MPPSC SES Electrical cheat sheet

Ohm's law: V = IR; power P = VI = I²R = V²/R. In AC, use RMS values and include power factor: P = VI cos φ.

Series: R_eq = ΣR, current common. Parallel: 1/R_eq = Σ(1/R), voltage common. Two resistors in parallel give R1R2/(R1+R2).

KCL: algebraic sum of currents at a node is zero. KVL: algebraic sum of voltages around any closed loop is zero.

Thevenin: any linear two-terminal network equals V_th (open-circuit voltage) in series with R_th (resistance seen with sources dead: voltage sources shorted, current sources opened).

Norton: I_N = short-circuit current, R_N = R_th; source conversion V_th = I_N·R_N.

Superposition applies only to linear circuits: response = sum of responses to each independent source acting alone; dependent sources are never deactivated.

Maximum power transfer: load R_L = R_th for DC; for AC, Z_L = Z_th* (conjugate match). Efficiency at match is 50% for resistive DC.

First-order transient: v(t) = v(∞) + [v(0⁺) − v(∞)]e^(−t/τ); τ = RC for RC, τ = L/R for RL. Capacitor voltage and inductor current cannot change instantaneously.

Series RLC resonance: f₀ = 1/(2π√(LC)); at resonance X_L = X_C, impedance is purely resistive, and Q = (1/R)√(L/C) = f₀/BW.

Transformer EMF: E = 4.44 f N Φ_max. Turns ratio a = N₁/N₂ = V₁/V₂; for an ideal transformer V₁I₁ = V₂I₂.

Transformer equivalent impedance referred to secondary: Z₂' = Z₁/a². Voltage regulation = (V_no-load − V_full-load)/V_full-load × 100%.

Core loss (hysteresis + eddy) is constant at fixed voltage and frequency; copper loss varies as I². Maximum efficiency occurs when copper loss equals core loss.

DC machine EMF: E = (PΦNZ)/(60A) for a generator; same expression is back EMF for a motor. Torque T = 0.159 PΦI Z/A (N·m with Φ in Wb).

DC shunt motor speed N ∝ (V − I_aR_a)/Φ; adding armature resistance or reducing terminal voltage reduces speed. A DC series motor must never start unloaded.

Synchronous speed N_s = 120f/P. Induction motor slip s = (N_s − N_r)/N_s; rotor frequency = sf; rotor EMF at standstill E₂, at slip s it is sE₂.

Induction motor torque T ∝ sE₂²R₂/(R₂² + (sX₂)²); maximum torque is independent of R₂, and the slip at maximum torque equals R₂/X₂.

Synchronous generator: E = V + I_a(R_a + jX_s); load angle δ governs power P = (EV/X_s)sin δ. Synchronous condensers supply leading reactive power for power-factor correction.

Parallel operation of transformers requires equal voltage ratio, equal polarity, and similar per-unit impedance and X/R ratio to share load in proportion to ratings.

Three-phase power: P = √3 V_L I_L cos φ; per-phase analysis uses V_phase = V_L/√3 and total power = 3 × per-phase power.

Transmission efficiency = P_received/P_sent; losses = I²R. Raising voltage for the same power reduces current and hence I²R loss by the square of the voltage ratio.

Short line: V_s = V_r + IZ. Medium line: nominal-π or nominal-T. Long line: ABCD parameters with A = D = cosh γl, B = Z_c sinh γl, C = (1/Z_c) sinh γl.

Surge impedance Z_c = √(L/C), about 400 Ω for overhead lines; surge impedance loading SIL = V_L²/Z_c.

Ferranti effect: receiving-end voltage exceeds sending-end voltage on a lightly loaded long line because of line charging capacitance.

Symmetrical components: any unbalanced three-phase set decomposes into positive, negative, and zero sequences. Zero-sequence currents need a neutral return path.

Per-unit: Z_pu = Z_actual/Z_base, with Z_base = V_base²/S_base; per-unit impedance is the same on either side of a transformer when bases follow the turns ratio.

