Derive the transfer function for an inverting op-amp with feedback resistor Rf and input resistor Rin.

• A. V_out = -(R_f/R_in) V_in
• B. V_out = (R_in/R_f) V_in
• C. V_out = V_in (1 + R_f/R_in)
• D. V_out = -V_in (R_in/R_f)

Answer: (A)

In the inverting op-amp, the non-inverting input is grounded, so the inverting input sits at a virtual ground (0 V). That means the node at the inverting input is held at zero volts, and current must flow from the input through Rin into that node and then through the feedback path to the output.

The current through the input resistor is Iin = Vin / Rin. The current through the feedback resistor, from the node toward the output, is Iout = (V- − Vout) / Rf = (0 − Vout) / Rf = −Vout / Rf. Applying Kirchhoff’s current law at the inverting node (sum of currents leaving the node equals zero, or equivalently the current through Rin must equal the current through Rf with opposite direction) gives Vin/Rin − Vout/Rf = 0. Solving for Vout yields Vout = −(Rf/Rin) · Vin.

This transfer function shows a linear, inverted relationship between input and output, with the gain magnitude set by the ratio of the feedback resistor to the input resistor. The negative sign reflects the inversion, and the result relies on the ideal-op-amp assumptions (infinite gain, zero input current, and operation in the linear region).