I am trying to find the -3dB freq. denoted \$f_H\$ in the following circuit, where \$R\$ has been replaced with a capacitor \$C\$. I tried doing that in two ways which yielded different results. I'd appreciate some feedback on which of the two is correct:

Applying Miller's theorem:

\$G_v=(A/sC)/(1/sC + R_{sig}(1-A))\$

First method:

\$f_H=1/(2\pi R_{sig}C(1-A)) = 1.58KHz\$

Second method:

I tried to determine the frequency at which the voltage gain decreases by a factor of \$\sqrt2\$, hence:

\$(A/sC)/(1/sC + R_{sig}(1-A)) = A/\sqrt 2\$

Substituting \$s=j\omega\$, yielded \$f_H=4.1KHz\$enter image description here

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    \$\begingroup\$ Your text talks about an opamp yet your schematic shows an amplifier (not an opamp). Is is unclear what the sign of A is which is quite essential as it changes the functionality of the circuit. \$\endgroup\$ – Bimpelrekkie Jun 7 '17 at 14:04
  • \$\begingroup\$ @Bimpelrekkie Sorry about that.. A is negative. \$\endgroup\$ – peripatein Jun 7 '17 at 17:51
  • \$\begingroup\$ @peripatein 3dB frequency is equal to \$f_H= \frac{1}{2\pi R_{sig}C(1+|A|)}\$ \$\endgroup\$ – G36 Jun 7 '17 at 18:54
  • \$\begingroup\$ @G36 Great, thanks! How may I now find the frequency of the unity gain? Must I first determine the BW? \$\endgroup\$ – peripatein Jun 7 '17 at 19:01
  • \$\begingroup\$ For the ideal integrator with op-amp the unity gain frequency is equal to \$F = \frac{1}{2 \pi R_{sig}C}\$. And your gain \$A\$ is frequency dependent or not ? \$\endgroup\$ – G36 Jun 7 '17 at 20:15

To be able to find the unity-gain frequency you can use this trick that will work for this practical circuit.


simulate this circuit – Schematic created using CircuitLab

Hence \$V_X = \frac{\frac{1}{s C1 (1+A)}}{R1+\frac{1}{s C1 (1+A)}}\$

And because we are interested only in magnitude we can write:

$$V_X = \frac{1}{\sqrt{1+(\frac{F}{F_H})^2}}$$

Where the \$F\$ - is a signal frequency and \$F_H\$ - is -3dB frequency.

And finally we have:

$$\frac{1}{\sqrt{1+(\frac{F}{F_H})^2}} * |A| = 1 $$

All you need is to solve for \$F\$

Hence the unity-gain frequency is

$$F_T = \frac{1}{2 \pi R_1 C_1} * \frac{\sqrt{A^2 -1}}{|A|+1}$$

Of-course the unity-gain frequency can also be find from transfer function directly.

  • \$\begingroup\$ how may I arrive at the same result using the transfer function? \$\endgroup\$ – peripatein Jun 10 '17 at 14:15
  • \$\begingroup\$ First you need do find the transfer function \$\endgroup\$ – G36 Jun 10 '17 at 14:36

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