Can you find the transfer function of two cascaded networks if you know their individual transfer functions?

Question

If you have network N1 with transfer function H1 and network N2 with transfer function H2, is there a way to find the transfer function H3 for the network produced when cascading N1 and N2 (and assuming you cant see inside N1 or N2)?

At first I thought you could just multiply H1 and H2, but that doesn't seem to give me a right answer.
If we consider a simple LPF:

The transfer function is: \$ H(s) = \left(\dfrac{1}{sC_{1}R_{1} + 1}\right) \$

Then if we cascade it with another LPF:

The transfer function is: \$ H(s) = \left(\dfrac{1}{s^2C_{1}R_{1}C_{2}R_{2} + sC_{1}R_{1} + sC_{2}R_{2} + sC_{2}R_{1} +1}\right) \$

which is not the same as \$ \left(\dfrac{1}{sC_{1}R_{1} + 1}\right) \$\$ \left(\dfrac{1}{sC_{2}R_{2} + 1}\right) \$

Is there a shortcut to get to H3 from H1 and H2 or would I just have to calculate H3 the long way using KCL or KVL? What about if I know \$R_1=R_2\$ and \$C_1=C_2\$?

Answer 1

In general you can't simply multiply the transfer functions together since the first section has a non-zero ouput impedance and the second section a finite input impedance. A transfer function of a network is only valid with it's output open-circuit.

Consider for example a section consisting of a single series resistor and a second section consisting of a single shunt resistor. Both have a transfer function of 1, but cascade them together and you have a potential divider. If you put a unity-gain buffer (with infinite input impedance and zero output impedance) between the sections then you could just multiply the transfer functions.

So knowing the individual sections' transfer functions is insufficient information to deduce the overall transfer function.