GATE EC Analog Circuits Formula Sheet

Organised by the official GATE 2027 EC syllabus, Section 4: Analog Circuits. Each worked example was checked in code. Practise these: GATE EC Analog Circuits previous year questions.

Diode circuits

Syllabus: Clipping, clamping and rectifiers.

Rectifier average and RMS output

\text{Half-wave: } V_{avg} = \frac{V_m}{\pi},\ V_{rms} = \frac{V_m}{2}; \qquad \text{Full-wave: } V_{avg} = \frac{2V_m}{\pi},\ V_{rms} = \frac{V_m}{\sqrt{2}}

\begin{aligned} &\text{Half-wave: } V_{avg} \\ &\quad = \frac{V_m}{\pi},\ V_{rms} \\ &\quad = \frac{V_m}{2}; \\ &\text{Full-wave: } V_{avg} \\ &\quad = \frac{2V_m}{\pi},\ V_{rms} \\ &\quad = \frac{V_m}{\sqrt{2}} \end{aligned}

Symbols

$V_m$ : peak voltage reaching the load (V)

Applies when

Ideal diodes, resistive load, sinusoidal input. With a real diode, use

V_{m}

minus the diode drops.

Common mistake

In a centre-tapped full-wave rectifier, taking

V_{m}

as the whole secondary's peak. Each diode sees only half the secondary.

Worked example

Full-wave rectifier with

V_{m}

= 10 V: average output: 6.366 V (half-wave: 3.183 V)

Capacitor-filter ripple

V_r \approx \frac{V_m}{f R C}\ \text{(half-wave)}, \qquad V_r \approx \frac{V_m}{2 f R C}\ \text{(full-wave)}

\begin{aligned} V_r &\approx \frac{V_m}{f R C}\ \text{(half-wave)}, \\ V_r &\approx \frac{V_m}{2 f R C}\ \text{(full-wave)} \end{aligned}

Symbols

$V_r$ : peak-to-peak ripple (V)
$f$ : input (mains) frequency (Hz)
$R$ : load resistance (Ω)
$C$ : filter capacitance (F)

Applies when

Small ripple,

V_{r}

much less than

V_{m}

, so the capacitor discharges almost linearly.

Common mistake

Using the half-wave formula for a full-wave rectifier: the capacitor recharges twice per cycle, halving the ripple.

Worked example

Full-wave,

V_{m}

= 10 V, f = 50 Hz, R = 1 kΩ, C = 1000 µF: 0.1 V

Clipper level

v_o = V_B + V_\gamma \quad \text{while the diode conducts}

\begin{gathered} v_o = V_B + V_\gamma \\ \text{while the diode conducts} \end{gathered}

Symbols

$V_B$ : bias source in series with the diode (V)
$V_\gamma$ : diode cut-in voltage, 0 for an ideal diode (V)

Applies when

A shunt diode-and-source clipper: above the level the diode conducts and holds the output.

Common mistake

Forgetting the cut-in voltage of a real diode, which moves the clipping level by about 0.7 V for silicon.

Worked example

A 3 V bias with a silicon diode (0.7 V): clipping level: 3.7 V

Clamper shift

v_o(t) \approx v_i(t) + V_m \quad \text{(positive clamper, ideal diode)}

\begin{gathered} v_o(t) \approx v_i(t) + V_m \\ \text{(positive clamper, ideal diode)} \end{gathered}

Symbols

$V_m$ : input peak (V)

Applies when

RC much longer than the input period, so the capacitor barely discharges between peaks.

Common mistake

Expecting the waveform's shape or peak-to-peak value to change. A clamper only shifts the DC level.

Worked example

A 5 V peak sine into a positive clamper: highest output: 10 V (it swings 0 to 10 V)

BJT and MOSFET amplifiers

Syllabus: Biasing, AC coupling, small signal analysis, frequency response. Current mirrors and differential amplifiers.

