Phasor Arithmetics

Consider a sinusoid and its corresponding phasor.

The derivative of the sinusoid in the time domain equals its phasor multiplied by j-omega in the phasor domain.

Similarly, when integrating a sinusoid in the time domain, it transforms into its phasor divided by j-omega in the phasor domain.

These transformations yield the sinusoid steady-state solution without knowing the initial values.

Now, consider two phasors in rectangular and polar forms. To add these two phasors, their rectangular forms are used.

The real part of the resultant phasor is the sum of the real parts of the two phasors, and its complex part is the sum of the complex parts of the individual phasors.

Similarly, to subtract two phasors, their rectangular forms are used. The real and complex parts of the resultant phasor are the differences of the real and imaginary parts of the individual phasors.

Polar forms are used to multiply and divide any two phasors, and the complex conjugate of a phasor can be expressed in both rectangular and polar forms.