Engineer Electricity cheat sheet

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

Thevenin equivalent: V_th = open-circuit voltage at the terminals, R_th = V_oc/I_sc; maximum power transfer occurs when R_load = R_th, giving efficiency of 50%.

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

Superposition applies only to linear circuits: solve with one independent source at a time and sum the responses; it cannot be used directly for power.

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

Three-phase power: P = √3·V_L·I_L·cos φ for a balanced load; delta phase current = line current/√3, star phase voltage = line voltage/√3.

First-order transient: v(t) = v_∞ + (v_0 − v_∞)e^(−t/τ), τ = RC or L/R; after 5τ the response is within about 1% of final value.

Mesh analysis uses KVL on independent loops; nodal analysis uses KCL and gives (n−1) independent equations for n nodes.

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

Lorentz force: F = q(E + v×B); magnetic force on a current-carrying conductor: F = BIL sin θ.

Biot–Savart and Ampère: B at distance r from a long straight conductor is B = μ₀I/(2πr); inside a solenoid B = μ₀nI.

Faraday's law: emf = −N dΦ/dt; Lenz's law fixes direction so induced current opposes the change in flux.

Intrinsic impedance of free space η₀ = √(μ₀/ε₀) ≈ 377 Ω; velocity v = 1/√(με) = c/√(ε_r μ_r).

Transmission-line reflection coefficient Γ = (Z_L − Z₀)/(Z_L + Z₀); VSWR = (1+|Γ|)/(1−|Γ|); matched load gives Γ = 0.

Skin depth δ = √(2/(ωμσ)); current crowds to the conductor surface as frequency rises, increasing effective resistance.

Poynting vector S = E×H gives power flow density in W/m²; wave impedance and phase constant β = 2π/λ determine propagation.

Transformer emf: E = 4.44·f·N·Φ_max; voltage ratio equals turns ratio V₁/V₂ = N₁/N₂, and current ratio is inverse.

Transformer tests: open-circuit test gives core (iron) loss and magnetizing branch; short-circuit test gives copper loss and series impedance.

DC machine emf: E = (pΦNZ)/(60A); separately excited torque T = kΦI_a; shunt motor speed falls with load, series motor speed varies widely.

Induction motor slip s = (n_s − n_r)/n_s, with synchronous speed n_s = 120f/p; rotor frequency = s·f.

Induction motor torque is proportional to slip at low slip and decreases beyond breakdown torque; starting current is typically 5–8 times rated.

Synchronous machine: emf E = V + I(R + jX_s); it can supply or absorb reactive power by adjusting field excitation (over-excited = lagging supply).

Losses: copper loss = I²R, core loss depends on flux density and frequency, friction and windage are mechanical; efficiency = output/(output + losses).

Single-phase induction motors are not self-starting; auxiliary windings, capacitors, or shaded poles provide the starting torque.

Per-unit value = actual value/base value; choose base power S_b and base voltage V_b, then Z_b = V_b²/S_b and per-unit impedance is the same on either side of an ideal transformer.

Short-line model: V_s = V_r + I·Z; medium-line uses nominal π or T; long lines use hyperbolic ABCD parameters with Z_c = √(z/y).

Surge impedance loading SIL = V_L²/Z_c; for overhead lines Z_c is roughly 400 Ω and SIL ≈ V_L²/400 MW.

Voltage regulation = (|V_r,noload| − |V_r,fullload|)/|V_r,fullload| × 100%; Ferranti effect raises receiving-end voltage on lightly loaded long lines.

Load flow: slack bus sets voltage and angle, PV buses hold voltage and real power, PQ buses specify load; Newton–Raphson converges in few iterations.

Transmission losses fall as voltage rises for fixed power: I = P/(√3·V_L·cos φ), so loss ∝ 1/V².

Symmetrical components: positive, negative, and zero sequence; zero-sequence current needs a return path and is blocked by delta windings.

Distribution: radial systems are simple but have poor reliability; ring or interconnected networks improve continuity; voltage drop ≈ √3·I·(R cos φ + X sin φ) for lines.

