Question

In a capacitive transducer used for the measurement of angular displacements and angular vibrations, the capacitance of the transducer, when all teeth are aligned is

• A. Directly proportional to gap between the teeth and inversely proportional to number of pairs of teeth
• B. Directly proportional to number of pairs of teeth and inversely proportional to length of each tooth
• C. Directly proportional to length of each tooth and inversely proportional to number of pairs of teeth
• D. Directly proportional to length of each tooth and inversely proportional to gap between the teeth

Hint:

Every capacitive sensor follows the relationship $C \propto \frac{A}{d}$. Therefore, capacitance is always directly proportional to the overlapping area dimensions (like tooth length) and inversely proportional to the separation gap ($d$).

Answer 1

The Correct Option is D

Concept: A variable-area multi-tooth capacitive transducer is commonly used to measure angular displacement, angular velocity, and mechanical vibrations. The sensor consists of a fixed stator disk and a rotating rotor disk, both machined with an identical pattern of radial teeth. The total capacitance $C$ of the system is governed by the parallel-plate capacitance equation: \[ C = \frac{\varepsilon_0 \varepsilon_r A}{d} \] Where $A$ is the total effective overlapping surface area between the stator and rotor teeth, and $d$ is the narrow air gap separating the two plates.

Step 1: Analyzing the geometry when all teeth are perfectly aligned.
When the angular position causes all stator and rotor teeth to line up exactly, the overlapping surface area reaches its maximum value ($A = A_{\max}$), which maximizes the total capacitance. The total overlapping area is directly proportional to the radial length of each individual tooth ($l$) and the number of teeth.

Step 2: Determining proportional relationships.
By substituting the geometric components into the parallel-plate equation:
• Capacitance $C$ is directly proportional to the effective overlapping area $A$, meaning it is directly proportional to the length of each tooth ($C \propto l$).
• Capacitance $C$ is inversely proportional to the physical gap separating the plates ($C \propto \frac{1}{d}$). Reviewing the options, Option (D) correctly states these proportional relationships, making it the correct choice.