The beta multiplier: a reference that ignores the supply

Measuring supply sensitivity

Give the dependence a number. The supply sensitivity is the fractional change of the current per fractional change of the supply: S = (ΔI/I)/(ΔV_DD/V_DD). The lab takes ΔI between 1.62 and 1.98 V, so S = 1 means "1 % of supply moves the current 1 %".

For the resistor bias, I = (V_DD − V_GS)/R. If V_GS held still, S would be V_DD/(V_DD − V_GS) = 1.8/(1.8 − 0.769) = 1.75. V_GS rises a little with the current, so the run gives 1.59. Either way it is above 1: the resistor sees only the part of the supply left over after V_GS.

A loop that sets its own current

A self-biased reference sets its current by the circuit itself. In the beta multiplier (also called the constant-g_m bias), N1 and N2 share a gate, N2 is K times wider, and R_S sits under N2. The PMOS mirror forces the two currents equal. So the loop settles where V_GS1 − V_GS2 = I·R_S.

The supply appears in neither condition. Raise VDD and the extra voltage lands across P1 and N2, whose currents barely depend on it. The circuit also balances with no current at all; E4.5 is about that second state.

Square law. With I = (β/2)(V_GS − V_th)² and β₂ = Kβ₁, the difference of the overdrives is √(2I/β₁)·(1 − 1/√K). Set it equal to I·R_S: I = 2(1 − 1/√K)²/(β₁R_S²). Then g_m1 = 2(1 − 1/√K)/R_S, which depends only on R_S and K, hence "constant g_m". For K = 4, g_m1R_S = 1.

Weak inversion. With a current exponential in V_GS/(nU_T), the difference is nU_T·ln K, so I = nU_T·ln K/R_S and g_m1R_S = ln K = 1.39. Here U_T = kT/q is the thermal voltage and n the slope factor from E2.

The run gives g_m1R_S = 0.896, with 91.7 mV across R_S. At 9.17 µA these devices are in moderate inversion, between weak and strong inversion, where neither formula holds. Two effects outside both formulas also lower it. N2's threshold ends 5.1 mV above N1's: its source sits 91.7 mV above its body, which raises the threshold (the body effect), and its wider channel takes back only part of that. And P1 and P2 sit at different V_DS, so the two currents are not quite equal.

What R_S, K and length do

The formulas still predict the direction of a change, and the lab tests them with five designs on one supply. The figure shows four of them against VDD.

Doubling R_S divides the current by 2.66: between the square law's 4 and weak inversion's 2, as moderate inversion should be. K = 8 lands between the two as well. Making every device longer at the same W/L changes the current by 14 %, although both formulas depend only on W/L. That is the measured device departing from both.

The beta multiplier's S is 0.241 against 1.59 for the resistor, 6.6 times less. It is not zero because P1's V_SD and N2's V_DS both follow VDD, and finite r_o lets their currents follow too. Lengthening only the PMOS takes S to 0.221; lengthening everything takes it to 0.133. The lever is E4.1's output resistance, and E4.3's cascodes raise it far more than length does.

Try it

Run it: five beta multipliers and the resistor bias share one supply sweep. Each prints its current at 1.8 V and its S; the last lines give the base design's R_S voltage, g_m1R_S, N2's threshold minus N1's and the supply where it reaches 90 % of its current.

Then design one. Change RSb until i18_b is 10 µA, and predict which way gm1_rs moves. Try K = 2 (N2 W=4): the square law says the current falls to (0.293/0.5)² = 0.34 of the base, weak inversion says 0.5. Then cascode the PMOS mirror and see what S does.

Check yourself

* a plain resistor from VDD to ground: its supply sensitivity
VDD vdd 0 1.8
R1 vdd 0 100k
.control
dc VDD 0.6 2 0.005
let i_r=v(vdd)/100k
* S between 1.62 V and 1.98 V (points 204 and 276); 1.8 V is point 240
let s_r=i_r*0+(i_r[276]-i_r[204])/i_r[240]/0.2
meas dc s_r FIND s_r AT=1.8
.endc
.end