Protection: overcurrent for feeders, differential for transformers and busbars, distance (mho/impedance) for long transmission lines, and Merz-Price circulating-current principle for generators.

Switchgear ratings: breaking capacity, making capacity (≈2.55 × symmetrical breaking current for peak asymmetrical), and short-time rating. A circuit breaker must interrupt at current zero.

Open-loop transfer function G(s)H(s); closed-loop T(s) = G/(1 + GH) for negative feedback. Characteristic equation is 1 + G(s)H(s) = 0.

First-order step response: c(t) = 1 − e^(−t/τ); time constant τ is the time to reach 63.2% of final value. Settling time (2%) ≈ 4τ.

Second-order standard form: ω_n²/(s² + 2ζω_n s + ω_n²). Overshoot % = 100e^(−ζπ/√(1−ζ²)); peak time t_p = π/(ω_n√(1−ζ²)).

Routh-Hurwitz: the number of sign changes in the first column equals the number of right-half-plane roots. A zero row is replaced by the derivative of the auxiliary polynomial.

Steady-state error: type 0 gives finite error to a step; type 1 gives zero step error and finite velocity error 1/K_v; type 2 gives zero velocity error and finite acceleration error.

Bode: a pole gives −20 dB/decade slope and −90° phase; a zero gives +20 dB/decade and +90°. Gain margin = −20 log|G(jω_pc)|; phase margin = 180° + ∠G(jω_gc).

Nyquist criterion: N = P − Z, where N is encirclements of −1, P is open-loop RHP poles, and Z is closed-loop RHP poles; for stability Z = 0.

Lead compensation adds phase (improves transient response and bandwidth); lag compensation raises low-frequency gain (improves steady-state accuracy).

State space: ẋ = Ax + Bu, y = Cx + Du. Stability requires all eigenvalues of A to have negative real parts.

SCR: turns on when anode is positive and a gate pulse is applied; it turns off only when anode current falls below the holding current. Latching current is the value needed to keep it on after the gate pulse ends.

Single-phase full-wave rectifier: average output V_dc = 2V_m/π for uncontrolled; with firing angle α, V_dc = (2V_m/π)cos α for a fully controlled bridge with continuous conduction.

Three-phase fully controlled bridge: V_dc = (3√3 V_m/π)cos α = 1.35 V_LL cos α. Inverter mode occurs when α > 90°.

Buck chopper: V_o = D·V_s; boost: V_o = V_s/(1 − D); buck-boost: V_o = −V_s·D/(1 − D), where D is the duty cycle.

Ripple frequency of a p-pulse converter is p·f_supply. A larger number of pulses gives a smoother DC output and a lower harmonic content.

Inverters: a voltage-source inverter uses a DC-link capacitor and antiparallel diodes; PWM controls both output voltage and frequency.

DC motor speed control: armature voltage control gives constant torque below base speed; field flux control gives constant power above base speed.

Induction motor V/f control keeps flux constant; below base speed torque capability is maintained, and above base speed the field weakens.

Cycloconverter converts fixed AC to variable-frequency AC in a single stage; it is suited to low-speed, high-torque drives.

PMMC instrument: deflection θ ∝ I, so the scale is uniform; it measures DC only and is used with a rectifier for AC.

Ammeter: low resistance, connected in series. Voltmeter: high resistance, connected in parallel. An ammeter shunt extends range: R_sh = R_m/(m − 1).

Wattmeter: current coil in series, potential coil in parallel. Reading = V I cos φ for a single-phase wattmeter; two-wattmeter method total power = W₁ + W₂, and tan φ = √3(W₁ − W₂)/(W₁ + W₂).

Energy meter: 1 kWh = meter constant (revolutions per kWh) × revolutions; creeping is prevented by a brake magnet.

Instrument transformer: CT secondary is rated 5 A or 1 A and must never be open-circuited; PT secondary is rated 110 V and must never be short-circuited.

Bridge balance: Wheatstone R_x = R₂R₃/R₁; Maxwell measures inductance, Schering measures capacitance and loss angle, Wien measures frequency.

Errors: systematic errors are correctable (calibration, zero, loading); random errors are treated statistically. Accuracy is closeness to true value; precision is repeatability.

