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>> This configuration is known as a summing amplifier.

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We are going to find that it takes

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its name from the fact that the output voltage is going to

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equal a scaled version of the sum of V_1 and V_2.

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To analyze this, we're going to do as we always have and that is write

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a node equation at the inverting terminal.

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Once again, we have the virtual short here

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that tells us that V sub n is equal to V sub p,

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and again in this case, the non-inverting terminal is tied to ground.

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So, V sub n is going to equal zero.

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But again, we'll leave V sub n in there to see what's happening,

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and then as we move along we'll replace V sub n with zero.

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So, let's sum the currents here at this terminal.

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Notice that we now have three branches one, two,

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three branches connected to the node in

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addition to this branch here coming to the inverting terminal,

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which is still once again going to have no current going in there.

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So, let's write the equations here,

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or the equation here.

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The current leaving V sub n going in this path is going to

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be V sub n minus V_1 divided by R_1,

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plus the current leaving this node going in

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this direction is going to be V sub n minus V_2,

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divided by R_2, plus the current in the feedback loop,

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which is going to be V sub n minus V-out divided by R sub f. As already mentioned,

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there is no current going into the amplifier itself,

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therefore the sum of those three terms equals zero.

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Again, noting that V sub n equals zero in each of these,

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we have then negative V_1 over R_1 plus negative V_2

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over R_2 equals V-out over R sub f. Now,

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if we multiply both sides of the equation by R sub f and factor out,

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this minus sign is there,

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we're going to have them that V-out is equal to

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negative R sub f over R_1 times V_1.

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Let's put a bracket around here,

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plus R sub f,

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over R_2 times V_2.

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That's supposed to be a plus sign there.

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As advertised, the output then is the sum of

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the two input voltages scaled by the ratios of the feedback resistance to either R_1,

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in the case of V_1, or the feedback resistor divided by R_2

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for the second input voltage.

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Now, it'll be pretty obvious that if we want the scaling factor to be the same,

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we simply make R_1 equal R_2.

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Under those circumstances, the output voltage then would be negative R sub f over,

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call it R_1 or call it R_2, I guess so.

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Let's just be specific and let's say R_1 equals R_2,

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equals R sub s. So,

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our gain term then would be R sub f over R sub s times V_1 plus V_2.

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So, we have a minus sign again that is

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an inverted output due to the fact that

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these two sources are connected to the inverting terminal.

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With R_1 equaling R_2,

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both are amplified by the same amount,

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and we end up with a scaled version of the sum of those two.

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If we don't want any gain out of it,

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we simply make R sub f equal R_1 equal to R_2,

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and then the output voltage would be just V_1 plus V_2.