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

Question

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.

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 !!

Vout and Vin are in volts.

Gain = Vout / Vin

dT is the phase shift in seconds.

Phase shift is in degrees.

Answer 1

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\$.

Answer 2

Signals and Systems is the topic you're looking for... it's a big field of study. If I recall correctly, a lot depends on being able to prove a system is "linear" and "time-invariant", but any system with an input and an output can be fully characterized by a "transfer function" - which is a function that varies with frequency of the input signal, and is essentially V_out(frequency) / V_in(frequency) and it's usually called 'H'.

If you have to systems and you chain them together, and system 1 has transfer function H1, and system 2 has transfer function H2, then the composed system has transfer function H1 * H2. In the time domain, systems are characterized by an analogous function called the "impulse response", usually called 'h(t)' and the composition operation is a nasty mathematical construct called "convolution."

This stuff is all wrapped up in Fourier and Laplace transforms, and I'm about out of steam remembering the nuts and bolts of it all. But that's the gist.