GATE E&C Engineering Analog Circuits Flashcards

24 sample cards from the Analog Circuits deck

Sampled from the end of the deck, so these are different cards from the ones shown on the syllabus page.

1. What is the function and key advantage of a differential amplifier (diff-pair)?

   It amplifies the difference between two input signals while rejecting signals common to both. Advantages: high common-mode rejection, good DC coupling without offset drift, and it forms the input stage of op-amps. $$v_o = A_d(v_1 - v_2)$$

2. Define Common-Mode Rejection Ratio (CMRR) and express it in dB.

   $$\text{CMRR} = \left|\frac{A_d}{A_{cm}}\right|, \qquad \text{CMRR(dB)} = 20\log_{10}\left|\frac{A_d}{A_{cm}}\right|$$ where $A_d$ is differential gain and $A_{cm}$ is common-mode gain. Higher CMRR means better rejection of common-mode noise.

3. For a BJT differential pair biased by tail current $I_{EE}$, give the differential-mode small-signal gain (single-ended and double-ended output).

   With $g_m = I_{EE}/(2V_T)$: double-ended output gain $A_d = g_m R_C$; single-ended output gain $A_d = \tfrac{1}{2} g_m R_C$. The tail current source sets the bias and provides high common-mode rejection.

4. Why is a current source (high $R_{tail}$) used as the tail in a differential amplifier?

   A high tail resistance minimizes the common-mode gain $A_{cm} \approx -R_C/(2R_{tail})$, thereby maximizing CMRR. An ideal current source ($R_{tail}\to\infty$) makes $A_{cm}\to 0$.

5. For an ideal op-amp, list the key assumptions used in circuit analysis.

   Infinite open-loop gain, infinite input impedance (zero input current), zero output impedance, infinite bandwidth, zero offset. With negative feedback, the 'virtual short' applies: $v_+ = v_-$ and no current flows into the inputs.

6. Derive the closed-loop gain of an inverting op-amp amplifier with input resistor $R_1$ and feedback resistor $R_f$.

   Using the virtual ground at the inverting input: $$A_v = -\frac{R_f}{R_1}$$ Input impedance equals $R_1$; output is phase-inverted.

7. Give the closed-loop gain of a non-inverting op-amp amplifier with feedback resistor $R_f$ and grounded resistor $R_1$.

   $$A_v = 1 + \frac{R_f}{R_1}$$ Gain is always $\geq 1$, output is in phase with input, and input impedance is very high (ideally infinite).

8. What is the gain of a voltage follower (op-amp buffer), and why is it useful?

   $A_v = 1$ (unity gain, non-inverting). It provides very high input impedance and very low output impedance, making it ideal for impedance buffering/isolation between a high-impedance source and a low-impedance load.

9. Write the output expression of an op-amp inverting summing amplifier with inputs $V_1, V_2, V_3$ through $R_1, R_2, R_3$ and feedback $R_f$.

   $$V_o = -\left(\frac{R_f}{R_1}V_1 + \frac{R_f}{R_2}V_2 + \frac{R_f}{R_3}V_3\right)$$ If $R_1=R_2=R_3=R_f=R$, then $V_o = -(V_1+V_2+V_3)$ (a unity-weighted inverting adder).

10. Give the output of an op-amp difference (subtractor) amplifier with $R_1=R_3$ and $R_2=R_4$.

    $$V_o = \frac{R_2}{R_1}(V_2 - V_1)$$ It amplifies the difference of the two inputs; balanced resistor ratios are required for good CMRR.

11. Write the ideal output of an op-amp differentiator and state its main practical drawback.

    $$v_o(t) = -R_f C \frac{dv_{in}}{dt}$$ Drawback: gain rises with frequency ($+20\,\text{dB/decade}$), amplifying high-frequency noise and risking instability. A series input resistor $R_1$ is added to limit high-frequency gain.

12. Write the ideal output of an op-amp integrator and state its main practical issue.

    $$v_o(t) = -\frac{1}{R C}\int v_{in}\,dt$$ Practical issue: DC offset and bias currents cause the capacitor to charge and the output to drift into saturation. A large feedback resistor $R_f$ across $C$ is added to provide a DC path and bound the gain.

13. For a practical (lossy) integrator with feedback resistor $R_f$ parallel to $C$, what is the DC gain and the frequency above which it integrates?

