Engineering: Circuits, Measurement Error and Dynamic Response
GATE Instrumentation Engineering (IN) tests engineering fundamentals through Engineering Mathematics, Electrical Circuits, Control Systems and Measurements. Numerical questions usually combine two of these: a bridge with an error budget, an op-amp with a first-order sensor, or a feedback loop with a Routh array.
Core rules
- Convert to base units before substitution: 4–20 mA to 0.004–0.020 A, µV to V, ppm to fraction.
- Network theorems: Thevenin/Norton reduce a transducer bridge to a source and resistance; maximum power transfer occurs at R_L = R_th.
- First-order step: y(t) = y∞ + (y0 − y∞)e^(−t/τ); at t = τ the response has covered 63.2% of the change; 2% settling ≈ 4τ, 5% settling ≈ 3τ.
- Negative feedback: closed-loop transfer function = G/(1 + GH). For positive feedback the denominator is 1 − GH.
- Error combination: for y = x1^a x2^b, worst-case relative error = |a|u1/x1 + |b|u2/x2; independent random errors combine as sqrt((a u1/x1)^2 + (b u2/x2)^2).
Worked example
A resistor is measured by R = V/I with V = 10.0 V ±1.0% and I = 2.00 A ±1.0%. The nominal R is 5.00 Ω. Worst-case relative error is 1% + 1% = 2%, giving ±0.10 Ω. If the question states the errors are independent random, use quadrature: sqrt(1^2 + 1^2) = 1.41%, giving ±0.071 Ω. The reasoning matters: systematic calibration errors add worst-case; random scatter is treated statistically, and averaging reduces it only as 1/sqrt(N).
Traps that cost marks
- Using 20 log10 for a voltage ratio; amplitude uses 20 log10, power uses 10 log10.
- Applying the small-unbalance bridge approximation ΔR/(4R) when ΔR is not small.
- Treating a fixed calibration offset as random error that averaging will remove.
- Dropping the sign in a feedback denominator and predicting the wrong stability.