BJT small-signal parameters

g_m = \frac{I_C}{V_T}, \qquad r_\pi = \frac{\beta}{g_m}, \qquad r_o = \frac{V_A}{I_C}

\begin{aligned} g_m &= \frac{I_C}{V_T}, \\ r_\pi &= \frac{\beta}{g_m}, \\ r_o &= \frac{V_A}{I_C} \end{aligned}

Symbols

$I_C$ : DC collector current (A)
$V_T$ : thermal voltage, about 26 mV at 300 K
$\beta$ : current gain
$V_A$ : Early voltage (V)

Applies when

Forward-active region, small signals (

v_{be}

much less than

V_{T}

Common mistake

Using the DC voltage instead of the bias current:

g_{m}

depends on

I_{C}

alone.

Worked example

I_{C}

= 1 mA,

V_{T}

= 26 mV: transconductance: 38.46 mS (

r_\pi

= 2.6 kΩ for β = 100)

MOSFET in saturation

I_D = \tfrac{1}{2}k_n (V_{GS} - V_t)^2, \qquad g_m = k_n (V_{GS} - V_t) = \frac{2 I_D}{V_{GS} - V_t}

\begin{aligned} I_D &= \tfrac{1}{2}k_n (V_{GS} - V_t)^2, \\ g_m &= k_n (V_{GS} - V_t) \\ &= \frac{2 I_D}{V_{GS} - V_t} \end{aligned}

Symbols

$k_n$ : $\mu_n C_{ox} W/L$ (A/V²)
$V_t$ : threshold voltage (V)
$V_{GS} - V_t$ : overdrive (V)

Applies when

Saturation:

V_{DS}

≥

V_{GS}

−

V_{t}

, channel-length modulation neglected.

Common mistake

Using the triode formula once

V_{DS}

exceeds the overdrive. Check the region at the new operating point.

Worked example

k_{n}

= 2 mA/V², overdrive 0.5 V: transconductance: 1 mS (

I_{D}

= 0.25 mA)

Common-emitter and common-source gain

A_v = -g_m (R_C \parallel r_o), \qquad A_v \approx -\frac{R_C}{r_e + R_E}\ \text{(unbypassed } R_E\text{)}

\begin{gathered} A_v = -g_m (R_C \parallel r_o) \\ A_v \approx -\frac{R_C}{r_e + R_E} \\ \text{(unbypassed } R_E\text{)} \end{gathered}

Symbols

$R_C$ : collector (or drain) load (Ω)
$r_e$ : 1/ $g_{m}$ for a BJT, roughly (Ω)
$R_E$ : unbypassed emitter resistance (Ω)

Applies when

Midband, coupling and bypass capacitors acting as shorts. For a MOSFET, replace

R_{C}

with

R_{D}

Common mistake

Forgetting that an unbypassed emitter resistor divides the gain by about 1 +

g_{m}

R_{E}

Worked example

g_{m}

= 40 mS,

R_{C}

= 5 kΩ,

r_{o}

very large: -200 V/V

Source and emitter follower

A_v = \frac{g_m R_S}{1 + g_m R_S}

Symbols

$R_S$ : resistance from the source (or emitter) to AC ground (Ω)

Applies when

Output taken at the source or emitter; body effect neglected.

Common mistake

Expecting a gain of exactly 1. It is just under 1, approaching 1 only as

g_{m}

R_{S}

grows.

Worked example

g_{m}

= 1 mS,

R_{S}

= 10 kΩ: 0.909 V/V

Miller effect

C_{in} = C_f\,(1 - A_v)

Symbols

$C_f$ : capacitance between input and output (F)
$A_v$ : voltage gain across it (negative for an inverting stage)

Applies when

A capacitor bridging an amplifier with gain

A_{v}

, at frequencies where

A_{v}

holds.

Common mistake

Using |

A_{v}

| − 1. With an inverting gain the factor is 1 + |

A_{v}

Worked example

C_{f}

= 2 pF across a stage with

A_{v}

= −100: 202 pF

Current mirror

I_{out} = I_{ref}\,\frac{(W/L)_2}{(W/L)_1}\ \text{(MOSFET)}, \qquad I_{out} = \frac{I_{ref}}{1 + 2/\beta}\ \text{(BJT, equal devices)}

\begin{gathered} \text{MOSFET: } I_{out} = I_{ref}\,\frac{(W/L)_2}{(W/L)_1} \\ \text{BJT, equal devices:} \\ I_{out} = \frac{I_{ref}}{1 + 2/\beta} \end{gathered}

Symbols

$I_{ref}$ : reference current (A)
$(W/L)$ : aspect ratios of the two transistors
$\beta$ : BJT current gain

Applies when

Both transistors in saturation or active region, channel-length modulation and Early effect neglected.

