Operational Amplifier Applications

Voltage Follower (Buffer)

Translates a voltage in a circuit ( $V_{in}$ ) without drawing any current, thus the power in the original circuit is unaffected

They are used as isolation buffers in circuits

v_{o} = - \frac{R _{f}}{R _{s}} v_{s}

Closed-loop gain: $G = v_{o} / v_{s}$

v_{o} = \frac{R _{s} + R _{f}}{R _{s}} v_{g}

v_{o} = \frac{R _{d} ( R _{a} + R _{b} )}{R _{a} ( R _{c} + R _{d} )} v_{b} - \frac{R _{b}}{R _{a}} v_{a}

Special case when $\frac{R _{a}}{R _{b}} = \frac{R _{c}}{R _{d}}$

v_{o} = \frac{R _{b}}{R _{a}} (v_{b} - v_{a})

v_{o} = - (\frac{R _{f}}{R _{a}} v_{a} + \frac{R _{f}}{R _{b}} v_{b} + \frac{R _{f}}{R _{c}} v_{c})

Special case if $R_{a} = R_{b} = R_{c} = R_{d}$

v_{o} = - \frac{R _{f}}{R _{s}} (v_{a} + v_{b} + v_{c})

v_{o} (t) = \frac{- 1}{R _{s} C _{f}} \int_{t_{o}}^{t} v_{s} d y + v_{o} (t_{o})

\frac{d ^{2} v _{o}}{d t ^{2}} = \frac{1}{R _{1} C _{1}} \frac{1}{R _{2} C _{2}} v_{g}

With Feedback:

\frac{d ^{2} v _{o}}{d t ^{2}} + (\frac{1}{τ _{1}} + \frac{1}{τ _{2}}) \frac{d v _{o}}{d t} + (\frac{1}{τ _{1} τ _{2}}) v_{o} = \frac{v _{g}}{R _{a} C _{1} R _{b} C _{2}}