[SOLVED] poles with negative frequency?

[wasserkasten]

Hello,

in textbooks (for example Sedra Smith, Microelectronic Circuits, Fifth Edition, page 334) one sometimes encounters a formula like:

Vo/Vi = ... * s / ( s + w )

where s is the Laplace variable (s = a + j*b) and w corresponds to the 3-db frequency of the high-pass filter.

Mathematically there should be a pole at "s = -w".
But the text describes it in the way:
"Readers will recognize f as the frequencies of the real pole of the amplifier."

where f = w/(2*pi)

Why doesn't the author mention the mathematically correct pole "-w" or at least "-f"?
(is it a convention to say "f" instead of "-w"?)

Thank you in advance.

 

[LvW]

Yes, for some applications the term "pole" is a bit misleading.
Writing a transfer function in terms of "s" you can find that the denominator vanishes for s=-w.
That means: The transfer function approaches infinity (it has a "pole") at a negative angular frequency s=sigma+jw=-w.
Thus: The pole frequency is wp=w=-sigma (negativ and real).
Hoever, this never can be measured because there are no negative frequencies in reality.
However, for some system theory purposes it is helpful to use the complex variable s leading to this result.

Now, you can ask: What can be measured at a frequency w=wp ?
You will see that the phase shift (between input and output) for this frequency is -45 deg and that the magnitude is 3 dB less than it's maximum value.
This is identical to the conditions which the definition of the 3dB cut-off frequency is based upon.
Therefore: The pole frequency w=wp for a first order RC circuit is identical to the cut-off frequency.
By the way: Fur 2nd-order Butterwoth filters the pole frequency also equals the 3dB cut-off.
For all other 2nd order responses the pole frequency is in the vicinity of the 3dB cut-off.

 

[wasserkasten]

Thank you for your quick reply.
So for transfer functions like
s / ( s - w )
and
s / ( s + w )

the pole frequency will always be mentioned as positive number
fp = wp / ( 2 * pi )
independent of the fact that the mathematically correct pole would be -wp or +wp ...(?)

 

[LvW]

Yes, in principle this is true, however please note that a transfer function like

s/(s-w)

cannot be realized since all coefficients must be positive.

 

[wasserkasten]

Thank you.