Pancyprian TS Elec. Eng. III ΘΚ cheat sheet

Ohm's law: V = I × R; power: P = V × I = I²R = V²/R. For AC, use RMS values and include power factor: P = V_rms I_rms cos φ.

Kirchhoff's laws: KCL - sum of currents entering a node equals sum leaving; KVL - sum of voltages around any closed loop is zero.

Series resistances add: R_total = R1 + R2 + …; parallel conductances add: 1/R_total = 1/R1 + 1/R2 + …; two resistors in parallel: R = R1R2/(R1+R2).

Thevenin's theorem: any linear two-terminal network can be replaced by an equivalent voltage source V_th in series with R_th; V_th is open-circuit voltage, R_th is resistance seen with sources killed (voltage sources shorted, current sources opened).

Norton's theorem: equivalent current source I_N = I_short in parallel with R_N = R_th. Source transformation: V_th = I_N × R_N.

Superposition: in a linear circuit with multiple independent sources, the response is the sum of responses to each source acting alone (other sources killed).

Impedance: Z = R + jX; for inductor X_L = ωL (ω = 2πf), for capacitor X_C = −1/(ωC). Admittance Y = 1/Z = G + jB.

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

Three-phase systems: in a balanced star (Y) connection, line voltage = √3 × phase voltage, line current = phase current; in delta (Δ), line voltage = phase voltage, line current = √3 × phase current. Total power P = √3 V_L I_L cos φ.

DC machine EMF equation: E = (p Φ N Z)/(60 A), where p = poles, Φ = flux per pole (Wb), N = speed (rpm), Z = total armature conductors, A = number of parallel paths (A = 2 for wave, A = p for lap).

DC motor torque: T = (P × 60)/(2π N) = k Φ I_a; for a shunt motor, speed N ∝ (V − I_a R_a)/Φ. Reducing flux increases speed.

Transformer EMF equation: E = 4.44 f N Φ_max; voltage ratio V₁/V₂ = N₁/N₂ = I₂/I₁. For an ideal transformer, input VA equals output VA.

Transformer equivalent circuit: referred to primary, secondary impedance is multiplied by (N₁/N₂)². Core loss is constant (hysteresis + eddy current); copper loss varies with load squared.

Induction motor slip: s = (n_s − n_r)/n_s, where n_s = 120 f / p (rpm). Rotor frequency f_r = s f. At standstill s = 1; at synchronous speed s = 0.

Three-phase induction motor torque: T = (3/ω_s) × [s E₂² R₂ / (R₂² + (s X₂)²)] (referred to stator). Maximum torque occurs at s = R₂/X₂.

Synchronous machine: synchronous reactance X_s; generated EMF E_f = V + j I X_s (per phase, neglecting R_a). Power-angle equation: P = (E_f V / X_s) sin δ.

Machine losses: copper loss = I²R, core loss (hysteresis ∝ f B_max^n, eddy current ∝ f² B_max² t²), mechanical loss (friction and windage), stray load loss. Efficiency = output / (output + losses).

Cable current-carrying capacity must satisfy I_b ≤ I_n ≤ I_z, where I_b = design current, I_n = protective device rating, I_z = cable tabulated capacity after applying correction factors (ambient temperature, grouping, thermal insulation).

Voltage drop limit for low-voltage installations: typically 3% for lighting and 5% for other uses (e.g., IEC 60364 / HD 60364). Voltage drop ΔV = (2 × L × I × (R cos φ + X sin φ)) for single-phase, and √3 × L × I × (R cos φ + X sin φ) for three-phase.

Earthing system types: TN-S (separate PE and N), TN-C (combined PEN), TN-C-S (combined then separate), TT (installation earth electrode independent of supply neutral), IT (isolated or high-impedance earthed source).

Protection against electric shock: automatic disconnection of supply requires disconnection times per system (e.g., TN: 0.4 s for 230 V final circuits ≤32 A; TT: 0.2 s) and disconnection time t = k²S²/I² for fault current.

Residual current devices (RCDs): rated residual operating current IΔn; 30 mA for additional protection (socket outlets ≤32 A, mobile equipment); 300 mA for fire protection. RCDs do not protect against overload.

