exp
Full Member level 1
Hi,
I am currently thinking about how to design a gm-C integrator and what my design tradeoffs are.
First of all, suppose infinite output resistance, then the integrator is ideal and I get
\[H(s) = -\frac{G_m}{C s}\]
But how I choose the integrator time constant Gm/C ? It's just a scalar but I guess it can't be arbitrary? Do I understand correctly that the choice is related to saturation? Suppose that I want to implement a finite integration:
\[
v_o = \int_0^{T_w} x(t) \, dt
\]
and suppose that both input x(t) and output v_o are limited to VDD volts. Then I could derive:
\[
V_{DD} > \frac{g}{C} T_w V_{DD} \Rightarrow \frac{g}{C} < T_w
\]
and I would have at least a restriction on the ratio. Does this make sense?
Suppose next that output resistance ro is not infinite, then is results in a leaky integrator and the transfer function becomes:
\[H(s) = -\frac{r_o G_m}{1 + s r_o C}\]
After the pole at 1/roC, the system behaves as an ideal integrator.
Since I integrate over a finite period Tw, is it valid to say that the lowest integrateable frequency is 1/Tw and hence 1/roC << 2π/Tw ? So given C, this defines the minimum output resistance I can have.
Finally, using a simple 1-stage Gm-cell, there is just the one dominant pole at 1/roC but I will also have finite transit frequency, i.e., gm is not constant over frequency. Is it reasonable to say that I pick the transistor in a way that the transit frequency fT is just 3-5x higher than the highest possible frequency I intent to integrate?
Furthermore, noise and distortion will be an issue but for a first-cut simulation I just want to understand how to pick gm, C and R (ro).
EDIT: is the integrator only "useable" for integration up to the unity gain frequency gm/C ? If no, then I am again confused what I need gm/C for and why I can't set it arbitrarily. If yes, the ratio between these two is fixed by the intrinsic gain gm*ro which is not a lot in current technologies.
I am currently thinking about how to design a gm-C integrator and what my design tradeoffs are.
First of all, suppose infinite output resistance, then the integrator is ideal and I get
\[H(s) = -\frac{G_m}{C s}\]
But how I choose the integrator time constant Gm/C ? It's just a scalar but I guess it can't be arbitrary? Do I understand correctly that the choice is related to saturation? Suppose that I want to implement a finite integration:
\[
v_o = \int_0^{T_w} x(t) \, dt
\]
and suppose that both input x(t) and output v_o are limited to VDD volts. Then I could derive:
\[
V_{DD} > \frac{g}{C} T_w V_{DD} \Rightarrow \frac{g}{C} < T_w
\]
and I would have at least a restriction on the ratio. Does this make sense?
Suppose next that output resistance ro is not infinite, then is results in a leaky integrator and the transfer function becomes:
\[H(s) = -\frac{r_o G_m}{1 + s r_o C}\]
After the pole at 1/roC, the system behaves as an ideal integrator.
Since I integrate over a finite period Tw, is it valid to say that the lowest integrateable frequency is 1/Tw and hence 1/roC << 2π/Tw ? So given C, this defines the minimum output resistance I can have.
Finally, using a simple 1-stage Gm-cell, there is just the one dominant pole at 1/roC but I will also have finite transit frequency, i.e., gm is not constant over frequency. Is it reasonable to say that I pick the transistor in a way that the transit frequency fT is just 3-5x higher than the highest possible frequency I intent to integrate?
Furthermore, noise and distortion will be an issue but for a first-cut simulation I just want to understand how to pick gm, C and R (ro).
EDIT: is the integrator only "useable" for integration up to the unity gain frequency gm/C ? If no, then I am again confused what I need gm/C for and why I can't set it arbitrarily. If yes, the ratio between these two is fixed by the intrinsic gain gm*ro which is not a lot in current technologies.
Last edited:
