for power is the watt (symbol: W). Apparent power is often expressed in volt-amperes (VA) since it is the product of RMS voltage and RMS current. The unit for reactive power is var, which stands for volt-ampere reactive. Since reactive power transfers no net energy to the load, it is sometimes called "wattless" power. It does, however, serve an important function in electrical grids and its lack has been cited as a significant factor in the Northeast blackout of 2003. Understanding the relationship among these three quantities lies at the heart of understanding power engineering. The mathematical relationship among them can be represented by vectors or expressed using complex numbers, S = P + j Q (where j is the imaginary unit). Calculations and equations in sinusoidal steady-state The formula for complex power (units: VA) in phasor form is: , where V denotes voltage in phasor form, with the amplitude as RMS, and I denotes current in phasor form, with the amplitude as RMS. Also by convention, the complex conjugate of I is used, which is denoted (or ), rather than I itself. This is done because otherwise using the product V I to define S would result in a quantity that depends on the reference angle chosen for V or I, but defining S as V I* results in a quantity that doesn't depend on the reference angle and allows to relate S to P and Q. Other forms of complex power (units in volt-amps, VA) are derived from Z, the load impedance (units in ohms, Ω). . Consequentially, with reference to the power triangle, real power (units in watts, W) is derived as: . For a purely resistive load, real power can be simplified to: .R denotes resistance (units in ohms, Ω) of the load. Reactive power (units in volts-amps-reactive, var) is derived as: . For a purely reactive load, reactive power can be simplified to: , where X denotes reactance (units in ohms, Ω) of the load. Combining, the complex power (units in volt-amps, VA) is back-derived as , and the apparent power (units in volt-amps, VA) as . These are simplified diagrammatically by the power triangle. Power factor The ratio of active power to apparent power in a circuit is called the power factor. For two systems transmitting the same amount of active power, the system with the lower power factor will have higher circulating currents