Discussion
Match the pairs :
| Set I | Set II |
|---|---|
| (a) Gear train rotational transformer | (i) \(V_{2}(s) = K_{t}\, w(s)\) |
| (b) Tachometer velocity sensor | (ii) \(\theta_{L} = n \theta_{m}\) or \(w_{L} = n w_{n}\) |
| (c) DC amplifier | (iii) \(\frac{X_{o}(s)}{X_{m}(s)} = \frac{-s^{2}}{s^{2} + (b/m)s + k/m}\) |
| (d) Acceleration sensor | (iv) \(K_{a}/(s\tau + 1)\) |
Answer: A
The correct code is (a)-(ii), (b)-(i), (c)-(iv), (d)-(iii).
A gear train acts as a rotational transformer: the load angle is the motor angle times the gear ratio, \(\theta_{L} = n\theta_{m}\) (ii). A tachometer produces a voltage proportional to shaft speed, \(V_{2}(s) = K_{t}\,\omega(s)\) (i). A dc amplifier is modelled as a gain with a single time constant, \(K_{a}/(s\tau + 1)\) (iv). An accelerometer is a mass-spring-damper, giving the second-order ratio with \(-s^{2}\) in the numerator (iii).
The other codes give the tachometer the first-order amplifier model or match the gear train to the second-order accelerometer equation.
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