Miller Texas A&M University NovemECEN 214 Linear Circuit Analysis Math Review - Complex Numbers of 11 Complex Numbers - Arithmetic Addition and Subtraction It is easiest to add and subtract complex numbers in their Carte- Im z sian from. Miller Texas A&M University NovemECEN 214 Linear Circuit Analysis Math Review - Complex Numbers of 11 Complex Numbers - Representation Examples: Convert each of the following complex numbers from Cartesian to polar form. Miller Texas A&M University NovemECEN 214 Linear Circuit Analysis Math Review - Complex Numbers of 11 Complex Numbers - Representation Examples: Convert each of the following complex numbers from polar to Cartesian form. The con- version from Cartesian to Polar works out to be y r = x 2 + y 2, = tan – 1 -.
The above equations show us how to convert from polar form to Cartesian form. Therefore, for a complex number with magnitude, r, and phase, , we write the complex number in its Polar form as z = re j and the Cartesian representation of the same com- plex number would be x = Re re j = r cos , y = Im re j = r sin . Multiplying both sides by a magnitude, r, produces 2 Re z re j = r cos + jr sin . The polar form of a complex 4 * number stems from Euler’s Identity, r e j = cos + j sin . Miller Texas A&M University NovemECEN 214 Linear Circuit Analysis Math Review - Complex Numbers of 11 Complex Numbers - Representation Polar Representation of Complex Numbers In its polar form, a complex number is written in terms of Im z magnitude, r, and phase, . 2 Re z We visualize complex numbers as points in a 2-D plane where the x-axis is the real part and the y-axis is the imag- inary part. The imaginary part is the number multiplying the j ). Example: z = 2 + 4j Im z Re 2 + 4j = 2 (Real part) 4 * Im 2 + 4j = 4 (Imaginary part) (Note: the j is not included in the imaginary part.
Cartesian Representation of Complex Numbers In its Cartesian form, a complex number is written in terms of a real part and an imagi- nary part. Note that most math books will use the letter i for this role, but in electrical engineering we prefer to use the letter j since i is typically used to represent current. ECEN 214 Linear Circuit Analysis Math Review - Complex Numbers of 11 Complex Numbers - Representation We construct complex numbers around the basic building block j = –1.