# Differential Amplifiers

Op-amps are differential amplifiers, designed to amplify the difference between two inputs. In an op-amp, the gain is usually very large, approaching in practice, and is modelled as infinite.

Op-amps are based on a circuit known as a long-tailed pair: • and are a matched pair of transistors, meaning they have the exact same electrical properties (
• The quiescent current is the current through the shared emitter resistor,
• If is large, then

## Biasing

When doing bias calculations, the two inputs and are assumed to be grounded, . As , . , the tail voltage, is always taken as negative. Using this, we can calculate :

sets the quiscent collector current.

When the two inputs are grounded, the output at the collectors and are the same. For matched transistors, .

## AC Analysis

The long-tailed pair can operate in two modes, depending upon how input is applied

• Differential mode amplifies the difference between the two input signals
• Common mode works similar to a regular BJT amplifier
• Better amplifiers have a high ratio of differential to common gain, called the Common Mode Rejection Ratio (CMRR)

### Differential Mode

The circuit below shows two AC sources connected, and , to give a differential input signal of . The differential output is the difference between the two outputs:

And the differential mode gain:

The way this circuit is usually used, however, is with one output referenced to ground: This gives a single-ended output, with a gain of:

The input and output resistances for differential mode inputs are:

### Common Mode

Common mode input is when the same signal is connected to both input terminals, . An ideal differential amplifier would reject common mode input, but this is often not the case. The performance of a differential amplifier is defined by it's CMRR, which would ideally be infinite, but is usually just very large in practice.

The common mode input resistance:

The generalised output of a differential amplifier, factoring in both common mode and differential mode input signals is:

## Example

Find to give , and for max AC swing, when . For max AC swing, :

Using , calculate the differential and common mode gains, and the CMRR of the circuit.

The common mode rejection ratio for this circuit is fairly low, because is low. as , so ideally is as large as possible. Replacing it with an ideal current source with infinite resistance can acheive this. 