How to compute gain margin for this simple example?

Question

I have the simple transfer function: $$\ L(s) = \frac{3}{s+1}$$

I am stuck computing the gain margin analytically. In Matlab, it tells me that the gain margin is infinite. This is how I am computing by hand:

1. Get $w_{pc}$ by computing L(jw) and making $arg(L(jw_{pc})) = -\pi$
2. Plug this in to $\frac1{|L(jw_{pc})|}$

Doing this, I get $g_m=\frac13$

Any ideas?

Answer 1

You have

$$ L(j \omega) = \frac{3}{j \omega + 1} = a + j b $$

with

$$ \begin{align} a &= \frac{3}{\omega^2 + 1} \\ b &= -\frac{3 \omega}{\omega^2 + 1} \end{align} $$

As you can see, the real part of this complex number is always positive. If this is the case, you can compute the argument of the complex number by

$$ \arg(a + j b) = \arctan(b/a) = -\arctan(\omega) $$

So, you end up with the equation

$$ \arctan(\omega) = \pi $$

However, this equation has no real solution because $\arctan : \mathbb{R} \mapsto (-\frac{\pi}{2}, \frac{\pi}{2})$ never reaches $\pi$.