Differential Pair

Analog Electronics · Differential Pair · Learn

1 — The Differential Pair as an Input Stage

1.1 — What It Does

Figure 1. BJT differential pair with collector loads and tail current source.

The differential pair takes two input voltages and amplifies only their difference. It is the universal front end of every op-amp, comparator, and instrumentation amplifier.

Two matched transistors share a single tail current source $I_{\text{TAIL}}$ . The differential input $V_{id} = V_{B1} - V_{B2}$ steers that current from one side to the other.

1.2 — The Tail Current Split

Figure 2. I_TAIL redistribution as V_id varies: equal at zero, steered left or right otherwise.

When $V_{id} = 0$ , both transistors see the same base voltage and each carries $I_{\text{TAIL}}/2$ . As $V_{id}$ swings positive, more current flows through Q1 and less through Q2.

The exact large-signal split is a hyperbolic tangent. For small signals the transfer is nearly linear:

i_{C1} = \frac{I_{\text{TAIL}}}{2}\left(1 + \frac{g_m \cdot v_{id}}{I_{\text{TAIL}}}\right)

1.3 — Small-Signal Differential Gain

Each collector current change appears across its $R_C$ , so the differential output is:

A_d = g_m \cdot R_C

where $g_m = I_{\text{TAIL}} / (2 V_T)$ . This is the gain you design for.

2 — Common-Mode Rejection

2.1 — What Common Mode Means

Figure 3. V_icm drives both bases equally; the high-impedance tail source blocks the common-mode signal.

A common-mode signal $V_{icm}$ is any voltage that appears identically on both inputs. It pushes both emitters up by the same amount.

2.2 — Why the Tail Source Rejects It

The tail current source has a very high output resistance $r_{o,\text{tail}}$ . Because $I_{\text{TAIL}}$ barely changes when $V_{icm}$ moves, neither collector current changes much either.

The common-mode gain is therefore tiny:

A_{cm} \approx -\frac{R_C}{2 \, r_{o,\text{tail}}}

2.3 — CMRR Definition

The Common-Mode Rejection Ratio is the ratio of differential gain to common-mode gain. Higher is better.

\text{CMRR} = \left|\frac{A_d}{A_{cm}}\right|

In decibels: $\text{CMRR}_{\text{dB}} = 20\log_{10}(\text{CMRR})$ . A good discrete diff pair achieves 60-80 dB; IC op-amps exceed 100 dB.

3 — Practical Considerations

3.1 — Matching

The diff pair works because Q1 and Q2 are matched: same $V_{BE}$ , same $\beta$ . Any mismatch converts common-mode input into a spurious differential signal.

Integrated circuits excel here because both transistors sit on the same die and track thermally.

3.2 — Where You See Diff Pairs

Every op-amp input stage, every comparator, every ADC front-end starts with a diff pair. It is also the core of ECL logic gates and Gilbert-cell mixers.

The differential pair amplifies only

V_{id} = V_{B1} - V_{B2}

with gain

A_d = g_m R_C

, while rejecting the common-mode signal thanks to the high-impedance tail source. The figure of merit is

\text{CMRR} = |A_d / A_{cm}|

Head to the Simulate tab to slide $V_{id}$ and watch the tail current redistribute in real time.