# Input impedance butterworth filter with multiple feedback

I have a problem finding the input impedance of a Butterworth filter. The circuit is shown in the figure below:

I have calculated the transfer function between input and output; but now I have to find the symbolic expression of the input impedance seen from the $V_{\text{in}}$ generator. I've tried a $V_{\text{in}}/I_{\text{in}}$ approach. Since

$$I_{\text{in}} = \frac{(V_{\text{in}} - V_{\text{x}})}{R_1}$$

and $$V_{\text{x}} = -V_{\text{o}}(sC_2R_2)$$

I find that $$Z_{\text{in}} = \frac{R_1}{1+ W(s) sC_2R_2}$$ Where $W(s)$ is the transfer function $$W(s) = \frac{V_{\text{o}}}{V_{\text{in}}}$$ This sounds wrong to me, because the input impedance should decrease at high frequency, and in my case it's increasing. Where am I wrong? Thank for the precious help!

EDIT : I've checked it with SAPWIN, and it looks like the expression above is correct. In the Picture there is the 1/Zin function.

Thanks to all for your help in solving the question!

• You seem to have a capacitor in your path which tend to conduct better at higher frequencies – PlasmaHH Nov 1 '16 at 16:05
• You're right. I've written the exact contrary of what I meant! – FataMadrina Nov 1 '16 at 16:06
• Vx is not −Vo(sC2R2) – Sunnyskyguy EE75 Nov 1 '16 at 17:39
• As an alternative, you could inject a current of 1A and measure the input voltage. – LvW Nov 1 '16 at 20:10

From my visual analysis,

Zin(dc)=R1+R2//R3

for f>>f-3dB

@ f = infinity Zin = R1

simulate this circuit – Schematic created using CircuitLab

DC gain is -R3/R1 where Vx(f=0)=0

For visual aid on Vx see Attenuation, phase shift of xy for VI plot using Java with sweep. You can change any parameter

• I totally agree. But what I want is a symbolic expression, in Laplace domain, of the input impedance. My approach is input voltage / input current. But I'm missing something. – FataMadrina Nov 1 '16 at 16:16
• KCL and KVL will find the correct result – Sunnyskyguy EE75 Nov 1 '16 at 16:19
• Sure. But where? – FataMadrina Nov 1 '16 at 16:27
• KCL on all input nodes and use Vin+-Vin-=0 with zero differential impedance ( virtual "floating" gnd) – Sunnyskyguy EE75 Nov 1 '16 at 16:32
• SAPWIN is a symbolic analyzer which can calculate input and output impedances as well as the transfer function - all in symbolic form and (if you wish) as a function of frequency. – LvW Nov 1 '16 at 17:57

I would consider using superposition and yes, you have to factor in the transfer function because R3 (and the output) imposes a significant degree of complexity on the input impedance. First the easy stuff; you can forget C2, and R2 can be set in parallel with C1 because the op-amp has a virtual earth. So, it boils down to: -

Now you have two voltage sources and a few fixed value impedances so use superposition to calculate what the voltage at "X" is then you can calculate the current through R1 and then you have the input impedance.