(a) The block diagram of a feedback system is shown in the figure: the forward path is G(s) = K/((s − 1)(s² + 4s + 7)) and the…

(a) Root locus, K at s = 0 and range of K for stability

The open-loop transfer function is G(s)H(s) = K(s + 4)/[(s − 1)(s² + 4s + 7)]. Its poles are s = 1 and s = −2 ± j√3 (−2 ± j1.73), and its zero is s = −4, so n = 3 and m = 1.

(i) Root locus. The locus has three branches starting at the poles.

1. Real axis: the locus lies between s = −4 and s = 1, because exactly one pole lies to the right of a point in this range.
2. Asymptotes: there are n − m = 2, at ±90°, meeting the real axis at σ = [(1 − 2 − 2) − (−4)]/2 = 0.5.
3. Break points: dK/ds = 0 gives 2s³ + 15s² + 24s + 19 = 0, whose only real root, s = −5.68, is not on the locus, so there are no real break-away points.
4. Angle of departure at p = −2 + j√3: 180° + 40.9° − 150° − 90° = −19.1°. The complex branches start heading to the right, cross the jω axis and then follow the ±90° asymptotes.
5. The branch starting at s = 1 moves left along the real axis and ends at the zero s = −4.

(ii) K at s = 0. The characteristic equation is (s − 1)(s² + 4s + 7) + K(s + 4) = s³ + 3s² + (3 + K)s + (4K − 7) = 0. At s = 0, 4K − 7 = 0, so K = 7/4 = 1.75. This is where the real branch crosses the origin into the left half-plane.

(iii) Stability range. Routh array: row s³: 1 and (3 + K); row s²: 3 and (4K − 7); row s¹: (16 − K)/3; row s⁰: (4K − 7). All first-column entries are positive when 4K − 7 > 0 and 16 − K > 0, so 7/4 < K < 16. At K = 16 the auxiliary equation 3s² + 57 = 0 gives the imaginary-axis crossing s = ±j√19 = ±j4.36, so the system is marginally stable there.

(b) Schering bridge

Connections. Let the four arms of the bridge be AB, BC, AD and DC, with the supply across AC and the detector across BD. Arm AB holds the capacitor under test, C₁, with its equivalent series resistance r₁. Arm BC holds a non-inductive resistor R₃. Arm AD holds a standard loss-free capacitor C₂. Arm DC holds a variable resistor R₄ in parallel with a variable capacitor C₄.

Balance equation. At balance, Z_AB Z_DC = Z_BC Z_AD:

[r₁ + 1/(jωC₁)] × R₄/(1 + jωC₄R₄) = R₃/(jωC₂) × 1, which gives [r₁ + 1/(jωC₁)] jωC₂R₄ = R₃(1 + jωC₄R₄).

Expanding: jωC₂R₄r₁ + C₂R₄/C₁ = R₃ + jωR₃R₄C₄.

Equating real parts: C₂R₄/C₁ = R₃, so C₁ = C₂R₄/R₃.

Equating imaginary parts: ωC₂R₄r₁ = ωR₃R₄C₄, so r₁ = R₃C₄/C₂.

Dissipation factor. tan δ = ωC₁r₁ = ω(C₂R₄/R₃)(R₃C₄/C₂) = ωC₄R₄. If R₄ is a fixed value, C₄ can be calibrated directly in dissipation factor. Because the standard capacitor is loss-free and independent of voltage, the bridge suits high-voltage measurements of cables and insulation.

(c) Linear delta modulator

Nyquist rate = 2 × 4 kHz = 8 kHz, so the sampling rate is f_s = 10 × 8 = 80 kHz and the sampling period is T_s = 12.5 µs. The step size is δ = 100 mV.

Slope overload is avoided if the maximum signal slope does not exceed the maximum slope the modulator can follow: A·2πf_m ≤ δ f_s. So A_max = δ f_s/(2π f_m) = 0.1 × 80 000/(2π × 1000) = 1.273 V.

Maximum power (normalised to a 1 Ω load) P_max = A_max²/2 = (1.273)²/2 = 0.81 W.