What is a transparent cell or structure in electronics? Most of us especially our colleagues involved in data converters, deal with such objects without noticing their transparent characters. The best example is the famous R-2R ladder one finds in many ADC and DACs. How the R-2R is built and are there other similar circuits?
What is a “transparent” cell? It is a structure, like clean glass, where you can look through it without any distortion. The “glass” thickness can be infinite, as depicted in the above figure! In electronics, the transparency character is the termination resistance RL seen from a source through a structure. Let’s illustrate it in our famous R-2R ladder in the figure below. Seen from the left, one sees always RL even when the number of cells is endless! But can that work without any restriction (i.e. RL value, cell structure)? Unfortunately no. There are conditions linking the values of RL and the transparent cell.
Even though the discussion can be very theoretical, there is a very well-known transparent structure called the R-2R ladder. It has fantastic useful characteristics such as the ability to produce binary-powered voltages that are used in DACs and ADCs. More about this later. What are the R-2R ladder characteristics? Let’s see more in detail the construction of the R-2R ladder: by putting R1=R, R2=2R, and terminating with RL=2R, you obtain a circuit that has very interesting and remarkable characteristics such as:
- The input sees the same RL independently of the number of cells; thus, it can be infinite); the whole R-2R structure is de-facto “transparent”!
- The input sees always one same equivalent impedance 2R independently of the number of cells in the series.
- If a voltage E is applied at the input, voltages on nodes appear as E/2, E/4, E/8, etc. making it ideal for a binary DAC construction.
To answer the question, see below the basic cell structure R1-R2 and terminated by RL:
The cell (in red) made by R1 and R2 can be called a “transparent” cell when the total equivalent resistor seen from the left side (Req) is equal to RL. In this case, you can then stack (i.e. put in series) as many red cells as you want: Req will see the same load resistor RL! This happens only under specific conditions linking R1, R2, and RL. To establish the condition, let’s write the equivalent resistor seen from the input. One can observe RL is in parallel with R2, then in series with R1.
Req = R2*RL/(R2+RL) + R1 and this must be equal to RL again. The general equation (and condition) to have “transparency” from input to RL is:
R2*RL/(R2+RL) + R1 = RL
By arranging the terms: RL² - R1RL – R1R2 = 0 that can be seen as a quadratic equation with RL as unknown. By solving it: RL = R1/2 +/- (R1² +4R1R2)1/2/2 : where only the positive solution can be considered:
When R1, R2, and RL are linked as above, you obtain a “transparent” cell! And there is an infinite number of solutions! Let’s verify on our famous R-2R ladder, with R1=R and R2=2R, RL is well equal to 2R
Other (other than R-2R) practical examples of transparent cell:
- With R1=R and R2=12R we have then RL to be equal to 4R.
- With R1=R and R2=3R, RL has to be equal to 2.303*R in order to get the transparent characteristics!
Reversing the problem:
Find R1 and R2 with RL fix, and one can put the problem in a different way. Fixing RL and computing the R1 and R2.
Let’s restart from the very first equation linking R1, R2, and RL:
RL² - R1RL – R1R2 = 0
Let’s now put R2 = kR1 with k being just a ratio
We have then RL² - R1RL – kR1² = 0
By renaming R1 = R we obtain simply: RL² - RRL – kR² = 0
By re-arranging the above second order equation with R as unknown: kR² + RLR - RL²= 0
We have explained how the R-2R ladder is built and how useful characteristics it brings to us. We have also established a mathematic tool to find other similar structures. These tools and formulas allow the reader to go further and to find a circuit cable, for example, to give a voltage series such as E, E/3, E/9, etc. This is to build a different type of DACs