GATE 2020 IN Q.24: solution and answer

GATE 2020 · INQ.24 · 1 mark · NAT

The capacitance $C_{x}$ of a capacitive type sensor is $(1000 x)$ pF, where $x$ is the input to the sensor. As shown in the figure, the sensor is excited by a voltage $10 sin (100 π t)$ V. The other terminal of the sensor is tied to the input of a high input impedance amplifier through a shielded cable, with shield connected to ground. The cable capacitance is 100 pF. The peak of the voltage $V_{A}$ at the input of the amplifier when $x = 0.1$ (in volts) is ____

GATE 2020 IN question 24: A sinusoidal source 10 sin(100 pi t) volts has one terminal grounded and the other connected to one plate of the variable… — Figure from the GATE 2020 IN paper. It is part of the question.

Answer and solution

ANSWER

The peak of $V_{A}$ is $5 V$ .

Official answer: 5Source: Official answer key, GATE 2020 IN (IIT Delhi)

The cable shield is grounded, so the cable capacitance $C_{c} = 100 pF$ sits between the amplifier input node $V_{A}$ and ground.
The amplifier has high input impedance, so it draws negligible current and can be ignored. The source, $C_{x}$ and $C_{c}$ then form a series loop, with $V_{A}$ the voltage across $C_{c}$ .
At $x = 0.1$ : $C_{x} = 1000 \times 0.1 = 100 pF$ .
The same charge $Q$ flows through both series capacitors, so $Q = V_{s} \frac{C _{x} C _{c}}{C _{x} + C _{c}}$ . The voltage across $C_{c}$ is $V_{A} = \frac{Q}{C _{c}} = V_{s} \frac{C _{x}}{C _{x} + C _{c}}$ .
The source peak is $10 V$ , so $V_{A, peak} = 10 \times \frac{100}{100 + 100} = 5 V$ .

KEY IDEA

A capacitive sensor feeding a high-impedance amplifier through a grounded-shield cable forms a capacitive divider: the cable capacitance acts as a shunt to ground, and $V_{A} = V_{s} \frac{C _{x}}{C _{x} + C _{c}}$ .

WATCH OUT FOR

Inverting the divider ratio to $\frac{C _{c}}{C _{x} + C _{c}}$ , as for resistors. For capacitors in series the voltage across one is proportional to the other's capacitance. Here the two values are equal, so the slip happens to give the same number. The ratio matters whenever $C_{x} \neq = C_{c}$ , and a changing $C_{x}$ is how the sensor produces its output.

From the GATE 2020 IN paper. Every question of this paper · Download the full paper · All 19 Sensing principles questions, with notes

GATE 2020 IN Q.24: solution and answer

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