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When complex number represent stationary AC signal, it is called phasor. Complex number and so phasor can be presented with two equivalent notifications:

  • Cartesian coordinates
  • Polar coordinates

cartesian coordinates

polar coordinates

Knowing exponential representative of complex number, we have relation between complex number and trigonometric functions as

trigonometric exponential

Phasor multiplication and phasor division are non-linear operations and so, different from addition and subtraction. We actually don’t use phasors to present product of two active AC signals, because due to non-linear nature of multiplication operation, product of two AC signals gives new signal with one DC component i.e. Active Power, and another component that is on frequency twice of mains i.e. Instant Power. But, by multiplying two complex numbers, we still get complex number with certain absolute value (distance from (0,0) point) and certain phase angle. It means that result of multiplying operation will produce complex number that represents AC signal with same frequency as input signals, and with it’s own amplitude and phase.

In that sense, we use phasor representation to calculate response of passive linear network (containing passive elements such as resistors, capacitors and inductors) on AC current input to produce AC voltage output by using multiplication of impedance and AC current phasors:

phasor i.e. complex number multiplication

And also we use phasor division to calculate AC current caused by AC voltage over passive linear network by dividing voltage and impedance phasor.

phasor i.e. complex number division

For steady state AC signals on passive linear network, response is far more easier to find by using complex number division and multiplication, rather then to solve linear differential equations.

External links:

Phasor multiplication and phasor division on Resonanceswavesandfields
Complex numbers on Dario Alpern’s site Home Page

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