The topic of this problem is operational amplifiers. And we're going to work with a circuit that is a summing amplifier. The problem is to find V out in terms of the input voltages. So if we look at our circuit that we have we can see that we have four input voltages, v sub 1 through v sub 4. We also have these resistors that are associated with the inputs as well, they're all given the same value of R. And we have an output voltage which is dropped across an output resistance, R sub L. We have a feedback resistor, which ties the output back to the input of the op amp, so R sub F and we have the op amp itself. So when we're solving these circuits with op amps in them, the first thing that we want to know Is what the general properties of an op-amp circuit are, and of an op-amp device are. First of all, we know that we have a current associated both the inverting input and a current associated with the non-inverting input of the op amp. We also have voltages associated with those as well. V+ and V-. We have an output and typically the symbol also includes a ground. So, the properties of the op amp which are important for linear circuit analysis are that. The input currents i minus and i plus for the inverting and non inverting are equal and they're 0. The other property that we need to utilize is that the voltage at the non inverting input Is equal to the voltage at the inverting input. So, we use these two properties of op amps and linear circuit analysis in order to be able to break our circuit down. Into something that we can solve using our tools that we have already learned. Those are things like nodal analysis, mesh analysis, voltage division, current division, super position and those type of analysis techniques. So let's see how we can do that in this circuit. So the first thing that we know is that the point at the ground gives us a zero volt reference in our circuit. And so if it's zero volts at this point in our circuit, it's also zero volts at this point in our circuit, because they are tied to one another. So we're at zero volts at the non-inverting input of our op-amp. We know that the voltages are equal at the inverting and non-inverting input of the op-amp. So that tells us that the voltage at this point in our circuit is also 0 volts. So we've used the property of the op-amp to find the voltage up at the node that ties R sub f with the inputs to the op amp circuit. So what we'll do now is we'll take let's just call this perhaps node 1. We're going to take the sum of the currents at node 1, using Kirchhoff's current law to come up with an equation that relates the input voltages to the output voltage. So we have our currents assigned for each one of the inputs. Let's sum the currents into node 1. And so for summing currents into node 1 then it would be topmost path, using the source V sub 1 the current's going to be V1 minus the voltage at this point, which is 0 Volts. Divided by R so it's V1 minus 0 over R for the top current. The second current associated with the source V sub 2 is going to be V sub 2 minus 0 over R because the voltage here is V2. Voltage at this point is 0 divided by R. Similarly we can write the expressions for the source V3 and the source V4, that's all the currents from those paths. The other currents that are flowing into node 1, or the current through the feedback path, which is going to be V out minus zero over Rf. And the current which is flowing directly out of the op app inverting terminal, and we know that current is equal to zero. Sum of all currents is equal to 0. So what we end up with is an equation which relates V out with all of the input voltages. If we rearrange this equation and solve for V out, then we can easily see that V out is going to be equal to minus R sub F, such that we're taking the R sub F and move it to the other side. Divided by R, since they all have the same resistance values at the input, times V1 plus V2 plus V3 plus V4. And so the output voltage is equal to the Inverted and amplified some of the input voltages. So this is a summing inverting op amp configuration, which relates to the input voltages to the output voltages, by the amplification factor that we can control in our circuit design.
