Why is an op-amp used in this summing circuit? Why can't I add the voltages without an op-amp? Current doesn't go into op-amp, so it goes through the Rf resistor either way.
Answer
The concept of the "op-amp virtual earth" is very important to understanding why the op-amp is used as a mixing/adding circuit: -
I've stolen the OP's picture and marked in red where the virtual earth is. Anything directly connected to the -input is called "virtual earth". OK that doesn't explain why so, here goes... If the voltage at the +input is 0V (aka earth or ground) then the voltage at the -input HAS to also be 0V.
This may seem a strange thing to say until you think about the general model of an op-amp - it has infinite gain (or at least very high gain) and, if the voltage difference between the two inputs were measurable, then the OP-AMP output would be end-stopped against one of the power rails.
But, this doesn't happen because of the process of negative feedback - the OP-AMP produces a voltage on its output that is just right (goldilocks value) for making the -input voltage (via Rf) exactly the same as +input. This is called negative feedback.
So how does this help the mixer - it means that all the inputs (V1 to V4) are connected to resistors that appear to go straight to 0V - this means the current thru each input resistor is NOT dependent on the other inputs and their currents - they all appear to go to ground or earthy or 0V.
This means it is a true mixer.
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