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 R2R ladder one finds in many ADC and DACs. How the R2R is built and are there other similar circuits?
About the Transparent Character of a Structure
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 R2R 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.
Why is the Transparent Cell Important?
Even though the discussion can be very theoretical, there is a very wellknown transparent structure called the R2R ladder. It has fantastic useful characteristics such as the ability to produce binarypowered voltages that are used in DACs and ADCs. More about this later. What are the R2R ladder characteristics? Let’s see more in detail the construction of the R2R 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 R2R structure is defacto “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.
How to find other Transparent Cells?
To answer the question, see below the basic cell structure R1R2 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 R2R ladder, with R1=R and R2=2R, RL is well equal to 2R
Other (other than R2R) 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:
R_{L}²  R_{1}R_{L} – R_{1}R_{2} = 0
Let’s now put R_{2} = kR_{1} with k being just a ratio
We have then R_{L}²  R_{1}R_{L} – kR_{1}² = 0
By renaming R_{1} = R we obtain simply: R_{L}²  RR_{L} – kR² = 0
By rearranging the above second order equation with R as unknown: kR² + R_{L}R  R_{L}²= 0
Conclusion
We have explained how the R2R 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
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