Digital-to-Analogue Converters, or DAC’s as they are more commonly known, are the ‘opposite of the Analogue-to-Digital Converters we looked at in a previous tutorial. DAC’s convert binary or non-binary numbers and codes into analogue ones with its output voltage (or current) being proportional to the value of its digital input number. For example, we may have a 4-bit digital logic circuit that ranges from 0000 to 11119, (Oto Fg) which a DAC converts to a voltage output ranging from 0 to 10V. Converting an “n’-bit digital input code into an equivalent analogue output voltage between 0 and some Viyax value can be done ina number of ways, but the most common and easily understood conversion methods uses a weighted resistors and a summing, amplifier, or a R-2R resistor ladder network and operational amplifier. Both digital-to- analogue conversion methods produce a weighted sum output, with the weights set by the resistive values used in the ladder networks contributing a different “weighted” amount tothe signals output. We saw in our tutorial section about Operational Amplifiers that an inverting amplifier Uses negative feedback to reduce its open-loop gain, Ao, and does so by feeding back a fraction of its output signal back to the input. We also saw that the input voltage Vi is connected directly to its inverting input via a resistor Riy and that the inverting amplifiers closed-loop voltage gain, Ayici) is determined by the ratio of these two resistors as shown.Inverting Operational Amplifier Circuit Then we can see that Vouris given as Vix mul by the closed-loop Gain (Ac), which is determined by the ratio of the feedback resistance, Rs to the input resistance, Riy.So by altering the values of either Rr or Riy we can change the closed-loop gain of the op-amp and therefore the value of Vout (Ir"Ry) for a given input signal. Here in this inverting operational amplifier example we have used a single input voltage signal, but what if we added another input resistor to combine two or more analogue signals into a single ‘output, what would be the effect on the circuit and its gain.ital-to-Analogue Converter Summing Am By connecting multiple inputs to the negative terminal of the operational amplifier, we can convert the single input circuit from above into a summing amplifier or to be more precise, “summing inverting voltage amplifier” circuit. As the negative feedback created by the feedback resistor, Rr biases the inverting input of the op-amp at zero potential, any input signals are effectively electrically isolated from each other with the output being the inverted sum of all the input signals combined. Thus summing amplifier in the inverting mode produces the negative sum of any number of lifter would produce the positive sum input voltages, whereas a no-inverting summing am of any number of input voltages. Consider the circuit below. Inverting Summing Amplifier Circuit Ie Re ov! V le tees.Inthe summing amplifier circuit above, the output voltage, (Vout) is proportional to the sum of the four input voltages, Vina, Vinz, Vina. and Ving and we can modify the original equation for the inverting amplifier configuration above to take account of these four new input values as follows: a+ +1+t, = Mis Me oR R. R, R. R, R, Moar = TE=Vin == REVin EM REM eM Vea = Be (Mra * Vina Vow Via) ‘Then we can see that the output voltage is an inverted, scaled sum of the four input voltages as each input voltage is multiplied by its corresponding gain and added to the next to produce the total output. Ifall the resistances are the same and of an equal value, that is: Re = Ry = Ro = Rg = Rg, then each input channel will have a closed-loop voltage gain of unity (1) so the output voltage is given simply by: Vout = ~(Vina + Vinz + Vina + Vina)If we now assume that the four inputs of the summing amplifier are binary inputs with voltage values of either 0 or 5 volts (LOW or HIGH, 0 or 1) and we double the resistive values of each input resistor with regards to the previous one, we can produce an output ion which would be the weighted sum of these four input voltages creating the basic circuit for a 4-bit binary weighted digital-to-analogue converter, or 4-bit weighted D/Aconverter. con Labelling the four summing inputs as A, B, C, D and making Rr = 1kO, with the four input resistors ranging from 1kQ to 8kQ (or multiples thereof), we can construct a simple 4-bit binary weighted analogue-to-digital converter circuit as shown. 