The classic Widlar bandgap reference uses two transistors with different current densities to generate a PTAT (Proportional To Absolute Temperature) voltage that cancels the CTAT (Complementary To Absolute Temperature) drift of a V_BE. It works, but it has a subtle flaw: the collector currents of the two transistors are set by a resistor ratio, and any mismatch in the base currents of Q1 and Q2 corrupts the ratio. The Wilson bandgap adds a third transistor to force equal collector currents through negative feedback, dramatically improving accuracy.

How it works: Two matched PNPs form a current mirror at the top of the circuit, forcing I_C1 = I_C2. Q1 runs at high current density (small emitter area, or single unit), Q2 runs at low current density (N parallel units, typically 8). The difference in V_BE across the two transistors appears across a resistor R1 in Q2's emitter:

• ΔV_BE = V_T · ln(N) where V_T = kT/q ≈ 26 mV at room temperature
• For N = 8: ΔV_BE = 26 mV · ln(8) ≈ 54 mV at 300 K
• This ΔV_BE/R1 sets the tail current, which is PTAT

The third transistor Q3 senses the output and feeds back to close the loop. The output voltage is V_BE3 + (R2/R1) · V_T · ln(N). Choose R2/R1 so the PTAT term contributes about +2 mV/°C at the output, exactly canceling the -2 mV/°C of V_BE3. You land at approximately 1.25 V — the silicon bandgap voltage extrapolated to 0 K.

Worked example: Target output 1.25 V at 300 K with N = 8. V_BE3 ≈ 0.65 V, so the PTAT term must add 0.60 V. With ΔV_BE = 54 mV, you need R2/R1 = 0.60/0.054 ≈ 11.1. If R1 = 10 kΩ, then R2 ≈ 111 kΩ. Tail current will be 54 mV / 10 kΩ = 5.4 µA.

Real-world example: The venerable REF200 and countless CMOS voltage references (LM4040, MAX6100 series) use Wilson-style bandgap topologies internally. The Wilson variant is preferred over the plain Widlar in modern IC processes because the added feedback loop makes the reference far less sensitive to β mismatch and Early voltage — you can hit 0.5% initial accuracy without laser trimming, and 50 ppm/°C over -40 to +85 °C with basic curvature correction.

Rule of thumb: If your bandgap output drifts more than you'd expect from V_BE curvature alone, suspect the current-mirror mismatch first. The Wilson topology's feedback fixes this at the cost of one transistor and a slightly higher minimum supply voltage (roughly 2·V_BE + V_CE(sat) ≈ 1.5 V).