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Fall 2004; E6316: Analog Systems in VLSI; HW4

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Fall 2004; E6316: Analog Systems in VLSI; HW4

Nagendra Krishnapura ([email protected]) due on 9 Nov. 2004

S/H A/D1 D/A G A/D2

-+

M bits

K bits Ve,amp Ve,A/D2[k]

Ve,A/D1[m]

Ve,D/A[m]

Ve,S/H

Vin

Σ

Vref Vref Vref

Figure 1:

1. Fig. 1 shows a two step flash converter. The overall resolution isN =M+Kbits. The error in each block is shown as an analog voltage referred to either the input or the output. i.e. Themthtransition of A/D1 occurs atmVLSB1+Ve,A/D1[m]and themthoutput of D/A ismVLSB1+Ve,D/A[m]. 0≤ m≤2M −1and0≤k≤2K−1.VLSB=Vref/2N is the LSB voltage of the overall converter.

(a) (2 pts.) Derive the value of the inputVin which corresponds to mth transition of A/D1. You should get an expression that combines the errors from different components.

(b) (2 pts.) Derive the value of the inputVinwhich corresponds tokth transition of A/D2. Assume that A/D1 is betweenmthand(m+ 1)th transitions.

(c) (2 pts.) In the result from (a) above, assume that the different terms contribute equally to the total error, which is constrained to0.5VLSB. Calculate the individual errors in terms ofVref. (d) (2 pts.) In the result from (b) above, assume that the different terms contribute equally to the

total error, which is constrained to0.5VLSB. Calculate the individual errors in terms ofVref. (e) (4 pts.) Calculate the allowable errors in each component for a 8 bit converter, forM = 5, K = 3

andM = 4, K = 4. Express the accuracy as an effective number of bits (A component with a voltage rangeVref has anLbit accuracy if its error magnitude is less thanVref/2L+1, i.e. half LSB atLbits). If (c) and (d) give different error constraints for the same component, use the more conservative constraint.

2. Assume that you have a 2 step flash A/D converter (no digital error correction) with 2 bits in each stage. All components other than the residue amplifier are ideal. Sketch the A/D characteristics for the following cases. Compare it with the ideal characteristics.

1

(2)

2

(a) (2 pts.) The amplifier has a gainG >4 (b) (2 pts.) The amplifier has a gainG <4

(c) (2 pts.) The amplifier has an input referred offsetVos>0 (d) (2 pts.) The amplifier has an input referred offsetVos<0

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