4-bit Binary Weighted Digital-to-Analogue Converter (M58) Rg= 1k ea Re= ko ead vo Digtal =e : ee ee 0 | tage veo AA foe we (use) For a 4-bit binary number there are 2* = 16 possible combinations or A, B, C, and D ranging from 0000, to 1111 which corresponds to decimal O to 15 respectively. If we an8-4- make the weight of each input bit double with respect to the other, we end up’ 2-1 binary code ratio corresponding to 23, 22, 2+ and 2°.So if we set the "D" input resistance at 1kQ, the “C” input resistance at 2kA (that is the double of D), the “B” input resistance at 4kQ (double C), and the "A’ input resistance at 8kQ (double B), with the feedback resistance Re set again at 1kO, then the transfer characteristic of the 4-bit binary weighted digital-to-analogue converter would be: 4-bit DAC Transfer Characteristic Rote Reve tee Vz +a= 1kQ Uy, eA y | Udy , Ike. Tk"? 2k CO" 4D FB” BEN Vet Vat 1kQ 1kQ = ~[ 10+ 5¥et ¥e* 3% | Sowe can see that ifa TTL voltage of +5 volts (logic 1) is applied to the summing amplifiers input, Vp which represents the most significant bit (MSB), the op-amp's gain will be Rp/Rg = 1kO/1kO = 4 (unity). Thus with a 4-bit binary code of 1000 applied, the output of the digital-to-analogue converter circuit will be -5 volts. Likewise, if +5 volts (logic 1) is applied to the summing amplifiers input Vc, the op-amp's gain will be Rp/Rg = 1kO/2kO = 4/2 (one half). So the 4-bit binary code of 0100 would produce an analogue output voltage of -2.5 volts. Again with a logic “1” applied to the summing amplifiers input Vp, the op-amp's gain will be Re/Rp = 1kO/4kA = 1/4 (one quarter) with the 4-bit binary code of 0010 producing an ‘output voltage of -1.25 volts, and finally a logic “1” applied to the summing amplifiers input, Va which represents the least significant bit (LSB), the op-amp's gain will therefore be Re/Ry = 1kO/8k0 = 1/8 (one eighth) with the 4-bit binary code of 0001 producing an ‘output voltage of -0.625 volts, (a 12.5% resolution). The resolution of this simple 8-4-2-1 binary weighted digital-to-analogue converter will produce an output voltage change of 0.625 volts per 1-bit change in the binary number, and we can express this output voltage change in the following table.4-bit Binary Weighted D/A Converter Output Poy a pic|s DV + YeVet UVa YeVa inVotts o}ol}o 05 +05 +05 +05 ° o}ol}o OS +05 10'S 4% 0.625 o}o}a OS +05 +5405 1.25 o}o}a OS 4054/25 +S 1.875 o}1}o 05 +4,°5+0°5 40'S -250 o}1}o 054,95 40'S +S -3.125 o}a}a OS 4,954 6540S -375 ofa}a OSHS HUSH N'S 4.375 1}ojo 1°5+0°5 +05 +0°5 -5.00 1}ojo 1540°540°5 +5 5.625 1}o|a 1540541 5+0°S 625 a}ol4 USHOS HUES HAS 6875 1}ajo 154,55 +0°5+0°5 -750 1}ajo 154 U5 4051/95 8.125qa aa lo A545 +425 +05 -875 aoa als PSH USN ES HES -9375 Where the output voltages are all negative due to the inverting input of the summing amplifier. By increasing the number of binary digits and the resistive summing network so that each resistor has a different weighting, the resolution of the analogue output voltage for a binary weighted digital-to-analogue converter can be increased. For example, an 8-bit DAC with TTL +5 inputs would produce a resolution of 0.039 (1/128*V) volts, while a 12- bit DAC would be 0.00244 (1/2048*V) volts per step (1. LSB) change of the input binary (or non-binary) code. Clearly then the disadvantage here is that a binary weighted resistor DAC requires a large range of high precision resistors (one per bit) for an “n’-bit DAC making practical (and expensive) for converters with more than a just a few bits of resolution. But we can expand on this i lea of a binary weighted di uses different value resistors one step further by converting it into a R-2R resistor ladder DAC which requires only two precision resistance values, namely Rand 2R. n which I-to-analogue circuit configuratiR-2R Resistive Ladder Network As its name implies, the “ladder” description comes from the ladder-like configuration of the resistors used the network. A R-2R resistive ladder network provides a simple means of converting digital voltage signals into an equivalent analogue output. Input voltages are applied to the ladder network at various points along its length and the more input points the better the resolution of the R-2R ladder. The output signal as a result of all these input voltage points is taken from the end of the ladder which is used to drive the inverting input of an operational amplifier. ladder networl Then a R-2R resis nothing more than long strings of parallel and series connected resistors acting as interconnected voltage dividers along its length, and whose output voltage depends soley on the interaction of the input voltages with each other. Consider the basic 4-bit R-2R ladder network (4-bits because ithas four input points) below. 