Electronic – How to calculate gain of two cascaded stages low pass filter (passive)

filterfrequency responsegainlow passphase shift

I built the following circuit and I would like to calculate the gain. I would like to predict the output voltage by knowing the input voltage. Also, I would like to predict the phase shift.

I got the transfer function from this question:Deriving 2nd order passive low pass filter cutoff frequency

I use this website to do math quickly instead of using my own calculator. The website provide the same transfer function of the previous question: http://sim.okawa-denshi.jp/en/CRCRtool.php

Here is how I use the website: I set the range of frequency from 999 to 1001 to get the gain and phase shift precisely at the frequency of 1000 Hz.

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My practical measurement of gain does not match the transfer function:

The transfer function says: The gain at 10 Hz would be -0.00125 dB that means the ratio between the output voltage and input voltage is 0.999 but mine is 0.5959 !!

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Vout and Vin are in volts.

Gain = Vout / Vin

dT is the phase shift in seconds.

Phase shift is in degrees.

Best Answer

The transfer function is: $$\small G(s)=\frac{1}{(RC)^2s^2+3RCs+1}$$

Hence \$\omega_n=\frac{1}{RC}=\small10^4\:rad/s\:(=1592\:Hz)\$, and \$\small\zeta=1.5\$, and it can be seen that the DC gain (\$\small s=0\$) is unity.

Converting this to the frequency domain, using \$ s\rightarrow j\omega\$:

$$\small G(j\omega)=\frac{1}{1-(\omega RC)^2+j3\omega RC}$$

At \$\small 10\:\small Hz\$, \$\small \omega RC=0.00628\$, hence the gain is almost unity and the phase angle is almost zero. At \$\small 1\:\small kHz\$, \$\small \omega RC=0.628\$, giving a gain of \$\small 0.505\$, and phase angle of \$\small \phi=-72^o\$.

So it seems that there's a problem with your experimental set-up. What's the input impedance of the instrument measuring Vout?

Let's do some detective work:

If the input impedance of the instrument were \$\small 3 \: k\Omega\$ resistive, then (i) the gain and phase at DC would be \$\small 0.6\$ and zero, respectively (i.e. same as your results); and (ii) the gain and phase at \$\small 1590 \:\small Hz\$ would be \$\small 0.29\$ and \$\small -79^o\$, which compares with your measurement of \$\small 0.31\$ and \$\small -73^o\$.