Common mistake

Ignoring the base currents in a BJT mirror, which take

2I_C/\beta

from the reference.

Worked example

BJT mirror,

I_{ref}

= 1 mA, β = 100: 0.98 mA

Differential pair

A_d = g_m R_D\ \text{(differential output)}, \qquad A_{cm} \approx -\frac{R_D}{2R_{SS}}, \qquad \text{CMRR} = \left|\frac{A_d}{A_{cm}}\right|

\begin{gathered} A_d = g_m R_D \\ \text{(differential output)} \\ A_{cm} \approx -\frac{R_D}{2R_{SS}} \\ \text{CMRR} = \left|\frac{A_d}{A_{cm}}\right| \end{gathered}

Symbols

$g_m$ : transconductance of each transistor (S)
$R_D$ : drain resistor on each side (Ω)
$R_{SS}$ : resistance of the tail current source (Ω)

Applies when

Matched transistors, small signals. A single-ended output has half the differential gain.

Common mistake

Using the full differential gain for a single-ended output.

Worked example

g_{m}

= 1 mS,

R_{D}

= 10 kΩ,

R_{SS}

= 50 kΩ, single-ended output: CMRR: 33.98 dB (

A_{d}

= 5, |

A_{cm}

| = 0.1)

Op-amp circuits

Syllabus: Amplifiers, summers, differentiators, integrators, active filters, Schmitt trigger and oscillators, dominant-pole (Miller) compensation, phase margin.

Inverting and non-inverting amplifiers

A_v = -\frac{R_f}{R_1}\ \text{(inverting)}, \qquad A_v = 1 + \frac{R_f}{R_1}\ \text{(non-inverting)}

\begin{aligned} A_v &= -\frac{R_f}{R_1}\ \text{(inverting)}, \\ A_v &= 1 \\ &\quad + \frac{R_f}{R_1}\ \text{(non-inverting)} \end{aligned}

Symbols

$R_f$ : feedback resistor (Ω)
$R_1$ : resistor from the inverting input to the source or ground (Ω)

Applies when

Ideal op-amp in negative feedback, not saturated.

Common mistake

Swapping the two: the non-inverting gain is never below 1.

Worked example

R_{f}

= 10 kΩ,

R_{1}

= 1 kΩ, non-inverting: 11 V/V (inverting: −10)

Summing amplifier

v_o = -R_f\left(\frac{v_1}{R_1} + \frac{v_2}{R_2} + \cdots\right)

Symbols

$v_k$ : input voltages (V)
$R_k$ : their input resistors (Ω)

Applies when

Inverting summer with an ideal op-amp; the inverting input is a virtual ground.

Common mistake

Forgetting the minus sign of the inverting summer.

Worked example

R_{f}

= 10 kΩ,

v_{1}

= 1 V through 10 kΩ,

v_{2}

= 0.5 V through 5 kΩ: -2 V

Integrator and differentiator

v_o = -\frac{1}{RC}\int v_i\,dt, \qquad v_o = -RC\,\frac{dv_i}{dt}

\begin{aligned} v_o &= -\frac{1}{RC}\int v_i\,dt, \\ v_o &= -RC\,\frac{dv_i}{dt} \end{aligned}

Symbols

$R, C$ : input and feedback elements (Ω, F)

Applies when

Ideal op-amp, not saturated. A practical integrator needs a resistor across C to stop it drifting.

Common mistake

Forgetting that a DC input saturates an integrator: the output ramps until it hits the rail.

Worked example

Integrator, R = 10 kΩ, C = 0.1 µF, input 1 V DC: output slope: -1000 V/s

First-order active filter corner

f_c = \frac{1}{2\pi R C}

Symbols

$R, C$ : the filter's resistor and capacitor (Ω, F)

Applies when

Single-pole RC section, ideal op-amp.

Common mistake

Dropping the 2π, which gives

ω_{c}

in rad/s instead of

f_{c}

in hertz.