Strain gauge gauge factor GF = (ΔR/R)/(ΔL/L), about 2 for metal foil gauges. A thermocouple uses the Seebeck effect; an RTD uses a positive temperature coefficient of resistance.

LVDT: linear variable differential transformer, an inductive displacement sensor whose output magnitude is proportional to displacement and whose phase indicates direction.

Diode: forward-biased when anode is positive; a silicon diode drops about 0.7 V. Zener diode works in reverse breakdown as a voltage regulator.

BJT regions: cutoff (both junctions off), active (BE forward, BC reverse), saturation (both forward). β = I_C/I_B; I_E = I_C + I_B.

Op-amp ideal properties: infinite gain, infinite input impedance, zero output impedance, infinite bandwidth. Virtual short: V₊ = V₋ in negative feedback.

Op-amp circuits: inverting gain = −R_f/R_in; non-inverting gain = 1 + R_f/R₁; integrator uses a capacitor in the feedback path; differentiator uses a capacitor at the input.

Boolean algebra: A + A'B = A + B; De Morgan: (A·B)' = A' + B' and (A + B)' = A'·B'.

Karnaugh map: group 1s in powers of two (1, 2, 4, 8), largest groups first; don't-care terms can be used to enlarge groups.

Flip-flops: SR has a forbidden state when both inputs are 1 (NOR type); JK removes it (toggles when both are 1); D is transparent data; T toggles.

A ripple counter with n flip-flops divides by 2ⁿ; a modulus-N counter needs at least ⌈log₂N⌉ flip-flops.

ADC resolution = V_ref/2ⁿ; a successive-approximation ADC is faster than a dual-slope ADC but slower than a flash ADC.

8085: 8-bit data bus, 16-bit address bus, 64 KB address space, 40 pins, 5 flags (S, Z, AC, P, CY), multiplexed AD0–AD7.

8085 clock is 3 MHz with a 6 MHz crystal (internally divided by 2); the instruction cycle is a fetch followed by execute, made of machine cycles and T-states.

8086: 16-bit data bus, 20-bit address bus, 1 MB memory, segmented architecture with CS, DS, SS, ES; physical address = segment × 16 + offset.

8051: 8-bit microcontroller with 4 KB ROM, 128 bytes RAM, 4 ports, two 16-bit timers, one serial port, 5 interrupt sources, and an 11.0592 MHz crystal for standard baud rates.

Interrupt priorities: in 8051, external interrupt 0 has the highest priority among the standard sources; the reset vector is 0000H.

Addressing modes: immediate, direct, indirect, register, register-indirect, and indexed. MOV A, #data is immediate; MOV A, R0 is register.

Embedded system characteristics: dedicated function, real-time constraints, limited memory and power, and a mix of hardware and firmware.

Communication buses: I²C uses two wires (SDA, SCL) with 7-bit addressing; SPI uses four wires (MOSI, MISO, SCK, SS) and is full duplex; UART is asynchronous.

Watchdog timer resets the processor if firmware fails to service it within a set period; it is a safety mechanism, not a timekeeping function.

Fourier series represents periodic signals as a sum of harmonics; a periodic signal has a line spectrum, while a non-periodic signal has a continuous spectrum.

Sampling theorem: f_s ≥ 2f_max. Aliasing occurs below this rate; an anti-aliasing low-pass filter is placed before the sampler.

LTI system: output = input * h(t). Convolution in time equals multiplication in frequency. Stability requires ∫|h(t)|dt < ∞.

AM: bandwidth = 2f_m; modulation index m = V_m/V_c, and m > 1 causes overmodulation. Total AM power P_t = P_c(1 + m²/2).

FM: Carson's rule gives bandwidth ≈ 2(Δf + f_m); FM has constant amplitude and better noise immunity than AM.

PCM: quantization noise decreases by about 6 dB per additional bit; SNR ≈ 6.02n + 1.76 dB for a full-scale sinusoid.

Shannon capacity C = B log₂(1 + S/N). Nyquist rate for a noiseless channel with M levels is 2B log₂M.

