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When you plug an electric guitar into an audio recorder, the raw signal is often just a few millivolts. If you amplify it upside-down, combining it with other audio tracks can cancel out the sound completely.
A non-inverting amplifier boosts the signal while keeping the peaks and troughs pointing in the exact same direction as the original wave.
How does the circuit set its exact amplification factor using just two resistors?

The Non-Inverting Gain Formula

In an ideal operational amplifier (op-amp), negative feedback forces the inverting terminal (βˆ’-) to match the non-inverting input (++), so Vβˆ’=VinV_- = V_{in}.
Because resistors RfR_f and R1R_1 form a voltage divider from VoutV_{out} to ground, Vβˆ’=Vout(R1R1+Rf)V_- = V_{out} \left(\frac{R_1}{R_1 + R_f}\right), which gives the closed-loop voltage gain (AvA_v):
Av=VoutVin=1+RfR1A_v = \frac{V_{out}}{V_{in}} = 1 + \frac{R_f}{R_1}
Notice that because both resistances are positive real numbers, the term RfR1\frac{R_f}{R_1} is always positive, meaning AvA_v is strictly greater than or equal to 11.
How do we calculate the precise output voltage when an AC or DC signal enters this circuit?

Calculating Gain and Output Voltage

To find the output voltage, multiply the input voltage by the gain: Vout=Avβ‹…VinV_{out} = A_v \cdot V_{in}. The output waveform is in-phase, meaning positive input swings produce positive output swings.
Suppose a circuit has R1=10Β kΞ©R_1 = 10\text{ k}\Omega, Rf=47Β kΞ©R_f = 47\text{ k}\Omega, and receives an input signal of Vin=0.40Β VV_{in} = 0.40\text{ V}.