Overcurrent relay grading: time multiplier and plug setting create discrimination; downstream relay must clear before upstream backup operates.

Distance relay zones: zone 1 covers about 80% of line impedance with no intentional delay; zone 2 covers 120–150% with a time delay.

Differential protection compares currents at both ends of a transformer, generator, or busbar; it operates on the internal fault current and is immune to through-faults.

CT saturation and ratio errors cause relay misoperation; the knee-point voltage and burden must be checked for through-fault stability.

Circuit breaker ratings: rated normal current, breaking capacity (symmetrical and asymmetrical), making capacity, and short-time withstand current.

Arc interruption: in an AC circuit the arc extinguishes at current zero; restriking voltage and rate of rise of recovery voltage determine whether it reignites.

SF6 breakers have high dielectric strength and arc-quenching ability; vacuum breakers suit medium voltage; air-blast suits high voltage.

Fuse: I²t characteristic must coordinate with the protected equipment; IDMT relay characteristic follows t = TMS·k/((I/I_s)^α − 1).

Single-phase full-wave rectifier: average output V_dc = 2V_m/π ≈ 0.637V_m; RMS = V_m/√2; ripple frequency is twice the supply frequency.

Three-phase full bridge: V_dc = 3√3·V_m/π ≈ 1.35·V_LL for an uncontrolled rectifier; firing angle control gives V_dc = 1.35·V_LL·cos α.

Thyristor conducts only when anode is positive and a gate pulse is applied; it turns off when current falls below the holding current.

Buck converter: V_o = D·V_in; boost: V_o = V_in/(1−D); buck-boost: V_o = −V_in·D/(1−D); D is duty cycle.

PWM inverters control output voltage and frequency; increasing the switching frequency reduces harmonic content but raises switching losses.

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

V/f control of induction motors keeps flux constant; below base frequency the voltage is reduced proportionally to avoid saturation.

Harmonics: a six-pulse converter produces 5th and 7th harmonics; a 12-pulse converter produces 11th and 13th; total harmonic distortion quantifies waveform distortion.

Transfer function G(s) = output/input with zero initial conditions; for a linear time-invariant system it is the ratio of Laplace transforms.

Second-order system: ω_n is natural frequency, ζ is damping ratio; percent overshoot = 100·e^(−ζπ/√(1−ζ²)); peak time = π/(ω_n√(1−ζ²)).

Steady-state error: type 0 has finite error to a step, type 1 has zero error to a step and finite error to a ramp, type 2 has zero error to a ramp.

Routh–Hurwitz array gives the number of right-half-plane roots; a sign change in the first column indicates instability.

Bode plot: gain margin is the dB value at the phase crossover where phase = −180°, phase margin is measured at the gain crossover where magnitude = 0 dB.

PID control: proportional action reduces rise time, integral action removes steady-state error but reduces stability, derivative action adds damping and amplifies noise.

Root locus starts at open-loop poles and ends at open-loop zeros; branches lie on the real axis where the number of poles plus zeros to the right is odd.

PLC scan cycle: read inputs, execute the program, update outputs; ladder logic uses contacts in series for AND and parallel for OR.

PMMC meter deflection is proportional to average current and suits DC; a rectifier-type meter reads average but is calibrated in RMS for sine waves.

Ammeter has low resistance and is connected in series; voltmeter has high resistance and is connected in parallel; the wattmeter current coil is in series and voltage coil in parallel.

Extend ammeter range with a shunt: R_sh = R_m/(m−1); extend voltmeter range with a series multiplier: R_se = R_m(m−1), where m is the multiplying factor.

Wattmeter reading in a two-wattmeter method: total power = W₁ + W₂; power factor = cos(tan⁻¹(√3(W₁−W₂)/(W₁+W₂))).

Energy meter: number of revolutions per kWh is the meter constant; error % = (measured − true)/true × 100.

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

Bridge methods: Wheatstone measures medium resistance, Kelvin double bridge measures low resistance, Megger measures insulation resistance.

Accuracy, precision, and resolution differ: accuracy is closeness to true value, precision is repeatability, resolution is the smallest detectable change.