Cable selection: consider conductor material (Cu/Al), insulation (PVC 70 °C, XLPE 90 °C), installation method (clipped direct, in conduit, buried), and derating factors. Minimum cross-sectional area for socket-outlet circuits is typically 2.5 mm² Cu.

Isolation and switching: every circuit must have a means of isolation; main switch must disconnect all live conductors. Protective devices must be selected so that I_n ≥ I_b and I_z ≥ I_n, and fault current I_f ≥ 1.45 × I_n for overload protection.

Conduit and trunking fill: maximum 40% fill for conduit; bending radius ≥ 6× cable diameter for fixed wiring (varies with standard).

Power diode: conducts when forward-biased; reverse recovery time t_rr affects switching losses. Average output of a half-wave rectifier with resistive load: V_dc = V_m/π; full-wave: V_dc = 2V_m/π.

Thyristor (SCR): triggered by gate pulse when forward-biased; turns off when current falls below holding current. Phase-control rectifier: V_dc = (2V_m/π) cos α for a full-wave fully controlled bridge with resistive load.

Buck converter (step-down): V_out = D × V_in, where D = duty cycle. Boost converter (step-up): V_out = V_in/(1 − D). Buck-boost: V_out = −V_in × D/(1 − D).

PWM inverter: output voltage fundamental V₁ = (√2/π) × V_dc × m_a for sinusoidal PWM (m_a = modulation index). Switching frequency determines harmonic order; dead time prevents shoot-through.

Three-phase inverter: 180° conduction mode gives line voltage waveform with six steps; 120° conduction mode gives three-phase balanced output. Space vector modulation (SVM) extends linear range by 15% compared to sinusoidal PWM.

DC motor drives: armature voltage control (constant torque region), field weakening (constant power region). Four-quadrant drive allows regenerative braking.

Variable frequency drive (VFD) for induction motor: V/f control maintains constant flux; at low speeds, boost voltage compensates stator resistance drop. Sensorless vector control estimates speed from motor model.

Soft starter: reduces starting current by phase control of thyristors; typical starting current 2–4 × I_n. Star-delta starter reduces starting current to 1/3 of direct-on-line and torque to 1/3.

Moving-coil (PMMC) instrument: measures DC only; deflection proportional to average current; scale linear. Moving-iron instrument: measures AC and DC (RMS); scale non-linear; reads RMS.

Electrodynamic (dynamometer) instrument: measures AC and DC; used for wattmeters and power factor meters; deflection ∝ I₁ I₂ cos φ.

Wattmeter connection: current coil in series with load, voltage coil in parallel. Two-wattmeter method for balanced three-phase: P_total = W₁ + W₂; power factor = cos[tan⁻¹(√3 (W₁ − W₂)/(W₁ + W₂))].

Measurement of resistance: Wheatstone bridge for medium resistance (1 Ω to 1 MΩ); Kelvin double bridge for low resistance (<1 Ω); megger (insulation tester) for high resistance (MΩ range).

Instrument transformers: current transformer (CT) secondary rated 5 A or 1 A; voltage transformer (VT) secondary rated 110 V or 100 V. Burden must not exceed rated VA; CT secondary must never be open-circuited under load.

Error and accuracy: absolute error = measured − true; relative error = (absolute error / true) × 100%. Accuracy class (e.g., 0.5) means maximum error as percentage of full-scale reading.

Oscilloscope: probes (1X, 10X), bandwidth, sampling rate. For accurate measurement, bandwidth ≥ 5 × signal frequency; rise time of scope ≤ 1/3 of signal rise time.

Insulation resistance testing: apply DC voltage (e.g., 500 V for 500 V installation) for 1 minute; minimum acceptable value per standard (e.g., 1 MΩ for LV installations). Polarization index = R_10min / R_1min; good if >2.

Overcurrent protection: fuse or circuit breaker must satisfy I_b ≤ I_n ≤ I_z and I_f ≥ 1.45 I_n (for overload). Short-circuit protection: I_cn ≥ I_sc (rated short-circuit capacity).

Fuse characteristics: gG (general purpose) for cable and line protection; aM for motor circuits; gR for semiconductor protection. Fuse time-current characteristic is inverse-time.

