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Brokaw Bandgap Gm Analysis

Cell 1

The basic cell has a structure as shown below:


We have:
Putting these in the 1st equation we have:
Also $[g_m(Δv-v_B)+g_m(Δv-v_A)]R_2=v_A$. From this we can solve for $v_A$ and we have:
Also $v_B-v_A=g_m(Δv-v_B)R_1$. From this and using the $v_A$ value from (2) we have:
Substituting $v_B$ from (3) and $v_A$ from (2) into (1) we have:
Dividing by $Δv$ and simplifying we have:
By substituting $g_m={\ln n}/R_1$ and taking approximations we have:
$$G_M≈1/{1/g_m(1+2R_2/R_1)+R_2}=1/{R_1/{\ln n}(1+2R_2/R_1)+R_2}≈1/{2R_2/{\ln n}+R_2}=1/R_2[{\ln n}/{2+\ln n}]$$

Useful numbers

Normally the n factor is set to 8 since that allows good layout. With n = 8 we have
Even if n=4 we have $G_M=0.4/R_2$ which is still close with about 25% error. So the number $1/{2R_2}$ is good to keep handy

Cell 2

This alternate topology has a structure as shown below:


Both topologies follow the same analysis:
Therefore we have:
If $g_mR_2≫1$ then we can approximate $G_M$ as:

Useful Numbers

Remember that the bandgap voltage is about 1.2V and Vbe is about 0.6V so in this case we have:
$${V_T\ln n}/R_1R_2≈0.6V$$
So for n=8 we have $R_2=11.1R_1$, putting this in the $G_M$ equation we have

Comparing the Transconductances


It is clearly seen than the $G_{M-I}>G_{M-II}$ because of the extra term $g_m^2R_2^2$ in the denominator. Hence the topology 1 always provides better accuracy by making the effects of mismatch and offset for the same bias currents lower.

From the numbers we also see that for n=8 the 1st topology provides 6 times more $G_M$.


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Copyright 2018 Milind Gupta