GATE Instrumentation Engineering (IN) Calculators
Calculators for the numerical questions in GATE Instrumentation Engineering, from circuits and control to sensors, measurements and communication, each showing the working.
54 calculators · 6 converters · common conversions
Engineering Mathematics
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Eigenvalue Calculator
det(A − λI) = 0
Σλ = trace A, Πλ = det A
Example: The matrix [[4, 1], [2, 3]]: Eigenvalues: λ₁ = 5, λ₂ = 2.
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LU Decomposition Calculator
A = LU, uᵢⱼ = aᵢⱼ − Σ lᵢₖuₖⱼ, lⱼᵢ = (aⱼᵢ − Σ lⱼₖuₖᵢ)/uᵢᵢ
Doolittle: L has 1s on the diagonal
Example: The matrix [[2, −1, −2], [−4, 6, 3], [−4, −2, 8]]: det A = product of U’s diagonal = 24.
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- Newton-Raphson Method Calculator – f(x), optional f′(x) and a starting guess → root, iteration table and convergence check.
- Numerical Integration Calculator – f(x), limits and intervals → trapezoidal, Simpson 1/3 and 3/8 results with a reference value.
- Euler Method Calculator – An ODE, initial value, step and end point → y by Euler, Heun, RK2 or RK4 with every step.
- Normal Distribution Calculator – Mean, standard deviation and bounds → normal probabilities, z-scores, density and percentiles.
- Bayes’ Theorem Calculator – Priors and likelihoods → total probability and posterior probabilities, with a Bayes table.
- Complex Number Calculator – Two complex numbers and an operation → result, modulus, argument and conjugate.
Electricity and Magnetism
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Coulomb's Law Calculator
F = k |q₁ q₂| / r²
k = 1/(4πε₀) = 8.9875517923 × 10⁹ N·m²/C², q the charges, r the distance between them
Example: Two +1 µC charges 10 cm apart in vacuum: Electrostatic force F = 0.898755 N.
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Electric Field Calculator
E = q/(4πε r²), λ/(2πε r), σ/(2ε)
ε = ε₀εᵣ; potential of a point charge V = q/(4πε r)
Example: A 1 nC point charge, 1 m away, in air: Electric field |E| = 8.98755 V/m.
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- Capacitance Calculator – Geometry and dielectric → capacitance in pF, nF and µF.
- Magnetic Field Calculator – Current and geometry → magnetic flux density B and field strength H.
- Inductance Calculator – Coil geometry, turns and core → inductance in µH, mH and H.
- Lorentz Force Calculator – Charge, speed and field (or current and length) → magnetic force, orbit radius and frequency.
Electrical Circuits and Machines
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Ohm's Law Calculator
V = I R, P = V I
V voltage, I current, R resistance, P power
Example: 12 V across 4 Ω drives 3 A and dissipates 36 W.
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Series and Parallel Calculator
R_s = R₁ + R₂ + …, 1/R_p = 1/R₁ + 1/R₂ + …
inductors combine like resistors; capacitors the other way round (parallel adds)
Example: Resistors of 100 Ω, 220 Ω and 470 Ω: Series equivalent = 790.
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- Voltage Divider Calculator – Input voltage and two resistors → output voltage, with load; or a current divider.
- Star-Delta Conversion Calculator – Three delta resistances → equivalent star, or star → delta.
- Maximum Power Transfer Calculator – Thevenin voltage and resistance → maximum load power, and power and efficiency at any load.
- RC & RL Time Constant Calculator – R and C (or L) → time constant and the voltage or current at any time t.
- RLC Resonance Calculator – R, L and C → resonant frequency, Q, bandwidth, half-power points and damping.
- Impedance Calculator – R, L, C and frequency → impedance magnitude, rectangular form, phase and power factor.
- AC Power Calculator – Voltage, current and power factor → kW, kVA, kVAR and complex power.
- Transformer Calculator – Rating, voltages and losses → turns ratio, currents, efficiency and maximum-efficiency load.
- DC-DC Converter Calculator – Input voltage, duty cycle (or target), L, C and switching frequency → output, ripple and mode.
- Rectifier Calculator – Transformer voltage, rectifier type and filter → DC output, ripple, PIV and efficiency.
Signals and Systems
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Sampling Theorem Calculator
fₛ ≥ 2f_max
alias frequency f_a = |f − k fₛ| with k the nearest integer to f/fₛ
Example: A 4 kHz signal sampled at 6 kHz, and a 5 kHz tone in the same system: Nyquist rate 2f_max = 8 kHz.
