Differential Pair

Part of The Transistor

The long-tail pair — amplifying the difference between two signals while rejecting common-mode noise.

Why This Matters

The differential pair is the most important multi-transistor building block in analog electronics. It amplifies the difference between two input signals (differential mode) while rejecting signals that appear identically on both inputs (common-mode noise). This common-mode rejection makes it immune to power supply noise, ground loops, and interference picked up by signal cables — the dominant problems in practical signal transmission.

Every operational amplifier, instrumentation amplifier, and comparator uses a differential input stage. The principle: if both inputs rise by the same amount (noise), the emitter current splits equally, both collector currents rise equally, and the differential output does not change. But if only one input rises, that transistor’s collector current increases while the other’s decreases — creating a differential output.

For a rebuilding civilization, the differential pair enables precision measurement over long distances (the noise-rejection property makes distant sensor readings reliable), low-distortion audio amplification (differential topology with negative feedback produces much lower distortion than single-ended stages), and the building of operational-amplifier-like circuits from discrete transistors.

Circuit Description

Long-tail pair: Two transistors (Q1, Q2) with emitters joined and connected to a shared emitter resistor (RE, often called the “tail”). The two bases are the input terminals. The two collectors (loaded by equal collector resistors RC1 and RC2) are the outputs.

Single-ended input: Apply signal to Q1 base; hold Q2 base at AC ground. Output taken as difference: V_out = VC2 - VC1. Or take only one output (single-ended output) and accept half the differential gain.

Differential input: Apply +V_in/2 to Q1 base, -V_in/2 to Q2 base. Differential output: V_out = VC2 - VC1.

Differential Mode Analysis

When a differential signal is applied (Q1 base up by v, Q2 base down by v):

• Q1 base-emitter voltage increases by v/2 (the emitter rises by v/2 due to the tail resistor)
• Q2 base-emitter voltage decreases by v/2
• Q1 collector current increases by Δi = gm × v/2
• Q2 collector current decreases by gm × v/2
• Differential output voltage: ΔV_out = 2 × Δi × RC = gm × v × RC

Differential gain: Ad = gm × RC = (IC/VT) × RC

This is identical to the single-transistor common-emitter gain (no emitter degeneration). The tail resistor does not appear in the differential gain expression — it does not affect differential signals, only common-mode signals.

For IC = 0.5 mA per transistor (1 mA total tail current), RC = 10 kΩ: Ad = (0.5×10^-3 / 0.026) × 10,000 = 19.2 × 10,000 = 192

Single-ended output gain: If output is taken from only one collector (e.g., VC2 with respect to Vcc): gain is Ad/2 = 96.

Common-Mode Rejection

A common-mode signal is one that appears identically on both inputs (both bases rise by v_cm). Both transistors’ VBE increases by the same amount; both collector currents tend to increase. But the emitter node (tied together through tail resistor RE_tail) rises, reducing VBE for both transistors:

V_emitter = v_cm - ΔV_BE_due_to_CM_current ≈ v_cm (for large RE_tail)

Since both collector currents change equally, the differential output (VC1 - VC2) does not change. The common-mode gain Ac → 0 as RE_tail → ∞.

Common-mode rejection ratio (CMRR): CMRR = |Ad / Ac| = gm × RC / (1/gm) × (1/(2RE_tail)) ≈ 2 × gm × RE_tail

For gm = 0.019 S and RE_tail = 47 kΩ: CMRR = 2 × 0.019 × 47000 = 1786 = 65 dB.

A CMRR of 65 dB means a common-mode noise signal must be 1786× larger than a differential signal to produce the same output. This makes differential stages essentially immune to noise signals that appear on both inputs simultaneously — exactly the property needed for long-distance sensor connections.

Replace tail resistor with current source: A current source in the tail provides near-infinite tail resistance (it is a high-impedance current source). This dramatically improves CMRR: > 100 dB (100,000:1) is achievable. Build a simple current mirror as the tail current source to improve common-mode rejection by 30-40 dB over using a resistor.

Practical Applications

Instrumentation amplifier: Two emitter followers as input buffers, followed by a differential pair output stage. The buffers provide high input impedance and prevent loading the sensor. The differential output rejects ground-loop noise between sensor and measurement point.

For a thermocouple measuring furnace temperature 100 meters away: the thermocouple generates 50 µV/°C. At 100 meters of wire, power line interference induces ~1-10 mV of common-mode noise. A differential input amplifier with 80 dB CMRR reduces this 1 mV noise to 0.1 µV — well below the 50 µV signal.

Audio balanced line driver/receiver: Professional audio uses differential (balanced) transmission on XLR connectors. Noise picked up by the cable appears on both conductors equally — common-mode. The differential receiver at the far end subtracts the two signals, canceling the noise while passing the audio. This allows 100-meter cable runs in noisy environments without degradation.

Comparator: A differential pair with large gain (or followed by multiple gain stages) acts as a comparator — if V_in+ > V_in-, the output swings high; if V_in+ < V_in-, output swings low. No external logic required: the nonlinearity of the high-gain differential stage converts the continuous comparison into a logic-level output.

Matching Requirements

The differential pair requires matched transistors. Mismatch in VBE causes a systematic DC offset at the output:

V_offset = ΔVBE × Ad (differential gain amplifies the mismatch)

For ΔVBE = 5 mV and Ad = 100: output offset = 500 mV. This is large and impractical.

To minimize offset: select transistors from the same batch with measured VBE within 2-5 mV of each other. Mount Q1 and Q2 on the same heatsink to maintain thermal matching (temperature gradients cause VBE gradients which degrade CMRR and increase offset over time).

Trimming offset: Add a small potentiometer between the emitters of Q1 and Q2. Adjusting it slightly imbalances the emitter resistances, producing a compensating offset that cancels the transistor VBE mismatch. Many precision op-amp designs include this offset null adjustment.

For critical applications (precision voltage measurement, high-CMRR instrumentation), a matched transistor pair from the same production lot, mounted in physical contact, gives better performance than the most elaborate single-transistor circuit.