Non Inverting Summing Amplifier, general case |

We start by finding the equations for $V_{+}$ and $V_{-}$:

\[V_{-} = V_{out} \frac{R_g}{R_g +R_f}\]

The currents to the $V_{+}$ node have the same subscripts as the voltages and resistances. We now have a bunch of equations:

\[V_{1} - V_{+} = I_{1}R_{1}\\

V_{2} - V_{+} = I_{2}R_{2} \]

and so on to:

\[V_{n} - V_{+} = I_{n}R_{n} \]

We also have

\[I_{1} + I_{2} +\cdots + I_{n} = 0 \]

So by arranging the equations in this form:

\[\frac{V_{n} - V_{+}}{R_{n}} = I_{n} \]

we get:

\[\frac{V_{1} - V_{+}}{R_{1}}+\frac{V_{2} - V_{+}}{R_{2}}+\cdots +\frac{V_{n} - V_{+}}{R_{n}} = 0 \]which can be simplified to:

\[V_{+}=\frac{V_{1}R_{1}R_{2}\cdots R_{n}/R_1+V_{2}R_{1}R_{2}\cdots R_{n}/R_2+\cdots +V_{n}R_{1}R_{2}\cdots R_{n}/R_n }{R_{1}R_{2}\cdots R_{n}/R_1+R_{1}R_{2}\cdots R_{n}/R_2+R_{1}R_{2}\cdots R_{n}/R_n} \]

(Hope I got that notation right...)

Using the golden op amp rules we now set $V_{+}= V_{-}$ and get:

\[V_{out} \frac{R_g}{R_g +R_f}=\frac{V_{1}R_{1}R_{2}\cdots R_{n}/R_1+V_{2}R_{1}R_{2}\cdots R_{n}/R_2+\cdots +V_{n}R_{1}R_{2}\cdots R_{n}/R_n }{R_{1}R_{2}\cdots R_{n}/R_1+R_{1}R_{2}\cdots R_{n}/R_2+R_{1}R_{2}\cdots R_{n}/R_n} \Rightarrow \\

V_{out} =\frac{R_g+R_f}{R_g } \frac{V_{1}R_{1}R_{2}\cdots R_{n}/R_1+V_{2}R_{1}R_{2}\cdots R_{n}/R_2+\cdots +V_{n}R_{1}R_{2}\cdots R_{n}/R_n }{R_{1}R_{2}\cdots R_{n}/R_1+R_{1}R_{2}\cdots R_{n}/R_2+R_{1}R_{2}\cdots R_{n}/R_n} \]

If we want to, we can simplify this drastically by setting one or both of these equations:

\[R_{1} = R_{2} = \ \cdots = R_{n} = R\\

R_{g} = R_{f} \]

Will give us:

\[ V_{out} = 2 \frac{V_{1}+V_{2}+\cdots +V_{n} }{n} \]

To get a unity gain summer we can set:

\[\frac{R_g+R_f}{nR_g } = 1 \Rightarrow \\

R_f = Rg(n-1) \]

That's it. By the way, the schematics I made using xcircuit. It was my first circuit using this tool, but it seems nice enough to try it some more...

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