1、3.3 Phasor According to the knowledge on the complex number and sinusoidal quantity in Section 3.2, one sinusoidal quantity can be expressed as ( ) m( ) c o s R e ( )x t X t t X= + = (1) where ( ) ( ) ()m m m( ) c o s s i n jtt X t j X t X e += + + + =X (2) is the complex number corresponding to the
2、 sinusoidal waveform x(t). From (2), X(t) is a vector whose amplitude is Xm and rotating angular frequency is . The complex numbers corresponding to the following two sinusoidal quantities are ( )( ) 12()1 1 m 1 1 1 m()2 2 m 2 2 2 m( ) co s ( )( ) co s ( )jtjtx t X t t X ex t X t t X e+= + = + =XX (
3、3) From (3), 1()Xt and 2()Xt are rotating vectors with as their angular frequency. Although 1()Xt and 2()Xt are rotating, they are static relatively for they have the same rotating angular frequency. Therefore, if we only consider the relationship between 1()1 1m() jtt X e +=X and 2()2 2m() jtt X e
4、+=X , the rotation can be ignored. Thus t in 1()Xt and 2()Xt can be removed. After removing t of 1()Xt and 2()Xt in (3), we obtain ( )( ) 121 1 m 1 1 1 m2 2 m 2 2 2 m( ) co s( ) co sjjx t X t X ex t X t X e= + = + =XX (4) For any sinusoidal quantity ( )m( ) c o sx t X t=+， its corresponding complex
5、number is ( )mm( ) c o s jx t X t X e = + =X (5) where m jXe=X is called the phasor corresponding to x(t). Alternatively, the phasor can be written as the polar form mX =X (6) The phasor form and the sinusoidal expression can be interconverted. ( )mmc o sX t X + (7) After introducing the concept of
6、the phasor, the difficult operation of the sinusoidal functions in time domain can be converted to the easy operation of the phasors, as shown in Table I. Table I: Time-domain sinusoidal functions and their phase-domain counterparts Time domain Phasor domain ()xt X ()k xt kX 12( ) ( )x t x t 12XX ()d xtdt jX where k is a constant. Since the phasor is independent of time, the advantage of introducing the concept of the phasor is that the circuit varying periodically (in time domain) can be converted to a static circuit (in phasor domain) which is easier to be analyzed.
