCV = 314.16 × 9×10⁻⁶ × 11 000 = 31.10 A I_R = I_C tan δ = 31.10 × 8.727×10⁻⁴ = 27.1 mA P = VI_R = V²ωC tan δ = (11 000)² × 314.16 × 9×10⁻⁶ × 8.727×10⁻⁴ = 298.6 W Equivalent loss resistance R = V/I_R = 11 000/0.02714 ≈ 405 kΩ. The insulation power factor is cos φ = sin δ ≈ 8.7×10⁻⁴.

(e) Constellation diagram and PSK constellations

Concept. A constellation diagram is the signal-space picture of a digital modulation. Each of the M symbols is written on two orthonormal basis functions, φ₁(t) = √(2/T) cos 2πf_ct and φ₂(t) = √(2/T) sin 2πf_ct, and plotted as a point (s_i1, s_i2). The distance of a point from the origin is √(symbol energy), and its angle is the carrier phase. The distance between points decides how easily noise causes one symbol to be mistaken for another, so the minimum distance d_min sets the error probability. Measured constellations are also used to diagnose noise, phase error and distortion (error vector magnitude).

M-PSK with equal energy Es. s_i(t) = √(2Es/T) cos(2πf_ct + θ_i) with θ_i = 2π(i − 1)/M. All points lie on a circle of radius √Es, and d_min = 2√Es sin(π/M).

• M = 2 (BPSK): two points at ±√Es on φ₁ (0° and 180°); d_min = 2√Es; 1 bit per symbol.
• M = 4 (QPSK): four points 90° apart (drawn at 45°, 135°, 225° and 315°); d_min = √2·√Es = 1.41√Es; 2 bits per symbol.
• M = 8 (8-PSK): eight points 45° apart; d_min = 2√Es sin 22.5° = 0.765√Es; 3 bits per symbol.

Consequence. For the same Es, a larger M carries more bits per symbol (log₂M) in the same bandwidth, but the points crowd together and the error rate rises. Per bit, BPSK and QPSK have the same error performance, while 8-PSK needs about 3.6 dB more Eb/N₀ than QPSK.