Differential Amplifier

A differential amplifier is one of the most foundational building blocks in analog circuit design. It amplifies the difference between two input signals while rejecting any signal that is common to both inputs. This property makes it indispensable in instrumentation, operational amplifiers, and communication systems where noise rejection is critical.

Core Concept Explanation

The circuit diagram below shows the fundamental BJT differential pair configuration with its characteristic tail current source that enables the differential amplification behavior.

In a differential amplifier, two transistors Q1 and Q2 share a common emitter node connected to a tail current source (Itail). The circuit responds to two modes of input: differential mode and common mode. In differential mode, the two inputs are equal in magnitude but opposite in sign (Vd = Vin1 - Vin2). In common mode, both inputs carry the same signal (Vc = (Vin1 + Vin2) / 2).

The differential pair amplifies only the difference signal. When Vin1 increases, Q1 draws more current from Itail, which reduces the current through Q2. This causes Vout1 to drop and Vout2 to rise, producing a differential output proportional to the input difference. The tail current source enforces that IC1 + IC2 = Itail at all times, which is the key to the circuit behavior.

The common mode rejection ratio (CMRR) quantifies how well the amplifier suppresses common signals while amplifying the difference. A high CMRR means the amplifier rejects noise, power supply hum, or any interference that appears equally on both inputs. This is why differential amplifiers are used at the input of op-amps and instrumentation amplifiers.

Mathematical Expression

The differential mode gain (Ad) is given by Ad = -gm x RC, where gm is the transconductance of each transistor and RC is the collector resistor. The common mode gain (Ac) depends on the tail resistance: Ac = -RC / (2 x REE), where REE is the equivalent resistance of the tail current source. A simple tail resistor gives finite CMRR, but a current source tail has very high output impedance, making Ac nearly zero.

CMRR is defined as the ratio of differential gain to common mode gain: CMRR = |Ad / Ac|. It is commonly expressed in decibels: CMRR(dB) = 20 x log10(|Ad / Ac|). For GATE problems, remember that CMRR increases as the tail current source impedance increases. A perfect current source gives infinite CMRR.

The transconductance gm = IC / VT, where IC is the quiescent collector current and VT is the thermal voltage (approximately 26 mV at room temperature). Each transistor in the pair operates at IC = Itail / 2 in the balanced condition.

Practical Understanding

In real circuits, mismatches between Q1 and Q2 (due to manufacturing variations) reduce CMRR. This is why matched transistor pairs or differential pair ICs are used in precision applications. The input offset voltage arises directly from such mismatches and is a key datasheet parameter for op-amps.

The tail current source is typically implemented using a third BJT biased in the active region. Its high output impedance (approximately beta x re) ensures that the tail current remains nearly constant despite common mode voltage variations, which is what gives the circuit its high CMRR in practice.

In op-amp design, the differential pair forms the first stage. The second stage provides additional voltage gain, and the output stage buffers the load. Understanding the differential pair therefore directly explains the internal working of every general-purpose op-amp such as the 741 or LM358.

Given:
RC = 5 kΩ, Itail = 1 mA, VT = 26 mV, REE (tail resistance) = 100 kΩ

Why this formula applies:
Each transistor biases at IC = Itail/2 = 0.5 mA. gm = IC/VT.
gm = 0.5 mA / 26 mV = 19.23 mA/V

Formula:
Ad = gm × RC
Ac = RC / (2 × REE)
CMRR = |Ad / Ac|

Substitution:
Ad = 19.23 × 10⁻³ × 5 × 10³ = 96.15
Ac = 5000 / (2 × 100000) = 0.025

Calculation:
CMRR = 96.15 / 0.025 = 3846

Final Answer:
CMRR = 3846 ≈ 71.7 dB

Exam Tip: In GATE, if the tail element is a current source (not a resistor), treat REE as infinite. This makes Ac = 0 and CMRR = infinity. If a tail resistor is given, always use Ac = RC / (2 x REE) and compute CMRR as |Ad/Ac|. Do not confuse differential gain with single-ended gain.

How the Differential Pair Works - Step by Step

• Itail splits equally between Q1 and Q2 when both inputs are equal (balanced condition).
• A differential input signal steers current from one transistor to the other, producing complementary output swings.
• Common mode input causes both collector currents to change by the same amount; the high impedance tail source resists this change strongly.
• The differential output voltage (Vout1 - Vout2) carries the amplified difference signal with gain Ad = gm x RC.
• Single-ended output (from one collector only) has half the differential gain and is used when a ground-referenced output is needed.

Quick Revision

• Differential amplifier amplifies Vd = Vin1 - Vin2 and rejects common mode signals.
• Differential gain: Ad = gm x RC, where gm = IC / VT = (Itail/2) / VT.
• Common mode gain: Ac = RC / (2 x REE); decreases as tail impedance increases.
• CMRR = |Ad / Ac| in linear scale; CMRR(dB) = 20 log10(|Ad / Ac|).
• Ideal current source tail makes REE infinite, giving CMRR = infinity and Ac = 0.
• Transistor mismatch introduces input offset voltage and degrades CMRR in real circuits.
• Trap: Do not use Ad = gm x RC for single-ended output without halving - single-ended gain is Ad/2 referred to one side.