    DC gain $= -R_f/R_1$. It behaves as an integrator for frequencies above the pole $$f = \frac{1}{2\pi R_f C}$$ and as an amplifier below it.

14. Classify active filters by their pass characteristic and name the four basic types.

    Low-pass (passes frequencies below $f_c$), high-pass (passes above $f_c$), band-pass (passes a band between $f_L$ and $f_H$), and band-stop/notch (rejects a band). Active filters use op-amps with R and C, avoiding bulky inductors.

15. Give the cutoff frequency formula for a first-order active RC filter and its roll-off rate.

    $$f_c = \frac{1}{2\pi R C}$$ A first-order filter rolls off at $-20\,\text{dB/decade}$ ($-6\,\text{dB/octave}$); an $n$th-order filter rolls off at $-20n\,\text{dB/decade}$.

16. What is the quality factor $Q$ of a band-pass filter, and how does it relate to bandwidth?

    $$Q = \frac{f_0}{\text{BW}} = \frac{f_0}{f_H - f_L}$$ where $f_0 = \sqrt{f_L f_H}$ is the center frequency. High $Q$ means a narrow, sharply selective passband.

17. Compare Butterworth, Chebyshev, and Bessel filter approximations.

    Butterworth: maximally flat passband, moderate roll-off, no ripple. Chebyshev: steeper roll-off but ripple in the passband (or stopband). Bessel: maximally flat group delay (best linear phase / least signal distortion) but gentlest roll-off.

18. For a Sallen-Key second-order low-pass filter with equal components $R$ and $C$, give the cutoff frequency.

    $$f_c = \frac{1}{2\pi R C}$$ The damping (and $Q$) is set by the op-amp gain $K = 1 + R_f/R_1$; for a Butterworth response the required gain gives $Q = 1/\sqrt{2} \approx 0.707$.

19. What is a Schmitt trigger and what key property distinguishes it from a normal comparator?

    A Schmitt trigger is a comparator with positive feedback that exhibits hysteresis: it has two different threshold voltages ($V_{TH}$ and $V_{TL}$). This noise immunity prevents false multiple transitions on slowly varying or noisy inputs, and it can square up signals.

20. For an inverting Schmitt trigger with output $\pm V_{sat}$ and feedback divider $R_1$ (to output) and $R_2$ (to ground at the $+$ input), give the upper and lower trip points and the hysteresis width.

    $$V_{TH} = +V_{sat}\frac{R_2}{R_1+R_2}, \quad V_{TL} = -V_{sat}\frac{R_2}{R_1+R_2}$$ Hysteresis width: $$V_H = V_{TH} - V_{TL} = \frac{2R_2}{R_1+R_2}V_{sat}$$

21. State the Barkhausen criterion for sustained sinusoidal oscillation.

    For a feedback oscillator with loop gain $A\beta$: the magnitude of the loop gain must be unity, $|A\beta| = 1$, and the total phase shift around the loop must be $0^\circ$ (or a multiple of $360^\circ$). To start, $|A\beta| \geq 1$ initially.

22. Give the frequency of oscillation and the gain condition for a Wien-bridge oscillator (equal $R$ and $C$).

    $$f_0 = \frac{1}{2\pi R C}$$ The amplifier must provide a gain of at least $3$ (i.e. $1 + R_f/R_1 = 3$) to satisfy the Barkhausen criterion.

23. Give the oscillation frequency for an RC phase-shift oscillator using three identical RC sections, and the minimum amplifier gain required.

    $$f_0 = \frac{1}{2\pi R C \sqrt{6}}$$ Each section contributes part of the needed $180^\circ$; the inverting amplifier must supply a magnitude gain of at least $29$ to sustain oscillation.

24. For an LC oscillator (Colpitts or Hartley), give the oscillation frequency. State the difference between the two.

    $$f_0 = \frac{1}{2\pi\sqrt{LC}}$$ Colpitts uses a capacitive tap ($C_1, C_2$ with $C = \frac{C_1 C_2}{C_1+C_2}$, single $L$). Hartley uses an inductive tap ($L_1, L_2$ with $L = L_1+L_2$, single $C$). Both feed back a fraction of the tank voltage for $0^\circ$ loop phase.

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