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What happens when you need an electrical signal not only made bigger, but completely flipped upside down?
The inverting amplifier does exactly that by scaling an input voltage while reversing its polarity.

The Inverting Gain Formula

Voltage gain (AvA_v) is the ratio of output voltage to input voltage: Av=VoutVinA_v = \frac{V_{out}}{V_{in}}.
For an inverting op-amp circuit, negative feedback sets this ratio strictly through external resistors: Av=βˆ’RfRinA_v = -\frac{R_f}{R_{in}} where RfR_f is the feedback resistor and RinR_{in} is the input resistor.
In 1941, Karl Swartzel designed the first operational amplifier at Bell Labs using vacuum tubes to perform mathematical operations for artillery aiming systems.
How do we use this equation to predict the exact voltage coming out of the circuit?

Calculating Output and Input Voltages

Rearranging the gain formula gives the direct output equation: Vout=βˆ’(RfRin)VinV_{out} = -\left(\frac{R_f}{R_{in}}\right) V_{in}
Let's work through an example: suppose Rin=10Β kΞ©R_{in} = 10\text{ k}\Omega, Rf=47Β kΞ©R_f = 47\text{ k}\Omega, and Vin=+1.5Β VV_{in} = +1.5\text{ V}.
First, calculate the voltage gain: Av=βˆ’47Β kΞ©10Β kΞ©=βˆ’4.7A_v = -\frac{47\text{ k}\Omega}{10\text{ k}\Omega} = -4.7.
Then, multiply by VinV_{in} to get the output: Vout=βˆ’4.7Γ—(+1.5Β V)=βˆ’7.05Β VV_{out} = -4.7 \times (+1.5\text{ V}) = -7.05\text{ V}.