Operational amplifiers: common configurations

Voltage follower is a versatile configuration to be used when one has to decouple impedance without affecting the voltage at a certain circuital node. It can be easily designed with just one operational amplifier and a few components (not shown in the picture). $$V_o = V_{in}$$ $$Z_o = 0Ohm$$ $$Z_{in} = \infty Ohm$$

The output voltage is amplified and negated with respect to the input voltage; if the component is supplied with positive rail only, the output will be clamped to 0V or so. $$V_o = -\frac{R_f}{R_n}V_{in}$$ $$Z_o = 0Ohm$$ $$Z_{in} = R_n$$

The output voltage is amplified by one plus the ratio between the feedback and the series resistor. If the gain is too high, the output will be clamped to rail voltage or less if the chosen component has some heavy nonidealities. $$V_o = \left( 1+\frac{R_f}{R_n} \right) V_{in}$$ $$Z_o = 0Ohm$$ $$Z_{in} = \infty Ohm$$

With this configuration, one can get an output voltage proportional to an input current: the multiplying factor is given by the feedback resistor. $$V_o = -R_f I_{in}$$ $$Z_o = 0Ohm$$ $$Z_{in} = \frac{V_{in}}{I_{in}}$$

The summing amplifier is a generic sum-subtraction circuit that can be analysed using superimposition effects law. Input impedance is not infinite but changes for each input at both + and - terminals. In the following expressions, R_x,p∼i means that the equivalent resistor must be calculated as the parallel of all 'x' resistors except for the resistor x,i; R_x,p is instead the parallel of all 'x' resistors. $$R_{n,p} = R_{n,1}//R_{n,2}//R_{n,3}//...//R_{n,n}$$ $$R_{p,p\tilde i} = R_{p,1}//R_{p,2}//R_{p,3}//...//R_{p,i-1}//R_{p,i+1}//...//R_{p,n}$$ $$\displaylines{V_o = \left(1+\frac{R_f}{R_{n,p}}\right) \left(V_{p,1}\frac{R_{p,p\tilde 1}}{R_{p,p\tilde 1} + R_{p,1}} + V_{p,2}\frac{R_{p,p\tilde 2}}{R_{p,p\tilde 2} + R_{p,2}} + ... + V_{p,n}\frac{R_{p,p\tilde n}}{R_{p,p\tilde n} + R_{p,n}}\right)+\\+ V_{n,1}\left(-\frac{R_f}{R_{n,1}}\right) + V_{n,2}\left(-\frac{R_f}{R_{n,2}}\right) + ... + V_{n,n}\left(-\frac{R_f}{R_{n,n}}\right)}$$ $$Z_o = 0Ohm$$ $$Z_{in,n,i} = R_{n,i}$$ $$Z_{in,p,i} = R_{p,i} + R_{p,p\tilde i}$$

This is a particular case of the generic summing amplifier: this configuration takes two analogue input signals and generates an output voltage, the difference between the two. A purely differential amplifier can be obtained by imposing $$R_f = R_{pl} = R_n = R_{ph}$$ The main properties of this configuration are shown below $$V_o = V_p \frac{R_{pl}}{R_{pl}+R_{ph}}\left( 1+\frac{R_f}{R_n} \right) - V_n \frac{R_f}{R_n} = V_p - V_n$$ $$Z_o = 0Ohm$$ $$Z_{in,n} = R_n$$ $$Z_{in,p} = R_{pl} + R_{ph}$$

Instrumentation amplifier is, ideally, identical to a normal operational amplifier, but it removes most of its nonidealities. It is made of three amplifiers connected as shown in the picture: as you can see, the two inputs go in two separate voltage follower stages, providing thus very high input impedance at both terminals, very low output impedance, very high gain, good rejection to common mode noise (CMRR very high). This configuration is preferred in systems where high gain in noisy environments is needed. $$V_o = V_p \left( 1+\frac{R_3}{R_2} \right) \frac{R_{pl}}{R_{ph} + R_{pl}} \left( 1+\frac{R_f}{R_n} \right) -V_n \left( 1+\frac{R_1}{R_2} \right) \frac{R_f}{R_n}$$ $$Z_o = 0Ohm$$ $$Z_{in} = \infty Ohm$$