GATE Electrical Engineering Analog and Digital Electronics Syllabus

What is the ripple frequency for half-wave and full-wave rectifiers fed from a $50\,\text{Hz}$ line?

Half-wave: same as line, $50\,\text{Hz}$. Full-wave (and bridge): twice the line, $100\,\text{Hz}$.

State the Peak Inverse Voltage (PIV) for a center-tapped full-wave rectifier and a bridge rectifier (ideal diodes).

Center-tapped: $\text{PIV} = 2V_m$. Bridge: $\text{PIV} = V_m$ (where $V_m$ is the peak across half the secondary / the load respectively).

Compare center-tapped and bridge full-wave rectifiers.

Center-tapped uses 2 diodes and a center-tapped transformer with $\text{PIV}=2V_m$. Bridge uses 4 diodes, no center tap, with $\text{PIV}=V_m$ but two diode drops per half-cycle. Bridge gives better transformer utilization.

Define the Transformer Utilization Factor (TUF) and give its full-wave bridge value.

$\text{TUF} = \dfrac{P_{dc}}{P_{ac(rated\,secondary)}}$. For a full-wave bridge $\text{TUF} \approx 0.812$; center-tapped $\approx 0.693$; half-wave $\approx 0.287$.

What is the purpose of a filter capacitor after a rectifier, and how does ripple depend on it?

It smooths the pulsating DC by charging on peaks and discharging into the load. For a full-wave rectifier with capacitor filter, ripple factor $r \approx \dfrac{1}{4\sqrt{3}\,fRC}$, so larger $C$ (or $R$) reduces ripple.

What is BJT biasing and why is it needed?

Biasing sets the DC operating (quiescent) point $Q$ — $I_{CQ}$ and $V_{CEQ}$ — so the transistor operates in the active region, ensuring faithful (undistorted) amplification of the AC signal.

For a BJT in the active region, give the relations between $I_C$, $I_B$, and $I_E$.

$I_C = \beta I_B$, $I_E = I_C + I_B = (\beta+1)I_B$, and $\alpha = \dfrac{\beta}{\beta+1}$, $I_C = \alpha I_E$.

Why is the voltage-divider (self) bias the most widely used BJT biasing scheme?

It makes the operating point nearly independent of $\beta$ by fixing the base voltage with a resistor divider and using an emitter resistor for negative feedback, giving good thermal/Q-point stability.

Define the stability factor $S(I_{CO})$ of a bias circuit.

$$S = \frac{\partial I_C}{\partial I_{CO}}$$ It measures how much collector current changes with reverse saturation current; a smaller $S$ (close to 1) means a more stable bias.

Why is fixed-base bias considered poor for stability?

Its stability factor is large, $S = \beta + 1$, so $I_C$ varies strongly with $\beta$ and temperature, easily driving the transistor toward saturation or cutoff (thermal runaway).

What is the role of the emitter resistor $R_E$ in biasing?

It provides series (current) negative feedback: a rise in $I_C$ raises $V_E$, reducing $V_{BE}$ and counteracting the increase, thereby stabilizing the Q-point against $\beta$ and temperature changes.

For voltage-divider bias, write the Thevenin base voltage and emitter current.

$V_{TH} = V_{CC}\dfrac{R_2}{R_1+R_2}$, $R_{TH}=R_1\Vert R_2$, and $I_E \approx \dfrac{V_{TH}-V_{BE}}{R_E + R_{TH}/(\beta+1)} \approx \dfrac{V_{TH}-V_{BE}}{R_E}$.

What is the hybrid-$\pi$ (small-signal) equivalent circuit of a BJT?

A linearized model of the transistor valid for small signals around the Q-point, with input resistance $r_\pi$, transconductance $g_m$, output resistance $r_o$, and dependent current source $g_m v_{be}$ (or $\beta i_b$).

Give the expressions for transconductance $g_m$ and $r_\pi$ in the small-signal model.

$g_m = \dfrac{I_C}{V_T}$ where $V_T \approx 26\,\text{mV}$ at room temperature; and $r_\pi = \dfrac{\beta}{g_m} = \dfrac{V_T}{I_B}$.

In the h-parameter model of a BJT, what do $h_{ie}$, $h_{fe}$, $h_{re}$, $h_{oe}$ represent?

$h_{ie}$ = input impedance, $h_{fe}$ = forward current gain ($\beta$), $h_{re}$ = reverse voltage transfer ratio, $h_{oe}$ = output admittance (common-emitter configuration).

