Complex Algebra and Impedance

Complex algebra is a powerful tool for solving problems in AC electric circuits, including sinusoidal systems. The complex number Z can be written in the rectangular form 

The complex number Z can also be expressed in the polar form 

Having introduced complex algebra, we are now able to go further, to the concept of electrical impedance or simply impedance. Electrical impedance extends the concept of resistance to AC circuits and therefore is also called AC impedance. As impedance is a complex quantity, the term complex impedance may also be used. Based on the definition of resistance described by Ohm s law, the current-voltage relationship in impedance can be expressed as 

For a sinusoidal system, the AC impedance of a resistor, X, . in the complex plane can be expressed as 

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