Linear Op-Amp circuits are those where a linear relationship exists between the input and output signals. A key principle in analyzing these circuits is the virtual short concept, which states that the voltage at the inverting input terminal is equal to the voltage at the non-inverting input terminal.

3.1 Basic Amplifier Configurations

• Inverting Amplifier: Amplifies and inverts the input signal (180° phase shift). The gain is determined by external resistors.
  • Gain Formula: \frac{V_0}{V_i}= \frac{-R_f}{R_1}
• Non-Inverting Amplifier: Amplifies the input signal without phase inversion.
  • Gain Formula: \frac{V_0}{V_i}=1+\frac{R_f}{R_1}
• Voltage Follower: A special case of the non-inverting amplifier with a gain of 1. It produces an output that directly follows the input voltage.
  • Gain Formula: \frac{V_0}{V_i}= 1 (since V_0 = V_i)

3.2 Arithmetic Circuits

• Adder (Summing Amplifier): Produces an output that is the inverted sum of the input voltages applied at its inverting terminal.
  • Output Voltage (for 2 inputs): V_{0}=-R_{f}\left(\frac{V_1}{R_1}+\frac{V_2}{R_2}\right). If all resistors are equal, V_{0}=-(V_{1}+V_{2}).
• Subtractor (Difference Amplifier): Produces an output equal to the amplified difference between the two input voltages.
  • Output Voltage: V_{0}=V_{1}\left(\frac{R_3}{R_2+R_3}\right)\left(1+\frac{R_f}{R_1}\right)-\left(\frac{R_f}{R_1}\right)V_{2}. If all resistors are equal, V_{0}=V_{1}-V_{2}.

3.3 Mathematical Operation Circuits

• Differentiator: Produces an output equal to the first derivative of the input signal. The output is an inverted, scaled version of the input’s derivative.
  • Output Voltage: V_{0}=-RC\frac{\text{d}V_{i}}{\text{d}t}
• Integrator: Produces an output equal to the integral of the input signal. The output is an inverted, scaled version of the input’s integral.
  • Output Voltage: V_{0}=-\frac{1}{RC}\int V_{i}{\text{d}t}

3.4 Electrical Quantity Converters

• Voltage-to-Current (V-to-I) Converter: Produces an output current proportional to the input voltage. The gain of this circuit is its Transconductance.
  • Output Current: I_{0}=\frac{V_i}{R_1}
• Current-to-Voltage (I-to-V) Converter: Produces an output voltage proportional to the input current. The gain of this circuit is its Transresistance.
  • Output Voltage: V_{0}=-R_{f}I_{i}