GATE 2023 EC Q.62: solution and answer

GATE 2023 · ECQ.62 · 2 marks · NAT

A sample and hold circuit is implemented using a resistive switch and a capacitor with a time constant of 1 μs. The time for the sampling switch to stay closed to charge a capacitor adequately to a full scale voltage of 1 V with 12-bit accuracy is ________ μs (rounded off to two decimal places).

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Answer and solution

ANSWER

The sampling switch must stay closed for $t = 12 τ ln 2 \approx 8.32 μ s$ .

Official answer: 8.30 to 8.34Source: Official answer key, GATE 2023 EC (IIT Kanpur)

While the switch is closed, the capacitor charges through the switch resistance as a first-order RC circuit: $v_{C} (t) = V_{F S} (1 - e^{- t / τ})$ , with $τ = 1 μ s$ and $V_{F S} = 1 V$ .
The error still left after time $t$ is $V_{F S} - v_{C} (t) = V_{F S} e^{- t / τ}$ .
12-bit accuracy means this error must be no bigger than one LSB: $LSB = \frac{V _{F S}}{2 ^{12}} = \frac{1}{4096} V \approx 0.244 mV$ .
Set the condition: $V_{F S} e^{- t / τ} \leq \frac{V _{F S}}{2 ^{12}} \Rightarrow e^{t / τ} \geq 2^{12}$ .
Take the natural log of both sides: $t \geq τ ln (2^{12}) = 12 τ ln 2 = 12 \times 0.6931 \times 1 μ s$
$t = 8.318 μ s \approx 8.32 μ s$ . The 1 V full-scale value cancels, so the answer depends only on $τ$ and the number of bits $N$ : $t = N τ ln 2$ .

KEY IDEA

An RC sample-and-hold settles exponentially. The capacitor has settled to $N$ -bit accuracy once the leftover error $V_{F S} e^{- t / τ}$ has fallen to one LSB, $V_{F S} / 2^{N}$ . That gives $t = τ ln (2^{N}) = N τ ln 2$ , which does not depend on the full-scale voltage.

WATCH OUT FOR

Using a half-LSB error limit, $V_{F S} / 2^{13}$ , gives $13 ln 2 \approx 9.01 μ s$ , which falls outside the key. The key here uses a one-LSB limit. Other slips: taking $lo g_{10}$ where the natural log is needed ( $12 lo g_{10} 2 \approx 3.61 μ s$ ), and setting $e^{- t / τ}$ equal to $1/12$ when it should be $1/ 2^{12}$ .

From the GATE 2023 EC paper. Every question of this paper · Download the full paper · All 18 ADC & DAC questions, with notes

GATE 2023 EC Q.62: solution and answer

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