Plot the magnitude and phase of the following complex functions:
G(w) = 1/(1+jw)
We never went over this in lecture but apparently I have homework on it. Looks pretty simple but just don't know where to begin. Could anyone maybe walk through the process of it rather than just answer?
edit: I apologize for my ambiguity here is the problem word for word:
Plot the magnitude and phase of the following complex functions:
(i) G(w) = 1/(1+jw)
(ii) G(ω) = −1/(1+jω)
(iii) G(ω) = j/(1+jω)
Best Answer
The \$j\$ is the same as \$i\$, the \$i\$ (imaginary) axis is perpendicular to the real axis. This means that we can use the Pythagorean theorem if we want to calculate the magnitude of a complex number.
So the magnitude of \$a+ib\$ is \$\sqrt{a^2+b^2}\$, and this is also the absolute value of any function. Just add some straight lines to denote absolute value, like this: \$|a+ib|\$.
So the \$|G(w)|\$ is \$|\frac{1}{1+jw}|=\frac{|1|}{|1+jw|}=\frac{1}{\sqrt{1+w^2}}\$. Now all you need to do is to insert real numbers into the w variable.
As mentioned before, the \$i\$ axis is perpendicular to the real axis. This means that we can retrieve our phase, \$\phi,=\tan^{-1}(\frac{b}{a})=\arctan(\frac{b}{a})\$.
The phase of two multiplied complex number is the same as adding their individual phases. \$i\$ is an example of this, \$i^2=-1 => 90+90=180\$. And the phase of two divided complex numbers is simply subtracting the phase of the denominator from the phase of the numerator.
So the \$\phi\$ of, say \$\frac{3+4i}{1+2i}\$ for an example is \$\phi=\arctan(\frac{4}{3})-\arctan(\frac{2}{1})\$.
The phase, \$\phi\$, is sometimes called the angle of a complex value, which is sometimes called argument, arg{}.
The \$\arg\{G(w)\}\$ is \$\arg\{\frac{1}{1+jw}\}=\arctan(\frac{0}{1})-\arctan(\frac{w}{1})=-\arctan(w)\$
Don't forget to multiply the \$\arg\{\}\$ with \$\frac{180}{\pi}\$ if you want it in regular degrees.
The w in your function is a frequency in radians per second which can be translated to regular Hertz by simply multiplying by \$2\pi\$. You can try 0 Hz, 0.1 Hz, 1 Hz, 10 Hz, 100 Hz. The actual values I recommend you to replace w with are 0, 0.628, 6.28, 62.8 and 628.
Those points will probably give you enough data to extrapolate the rest of the function.