Differential pair

How it works

A differential pair is a common input stage for amplifiers and comparators. Matched devices reject a shared input change while the bias devices retain enough headroom. Strong-inversion square-law estimates and weak-inversion exponential steering describe different operating regimes; neither is an exact compact-model equation.

Mt copies the testbench's I_tail (20 µA by default), each side carries half at balance, and each drain sits at V_DD − I_tail R_D/2; V_CM, 0.7 V by default, must keep the tail node high enough for Mt to stay saturated. The ground port returns the bias diode and tail current and ties the NMOS bodies.

Signal path

• Tail mirror (Mb, Mt): Mb, diode-connected, takes the testbench's I_tail; Mt copies it into the tail node the pair shares.
• Input pair (M1, M2): A differential input steers current between the two sides. A shared input change mainly moves the tail node; ideal matched sides keep an equal split while all devices retain headroom.
• Loads (Rd1, Rd2): Turn the two drain currents into voltages below V_DD; v(d2) − v(d1) is the differential output.

Key relations

• Transconductance: g_m = √(µ_n C_ox (W/L) I_tail). each device at balance, strong inversion; nearer weak inversion, as here, it tends to I_tail/(2 n V_T)
• Differential gain: A_d = (v_d2 − v_d1) / (v_inp − v_inn) ≈ g_m (R_D ∥ r_o). g_m R_D when r_o ≫ R_D; each output alone carries half of it
• Full steering: |V_id| ≈ √2 · V_ov. strong inversion; biased nearer weak inversion, the currents approach their limits exponentially instead
• Tail headroom: V_CM − V_GS1 > V_ov(Mt). below it Mt enters triode and I_tail, and g_m with it, collapses
• Drain headroom: V_DD − I_tail R_D/2 > V_CM − V_TH. keeps M1 and M2 saturated at balance; raising R_D for gain is what breaks it

Trade-offs

• I_tail: more current raises g_m and drops more across R_D. In strong inversion it also widens the steering range; in ideal weak inversion the normalized steering range is mainly set by thermal voltage, not current.
• R_D: gain rises with it, but the drains fall by I_tail R_D/2 and push M1 and M2 towards triode.
• L_tail: a longer tail device has more output resistance, holding I_tail steady while the pair steers and the tail node moves.
• V_CM: too low and Mt leaves saturation and I_tail collapses; too high and M1 and M2 run out of drain headroom into triode.

Testbenches and limits

• Differential sweep

Design variables and defaults

Ports

• inp input (+)
• inn input (-)
• d1 output
• d2 output
• bn bias
• vdd supply
• 0 ground: NMOS substrate and tail/bias-current return

Reference

B. Razavi, Design of Analog CMOS Integrated Circuits, 2nd ed., McGraw-Hill, 2017. Differential Amplifiers - the basic MOS differential pair and its large-signal behaviour.

The topology and the result that the pair fully steers its tail current at a differential input of about sqrt(2)*V_ov.