Cable current-carrying capacity depends on conductor size, insulation, installation method, ambient temperature, and grouping; derating factors apply multiplicatively.

Voltage drop for a single-phase circuit is about 2·I·L·(R cos φ + X sin φ); for three-phase it is √3·I·L·(R cos φ + X sin φ).

Earthing systems: TN-S uses a separate protective conductor, TN-C combines neutral and protective functions, TT uses a local earth electrode, IT has an isolated or high-impedance neutral.

Protection by disconnection requires the earth-fault loop impedance to be low enough for the protective device to trip within the specified disconnection time.

Residual current devices detect imbalance between live and neutral; a 30 mA RCD is used for supplementary protection against electric shock.

Conductor colour coding and terminal identification must follow the applicable national code; protective earth is green-yellow and must never be used as a live conductor.

Cable sizing must satisfy current capacity, voltage drop limits, short-circuit withstand (I²t ≤ k²S²), and thermal constraints.

Hazardous areas use zoning and equipment protection concepts such as flameproof, increased safety, and intrinsic safety.

Time value of money: F = P(1+i)^n; present worth P = F/(1+i)^n; for an annuity, P = A[(1+i)^n − 1]/[i(1+i)^n].

Net present value discounts all cash flows at the required rate of return; accept if NPV > 0; IRR is the discount rate where NPV = 0.

Payback period is the time to recover the initial investment; it ignores cash flows after payback and the time value of money unless discounted.

Depreciation: straight line = (cost − salvage)/life; declining balance applies a constant rate to the book value; MACRS uses prescribed recovery periods.

Benefit–cost ratio = present worth of benefits/present worth of costs; a ratio greater than 1 indicates an economically acceptable project.

Critical path method: the critical path has zero total float and determines the shortest project duration; crashing reduces duration at extra cost.

Earned value: CPI = EV/AC, SPI = EV/PV; CPI below 1 means over budget, SPI below 1 means behind schedule.

Professional ethics: hold public safety and welfare paramount, practise only within competence, disclose conflicts of interest, and maintain confidentiality.

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

Thevenin equivalent: V_th = open-circuit voltage at the terminals, R_th = V_oc/I_sc; maximum power transfer occurs when R_load = R_th, giving efficiency of 50%.

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

Superposition applies only to linear circuits: solve with one independent source at a time and sum the responses; it cannot be used directly for power.

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

Three-phase power: P = √3·V_L·I_L·cos φ for a balanced load; delta phase current = line current/√3, star phase voltage = line voltage/√3.

First-order transient: v(t) = v_∞ + (v_0 − v_∞)e^(−t/τ), τ = RC or L/R; after 5τ the response is within about 1% of final value.

Mesh analysis uses KVL on independent loops; nodal analysis uses KCL and gives (n−1) independent equations for n nodes.

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

Lorentz force: F = q(E + v×B); magnetic force on a current-carrying conductor: F = BIL sin θ.

Biot–Savart and Ampère: B at distance r from a long straight conductor is B = μ₀I/(2πr); inside a solenoid B = μ₀nI.

Faraday's law: emf = −N dΦ/dt; Lenz's law fixes direction so induced current opposes the change in flux.

Intrinsic impedance of free space η₀ = √(μ₀/ε₀) ≈ 377 Ω; velocity v = 1/√(με) = c/√(ε_r μ_r).

Transmission-line reflection coefficient Γ = (Z_L − Z₀)/(Z_L + Z₀); VSWR = (1+|Γ|)/(1−|Γ|); matched load gives Γ = 0.

Skin depth δ = √(2/(ωμσ)); current crowds to the conductor surface as frequency rises, increasing effective resistance.

Poynting vector S = E×H gives power flow density in W/m²; wave impedance and phase constant β = 2π/λ determine propagation.

Transformer emf: E = 4.44·f·N·Φ_max; voltage ratio equals turns ratio V₁/V₂ = N₁/N₂, and current ratio is inverse.

Transformer tests: open-circuit test gives core (iron) loss and magnetizing branch; short-circuit test gives copper loss and series impedance.

