GATE 2025 IN Q.31: solution and answer

Official answer: B, C, DSource: Official answer key, GATE 2025 IN (IIT Roorkee)

1. Test for linearity (superposition and homogeneity). A circuit is linear only if its output is a fixed linear combination of its inputs, e.g. $v_o=k{v}_i$, for every input. If scaling the input by $a$ does not scale the output by $a$, or if the output depends on the sign or the past history of the input, the operation is non-linear. All op-amps are assumed ideal.

2. (A) Instrumentation amplifier. It uses negative feedback, so the ideal op-amps stay in their linear region. For the standard three-op-amp circuit (gain resistor $R_G$, input-stage resistors $R$, difference-stage resistors $R_2$ and $R_3$), $v_o=A_d(v_2-v_1)$ with $A_d=\left(1+\frac{2R}{R_G}\right)\frac{R_3}{R_2}$. That is a fixed linear combination of the inputs, so superposition holds and the circuit is linear.

3. (B) Schmitt trigger. It uses positive feedback, so the op-amp saturates and $v_o$ takes only the values $+V_{sat}$ and $-V_{sat}$. The output switches at the upper threshold $V_{UT}$ when the input is rising and at the lower threshold $V_{LT}$ when it is falling (hysteresis). A two-level output that depends on the input's history cannot be written as $k{v}_i$, so the circuit is non-linear.

4. (C) Logarithmic amplifier. A diode (or a BJT) sits in the feedback path of an inverting stage with input resistor $R$. The inverting input is a virtual ground, so $I_D=v_i/R$ (for $v_i>0$) and $v_o=-V_D$. The diode law $I_D=I_S{e}^{V_D/V_T}$ (the $-1$ term is neglected in forward conduction) gives $$v_o=-V_T\ln \left(\frac{v_i}{I_{S}R}\right)$$ If the input is scaled by $a$, the output becomes $v_o-V_T\ln a$: a constant is added, the output is not multiplied by $a$. Homogeneity fails, so the circuit is non-linear.

5. (D) Precision rectifier. The diode inside the feedback loop conducts for only one polarity of input. For a typical half-wave circuit, $v_o=v_i$ for $v_i>0$ and $v_o=0$ for $v_i<0$; for a full-wave circuit, $v_o=|v_i|$. A piecewise function like this breaks homogeneity (for example, $|-v|\ne -|v|$), so the circuit is non-linear. The high loop gain of the op-amp only removes the effect of the diode's cut-in voltage (about $0.7\text{V}$), so that small signals are rectified too. It does not make the operation linear.

6. Result. The non-linear circuits are the Schmitt trigger, the logarithmic amplifier and the precision rectifier: options B, C, D.