What is the small-signal output resistance $r_o$ and how does it relate to the Early voltage?

$$r_o = \frac{V_A + V_{CE}}{I_C} \approx \frac{V_A}{I_C}$$ where $V_A$ is the Early voltage; it models the slope of the output characteristics in the active region.

What is meant by the frequency response of an amplifier?

The variation of gain (magnitude and phase) with frequency, typically plotted on a Bode (log) scale, showing a flat midband bounded by lower and upper cutoff (half-power) frequencies.

Define the lower and upper cutoff frequencies and bandwidth of an amplifier.

The cutoff (3-dB) frequencies $f_L$ and $f_H$ are where gain falls to $\dfrac{1}{\sqrt{2}}$ (i.e. $-3\,\text{dB}$) of midband. Bandwidth $BW = f_H - f_L \approx f_H$ for $f_H \gg f_L$.

What limits the low-frequency and high-frequency response of a BJT amplifier?

Low frequency: coupling and bypass capacitors (their reactance rises, reducing gain). High frequency: internal junction/parasitic capacitances ($C_\pi$, $C_\mu$) and the Miller effect shunt the signal.

State the Miller effect and its effect on input capacitance.

A feedback capacitance $C_\mu$ between input and output of an inverting amplifier of gain $A_v$ appears at the input as $C_{in} = C_\mu(1+|A_v|)$, greatly lowering the high-frequency cutoff.

Define the gain-bandwidth product and the transition frequency $f_T$.

The gain-bandwidth product is approximately constant; $f_T$ is the frequency where the short-circuit current gain falls to unity: $f_T = \dfrac{g_m}{2\pi(C_\pi + C_\mu)}$.

On a Bode plot, what is the asymptotic slope of a single-pole roll-off?

$-20\,\text{dB/decade}$ (equivalently $-6\,\text{dB/octave}$), with a phase shift approaching $-90^\circ$.

Describe the output characteristics of a BJT in common-emitter configuration.

A plot of $I_C$ versus $V_{CE}$ for various $I_B$. It shows a steep saturation region at low $V_{CE}$, a nearly flat active region (slight upward slope due to the Early effect), and cutoff at $I_B=0$.

Describe the input characteristics of a common-emitter BJT.

A plot of $I_B$ versus $V_{BE}$ (for fixed $V_{CE}$), resembling a forward-biased diode curve with a cut-in voltage around $0.6$–$0.7\,\text{V}$ for silicon.

What does the I–V characteristic of a forward-biased diode follow (Shockley equation)?

$$I = I_S\left(e^{\,V/(\eta V_T)} - 1\right)$$ where $I_S$ is reverse saturation current, $\eta$ the ideality factor, and $V_T = kT/q \approx 26\,\text{mV}$ at $300\,\text{K}$.

What is a Sallen-Key filter?

A second-order active filter topology using one op-amp (as a non-inverting unity or finite-gain amplifier) with two resistors and two capacitors, giving a 2-pole low-pass, high-pass, or band-pass response.

For a unity-gain Sallen-Key low-pass filter, give the cutoff frequency and quality factor.

$$f_0 = \frac{1}{2\pi\sqrt{R_1 R_2 C_1 C_2}}, \qquad Q = \frac{\sqrt{R_1 R_2 C_1 C_2}}{C_2(R_1+R_2)}$$

What is one practical advantage of the Sallen-Key topology?

It realizes a second-order (12 dB/octave) response with a single op-amp and few passive components, has high input impedance, low output impedance, and needs no inductors.

What defines a Butterworth filter?

A filter with a maximally flat magnitude response in the passband (no ripple), whose magnitude is $$|H(j\omega)| = \frac{1}{\sqrt{1+(\omega/\omega_c)^{2n}}}$$ for order $n$.

What is the roll-off rate of an $n$-th order Butterworth filter, and what is the $Q$ of a 2nd-order Butterworth?

Roll-off is $-20n\,\text{dB/decade}$ (i.e. $-20\,\text{dB/decade}$ per pole). A second-order Butterworth has $Q = \dfrac{1}{\sqrt{2}} \approx 0.707$, giving the maximally flat response.

Compare Butterworth, Chebyshev, and Bessel filters in terms of passband response.

Butterworth: maximally flat passband, moderate roll-off. Chebyshev: passband ripple but steeper roll-off. Bessel: gentle roll-off but maximally flat (linear) phase / constant group delay.