Worked example

R = 10 kΩ, C = 10 nF: 1591.5 Hz

Schmitt trigger thresholds

V_{TH,TL} = \pm V_{sat}\,\frac{R_1}{R_1 + R_2}\ \text{(inverting)}

\begin{gathered} V_{TH,TL} = \pm V_{sat}\,\frac{R_1}{R_1 + R_2} \\ \text{(inverting)} \end{gathered}

Symbols

$V_{sat}$ : output saturation voltage (V)
$R_1$ : resistor from the + input to ground (Ω)
$R_2$ : feedback resistor from output to the + input (Ω)

Applies when

Inverting Schmitt trigger with the reference at ground; the hysteresis is twice the threshold.

Common mistake

Using the non-inverting formula, where the thresholds are ±

V_{sat}

R_{1}

R_{2}

measured at the input.

Worked example

V_{sat}

= 12 V,

R_{1}

= 10 kΩ,

R_{2}

= 40 kΩ: upper threshold: 2.4 V

Wien-bridge and RC phase-shift oscillators

\text{Wien: } f = \frac{1}{2\pi R C},\ A = 3; \qquad \text{phase-shift (3 RC sections): } f = \frac{1}{2\pi\sqrt{6}RC},\ |A| = 29

\begin{gathered} \text{Wien: } f = \frac{1}{2\pi R C},\ A = 3 \\ \text{Phase-shift (3 RC sections):} \\ f = \frac{1}{2\pi\sqrt{6}RC},\ |A| = 29 \end{gathered}

Symbols

$A$ : amplifier gain needed for sustained oscillation

Applies when

Barkhausen criterion: loop gain exactly 1 with zero total phase at f.

Common mistake

Mixing them up: 29 and √6 belong to the three-section phase-shift oscillator, 3 and 1/(2πRC) to the Wien bridge.

Worked example

Wien bridge with R = 10 kΩ, C = 10 nF: 1591.5 Hz

Gain-bandwidth product

f_{3dB} = \frac{\text{GBW}}{1 + R_f/R_1}

Symbols

$\text{GBW}$ : unity-gain bandwidth of a dominant-pole op-amp (Hz)

Applies when

A dominant-pole (internally compensated) op-amp; the noise gain 1 +

R_{f}

R_{1}

sets the bandwidth for both configurations.

Common mistake

Dividing by the inverting gain |

R_{f}

R_{1}

| instead of the noise gain 1 +

R_{f}

R_{1}

Worked example

GBW = 1 MHz, non-inverting gain 11: 90.9 kHz

Phase margin

\text{PM} = 180° + \angle L(j\omega_{gc}), \qquad |L(j\omega_{gc})| = 1

\begin{aligned} \text{PM} &= 180° + \angle L(j\omega_{gc}), \\ |L(j\omega_{gc})| &= 1 \end{aligned}

Symbols

$L(s)$ : loop gain
$\omega_{gc}$ : gain crossover frequency (rad/s)

Applies when

Stable open loop. Dominant-pole compensation pushes

ω_{gc}

below the second pole to raise PM.

Common mistake

Reading the phase at the frequency where the phase is −180° (that gives the gain margin, not PM).

Worked example

L(s) = 10⁵ / ((1 + s/10)(1 + s/10⁶)): phase margin: 51.8 degrees

Practise these: GATE EC Analog Circuits previous year questions

Source for the section headings: the official GATE 2027 EC syllabus, IIT Madras. The formulas, examples and notes are Studyloaf's own.

GATE EC Analog Circuits Formula Sheet

Diode circuits

Rectifier average and RMS output

Capacitor-filter ripple

Clipper level

Clamper shift

BJT and MOSFET amplifiers

BJT small-signal parameters

MOSFET in saturation

Common-emitter and common-source gain

Source and emitter follower

Miller effect

Current mirror

Differential pair

Op-amp circuits

Inverting and non-inverting amplifiers

Summing amplifier

Integrator and differentiator

First-order active filter corner

Schmitt trigger thresholds

Wien-bridge and RC phase-shift oscillators

Gain-bandwidth product

Phase margin

Other GATE EC formula sheets