Complex Modulus (HW1, ECE301, Fall 2008)

Complex Modulus, also known as the "Norm" of a complex number, is represented as $ |z| $.

$ |x + iy| = \sqrt{x^2 + y^2} $


In exponential form for $ |z| $

$ |re^{i\phi}| = r $

(This format is used when dealing with Phasors)

Basics

  • $ \frac{|Ae^{i\phi_{1}}|}{|Be^{i\phi_{2}}|} = \frac{A}{B}\frac{|e^{i\phi_{1}}|}{|e^{i\phi_{2}}|} = \frac{A}{B} $


  • $ |\frac{Ae^{i\phi_{1}}}{Be^{i\phi_{2}}}| = \frac{A}{B}|e^{i(\phi_{1}-\phi_{2})}| = \frac{A}{B} $


  • $ |\frac{Ae^{i\phi_{1}}}{Be^{i\phi_{2}}}| = \frac{|Ae^{i\phi_{1}}|}{|Be^{i\phi_{2}}|} $


  • $ |Ae^{i\phi_{1}}||Be^{i\phi_{2}}| = {A}{B}|e^{i\phi_{1}}||e^{i\phi_{2}}| = {A}{B} $


  • $ |(Ae^{i\phi_{1}})(Be^{i\phi_{2}})| = {A}{B}|e^{i\phi_{1}+i\phi_{2}}| = {A}{B} $


  • $ |Ae^{i\phi_{1}}||Be^{i\phi_{2}}| = |(Ae^{i\phi_{1}})(Be^{i\phi_{2}})| $


  • $ |z^n|=|z|^n $


  • $ |z|^2 $ of $ |z| $ is known as the Absolute Square.


  • $ z\overline z=|z|^2 $
     Where $ z $ is a complex number and $ \overline z $ is the complex conjugate.
     
     $ z = x + iy $
     
     $ \overline z=x-iy $

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