Phasors, impedances, and ac transfer functions belong to a class of variables known as complex variables, so called because they involve real numbers as well as the imaginary unit .

• A complex variable Z can be represented in either one of three forms:

  1. The shorthand form, also called the polar form:

     
     

  2. The rectangular form, also called the cartesian form:

     
     

  3. The exponential form:

     
     

• In the above expressions, Z _m is the length or magnitude of Z, and ? is the angle or argument of Z. The are denoted as

  
  

  and are called the polar coordinates or Z. Moreover, Z _r is the real part of Z, and Z _i is the imaginary part of Z, expressed as

  
  

  They are called the rectangular or cartesian coordinates of Z.

• Given Z _m or ?, Z _r and Z _i are found as

  
  

  Conversely, given Z _r and Z _i, Z _m and ? are found as

  
  
  
  
  
  

  The ? sign is chosen in such a way as to keep ?180 ? ? ? 180 .

• Given two complex variables, X = X _r + jX _i and Y = Y _r + jY _i, their sum and difference are

  
  

• Given two complex variables, and , their product and...