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DFT Calculator
X[k] = Σ x[n] e^(−j2πkn/N)
k = 0 … N − 1; X[0] is the sum of the samples
Example: The 4-point sequence x[n] = {1, 2, 3, 4}: X[0] (sum of the samples) = 10.
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Convolution Calculator
y[n] = Σₖ x[k] h[n − k]
length of linear convolution = L_x + L_h − 1; circular wraps modulo N
Example: x[n] = {1, 2, 3} through h[n] = {1, 1, 1}: Linear convolution y[n] = {1, 3, 6, 5, 3}.
Control Systems
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Second-Order System Calculator
Mₚ = e^(−ζπ/√(1−ζ²)), tₚ = π/ω_d
ω_d = ωₙ√(1 − ζ²), t_r = (π − arccos ζ)/ω_d, t_s ≈ 4/(ζωₙ) (2 %)
Example: A system with ζ = 0.5 and ωₙ = 4 rad/s (s² + 4s + 16): Peak overshoot Mₚ = 16.3034 %.
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Routh-Hurwitz Calculator
bᵢ = (a₁a₂ᵢ − a₀a₂ᵢ₊₁)/a₁
number of sign changes in the first column = number of roots in the right half-plane
Example: The characteristic equation s⁴ + 2s³ + 3s² + 4s + 5 = 0: Result: Unstable: 2 roots in the right half-plane.
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Steady-State Error Calculator
e_ss = A/(1 + Kₚ), A/K_v, A/Kₐ
Kₚ = lim G(s), K_v = lim sG(s), Kₐ = lim s²G(s) as s → 0
Example: G(s) = 10 / [s(s + 1)(s + 5)] with a unit ramp input: Steady-state error (ramp) e_ss = 0.5.
Analog Electronics
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Op-Amp Gain Calculator
Inverting −R_f/Rᵢₙ, non-inverting 1 + R_f/Rᵢₙ
difference (R_f/Rᵢₙ)(V₂ − V₁); in-amp 1 + 2R/R_G; summing −(R_f/Rᵢₙ)ΣV
Example: An inverting amplifier with Rᵢₙ = 10 kΩ, R_f = 100 kΩ and 0.2 V in: Voltage gain = -10.
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555 Timer Calculator
f = 1/(ln 2 (R₁ + 2R₂)C) ≈ 1.44/((R₁ + 2R₂)C)
duty = (R₁ + R₂)/(R₁ + 2R₂); monostable T = ln 3 × RC ≈ 1.1RC
Example: An astable 555 with R₁ = 1 kΩ, R₂ = 10 kΩ and C = 10 µF: Frequency f = 6.86998 Hz.
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- Decibel Calculator – Power or voltage ratio ↔ dB, and watts ↔ dBm.
- LED Resistor Calculator – Supply voltage, LED forward voltage and current → series resistor, standard value and power.
Digital Electronics
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ADC Resolution Calculator
LSB = Vref / 2ⁿ, SNR = 6.02n + 1.76 dB
code = floor(Vᵢₙ/LSB); ENOB = (SINAD − 1.76)/6.02
Example: A 12-bit ADC with a 3.3 V reference converting 1.65 V: Resolution (1 LSB) = 0.805664 mV.
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K-Map Solver
F = Σm(…) + d(…) → minimal SOP / POS
Quine–McCluskey with an exact minimum cover
Example: F(A, B, C, D) = Σm(0, 1, 2, 5, 6, 7, 8, 9, 10, 14): Minimal SOP: F(A, B, C, D) = B′C′ + CD′ + A′BD.
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- Number Base Converter – A number in any base 2–36 → binary, octal, decimal and hex, with fractional parts.
- Two’s Complement Calculator – Decimal ↔ two’s complement, one’s complement and sign-magnitude bit patterns, with hex.
Measurements
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Wheatstone Bridge Calculator
Rₓ = R₂R₃ / R₁ at balance
Vₒᵤₜ = V_s [R₂/(R₁ + R₂) − Rₓ/(R₃ + Rₓ)]
Example: A bridge with equal 1 kΩ ratio arms balanced with R₃ = 470 Ω: Unknown resistance Rₓ = 470 Ω.
