--- title: "Inverted R-2R DAC output and advantages" subject: "Advanced Electronics" chapter: 3 key: "semester_4/advanced_electronics:inverted-r2r-dac-advantages" source: https://www.bctengineer.com/q/semester_4/advanced_electronics/inverted-r2r-dac-advantages site: BCT Engineering --- # Inverted R-2R DAC output and advantages Advanced Electronics · Chapter 3: Digital-to-Analog and Analog-to-Digital Conversion - Asked in: 2074 Asoj, 5+2 marks; 2070 Chaitra, 2+5 marks; 2068 Chaitra, 5+2 marks (asked 3 times) - Video: [R-2R DAC and Inverted R-2R DAC](https://www.youtube.com/watch?v=lxbpErV7qVs) — Hema Malini Amirthalingam - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_4/advanced_electronics/inverted-r2r-dac-advantages - Whole chapter: https://www.bctengineer.com/md/semester_4/advanced_electronics/chapter-3 ## Question Derive the expression for output voltage in inverted R-2R ladder type Digital to Analog converter (DAC). What are the advantages of R-2R type DAC? ## Short answer In the inverted R-2R ladder, each 2R leg current is I/2^k, and bits switch currents to virtual ground or ground. The output is Vo=−(Rf/R)Vref(b1/2+b2/4+...+bn/2^n)=−(Rf/R)Vref·D/2^n. Advantages: only two resistor values, easy expansion, constant currents, high speed, and good accuracy. ## Answer An **inverted (current-mode) R-2R ladder DAC** uses only two resistor values, R and 2R. Each 2R leg is switched either to the virtual ground of the op-amp (bit = 1) or to real ground (bit = 0). Since both positions are at 0 V, the ladder currents never change. ```text Vref R R R o--+--/\/\--+--/\/\--+--/\/\--+-----+ | | | | | 2R 2R 2R 2R 2R | I/2 | I/4 | I/8 | I/16 | [S1] [S2] [S3] [S4] GND (MSB) (LSB) bit = 1: switch to Iout bus bit = 0: switch to GND Rf Iout +--/\/\--+ bus --+--(-) | >---+--- Vo GND ---(+) ``` #### Derivation (n-bit) 1. Looking right from any node, the resistance is 2R in parallel with 2R = R, plus R in series gives 2R again. So the total resistance seen by Vref is R, and ```text I = Vref / R ``` 2. At each node the current splits equally between the 2R leg and the rest of the ladder (both are 2R to ground potential). So the leg currents are ```text I1 = I/2, I2 = I/4, I3 = I/8, ... In = I/2ⁿ ``` 3. Bit bk = 1 sends Ik to the summing bus (virtual ground); bk = 0 sends it to ground. ```text Iout = (Vref/R) [b1/2 + b2/4 + b3/8 + ... + bn/2ⁿ] ``` 4. The op-amp converts this current to voltage: ```text Vo = -Rf · Iout = -(Rf/R) · Vref · [b1·2⁻¹ + b2·2⁻² + ... + bn·2⁻ⁿ] = -(Rf/R) · Vref · D/2ⁿ ``` where D is the decimal value of the input (b1 = MSB). With Rf = R, 4-bit, Vref = 10 V and input 1010: Vo = −10 × 10/16 = −6.25 V. #### Advantages of the R-2R type DAC - **Only two resistor values** (R and 2R), easy to make and match accurately in ICs, regardless of the number of bits. - **Easily expanded** to more bits by adding R-2R sections. - **Constant current in each leg** and constant load on Vref (inverted type), since switches only move current between two 0 V points. - **High speed and fewer glitches:** node voltages do not change, so stray capacitances are not charged and discharged. - Good accuracy and temperature tracking; resistor values stay in a practical range (unlike the weighted-resistor DAC). ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. In the inverted R-2R DAC, why do the ladder currents not change with the input code? A. Each switch connects to Vref or ground B. Each 2R leg goes to virtual ground or real ground, both at 0 V C. The op-amp has infinite gain only for bit 1 D. The ladder is open-circuited for bit 0 Answer: B. Both switch positions are at 0 V, so the currents are only steered, not changed. (Key point: Switch positions) 2. What is the resistance seen by Vref in the inverted ladder, and the total current? A. 2R, so I = Vref/2R B. R/2, so I = 2Vref/R C. R, so I = Vref/R D. 2R, so I = Vref/R Answer: C. At each node 2R||2R = R, plus series R gives 2R, repeating to a total of R. (Key point: Ladder resistance) 3. The output of an n-bit inverted R-2R DAC is: A. Vo = (Rf/R) Vref D x 2^n B. Vo = -(R/Rf) Vref D/2^n C. Vo = -Vref D/2^(n-1) D. Vo = -(Rf/R) Vref D/2^n Answer: D. Iout = (Vref/R) D/2^n and Vo = -Rf Iout. (Key point: Output expression) 4. Which is an advantage of the R-2R DAC over the weighted resistor DAC? A. Only two resistor values, R and 2R, regardless of bits B. It needs resistors up to 2^(n-1) R C. It needs more precise MSB resistors D. It is slower because nodes charge Answer: A. Two values are easy to match in ICs and the ladder is easily extended. (Key point: Advantages) 5. Why is the inverted ladder fast with few glitches? A. Its resistors are very small B. Node voltages stay constant, so stray capacitances are not charged and discharged C. It uses no op-amp D. The clock is faster Answer: B. Constant node voltages remove charging delays. (Key point: Speed) ## Close variants set in other exams - [Non-inverted R-2R ladder DAC output](https://www.bctengineer.com/question.md?key=semester_4%2Fadvanced_electronics%3Anon-inverted-r2r-dac) — Advanced Electronics, chapter 3, asked 4× ## Related in Advanced Electronics - [4-bit inverted R-2R DAC and why better](https://www.bctengineer.com/question.md?key=semester_4%2Fadvanced_electronics%3Ainverted-r2r-4bit-better) — Advanced Electronics, chapter 3 - [Design 4-bit R-2R DAC; unipolar vs bipolar](https://www.bctengineer.com/question.md?key=semester_4%2Fadvanced_electronics%3Adesign-4bit-r2r-unipolar-bipolar) — Advanced Electronics, chapter 3 - [5-bit R-2R DAC output for 10110](https://www.bctengineer.com/question.md?key=semester_4%2Fadvanced_electronics%3Ar2r-5bit-10110) — Advanced Electronics, chapter 3 - [4-bit R-2R ladder bit voltages and 1011 output](https://www.bctengineer.com/question.md?key=semester_4%2Fadvanced_electronics%3Ar2r-4bit-12v-1011) — Advanced Electronics, chapter 3 - [Current-mode R-2R preference; bipolar DAC](https://www.bctengineer.com/question.md?key=semester_4%2Fadvanced_electronics%3Acurrent-mode-r2r-bipolar-dac) — Advanced Electronics, chapter 3 - [Bipolar DAC output and 3-bit transfer curve](https://www.bctengineer.com/question.md?key=semester_4%2Fadvanced_electronics%3Abipolar-dac-transfer-curve) — Advanced Electronics, chapter 3 ## Related in other subjects - [R-2R ladder DAC versus weighted resistor](https://www.bctengineer.com/question.md?key=semester_4%2Finstrumentation_i%3Ar-2r-ladder-dac) — Instrumentation I, chapter 5