What is a 3-bit Digital to Analogue Converter and How Does it Work?

[Michael Neo]

Homework Statement

For the digital to analogue convertor circuit shown below and taking C as the most significant bit, construct a table of all the possible output voltages given a supply voltage of 24 volts.
Show all calculations.

Relevant Equations

None.

There is an example of a two bit DAC in the notes but not a three bit DAC.

Three digital inputs results in 8 analogue outputs:

2³ = 8

Digital Inputs_________Analogue Outputs
CBA
000____________________0
001____________________1/12 x 24 = 2 V
010____________________2/12 x 24 = 4 V
011____________________4/12 x 24 = 8 V
100____________________
101____________________
110____________________
111____________________

The incomplete table is included in the notes.
I obviously understand the fraction must be multiplied by the supply voltage.
The 2 bit DAC in the notes is solved by using circuit analysis and constructing equivalent pieces of the circuit.
That is straightforward but the fractions they derive from it aren't explained clearly.
Any advice to would be much appreciated.

I have searched online but there are no R/2R circuits like this.

 

[Michael Neo]

A small piece of advice to get me started would be much appreciated!

 

[berkeman]

Well, I would just make the table and figure out what the equivilent resistances are from point "D" to 24V and Ground. I haven't tried it yet, but that's how I'd approach it.

I'd also probably fill in the table for the entries 000, 111, 100, 010, and 001 first, to get a feel for how the other entries would probably play out...

 

[Michael Neo]

I thought this method might work, but when you sub the numbers into the formula it doesn't match the outputs in the table above.

https://uk.tek.com/blog/tutorial-digital-analog-conversion-–-r-2r-dac

 

[DaveE]

The 011 value in your table is incorrect. This is easy to see if you have studied superposition in linear circuits. Basically, in a linear circuit with an output function f(x), then f(a+b) = f(a) + f(b), where a and b represent different inputs to the circuit. That could be two different values at a single input, or it could also be the output from two different inputs.

The key insight into the analysis of all forms of the R-2R ladder circuits is that the resistance seen looking away from any of the central nodes is the same fixed value. This applies for either direction from the node. After you have figured that out the analysis becomes a simple exercise in voltage dividers.

 

[berkeman]

But as you mentioned before, this R-2R ladder DAC has a slightly different configuration than normal... Can you show us the values you got (and the resistor pullup/pulldown numbers that led to the output voltages)?

 

[Michael Neo]

Thank you both for taking the time to reply.

The three bit DAC is exactly the same as the one in the problem statement.

Here are the notes:

 

[Michael Neo]

@DaveE

The table is from the notes.

Ah!

In binary, the sequence in the table is 1, 2, 4.

3 has been ommitted.

Since they increase by a fixed amount, 011 (i.e. 3) gives 3/12 of V output.