(→Complex Modulus) |
|||
| (10 intermediate revisions by 2 users not shown) | |||
| Line 1: | Line 1: | ||
| − | == Complex Modulus == | + | [[Category:Complex Number Magnitude]] |
| + | [[Category:ECE301]] | ||
| + | |||
| + | == Complex Modulus ([[Homework_1_ECE301Fall2008mboutin|HW1]], [[ECE301]], [[Main_Page_ECE301Fall2008mboutin|Fall 2008]])== | ||
Complex Modulus, also known as the "Norm" of a complex number, is represented as <math>|z|</math>. | Complex Modulus, also known as the "Norm" of a complex number, is represented as <math>|z|</math>. | ||
<math>|x + iy| = \sqrt{x^2 + y^2}</math> | <math>|x + iy| = \sqrt{x^2 + y^2}</math> | ||
| + | |||
In exponential form for <math>|z|</math> | In exponential form for <math>|z|</math> | ||
<math>|re^{i\phi}| = r</math> | <math>|re^{i\phi}| = r</math> | ||
| + | |||
| + | (This format is used when dealing with Phasors) | ||
== Basics == | == Basics == | ||
| − | <math>|z|^2</math> of <math>|z|</math> is known as the '''Absolute Square'''. | + | *<math>\frac{|Ae^{i\phi_{1}}|}{|Be^{i\phi_{2}}|} = \frac{A}{B}\frac{|e^{i\phi_{1}}|}{|e^{i\phi_{2}}|} = \frac{A}{B}</math> |
| + | |||
| + | |||
| + | *<math>|\frac{Ae^{i\phi_{1}}}{Be^{i\phi_{2}}}| = \frac{A}{B}|e^{i(\phi_{1}-\phi_{2})}| = \frac{A}{B}</math> | ||
| + | |||
| + | |||
| + | *<math>|\frac{Ae^{i\phi_{1}}}{Be^{i\phi_{2}}}| = \frac{|Ae^{i\phi_{1}}|}{|Be^{i\phi_{2}}|}</math> | ||
| + | |||
| + | |||
| + | *<math>|Ae^{i\phi_{1}}||Be^{i\phi_{2}}| = {A}{B}|e^{i\phi_{1}}||e^{i\phi_{2}}| = {A}{B}</math> | ||
| + | |||
| + | |||
| + | *<math>|(Ae^{i\phi_{1}})(Be^{i\phi_{2}})| = {A}{B}|e^{i\phi_{1}+i\phi_{2}}| = {A}{B}</math> | ||
| + | |||
| + | |||
| + | *<math>|Ae^{i\phi_{1}}||Be^{i\phi_{2}}| = |(Ae^{i\phi_{1}})(Be^{i\phi_{2}})|</math> | ||
| + | |||
| + | |||
| + | *<math>|z^n|=|z|^n</math> | ||
| + | |||
| + | |||
| + | *<math>|z|^2</math> of <math>|z|</math> is known as the '''Absolute Square'''. | ||
| + | |||
| + | |||
| + | *<math>z\overline z=|z|^2</math> | ||
| + | |||
| + | Where <math>z</math> is a complex number and <math>\overline z</math> is the complex conjugate. | ||
| + | |||
| + | <math>z = x + iy</math> | ||
| + | |||
| + | <math>\overline z=x-iy</math> | ||
| + | ---- | ||
| + | [[Main_Page_ECE301Fall2008mboutin|Back to ECE301 Fall 2008 Prof. Boutin]] | ||
| + | |||
| + | [[ECE301|Back to ECE301]] | ||
| + | |||
| + | [[More_on_complex_magnitude|Back to Complex Magnitude page]] | ||
| + | |||
| + | Visit the [[ComplexNumberFormulas|"Complex Number Identities and Formulas" page]] | ||
Latest revision as of 05:35, 23 September 2011
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 $
Back to ECE301 Fall 2008 Prof. Boutin
