Ceaseless Student

OK. I’m going to go through two examples. The first one is a first-order low pass Gm-C filter. I’ll just show the circuit and then we can analyze and note that it works. Then I’ll show you the cooler part - design.

We’ll write a transfer function that we want to implement. Then map that to differential equations and go from the ODEs to circuits. I think I’ll finish out 1 way of implementing the transfer function and leave the other one at the diffEQ step in case someone actually wants to try their hand at a little bit of circuit fun. I’ll put up a link to the solution too.

OK. So here’s what a 1st order Gm-C low pass filter looks like. Why is it called a Gm-C filter? Well, you know how in an R-C filter, the R and C determine the time constant (tau)? Gm is a conductance (the inverse of resistance) so it similarly sets tau.

Gm-C Low Pass Filter

The Gm comes from the OTA. Recall that an OTA outputs a current proportional to its differential input voltage? In math: I_out=G_m(V₊-V_-). Now we note that V_out=V_- and V_in=V₊ in our circuit. We also note that all of the output current has to go into the cap and that the cap follows I_out=CV’_out. We combine our equations and get CV’_out=G_m(V_in-V_out). Let’s use τ=G_m/C. S’more rearranging gets us τV’_out+V_out=V_in.

Unless you’ve taken a class that teaches it, you’ll have to trust me that you can effectively call a derivative s and all the normal rules of multiplication etc apply (a friend of mine would tell you it’s because Liebniz’s notation is a strong notation). So we go ahead and do that and get V_in/V_out=1/(τs+1) which is the canonical form of the transfer function for a first-order low pass. Nice.

Hmmm… this is being longer than I thought it would be so I’ll be splitting it up. Next time we’ll look at a second-order filter that selects for a frequency: V_in/V_out=s/(τs+1)².