1. Capacitance VariationsL W

2. y W = W + D
W(x) N L = L + D
L(y) N D
W(x) , D
L(y)
are random x

4. y x

10. HW: Perform similar development as we did for capacitors
to derive similar conclusions.

11. Device Matching Device accuracy can't be made very accurate due to
Deterministic process errors
Gradient errors
Random errors
Typical achievable accuracy
most cases
Up to in some cases
But matching accuracy can be much better

12. Example: Capacitor Matching

17. HW: show that when a capacitor with area A is repeated M times and connected in either series or parallel connection to get a total capacitance of MC or C/M, its random error variance is reduced to:
Show that when a capacitor with area A is repeated MN times and connected in either series/parallel connection to get a total capacitance of MC/N, its random error variance is reduced to:

18. If we do this to both cap A and cap B, then we have:
Suppose the total area for both cap A and cap B is given, show that the ratio error is minimized is cap A and cap B are allowed the same area, i.e.,

19. k W L
HW: State similar results regarding resistors and prove your statements.
Note that Rtot = Rsh + 2Rc/(kW/t)

23. This applies to 1 to 1 match as well as non unity matches

24. Capacitance is an “integratible” circuit parameter
Any integratible parameter can achieve matching against linear gradient by maintaining common centroid
But some parameters do not have the integratible property
Resistance is an example
Mere common centroid of the total area is not sufficient for achieving matching.

25. Use of unit cells
Unit cell shape can be arbitrary, but regular shape is recommended
Unit cell largest dimension should be sufficiently small
So that over the cell area gradient errors are much smaller than random errors
Use translation to make copies of unit cells
Translation maintains photolithographic invariance
No rotations or reflections for creating identical copies
But can use reflection to get two-component or four-component unit cells
This helps achieve symmetry
To create copies of the unit cell, translate all components of each cell
Each component can have different translation

26. Several copies of the unit cell are connected to form device A
Another several copies of the unit cell are connected to form device B
The connectivity topology for the two devices should be the same
Can layout half or quarter of each device and flip to get the whole device but should just flip the “pattern” (including the connectivity) and still translate the cells