Operational Amplifiers

High-gain voltage amplifiers with differential inputs and single-ended output.

Definition

An operational amplifier (op-amp) is a DC-coupled high-gain electronic voltage amplifier with a differential input and usually a single-ended output. Op-amps are fundamental building blocks in analog circuits.

Ideal Op-Amp Rules

With negative feedback, an ideal op-amp obeys two rules:

  1. No current flows into the input pins — the input impedance is effectively infinite, so input current is diverted through the external feedback network.
  2. The output voltage adjusts to bring the input pins to the same voltage — the voltage difference between the inverting ($V_-$) and non-inverting ($V_+$) terminals is driven to zero. This is the origin of the virtual short (and virtual ground when $V_+$ is grounded).

Inverting Amplifier

In an inverting amplifier, the input signal is applied to the inverting input (-) through an input resistor $R_1$, while the non-inverting input (+) is connected to ground.

Virtual Ground

Because the non-inverting input is at 0 V and the ideal op-amp rules force both inputs to the same potential, the inverting input node is held at 0 V. This point is called virtual ground — it is not physically grounded, but sits at ground potential.

Gain Derivation

Since no current enters the op-amp input (Rule 1), the current through $R_1$ must equal the current through the feedback resistor $R_f$:

$$I = \frac{V_{in}}{R_1} = -\frac{V_{out}}{R_f}$$

Rearranging gives the inverting amplifier output:

$$V_{out} = -\frac{R_f}{R_1} V_{in}$$

The voltage gain is:

$$A_v = -\frac{R_f}{R_1}$$

The negative sign indicates a 180° phase inversion between input and output.

Example

Given $R_1 = 100 \text{ k}\Omega$ and $R_f = 500 \text{ k}\Omega$, find $V_{in}$ required for $V_{out} = -10 \text{ V}$:

$$-10 = -\frac{500}{100} V_{in} \implies V_{in} = 2 \text{ V}$$

Key Concepts

  • Ideal Op-Amp Properties — infinite gain, infinite input impedance, zero output impedance
  • Virtual Short — $V_+ = V_-$ due to infinite gain with negative feedback
  • Virtual Ground — inverting input held at 0 V when non-inverting input is grounded
  • Golden Rules — no current into inputs, inputs at same voltage (with feedback)
  • Negative Feedback — output fed back to inverting (-) input for stable linear operation
  • Positive Feedback — output fed back to non-inverting (+) input
  • Inverting Amplifier — input to (-), (+) grounded, negative gain
  • Non-inverting Amplifier — input to (+), feedback to (-), positive gain > 1
  • Voltage Follower — unity gain buffer, high input impedance
  • Summing Amplifier — multiple inputs added
  • Differential Amplifier — amplifies difference between inputs
  • Comparator — open-loop, digital output
  • Frequency Response — gain-bandwidth product

Key Formulas

Formula Description
$V_{out} = A(V_+ - V_-)$ Open-loop output
$V_{out} = -\frac{R_f}{R_1} V_{in}$ Inverting amplifier
$A = 1 + \frac{R_f}{R_1}$ Non-inverting gain
$V_{out} = -R_f\left(\frac{V_1}{R_1} + \frac{V_2}{R_2} + ...\right)$ Summing amplifier
$V_{out} = \frac{R_2}{R_1}(V_2 - V_1)$ Differential amplifier
$f_t = A_{CL} \cdot f_{CL}$ Gain-bandwidth product

Non-Inverting Amplifier

In a non-inverting amplifier, the input signal is applied to the non-inverting (+) input terminal. Negative feedback is provided from the output to the inverting (-) input through a feedback resistor $R_f$, while a resistor $R_1$ connects the inverting input to ground.

Because of the virtual short (ideal op-amp Rule 2), the voltage at the inverting (-) pin equals the voltage at the non-inverting (+) pin: $$V_- = V_+ = V_{in}$$

Since no current enters the op-amp inputs, the current through $R_1$ is: $$I = \frac{V_{in}}{R_1}$$

This same current flows through $R_f$. The output voltage is the sum of the voltage at the inverting pin and the voltage drop across $R_f$:

$$V_{out} = V_{in} + I \cdot R_f = V_{in} + V_{in}\frac{R_f}{R_1} = V_{in}\left(1 + \frac{R_f}{R_1}\right) = \left(\frac{R_1 + R_f}{R_1}\right)V_{in}$$

Characteristics:

  • Amplifies both DC and AC signals.
  • Output is not inverted — it remains in phase with the input.
  • Voltage gain is always greater than 1 (or equal to 1 in the buffer case).

Lecture Example (L38): For $V_{in} = 2\text{ V}$, $R_1 = 100\text{ kΩ}$, and $R_f = 500\text{ kΩ}$: $$V_{out} = \left(\frac{100\text{ kΩ} + 500\text{ kΩ}}{100\text{ kΩ}}\right)(2\text{ V}) = \mathbf{+12\text{ V}}$$

Voltage Follower (Unity-Gain Buffer)

A special case of the non-inverting amplifier occurs when no feedback resistor is used ($R_f = 0$) and the inverting input is connected directly to the output, with $R_1$ effectively open (infinite). The gain becomes:

$$A = 1 + \frac{R_f}{R_1} \approx 1$$

In this configuration, called a voltage follower or unity-gain buffer:

  • $V_{out} = V_{in}$ with a gain of exactly 1.
  • The output is not inverted.
  • The circuit provides very high input impedance and very low output impedance.
  • It is used to isolate stages, prevent loading effects, and drive heavy loads without drawing current from the source.

Related Concepts

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