JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD B.Tech III MECHANICAL MEASUREMENTS AND CONTROL SYSTEMS Previous Question Paper (September - 2021 - R16)

A zero-order system is characterized by an output that follows the input instantaneously without any time lag or distortion. The relationship is defined by the equation a₀q_o = b₀q_i, where q_o is the output and q_i is the input. A potentiometer is a typical example.

A first-order system includes a single energy storage element and a dissipation element, resulting in a time-dependent response. Its behavior is governed by the differential equation a₁(dq_o/dt) + a₀q_o = b₀q_i. When subjected to a step input, the output follows an exponential curve defined by the time constant τ, reaching 63.2% of its final value in one time constant. Examples include a mercury-in-glass thermometer or an RC circuit.

A second-order system involves two energy storage elements, leading to oscillatory behavior. The governing equation is a₂(d²q_o/dt²) + a₁(dq_o/dt) + a₀q_o = b₀q_i. The response is characterized by the natural frequency ω_n and the damping ratio ζ. Depending on the value of ζ, the system can be underdamped (oscillatory), critically damped (fastest response without oscillation), or overdamped (sluggish response). A spring-mass-damper system or a pressure transducer with a diaphragm are common examples.

Capacitive transducers operate on the principle of a parallel plate capacitor, where the capacitance C is given by C = εA/d, where ε is the permittivity of the dielectric, A is the overlapping area of plates, and d is the distance between them. The three primary principles of operation are:

1. Variation of distance between plates: In this method, one plate is fixed while the other is movable. As the distance d changes due to a physical displacement, the capacitance changes inversely. This is highly sensitive for small displacements but exhibits a non-linear relationship.
2. Variation of overlapping area: By moving one plate parallel to the other, the effective area A changes. Since the capacitance is directly proportional to the area, this method provides a linear relationship between displacement and capacitance, making it suitable for measuring larger linear or angular displacements.
3. Variation of dielectric constant: This method involves moving a dielectric material between the fixed plates of a capacitor. As the dielectric material enters or leaves the gap, the effective permittivity ε changes, thereby altering the capacitance. This principle is commonly used for measuring liquid levels or the thickness of non-conductive materials.