# Correct Phase (Bode) plot of transfer function

I've written the following question here:

I've learned some basic rules to draw Bode plots and when it comes to Phase plot, some of them are:

• a single zero located in the left half of s-plane has a contribution of 0 degrees near origin and +90 degrees on high frequencies;
• a single pole located in the left half of s-plane has a contribution of 0 degrees near origin and -90 degrees on high frequencies;
• a single zero located in the right half of s-plane has a contribution of +180 degrees near origin and +90 degrees on high frequencies;
• a single pole located in the right half of s-plane has a contribution of -180 degrees near origin and -90 degrees on high frequencies;

Note that if you do the math to find the phase by the transfer function, you may find that would be more intuitive to think:

• a single zero located in the right half of s-plane has a contribution of 0 degrees near origin and -90 degrees on high frequencies;
• a single pole located in the right half of s-plane has a contribution of 0 degrees near origin and +90 degrees on high frequencies;

In the first set of rules, all poles would end up in -90 degrees for high frequencies and all zeroes would end up with +90 degrees for high frequencies while with the second way of thinking, all of them would start from zero.

It may also makes sense to think that these last two rules are just a matter of adding +180 or -180 degrees in the cases above which may not change the results since the phase comes from the tangent function which allows us to do that (since $tg(10^o)=tg(190^o)$).

However, making the second way, a zero in the right half of s-plane would have the same phase plot of a pole in the left halft of the s-plane.

So, after all, is there a correct and unique rule?

And in a few seconds I deleted it because I came to this conclusion:

• $H(jw)$ is represented by a complex number which has a magnitude and phase. And even though $tg(\phi)=tg(\phi + 180^o)$, rotating this complex number by $180^o$ would give me a different complex number.

However, when we have to calculate what is the magnitude and phase of this complex number $Z$, we have:

$$Z=a+jb , \space\space j^2=-1$$ $$|H(jw)|=|Z|=\sqrt{a^2+b^2}$$ $$arg(H(jw))=arg(Z)=\frac{b}{a}$$

So, if I rotate $Z$ +180 degrees, then:

$$Z=-a-jb$$ $$arg(Z)=\frac{-b}{-a}=\frac{b}{a}$$

So, again the same problem. Even though we can have two different complex numbers, Bode plot might have more than one result if we follow these mathematical procedures. How to distinguish between both?

• Without telling us what you originally wrote (and then supposedly deleted), what's your actual question? May 19, 2014 at 7:43
• Your coordinate conversion formula is valid only if $a>0$. see answer below May 19, 2014 at 8:35

Any complex number $Z$ can be represented in rectangular coordinates or polar coordinates $$Z = x + jy = A\angle\phi$$ where, $A = \sqrt{x^2+y^2}$ and $\phi = atan2(y,x)$ where, $atan2(y,x)$ is defined as
As you can see your conversion formula is valid only if $x>0$.
Hence $\phi$ for $a+jb$ and $\phi$ for $-a-jb$ will be different.