GATE 2025 IN Q.34: solution and answer

GATE 2025 · INQ.34 · 1 mark · NAT

A dual-slope ADC has a fixed integration time of 100 ms. The reference voltage of the ADC is $- 5 V$ . The time taken by the ADC to measure an input voltage of $1.25 V$ is __________ ms (rounded off to the nearest integer).

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Answer and solution

ANSWER

The ADC takes $T_{1} + T_{2} = 100 + 25 = 125 ms$ to measure $1.25 V$ .

Official answer: 125Source: Official answer key, GATE 2025 IN (IIT Roorkee)

Phase 1 (run-up, fixed time): the integrator integrates the unknown input $V_{in}$ for a fixed time $T_{1} = 100 ms$ . Its output reaches $∣ V_{1} ∣ = \frac{V _{in} T _{1}}{R C}$ .
Phase 2 (run-down, variable time): the integrator input is switched to the reference $V_{r e f} = - 5 V$ , which has the opposite polarity to $V_{in}$ . The integrator output therefore ramps back towards zero at slope $\frac{∣ V _{r e f} ∣}{R C}$ , taking a time $T_{2}$ to reach zero.
Charge balance (the dual-slope rule): the rise in phase 1 equals the fall in phase 2, $\frac{V _{in} T _{1}}{R C} = \frac{∣ V _{r e f} ∣ T _{2}}{R C} \Rightarrow T_{2} = T_{1} \frac{V _{in}}{∣ V _{r e f} ∣}$ The $R C$ cancels, so the result does not depend on the component values.
Run-down time: $T_{2} = 100 ms \times \frac{1.25}{5} = 25 ms$
Total measurement (conversion) time: one measurement needs both phases, so $T_{co n v} = T_{1} + T_{2} = 100 + 25 = 125 ms$

KEY IDEA

In a dual-slope ADC the run-down time is $T_{2} = T_{1} V_{in} /∣ V_{r e f} ∣$ , independent of $R$ and $C$ . The time to measure one input is the fixed run-up time plus that run-down time: $T_{co n v} = T_{1} + T_{2}$ .

WATCH OUT FOR

Giving only the run-down time, $T_{2} = 25 ms$ , which is the part proportional to the input. The measurement also includes the fixed 100 ms integration of the input, so the answer is 125 ms. A second slip is carrying the minus sign of $V_{r e f}$ into the ratio and getting a negative time. The reference is negative only so that it opposes the input, and the magnitude is what sets the slope.

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GATE 2025 IN Q.34: solution and answer

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