Sorry for my ignorance in this field but I have failed miserably trying to figure the gain of this circuit:

^{simulate this circuit – Schematic created using CircuitLab}

Simulation shows that the input impedance of the first stage is 509.4k I do not understand why not 504.7. Where this second 4k7 comes from?

## Best Answer

If we make use of the equations for inverting amplifier, then we can see that \$V_{out}=-\frac{R_{f}}{R_{in}}V_{in}\$.

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

I will work in absolute, so amplification \$= |-\frac{R_{f}}{R_{in}}|=\frac{R_{f}}{R_{in}}\$

As @Transistor pointed out, OA1 IN- is a virtual earth, this means that \$R_3\$ is in parallel with \$R_2\$.

If we look back to the expression above and substitute we can understand that

Amplification for OA1, is \$0.0134≃-37.4\text{ dB}\$, the gain is less than 1.

Amplification for OA2 is \$1.44≃3.2\text{ dB}\$.

Since everything is cascaded, we can just multiply the amplifications, logarithmic addition is the same as linear multiplication.

Amplification from \$V_{in}\$ to \$V_{out}\$ = \$\frac{V_{out}}{V_{in}}=0.0134×1.44 = -37.4\text{ dB}+3.2\text{ dB}\$

That's \$-34.2\text{ dB}\$, not that good. Or \$0.019296\$. This means that if \$V_{in}=\frac{1}{0.019296}≃52\text{ V}\$, then you will read \$1\text{ V}\$ at the output.