Phasors, complex power and three-phase engineering models
Power-system calculations need a clearly defined electrical model. Establish whether voltages and currents are instantaneous, peak or RMS, whether the circuit is sinusoidal, and whether a three-phase load is balanced. A formula valid for one case does not automatically apply to distorted or unbalanced operation. Draw the reference directions and identify line and phase quantities before inserting numbers.
For sinusoidal steady state, represent resistance and reactance as complex impedance. Series impedances add as complex quantities, while parallel branches are conveniently combined through admittance. With the usual engineering phasor convention, inductive reactance contributes positive imaginary impedance and capacitive reactance negative imaginary impedance. The magnitude of a sum is not usually the sum of magnitudes. Kirchhoff equations can be solved in complex form just as DC equations use real quantities, while respecting the actual frequency and source relationships.
Complex power for a single-phase RMS phasor model is voltage multiplied by the complex conjugate of current. Its real part is active power and its imaginary part reactive power under the chosen sign convention. Apparent power is the magnitude. For a balanced sinusoidal three-phase load, active power can be calculated as square root of three times line voltage times line current times power factor. Star and delta connections have different line-to-phase relationships, so a memorized ratio without the connection diagram can give a plausible wrong result.
Power-factor correction changes reactive demand in a stated model. It does not erase the active energy used by an unchanged load. Correcting a fundamental-frequency displacement factor is not the same as eliminating waveform distortion. Capacitor behavior, harmonics and resonance require actual system assessment; a classroom reactive-power result is not a capacitor installation specification. Similarly, equal active kilowatts do not imply equal line current when voltage or apparent power differs.
Per-unit calculations normalize quantities to consistent bases. Define power and voltage bases, derive the corresponding impedance or current base, and convert equipment impedances before combining them. A percentage impedance quoted on one equipment rating cannot be inserted unchanged into every network base. After solving, convert back to physical units and check whether the result has the expected order of magnitude. These are genuine engineering foundations, not current regional wiring limits, equipment approval or permission to energize a modeled network.