# Low-Pass and High-Pass Filters Outputs of Summing Amplifier

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?)

• Ground symbols indicate the common (reference) points. Also, you should consider the impedances of the reactive elements (e.g. $Z_C=-j/(\omega \ C)$ ) and apply KVL/KCL in the loops (source/input is $V_{sum}(t)$ with zero impedance and output is $V_{o1}(t)$ for the top and $V_{o2}(t)$ for the bottom loop. Simple voltage divider). Commented Nov 29, 2017 at 5:45
• So I can apply KVL normally even though the loops are open (since current flows through the ground)? And KCL is still not possible since the output current of the op-amp is unknown? Commented Nov 29, 2017 at 11:09
• Who says the loops are open? Connect the ground symbols together and you'll see the loops are closed. KCL is possible because each loop is the load of $V_{sum}$. Commented Nov 29, 2017 at 11:23
• Thanks anyway! This helped me understand grounds: electronics.stackexchange.com/questions/148675/…. Commented Nov 29, 2017 at 12:30

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: