Difference between revisions of "Complex Number Review"

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(Finding the magnitude)
(Finding the magnitude)
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In exponential form, we can see that the magnitude of <math>Ae^{j \theta}</math> is:
 
In exponential form, we can see that the magnitude of <math>Ae^{j \theta}</math> is:
 
:<math>|Ae^{j \theta}| = A</math>
 
:<math>|Ae^{j \theta}| = A</math>
[[Image:ComplexExponentialForm.png|thumb|center|250px]]
+
[[Image:ComplexExponentialForm.png|thumb|center|230px]]
 
Thus, when two complex numbers are multiplied, their magnitudes multiply. Similarly, when two complex numbers are divided (as we often see in transfer functions), their magnitudes are divided. For example, <math>|{Ae^{j \theta} \over Be^{j \theta}}| = {A \over B}</math>
 
Thus, when two complex numbers are multiplied, their magnitudes multiply. Similarly, when two complex numbers are divided (as we often see in transfer functions), their magnitudes are divided. For example, <math>|{Ae^{j \theta} \over Be^{j \theta}}| = {A \over B}</math>
  

Revision as of 15:10, 12 August 2016

20.309: Biological Instrumentation and Measurement

ImageBar 774.jpg

Overview

Complex numbers include a real component and an imaginary component, which involves the imaginary number $ j = \sqrt{-1} $. We can represent a complex number like this: $ z = x + jy $, where $ x $ is the real part and $ y $ is the imaginary part. Complex numbers can be represented on a plot as shown on the right, where the horizontal axis corresponds to the real part and the vertical axis corresponds to the imaginary part.
ComplexPlane.png

Since we are often dealing with sinusoidal waveforms, it can be helpful to think of complex numbers also in exponential form. According to Euler's identities,

$ e^{j \theta} = cos(\theta) + j sin(\theta) $
$ cos(\theta) = {e^{j\theta} + e^{-j\theta}\over 2} $
$ sin(\theta) = {e^{j\theta} - e^{-j\theta}\over 2} $

When complex numbers are added together, the real components add to real components, and the imaginary components add to imaginary components. Similarly, when complex numbers are subtracted, the real parts are subtracted from each other, and the imaginary parts are subtracted from each other.

Finding the magnitude

In the rectangular form of the complex number $ z = x + jy $, we can see that the magnitude of $ z $ (the length of the blue arrow in the plot below) is:

$ |z| = \sqrt{x^2 + y^2} $
Magnitude.png

In exponential form, we can see that the magnitude of $ Ae^{j \theta} $ is:

$ |Ae^{j \theta}| = A $
ComplexExponentialForm.png

Thus, when two complex numbers are multiplied, their magnitudes multiply. Similarly, when two complex numbers are divided (as we often see in transfer functions), their magnitudes are divided. For example, $ |{Ae^{j \theta} \over Be^{j \theta}}| = {A \over B} $

Finding the phase

To the find the phase of a complex number, we find the value for $ \theta $ as demarcated in the plot below.

We can see that $ \theta = arctan({y \over x}) $
Phase.png

When multiplying or dividing complex numbers, one can more clearly see what happens to the phase when using exponential form. As you know, when multiplying exponentials, the exponents add; and when dividing exponentials, the exponents subtract. For example, $ {Ae^{j \theta_1} \over Be^{j \theta_2}}= {A \over B} e^{j(\theta_1 - \theta_2)} $. You can apply this fact to transfer functions and Bode plots.