Operational Amplifier

A high-gain electronic voltage amplifier with a differential input and, usually, a single-ended output. An active circuit element. Voltage-controlled voltage source. Aka. op-amp.

• Has 2 inputs:
  • Inverting input (-)
  • Non-inverting input (+)
• Has 1 output
• 2 supply terminals (usually DC):
  • Positive
  • Negative
• Reference/ground

Gain

The ratio between the output and the difference between the input voltages.

$$A = \frac{V_0}{V_d}$$

Equivalent circuit

Here:

• $R_\text{in}$ - Input resistance, very high
• $R_\text{out}$ - Output resistance, very low
• $A$ - Gain, ranges from $10^5$ to $10^7$
• $V_{d}$ - voltage difference between input terminals
• $V_{in}$ - voltage input, ranges between $5V$ and $18V$

Ideal op-amp

• Infinite input impedance $R_\text{in} = \infty$
• Zero output impedance $R_\text{out} = 0$
• Infinite gain for differential input signal $A = \infty$ ($V_d = 0$)
• Infinite bandwidth $B=\infty$

Summing-point constraint

When the op-amp is negative feedback, the voltage at the inverting input terminal is equal to the voltage at the non-inverting input terminal.

Feedback types

Usually op-amp circuits are designed with feedback. A feedback is a connection from the output to the input of the op-amp.

Negative feedback

The output signal is fed back to the inverting input terminal through a resistor. Gain decreases.

Positive feedback

The output signal is fed back to the non-inverting input terminal through a resistor.

Uses

Can be used to perform a variety of operations on signals, such as:

• Amplification
• Addition
• Subtraction
• Multiplication
• Division
• Integration
• Differentiation

Types

Inverting amplifier

Inverts the input signal. Input signal is fed to inverting input through $R_1$. Non-inverting input is grounded. Negative feedback is used with $R_2$ resistor.

Image from Spiceman

$$A_v = \text{Closed-loop voltage gain} = \frac{v_0}{v_\text{in}} = -\frac{R_2}{R_1}$$

Non-inverting amplifier

Input signal is fed to non-inverting input terminal. Inverting input terminal is grounded through $R_1$ resistor. Negative feedback is used with $R_2$ resistor. Negative feedback is used with $R_2$ resistor.

$$A_v = \frac{v_0}{v_\text{in}} = 1 + \frac{R_2}{R_1}$$

Here:

• $A_v$ - Closed-loop voltage gain

Summing amplifier

An extension of the inverting amplifier. Multiple input signals are added together.

$$v_0 = -\left(\frac{R_f}{R_1}v_1 + \frac{R_f}{R_2}v_2 + \ldots + \frac{R_f}{R_n}v_n\right)$$

Differential amplifier

Used to amplify the difference between two input signals.

When $\frac{R_4}{R_3} = \frac{R_2}{R_1}$: $$

$$v_0 = \frac{R_2}{R_1}(v_1 - v_2)$$

Otherwise, the output must be derived.

Integrator

Input is fed to inverting input terminal through a resistor $R$. Non-inverting input terminal is grounded. Negative feedback through a capacitor. Includes a reset switch. Output is the integral of the input signal. $$

$$v_0 = -\frac{1}{RC}\int_0^t v_\text{in}\text{d}t$$

Differentiator

Input is fed to inverting input terminal through a capacitor $C$. Non-inverting input terminal is grounded. Negative feedback through a resistor $R$. Output is the derivative of the input signal.

$$v_0 = -RC\frac{\text{d}v_\text{in}}{\text{d}t}$$