filterhomeworkoperational-amplifierpassive-filter
Consider the following figure and questions.
Using KCL, I found vsum(t) = −v1(t) − v2(t). The transfer functions are TLPF(jw) = vo1(t)/vsum(t) and THPF(jw) = vo2(t)/vsum(t). However, I am stuck while finding vo1(t) and vo2(t). I am not sure how to apply KCL at the common filter node (since the output current of the amplifier is unknown) and I don't think I can apply KVL to the right LPF + HPF loop since it's open (or is it? What do the ground symbols indicate?)
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.
I can't really speak for signal flow graphs, simply because I don't use them. I've always seem them as useful for planning an algorithm in assembly, but for non discrete time signals, they're more confusing than helpful.
It is possible to extract transfer functions directly from a Sallen-Key circuit, but you need to know what you're looking for, and what the basic transfer functions look like. For example:
Here we have an assumed Sallen-Key second order low pass, where \$R=R_1=R_2,\ C=C_1=C_2\$. The generic transfer function is known, as well as \$\alpha=3-\dfrac{R_4}{R_3}\$, and \$\omega_c=\dfrac{1}{RC}\$. From there, it can be reconstructed.
Best Answer
Alright. I'll open the gate to the path. And you're alone from this point on.
You can think of \$V_{sum}(t)\$ as a voltage source, because the summing amplifier has zero output impedance (in theory). So you can draw the two loops:
simulate this circuit – Schematic created using CircuitLab
Now you can put impedances and analyze the circuit. Note that the output voltages can be calculated from either KVL or KCL.