Circuit breaker types: MCB (miniature, ≤125 A), MCCB (moulded case, up to 1600 A), ACB (air circuit breaker, up to 6300 A). Breaking capacity I_cn must exceed prospective fault current.

Earth fault loop impedance Z_s must be low enough to allow disconnection within required time: Z_s ≤ U_0 / I_a, where U_0 = nominal voltage to earth (230 V), I_a = current causing disconnection in specified time.

Earthing electrode resistance: R = ρ/(2πL) × [ln(4L/d) − 1] for a vertical rod (ρ = soil resistivity, L = length, d = diameter). Multiple rods in parallel reduce resistance but with mutual coupling.

Protective earthing conductor sizing: cross-section S_PE = S_phase for S_phase ≤ 16 mm²; S_PE = 16 mm² for 16 < S_phase ≤ 35 mm²; S_PE = S_phase/2 for S_phase > 35 mm² (copper).

Residual current monitoring (RCM) and RCD: RCD detects residual current (difference between live and neutral). Type AC responds to AC only; Type A also to pulsating DC; Type B also to smooth DC.

Safety extra-low voltage (SELV): voltage ≤ 50 V AC or 120 V ripple-free DC; separated from earth and other circuits. PELV: earthed but otherwise like SELV.

Power station types: thermal (coal, gas, oil), hydro, nuclear, wind, solar. Base load: nuclear and large coal; peak load: gas turbines and hydro. Load factor = average load / maximum demand.

Transmission voltages: high voltage (HV) 110–220 kV, extra-high voltage (EHV) 275–765 kV, ultra-high voltage (UHV) >765 kV. Higher voltage reduces current and I²R losses for same power.

Per-unit system: base quantities (S_base, V_base, Z_base = V_base²/S_base). Per-unit impedance of transformer = (Z% / 100) × (S_base / S_rated).

Transmission line parameters: resistance R, inductance L, capacitance C, conductance G. Surge impedance Z_c = √(L/C); for overhead line ~400 Ω, for cable ~50 Ω. SIL = V²/Z_c.

Short transmission line (<80 km): capacitance neglected; medium (80–250 km): nominal π or T model; long (>250 km): distributed parameters. Voltage regulation = (|V_no-load| − |V_full-load|)/|V_full-load| × 100%.

Distribution systems: radial, ring, interconnected. Voltage drop in distributor: ΔV = I × R × cos φ + I × X × sin φ (per phase).

Fault levels: symmetrical fault current I_f = V_phase / Z_total (per phase). Asymmetrical fault includes DC offset; peak factor = 1.8 for 50 Hz (X/R ratio dependent).

Renewable generation: wind turbine power P = ½ ρ A v³ C_p (Betz limit C_p ≤ 0.593). Solar PV: I-V curve, maximum power point (MPP); fill factor = P_max / (V_oc × I_sc).

IEC 60617 graphical symbols for electrical diagrams: resistor (rectangle), capacitor (two parallel lines), inductor (coils), diode (triangle with bar), thyristor (diode with gate), earth (three horizontal lines).

Types of diagrams: block diagram (functional blocks), circuit diagram (schematic, shows connections), wiring diagram (physical layout), ladder diagram (PLC), single-line diagram (power systems).

Single-line diagram (SLD): represents three-phase system with one line; shows generators, transformers, busbars, circuit breakers, and protection devices. Used for power system analysis.

Cable and core identification: phase colors (L1 brown, L2 black, L3 grey, N blue, PE green-yellow) per IEC 60446. Older UK: red, yellow, blue, black.

Drawing scales: for electrical installation plans, common scales 1:50, 1:100. Site plans 1:500 or 1:1000. Location plans 1:2500.

Line types: continuous thick for main circuits, continuous thin for control circuits, dashed for mechanical linkage or enclosure, dotted for screened or spare cores.

Reference designation: components identified by letter code (e.g., Q for circuit breaker, F for fuse, K for relay, M for motor) followed by number, per IEC 61346.

Documentation: bill of materials (BOM), test certificates, schedule of circuits, as-built drawings. Cable schedule lists cable tag, type, size, length, from/to.