Digital modulation: ASK, FSK, and PSK. QPSK carries 2 bits per symbol; 16-QAM carries 4 bits per symbol.

Antenna: half-wave dipole length ≈ λ/2; free-space wavelength λ = c/f, so 300 MHz corresponds to about 1 m.

Coulomb's law: F = q₁q₂/(4πε₀r²); ε₀ = 8.854 × 10⁻¹² F/m. Electric field intensity E = F/q.

Gauss's law: ∮D·dS = Q_enclosed. Inside a hollow charged conductor the field is zero; charge resides on the outer surface.

Capacitance: parallel plate C = εA/d; energy stored = ½CV². Coaxial cable C = 2πεl/ln(b/a).

Biot–Savart and Ampere's law: ∮H·dl = I_enclosed. Force on a conductor F = BIl sin θ.

Faraday's law: e = −dΦ/dt; Lenz's law gives the direction opposing the change. Induced EMF in a conductor moving in a field e = Blv.

Maxwell's equations: ∇·D = ρ, ∇·B = 0, ∇×E = −∂B/∂t, ∇×H = J + ∂D/∂t.

Breakdown of gases: Townsend's first ionization coefficient α, with the sparking criterion γ(e^(αd) − 1) = 1; Paschen's law states that breakdown voltage depends on the product of pressure and gap length.

High-voltage generation: a Cockcroft–Walton multiplier gives a DC output with a small ripple; a Van de Graaff generator produces very high DC voltage; a Tesla coil produces high-frequency AC.

HV measurement: a sphere gap measures peak voltage; a potential divider with a peak voltmeter or an electrostatic voltmeter is used for impulse and AC measurement.

Illumination: luminous flux in lumens, illuminance in lux (lm/m²), luminous intensity in candela. Inverse-square law: E = I/d²; cosine law: E = (I/d²)cos θ.

Lamps: incandescent about 10–15 lm/W; fluorescent about 50–80 lm/W; LED 80–150 lm/W. Sodium vapour lamps are efficient but have poor colour rendering.

Traction: tractive effort F = 277.8·α·W (N) for acceleration α in km/h/s and W in tonnes, including rotational inertia.

Adhesive weight limits tractive effort: F_max = μ·W_adhesive. AC series motors are preferred for traction because of high starting torque and simple speed control.

Traction power supply: 25 kV, 50 Hz single-phase AC for main lines; 750 V DC third rail for metros; 1500 V DC overhead for suburban systems.

Electric heating: resistance heating uses I²R, with efficiency near 100% at the load; induction heating uses eddy currents; dielectric heating uses high-frequency electric fields for non-conductors.

Arc furnace: power factor is low and fluctuates, so static VAR compensators or series capacitors are used for compensation.

Welding: arc welding needs a drooping (constant-current) characteristic; resistance welding needs a low-voltage, high-current supply.

Refrigeration and air conditioning: coefficient of performance COP = refrigerating effect/work input; a heat pump's COP = heating effect/work input.

Engineering ethics principles: public safety and welfare first, competence within one's area, honesty in reporting data, confidentiality, and avoidance of conflicts of interest.

Whistleblowing is justified when there is a serious threat to public safety, internal channels have failed, and documented evidence exists.

Sustainable development balances economic growth, environmental protection, and social equity; environmental impact assessment precedes major project approval.

Indian constitutional values: justice, liberty, equality, and fraternity; fundamental duties under Article 51A include protecting the environment.

Numerical aptitude: work and time uses 1/A + 1/B = 1/T; mixtures use alligation; speed–time–distance uses relative speed for opposite and same directions.

Data interpretation: percentage change = (new − old)/old × 100; read the units and the base year before computing any ratio.

Logical reasoning: syllogisms follow the Venn-diagram method; for series and coding questions, identify the operation pattern before applying it.

Renewable energy targets: India's non-fossil capacity goal and the National Solar Mission are policy items frequently asked in general studies.

Disaster management cycle: mitigation, preparedness, response, and recovery; the National Disaster Management Authority is the apex body.