Electrical Formulas
A digital handbook for electrical engineers.
Ohm's Law
V = I × R
V: Voltage (Volts)
I: Current (Amps)
R: Resistance (Ohms)
Power (DC)
P = V × I
P: Power (Watts)
V: Voltage (Volts)
I: Current (Amps)
Power (AC, Single-Phase)
P = V × I × cos(φ)
P: Active Power (Watts)
V: Voltage (Volts)
I: Current (Amps)
cos(φ): Power Factor
Energy
E = P × t
E: Energy (Watt-hours)
P: Power (Watts)
t: Time (hours)
Apparent Power
S = V × I
S: Apparent Power (Volt-Amps)
V: Voltage (Volts)
I: Current (Amps)
Reactive Power
Q = V × I × sin(φ)
Q: Reactive Power (VAR)
V: Voltage (Volts)
I: Current (Amps)
sin(φ): Sine of the angle between V and I
Power Relationship
S² = P² + Q²
S: Apparent Power (VA)
P: Active Power (W)
Q: Reactive Power (VAR)
Impedance
Z = √(R² + (XL – XC)²)
Z: Impedance (Ohms)
R: Resistance (Ohms)
XL: Inductive Reactance (Ohms)
XC: Capacitive Reactance (Ohms)
Voltage Drop (3-Phase)
ΔV = (√3 × I × L × (Rcos(φ) + Xsin(φ))) / 1000
ΔV: Voltage Drop (Volts)
I: Current (Amps)
L: Length (km)
R: Resistance (Ω/km)
X: Reactance (Ω/km)
cos(φ): Power Factor
Current Density
J = I / A
J: Current Density (A/mm²)
I: Current (Amps)
A: Cross-sectional Area (mm²)
Capacitance of Single-Core Cable
C = (2πε) / ln(D/r)
C: Capacitance per unit length (F/m)
ε: Permittivity of the dielectric (F/m)
D: Outer diameter of dielectric
r: Conductor radius
Dielectric Loss
Pd = V²ωC tan(δ)
Pd: Dielectric Loss (W)
V: RMS Voltage (V)
ω: Angular frequency (rad/s)
C: Capacitance (F)
tan(δ): Loss tangent of the dielectric
Skin Depth
δ = √(2 / (ωμσ))
δ: Skin Depth (m)
ω: Angular frequency (rad/s)
μ: Permeability of the conductor (H/m)
σ: Conductivity of the conductor (S/m)
Transformer Turns Ratio
Vp/Vs = Np/Ns
Vp: Primary Voltage
Vs: Secondary Voltage
Np: Primary Turns
Ns: Secondary Turns
Transformer Voltage Regulation
VR = ((IRcos(φ) ± IXsin(φ)) / V) * 100
VR: Voltage Regulation (%)
I: Full load current (A)
R: Equivalent resistance (Ω)
X: Equivalent reactance (Ω)
V: Full load secondary voltage (V)
cos(φ): Power Factor (Lagging +, Leading -)
Synchronous Machine Power
P = (EV / Xs) * sin(δ)
P: Power Output (W)
E: Generated EMF (V)
V: Terminal Voltage (V)
Xs: Synchronous Reactance (Ω)
δ: Load angle
Induction Motor Slip
s = (Ns - Nr) / Ns
s: Slip
Ns: Synchronous Speed (RPM)
Nr: Rotor Speed (RPM)
Slip at Maximum Torque
s_max = R2 / X2
s_max: Slip at maximum torque
R2: Rotor resistance (Ω)
X2: Rotor reactance (Ω)
Per Unit System
Z_pu = Z_actual / Z_base, where Z_base = (V_base)² / S_base
Z_pu: Per-unit impedance
Z_actual: Actual impedance (Ω)
Z_base: Base impedance (Ω)
V_base: Base voltage (V)
S_base: Base apparent power (VA)
Line Efficiency
η = (P_receiving / P_sending) * 100%
η: Efficiency (%)
P_receiving: Power at receiving end (W)
P_sending: Power at sending end (W)
Corona Power Loss (Peek's Formula)
P_c = (241/δ) * (f+25) * √(r/D) * (V-V_c)² * 10⁻⁵
P_c: Corona power loss per km per phase (kW)
f: Frequency (Hz)
δ: Air density correction factor
r: Conductor radius (cm)
D: Spacing between conductors (cm)
V: Phase-to-neutral voltage (kV)
V_c: Critical disruptive voltage (kV)
Sag in Transmission Lines
Sag = wL² / (8T)
Sag: Sag (m)
w: Weight of conductor per unit length (N/m)
L: Span length (m)
T: Tension in the conductor (N)
Symmetrical Components
Ia = I1+I2+I0, Ib = a²I1+aI2+I0, Ic = aI1+a²I2+I0
Ia, Ib, Ic: Phase currents
I0, I1, I2: Zero, positive, and negative sequence currents
a: Phase operator (e^j120°)
Fault Current (3-phase short circuit)
I_sc = E / Z_th
I_sc: Short-circuit current (A)
E: Thevenin equivalent voltage (V)
Z_th: Thevenin equivalent impedance (Ω)
Relay Operating Time (IDMT)
t = 0.14 / ((I/I_set)^0.02 - 1)
t: Operating time (s)
I: Fault current (A)
I_set: Relay pickup current setting (A)
Arc Energy
E = ∫V_arc * I dt
E: Arc Energy (Joules)
V_arc: Arc voltage (V)
I: Arc current (A)
t: Time (s)
Total Harmonic Distortion (THD)
THD = (√(Σ I_n²)) / I_1 * 100 (for n=2 to ∞)
THD: Total Harmonic Distortion (%)
I_n: RMS current of the nth harmonic
I_1: RMS current of the fundamental frequency
Power Factor with Harmonics
PF = P / (V_rms * I_rms)
PF: True Power Factor
P: Average power (W)
V_rms: Total RMS voltage (V)
I_rms: Total RMS current (A)
Flicker Severity (IEC Standard)
P_st = (1/N * Σ (ΔV/V)²) ^ (1/2)
P_st: Short-term flicker severity
N: Number of voltage changes
ΔV/V: Relative voltage change
Swing Equation
(2H/ω_s) * (d²δ/dt²) = P_m - P_e
H: Inertia constant (s)
ω_s: Synchronous speed (rad/s)
δ: Rotor angle (rad)
P_m: Mechanical power input (pu)
P_e: Electrical power output (pu)
Transfer Function (General Form)
G(s) = Output(s) / Input(s)
G(s): Transfer function in the s-domain
Output(s): Laplace transform of the output
Input(s): Laplace transform of the input