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For some active filter circuit I have derived the following transfer function: $$T(s) = \frac12\cdot{s^2+{1\over2R_1^2C_1^2}\over s^2+{1\over R_1C_1}s+{1\over2R_1^2C_1^2}}$$

I need to show that this circuit implements a notch filter, for which the transfer function is given in the following form: $$T(s)_\text{NF}={s^2+\omega_0^2\over s^2+{\omega_0\over Q}s+\omega_0^2}$$

It is easy to see that my transfer functions almost corresponds to the given form except for the constant factor of \$\frac12\$ before the fraction. Should I be bothered by this factor, i.e, does it influence the derived TF to be not of a notch filter TF form?

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    \$\begingroup\$ As you can see, the first function approaches a value of 1/2 for w=0 (at DC) - where as the 2nd function is unity at w=0. That is the only difference - a constant scaling factor. \$\endgroup\$ – LvW Dec 6 '14 at 16:27
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If \$\omega_0\$ equals \$\dfrac{1}{RC\sqrt2}\$ then what's the problem?

This would make Q = \$\dfrac{1}{\sqrt2}\$.

It all sounds very reasonable to me. DC transfer function will be 0.5 as indicated by @LvW.

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  • \$\begingroup\$ Indeed, it all sounds reasonable, I indicated it answering to LvW's comment. Still, thanks for the answer. \$\endgroup\$ – Dmitry Kazakov Dec 6 '14 at 17:07
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I think that the constant 1/2 factor doesn't matter as much as the first order term at the denominator, \$\dfrac{1}{R_1C_1}s\$. Depending on the requirements, that might violate them.

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