DC machine emf: E = (pΦNZ)/(60A); separately excited torque T = kΦI_a; shunt motor speed falls with load, series motor speed varies widely.

Induction motor slip s = (n_s − n_r)/n_s, with synchronous speed n_s = 120f/p; rotor frequency = s·f.

Induction motor torque is proportional to slip at low slip and decreases beyond breakdown torque; starting current is typically 5–8 times rated.

Synchronous machine: emf E = V + I(R + jX_s); it can supply or absorb reactive power by adjusting field excitation (over-excited = lagging supply).

Losses: copper loss = I²R, core loss depends on flux density and frequency, friction and windage are mechanical; efficiency = output/(output + losses).

Single-phase induction motors are not self-starting; auxiliary windings, capacitors, or shaded poles provide the starting torque.

Per-unit value = actual value/base value; choose base power S_b and base voltage V_b, then Z_b = V_b²/S_b and per-unit impedance is the same on either side of an ideal transformer.

Short-line model: V_s = V_r + I·Z; medium-line uses nominal π or T; long lines use hyperbolic ABCD parameters with Z_c = √(z/y).

Surge impedance loading SIL = V_L²/Z_c; for overhead lines Z_c is roughly 400 Ω and SIL ≈ V_L²/400 MW.

Voltage regulation = (|V_r,noload| − |V_r,fullload|)/|V_r,fullload| × 100%; Ferranti effect raises receiving-end voltage on lightly loaded long lines.

Load flow: slack bus sets voltage and angle, PV buses hold voltage and real power, PQ buses specify load; Newton–Raphson converges in few iterations.

Transmission losses fall as voltage rises for fixed power: I = P/(√3·V_L·cos φ), so loss ∝ 1/V².

Symmetrical components: positive, negative, and zero sequence; zero-sequence current needs a return path and is blocked by delta windings.

Distribution: radial systems are simple but have poor reliability; ring or interconnected networks improve continuity; voltage drop ≈ √3·I·(R cos φ + X sin φ) for lines.

Overcurrent relay grading: time multiplier and plug setting create discrimination; downstream relay must clear before upstream backup operates.

Distance relay zones: zone 1 covers about 80% of line impedance with no intentional delay; zone 2 covers 120–150% with a time delay.

Differential protection compares currents at both ends of a transformer, generator, or busbar; it operates on the internal fault current and is immune to through-faults.

CT saturation and ratio errors cause relay misoperation; the knee-point voltage and burden must be checked for through-fault stability.

Circuit breaker ratings: rated normal current, breaking capacity (symmetrical and asymmetrical), making capacity, and short-time withstand current.

Arc interruption: in an AC circuit the arc extinguishes at current zero; restriking voltage and rate of rise of recovery voltage determine whether it reignites.

SF6 breakers have high dielectric strength and arc-quenching ability; vacuum breakers suit medium voltage; air-blast suits high voltage.

Fuse: I²t characteristic must coordinate with the protected equipment; IDMT relay characteristic follows t = TMS·k/((I/I_s)^α − 1).

Single-phase full-wave rectifier: average output V_dc = 2V_m/π ≈ 0.637V_m; RMS = V_m/√2; ripple frequency is twice the supply frequency.

Three-phase full bridge: V_dc = 3√3·V_m/π ≈ 1.35·V_LL for an uncontrolled rectifier; firing angle control gives V_dc = 1.35·V_LL·cos α.

Thyristor conducts only when anode is positive and a gate pulse is applied; it turns off when current falls below the holding current.

Buck converter: V_o = D·V_in; boost: V_o = V_in/(1−D); buck-boost: V_o = −V_in·D/(1−D); D is duty cycle.

PWM inverters control output voltage and frequency; increasing the switching frequency reduces harmonic content but raises switching losses.

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

V/f control of induction motors keeps flux constant; below base frequency the voltage is reduced proportionally to avoid saturation.

Harmonics: a six-pulse converter produces 5th and 7th harmonics; a 12-pulse converter produces 11th and 13th; total harmonic distortion quantifies waveform distortion.

