Answer to Question #199858 in Electrical Engineering for richard gate

Question

i. A computer manufacturing company has approached you to assist it to design a summer circuit for its new type of computers. You are provided with resistors R1, R2, R3, R4 and a feedback resistor Rf. Design and draw the summer circuit

ii. Derive the formula for the output voltage Vo of the summer

Answer 1

Let us consider the diagram below.

Op-amps excel at delivering a wide range of helpful and creative solutions to regular or complex circuit operations. An excellent example is a summing circuit. A summing circuit is commonly employed when many inputs or channels must be summed or subtracted to generate a composite output. This circuit is ideal for combining numerous audio channels.

Negative Feedback is the process of "feeding back" a percentage of the output signal to the input. Still, to make the feedback negative, we must send it back to the op-negative amp's or "inverting input" terminal through an external Feedback Resistor named R. The feedback link between the output and the inverting input terminal drives the differential input voltage to zero.

Because this effect creates a closed-loop circuit in the amplifier, the gain of the amplifier is now referred to as its Closed-loop Gain. Then, a closed-loop inverting amplifier employs negative feedback to precisely adjust the amplifier's overall gain, but at the expense of lowering the amplifier's gain.

As a result of the negative feedback, the inverting input terminal has a different signal than the actual input voltage since it is the total of the input voltage plus the negative feedback voltage, earning it the label or term of a Summing Point. As a result, we must use an Input Resistor, Rin, to isolate the true input signal from the inverted input.

$i= \frac{V_{in}-V_{out}}{R_{in}-R_{f}} \implies i= \frac{V_{in}-V_{2}}{R_{in}}= \frac{V_{2}-V_{out}}{R_{f}}$

$i= \frac{V_{in}}{R_{in}}-\frac{V_{2}}{R_{in}}=\frac{V_{2}}{R_{f}}-\frac{V_{out}}{R_{f}}$

So $\frac{V_{in}}{R_{in}}= V_2[\frac{1}{R_{in}}+\frac{1}{R_{f}}]-\frac{V_{out}}{R_{f}}$

$i=\frac{V_{in}-0}{R_{in}}=\frac{0-V_{out}}{R_{f}} ; \frac{R_{f}}{R_{in}}=\frac{0-V_{out}}{V_{in}-0}$

The closed loop gain $\frac{V_{out}}{V_{in}}=-\frac{R_{f}}{R_{in}}$

Then, the Closed-Loop Voltage Gain of an Inverting Amplifier is given as.

$Gain(Av)=\frac{V_{out}}{V_{in}}=-\frac{R_{f}}{R_{in}}$

and this can be transposed to give Vout as:

$V_{out}=-\frac{R_{f}}{R_{in}} \times V_{in}$