GATE E&C Engineering Analog Circuits Syllabus

What limits the high-frequency response of a CE amplifier, and which effect dominates?

The internal junction capacitances $C_\pi$ and $C_\mu$ limit high-frequency gain. The Miller effect multiplies $C_\mu$ by $(1+|A_v|)$, giving an effective input capacitance $C_{in} = C_\pi + C_\mu(1+|A_v|)$, which dominates and sets the upper cutoff frequency.

State the Miller effect: how an impedance $Z$ bridging input and output of an inverting amplifier of gain $-A$ appears at the input.

The bridging impedance appears at the input as $$Z_{in} = \frac{Z}{1+A}$$ For a capacitor $C$, the effective input capacitance becomes $C_{in} = C(1+A)$, greatly reducing bandwidth.

Define the gain-bandwidth product (GBW) and the unity-gain frequency $f_T$ of a transistor.

GBW is the product of midband gain and bandwidth, approximately constant for a given device. $f_T$ is the frequency where the short-circuit current gain falls to unity: $$f_T = \frac{g_m}{2\pi (C_\pi + C_\mu)}$$

A single-pole amplifier has DC gain $A_0$ and pole at $f_p$. Write its frequency response magnitude and the $-3\,\text{dB}$ bandwidth.

$$A(f) = \frac{A_0}{1 + jf/f_p}, \qquad |A(f)| = \frac{A_0}{\sqrt{1+(f/f_p)^2}}$$ The $-3\,\text{dB}$ bandwidth equals the pole frequency $f_p$; gain rolls off at $-20\,\text{dB/decade}$ beyond it.

What is a current mirror, and what is its basic function?

A current mirror is a circuit that copies (mirrors) a reference current into one or more output branches, ideally independent of the output voltage. It provides active loads and bias currents in IC amplifiers.

For a basic two-transistor BJT current mirror with reference current $I_{ref}$, write the output current including the finite-$\beta$ error.

$$I_{out} = \frac{I_{ref}}{1 + \dfrac{2}{\beta}}$$ For large $\beta$, $I_{out} \approx I_{ref}$. The factor accounts for base currents stealing from the mirror.

Why does a Wilson or cascode current mirror outperform the basic current mirror?

They greatly increase the output resistance (improving current-source ideality and reducing dependence on output voltage / Early effect) and reduce the base-current ($\beta$) error, giving much better matching of $I_{out}$ to $I_{ref}$.

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. $$v_o = A_d(v_1 - v_2)$$

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.

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.

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 $A_{cm}\to 0$.

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.

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.

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).

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.

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).

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.

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.

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.

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.

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.

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 $-20n\,\text{dB/decade}$.

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.

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.

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 $Q = 1/\sqrt{2} \approx 0.707$.

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.

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

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.

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.

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.

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.