# Phasor

Phasors in an RLC series connection in the complex plane

The phasor or the complex amplitude is used in the complex representation of sinusoidal time-dependent quantities. It combines the amplitude and the zero phase angle into one complex variable . The letter j is used here for the imaginary unit and the symbol of a complex variable is identified by an underscore (in accordance with DIN 1304-1 and DIN 5483-3). ${\ displaystyle {\ hat {a}}}$ ${\ displaystyle \ varphi}$ ${\ displaystyle {\ underline {\ hat {a}}} = {\ hat {a}} \ cdot \ mathrm {e} ^ {\ mathrm {j} \ varphi}}$

A harmonic oscillating time-dependent physical quantity of the general form

${\ displaystyle a (t) = {\ hat {a}} \ cdot \ cos (\ omega t + \ varphi)}$is described by means of the phasor .${\ displaystyle {\ underline {a}} (t) = {\ underline {\ hat {a}}} \ cdot \ mathrm {e} ^ {\ mathrm {j} \ omega t}}$

With the conversion

${\ displaystyle {\ underline {\ hat {a}}} \ cdot \ mathrm {e} ^ {\ mathrm {j} \ omega t} = {\ hat {a}} \ cdot \ mathrm {e} ^ {\ mathrm {j} (\ omega t + \ varphi)} = {\ hat {a}} \ cdot (\ cos (\ omega t + \ varphi) + \ mathrm {j} \, \ sin (\ omega t + \ varphi)) }$

the original size of it is the real part . ${\ displaystyle a (t)}$

The use of such complex variables is used, for example, in the context of the complex AC calculation. This representation has the advantage that analytical operations such as differentiation and integration can be carried out much more easily than when using the trigonometric functions.

The phasor has the particular advantage that the (sinusoidal) time dependency does not appear in it. While rotating in the complex plane as a rotary pointer, the phasor is stationary. Its orientation can be determined arbitrarily like the time zero point or the zero phase angle, but it is uniform for all phasors of a connection. It is only a matter of expediency to place a reference quantity in the positive direction of the real axis. When impedances are connected in series, as shown in the picture, the current flowing through all partial resistances can be used. ${\ displaystyle {\ underline {a}} (t)}$

The use of the phasor in the exponential form is also helpful for multiplication and division, while the use of the algebraic form is appropriate for addition and subtraction. ${\ displaystyle {\ underline {\ hat {a}}} = {\ hat {a}} \ cdot \ mathrm {e} ^ {\ mathrm {j} \ varphi}}$ ${\ displaystyle {\ underline {\ hat {a}}} = x + \ mathrm {j} y}$