# Poles with Increasing Phase

#### jegues

Joined Sep 13, 2010
733
I'm having trouble understanding the bode plot for,

$$G(s) = \frac{1}{2s-1}$$

Usually when I am drawing bode plots for either real poles or zeros, I simply solve for the break frequency knowing that a zero/pole will give me a +/-20dB/decade asymptote after the break frequency on my magnitude plot, and a +/-90° phase shift across two decades, with the center (i.e. the +/-45° point) at my break frequency on my phase plot.

This has always worked when I have,

$$(s+a) \quad a>0$$

but it seems to change when,

$$(s - a) \quad a>0 \quad \text{or, } \quad (-s+a) \quad a>0$$

I can't wrap my head around why a pole has an increasing phase that starts at -180°. I was expecting a phase decrease from 0° to -90°.

Can someone explain what I am misunderstanding?

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#### t_n_k

Joined Mar 6, 2009
5,455
I'm having trouble understanding the bode plot for,

$$G(s) = \frac{1}{2s-1}$$

Usually when I am drawing bode plots for either real poles or zeros, I simply solve for the break frequency knowing that a zero/pole will give me a +/-20dB/decade asymptote after the break frequency on my magnitude plot, and a +/-90° phase shift across two decades, with the center (i.e. the +/-45° point) at my break frequency on my phase plot.

This has always worked when I have,

$$(s+a) \quad a>0$$

but it seems to change when,

$$(s - a) \quad a>0 \quad \text{or, } \quad (-s+a) \quad a>0$$

I can't wrap my head around why a pole has an increasing phase that starts at -180°. I was expecting a phase decrease from 0° to -90°.

Can someone explain what I am misunderstanding?
Perhaps if you re-cast the transfer function in the complex frequency domain

$$G_{(j \omega)}=\frac{1}{(-1+j2\omega)}$$

Then rationalise the function ....

$$G_{(j \omega)}=-\frac{1}{(1+4 {\omega}^2)}-j \frac{2\omega}{(1+4{\omega}^2)}$$

What is the initial phase as ω approaches zero?

#### LvW

Joined Jun 13, 2013
1,187
jegues, do you realize that the pole is real and in the right half of the s-plane?
Thus, it belongs to an unstable non-realizable system.