What is the closed-loop gain of a non-inverting amplifier and how does it relate to the feedback resistor network?

What is the closed-loop gain of a non-inverting amplifier and how does it relate to the feedback resistor network?

• A. Gain = -Rf/Rin
• B. Gain = 1 - (Rf/Rin)
• C. Gain = 1 + (Rf/Rin)
• D. Gain = Rf/Rin

Answer: (C)

In a non-inverting amplifier, negative feedback through the resistor network sets the output so that the inverting input sits at the same voltage as the non-inverting input (a virtual short). The input signal appears at the non-inverting terminal, so the output must be scaled up by a factor that depends on the feedback network formed by the resistors. Let Vin be the voltage at the non-inverting input, and let Rin be the resistor from the inverting input to ground, with Rf as the feedback resistor from the output to the inverting input. Because the op-amp input current is ~0, the current through Rin equals the current through Rf. The node voltage at the inverting input is Vin, so the current through Rin is Vin/Rin, and the current through Rf is (Vout − Vin)/Rf. Setting these equal: (Vout − Vin)/Rf = Vin/Rin. Solve for Vout: Vout = Vin + (Rf/Rin)·Vin = Vin[1 + (Rf/Rin)]. Thus the closed-loop gain is 1 + (Rf/Rin). This is a positive gain greater than 1 (assuming Rf and Rin are positive). The other forms correspond to different configurations (for example, an inverting amplifier yields −Rf/Rin), which is why the correct expression here is the one that adds 1 to the ratio.

In a non-inverting amplifier, negative feedback through the resistor network sets the output so that the inverting input sits at the same voltage as the non-inverting input (a virtual short). The input signal appears at the non-inverting terminal, so the output must be scaled up by a factor that depends on the feedback network formed by the resistors.

Let Vin be the voltage at the non-inverting input, and let Rin be the resistor from the inverting input to ground, with Rf as the feedback resistor from the output to the inverting input. Because the op-amp input current is ~0, the current through Rin equals the current through Rf. The node voltage at the inverting input is Vin, so the current through Rin is Vin/Rin, and the current through Rf is (Vout − Vin)/Rf. Setting these equal: (Vout − Vin)/Rf = Vin/Rin. Solve for Vout: Vout = Vin + (Rf/Rin)·Vin = Vin[1 + (Rf/Rin)].

Thus the closed-loop gain is 1 + (Rf/Rin). This is a positive gain greater than 1 (assuming Rf and Rin are positive). The other forms correspond to different configurations (for example, an inverting amplifier yields −Rf/Rin), which is why the correct expression here is the one that adds 1 to the ratio.