Analysis for Challenge 9
The derivation of CX proceeds as follows: We first ask, for Figure 1, what total capacitance is “seen” looking to the left of 16C (MSB-3)? The answer is 16C. For equivalency in Figure 2, the total capacitance looking into (and from the right of) CX must be 1C. CX is in series with 16C, so we can write
1/CX + 1/16C = 1/1C
1/CX = 1/1C – 1/16C
1/CX = (16/16 – 1/16)/C
1/CX = 15/16C
CX = (16/15)C
which is precisely the value used by Culurciello and Andreou (see Figure 2, p. 859, at https://engineering.purdue.edu/elab/data/papers/TCASIIadc06.pdf). Also, please notice that the total required C is nearly a factor of 8 less when compared with the Σ=256C version.
For another tutorial, start your reading on page 459 of the Motorola M68HC11 Reference Manual. You will notice (Figure 12-3, p. 471) that CX has taken on a slightly different value than what was derived in this circuit challenge. This is because of the introduction of a half-unit-value (C/2) capacitor. The use of C/2 is optional; the Motorola write-up (beginning on page 467) explains their reason for including it. The derivation of CX proceeds in the same manner as above, and will (when C/2 is introduced) yield CX=1.1C.
Is There a Limit to the Number of Bridge Capacitors?
Actually, there can be a bridge capacitor placed between each “bit”. The result is a C-2C ladder (analogous to the R-2R ladder). This writer has determined the circuit would look like the figure below (without the switches, comparator circuitry, etc.).
This gives a total of 23C. And what’s especially attractive about it is that each 2C bridge capacitor can be constructed from two 1C capacitors. Thus all capacitors may be identically sized (perfect for matching in an integrated circuit environment). Whether or not this topology can be used in an integrated circuit, however, is a strong function of parasitics. Assuming a process is well-modeled, a would-be user of the C-2C ladder would be well advised to put together a rigorous simulation matrix before committing resources to a mask set.
But, as is the case with many good ideas, someone else thought of it first. A couple of references:
• US Patent Number 4,028,694
• A drawing may be viewed here (see Figure 6.11, p. 112)