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Error Propagation Calculator
Sums: ΔA + ΔB; products: ΔA/A + ΔB/B
powers: Δy/y = |n| ΔA/A; statistical errors combine as √(Σ squares)
Example: A = 100 ± 1 and B = 20 ± 0.4 multiplied, limiting errors: Result y = 2000 ± 60.
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- Ammeter Shunt Calculator – Meter full-scale current and resistance → shunt for a current range or multiplier for a voltage range.
- RMS Value Calculator – Waveform and peak value → RMS, average, form factor and crest factor.
Sensors and Industrial Instrumentation
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RTD Calculator
R = R₀(1 + AT + BT²) for T ≥ 0 °C
A = 3.9083 × 10⁻³, B = −5.775 × 10⁻⁷; below 0 °C add C(T − 100)T³, C = −4.183 × 10⁻¹²
Example: A Pt100 at 100 °C: Resistance R = 138.5055 Ω.
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Thermistor Calculator
R = R₀ exp[B(1/T − 1/T₀)]
T in kelvin; inverse 1/T = 1/T₀ + ln(R/R₀)/B
Example: A 10 kΩ (at 25 °C) NTC thermistor with B = 3950 K at 0 °C: Resistance R = 33.6206 kΩ.
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- Strain Gauge Calculator – Gauge factor, strain, resistance and excitation → ΔR and bridge output voltage.
- 4-20 mA Calculator – Range and loop current ↔ process value, percent of span and 1–5 V signal.
- Reynolds Number Calculator – Velocity, diameter, density and viscosity in any units → Re and flow regime.
Communication and Optical Instrumentation
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AM Modulation Calculator
m = (V_max − V_min)/(V_max + V_min), Pₜ = P_c(1 + m²/2)
each sideband P_c m²/4; efficiency m²/(2 + m²); bandwidth 2f_m
Example: An AM envelope swinging between 15 V and 5 V on a 100 W carrier with 5 kHz audio: Modulation index m = 0.5.
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FM Bandwidth Calculator
B ≈ 2(Δf + f_m), β = Δf/f_m
sideband n has amplitude Jₙ(β) of the unmodulated carrier
Example: Broadcast FM with 75 kHz peak deviation and 15 kHz audio: Carson bandwidth = 180 kHz.
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- PCM Bit Rate Calculator – Bits per sample, sampling rate and channels → bit rate, levels, SQNR and bandwidth.
- Photon Energy Calculator – Wavelength, frequency or energy → the other two, per photon and per mole.
- Wave Speed Calculator – Any two of wave speed, frequency and wavelength → the third, plus the period.
Unit Converters for GATE Instrumentation (IN)
- Temperature Converter – °C, °F, K, °R
- Pressure Converter – Pa, kPa, psi, bar, inH₂O, mmHg, torr, atm
- Flow Rate Converter – CFM, m³/h, L/min, L/s, GPM, sccm, SLPM
- Frequency & RPM Converter – Hz, kHz, MHz, GHz, rpm, rad/s
- Power Converter – W, kW, MW, hp, PS, BTU/h, TR, kcal/h
- Electrical Resistivity Converter – Ω·m, Ω·cm, µΩ·cm, Ω·mm²/m, MΩ·cm
Common GATE Instrumentation (IN) Conversions
- Celsius to Fahrenheit
- Fahrenheit to Celsius
- pascal to psi
- psi to pascal
- cubic foot per minute to cubic meter per hour
- cubic meter per hour to cubic foot per minute
- revolution per minute to hertz
- hertz to revolution per minute
- kilowatt to horsepower
- horsepower to kilowatt
- microohm centimetre to ohm metre
- ohm metre to microohm centimetre
How to use these GATE Instrumentation calculators
GATE Instrumentation questions mix circuit theory, control and electronics with the specific maths of sensors and measurement. Alongside the shared circuit, signals and control tools, this hub has calculators for the platinum RTD (IEC 60751 Callendar–Van Dusen), NTC thermistors, strain gauges in quarter, half and full bridges, 4–20 mA scaling, Wheatstone bridges, meter shunts and multipliers, error propagation, op-amp configurations, ADC resolution and SNR, and AM, FM and PCM.
The sensor calculators use the published coefficients rather than linear approximations, but also show the linear figure most questions use, so you can match the convention in the question. The measurement tools give both limiting and statistical errors, because questions ask for either.
Use them to check practice answers and to build intuition: change one input and watch how a bridge output, a quantisation step or a modulation bandwidth responds.