Transfer function G(s) = output/input with zero initial conditions; for a linear time-invariant system it is the ratio of Laplace transforms.

Second-order system: ω_n is natural frequency, ζ is damping ratio; percent overshoot = 100·e^(−ζπ/√(1−ζ²)); peak time = π/(ω_n√(1−ζ²)).

Steady-state error: type 0 has finite error to a step, type 1 has zero error to a step and finite error to a ramp, type 2 has zero error to a ramp.

Routh–Hurwitz array gives the number of right-half-plane roots; a sign change in the first column indicates instability.

Bode plot: gain margin is the dB value at the phase crossover where phase = −180°, phase margin is measured at the gain crossover where magnitude = 0 dB.

PID control: proportional action reduces rise time, integral action removes steady-state error but reduces stability, derivative action adds damping and amplifies noise.

Root locus starts at open-loop poles and ends at open-loop zeros; branches lie on the real axis where the number of poles plus zeros to the right is odd.

PLC scan cycle: read inputs, execute the program, update outputs; ladder logic uses contacts in series for AND and parallel for OR.

PMMC meter deflection is proportional to average current and suits DC; a rectifier-type meter reads average but is calibrated in RMS for sine waves.

Ammeter has low resistance and is connected in series; voltmeter has high resistance and is connected in parallel; the wattmeter current coil is in series and voltage coil in parallel.

Extend ammeter range with a shunt: R_sh = R_m/(m−1); extend voltmeter range with a series multiplier: R_se = R_m(m−1), where m is the multiplying factor.

Wattmeter reading in a two-wattmeter method: total power = W₁ + W₂; power factor = cos(tan⁻¹(√3(W₁−W₂)/(W₁+W₂))).

Energy meter: number of revolutions per kWh is the meter constant; error % = (measured − true)/true × 100.

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

Bridge methods: Wheatstone measures medium resistance, Kelvin double bridge measures low resistance, Megger measures insulation resistance.

Accuracy, precision, and resolution differ: accuracy is closeness to true value, precision is repeatability, resolution is the smallest detectable change.

Cable current-carrying capacity depends on conductor size, insulation, installation method, ambient temperature, and grouping; derating factors apply multiplicatively.

Voltage drop for a single-phase circuit is about 2·I·L·(R cos φ + X sin φ); for three-phase it is √3·I·L·(R cos φ + X sin φ).

Earthing systems: TN-S uses a separate protective conductor, TN-C combines neutral and protective functions, TT uses a local earth electrode, IT has an isolated or high-impedance neutral.

Protection by disconnection requires the earth-fault loop impedance to be low enough for the protective device to trip within the specified disconnection time.

Residual current devices detect imbalance between live and neutral; a 30 mA RCD is used for supplementary protection against electric shock.

Conductor colour coding and terminal identification must follow the applicable national code; protective earth is green-yellow and must never be used as a live conductor.

Cable sizing must satisfy current capacity, voltage drop limits, short-circuit withstand (I²t ≤ k²S²), and thermal constraints.

Hazardous areas use zoning and equipment protection concepts such as flameproof, increased safety, and intrinsic safety.

Time value of money: F = P(1+i)^n; present worth P = F/(1+i)^n; for an annuity, P = A[(1+i)^n − 1]/[i(1+i)^n].

Net present value discounts all cash flows at the required rate of return; accept if NPV > 0; IRR is the discount rate where NPV = 0.

Payback period is the time to recover the initial investment; it ignores cash flows after payback and the time value of money unless discounted.

Depreciation: straight line = (cost − salvage)/life; declining balance applies a constant rate to the book value; MACRS uses prescribed recovery periods.

Benefit–cost ratio = present worth of benefits/present worth of costs; a ratio greater than 1 indicates an economically acceptable project.

Critical path method: the critical path has zero total float and determines the shortest project duration; crashing reduces duration at extra cost.

Earned value: CPI = EV/AC, SPI = EV/PV; CPI below 1 means over budget, SPI below 1 means behind schedule.

Professional ethics: hold public safety and welfare paramount, practise only within competence, disclose conflicts of interest, and maintain confidentiality.