Page 221 - DCAP108_DIGITAL_CIRCUITS_AND_LOGIC_DESIGNS

Digital Circuits and Logic Design

Notes
3. The full-scale output voltage is equal to the positive voltage of the digital input signal. (The
divider would work equally well with input voltages of 0 and V.)
4. The LSB has a weight of 1/(2 – 1), where n is the number of input bits.
n
n
5. The change in output voltage due to a change in the LSB is equal to V/(2 – 1), where V is
the digital input voltage level.
6. The output voltage V can be found for any digital input signal by using the following
A
modified form of Millman’s theorem:
3
0
2
1
V 2 + V 2 + V 2 + V 2 + ... + V 2 n − 1
V = 0 1 2 3 n − 1 ...(12.1)
A n
2 − 1
Where V , V , V , V ,………, V n – 1 are the digital input voltage levels (0 or V) and n is the number
0
3
2
1
of input bits.
12.3:
For a 5-bit resistive divider, determine the following: (a) the weight assigned to the LSB; (b) the
weight assigned to the second and third LSB; (c) the change in output voltage due to a change
in the LSB, the second LSB, and the third LSB; (d) the output voltage for a digital input of 10101.
Assume 0 = 0 V and 1 = + 10 V.
Solution:
(a) The LSB weight is 1/(2 – 1) = 1/31.
5
(b) The second LSB weight is 2/31, and the third LSB weight is 4/31.
(c) The LSB causes a change in the output voltage of 10/31 V. The second LSB causes an output
voltage change of 20/31 V, and the third LSB causes an output voltage change of 40/31 V.
(d) The output voltage for a digital input of 10101 is
0
1
2
3
*
*
*
*
*
V = 10 2 + 102 + 10 2 + 0 2 + 102 4
A 5
2 − 1
(
4
10 1 ++ 16) 210
= = = +6.77 V
32 − 1 31
This resistive divider has two serious drawbacks. The first is the fact that each resistor in the
network has a different value. Since these dividers are usually constructed by using precision
resistors, the added expense becomes unattractive. Moreover, the resistor used for the MSB is
required to handle a much greater current than that used for the LSB resistor. For example, in
a 10-bit system, the current through the MSB resistor is approximately 500 times as large as the
current through the LSB resistor. For these reasons, a second type of resistive network, called a
ladder, has been developed.
12.2 Binary Ladders
The binary ladder is a resistive network whose output voltage is a properly weighted sum of
the digital inputs. Such a ladder, designed for 4 bits, is shown in Figure 12.6. It is constructed
of resistors that have only two values and thus overcomes one of the objections to the resistive
divider previously discussed. The left end of the ladder is terminated in a resistance of 2R, and
we shall assume for the moment that the right end of the ladder (the output) is open-circuited.
Let us now examine the resistive properties of the network, assuming that all the digital inputs
are at ground. Beginning at node A, the total resistance looking into the terminating resister is 2R.

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