Page 224 - DCAP108_DIGITAL_CIRCUITS_AND_LOGIC_DESIGNS

Unit 12: A/D and D/A Converters

the binary ladder are summarized in Figure. 12.10 notice that each digital input is transformed Notes
into a properly weighted binary output voltage.
12.4:
What are the output voltages caused by each bit in a 5-bit ladder if the input levels are 0 = 0V
and l = +l0 V?
Solution:

The output voltages can be easily calculated by using Figure. 12.10. They are
V
First MSB V = = +10/2 = +5V
A 2
V
Second MSB V = = +10/4 = +2.5V
A 4
V
Third MSB V = = +10/8 = +1.25V
A 8
V
Fourth MSB V = = +10/16 = +0.625V
A 16
V
LSB = fifth MSB V = = +10/32 = +0.3125V
A 32
Since this ladder is composed of linear resistors, it is a linear network and the principle of
superposition can be used. This means that the total output voltage due to a combination of input
digital levels can be found by simply taking the sum of the output levels caused by each digital
input individually.
Figure 12.10: Binary Ladder Output Voltages

Bit position Binary weight Output voltage
MSB 1/2 V/2
2nd MSB 1/2 V/4
3rd MSB 1/2 V/8
4th MSB 1/2 V/16
5th MSB 1/2 V/32
6th MSB 1/2 V/64
7th MSB 1/2 V/128
• • •
• • •
• • •
Nth MSB 1/2 N V/2 N

In equation form, the output voltage is given by
V V V V V
V = + + + + + .... ...(12.2)
A 2 4 8 16 2 n
where n is the total number of bits at the input.

This equation can be simplified somewhat by factoring and collecting terms, the output voltage
can then be given in the form
1
2
3
0
V 2 + V 2 + V 2 + V 2 + ..... + V 2 n − 1
V = 0 1 2 3 n − 1 ...(12.3)
A n
2
where V , V , V , V ,…….,V n – 1 are the digital input voltage levels. Equation (12.3) can be used to
1
3
2
0
find the output voltage from the ladder for any digital input signal.
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