4-bit R-2R Resistive Ladder NetworkThis 4-bit resistive ladder circuit may look complicated, but its all about connecting resistors together in parallel and series combinations and working back to the input source using simple circuit laws to find the proportional value of the output. Lets assume all the binary inputs are grounded at O volts, that is: Vq = Vg = Vc = Vp = OV (LOW). The binary code corresponding to these four inputs will therefore be: 0000. Starting from the left hand side and using the simplified equation for two parallel resistors and series resistors, we can find the equivalent resistance of the ladder network as: Resistors Ry and R2 are in “parallel” with each other but in “series” with resistor R. Then we can find the equivalent resistance of these three resistors and call it Ra for simplicity (or any other form of identification you want). [ OROR © ——— =R+R=2R 2R+2RThen Ra is equivalent to “2R”. Now we can see that the equivalent resistance "Ra" isin parallel with R4 with the parallel combination in series with Rs. Parallel Circuit Again we can find the equivalent resistance of this combination and call it Ra. Ladder Resistor Rb ‘So Rg combination is equivalent to "2R", Hopefully we can see that this equivalent resistance Rg isin parallel with Rg with the parallel combination in series with Ry as shown.As before we find the equivalent resistance and call it Rc. R,xR 2Rx 2R R,=R,+>2 —6§ =R+ > =R+R=2R R,+R, 2R+2R Again, resistor combination Rc is equivalent to “2R" which is in parallel with Rg as shown. Ro RSA WSR) F- Parallel Circuit‘As we have shown above, when two equal resistor values are paralled together, the resulting value is one-half, so 2! parallel with 2R equals an equivalent resistance of R. So the whole 4-bit R-2R resistive ladder network comprising of individual resistors connected together in parallel and series combinations has an equivalent resistance (Req) of *R" when a binary code of *0000" is applied to its four inputs. ‘Therefore with a binary code of “0000” applied as inputs, our basic 4-bit R-2R digital-to- analogue converter circuit would look something like thi: R-2R DAC Circuit with Four Zero (LOW) Inputs ‘The ouput voltage for an inverting operational amplifier is given as: (Rr/Riy)"Vin. If we make Rr equal to R, that is Rr= R= 1, and as R is terminated to ground (OV), then there is no Vin voltage value, (Vix = 0) so the output voltage would be: (1/1)*0 = O volts. So for a4- bit R-2R DAC with four grounded inputs (LOW), the output voltage will be “zero” volts, ‘thus a 4-bit digital input of 0000 produces an analogue output of 0 volts. So what happens now if we connect input bit Va HIGH to +5 volts. What would be the equivalent resistive value of the R-2R ladder network and the output voltage from the op- amp.R-2R DAC with Input Va, Vars5v Input Va is HIGH and logic level “1” and all the other inputs grounded at logic level “0”. As the R/2R ladder network isa linear circuit we can find Thevenin’s equivalent resistance using the same parallel and series resistance calculations as above to calculate the expected output voltage. The output voltage, Vouris therefore calculated at 312.5 mi volts (312.5 mV). ‘As we havea 4-bit R-2R resistive ladder network, this 312.5 mV voltage change is one- sixteenth the value of the +5V input (5/0.3125 = 16) voltage so is classed as the Least Significant Bit, (LSB). Being the least significant bit, input Va will therefore determine the it digital-to-analogue converter, as the smallest voltage inputs. “resolution’ of our simple 4+ change in the analogue output corresponds toa single step change of the di Thus for our 4-bit DAC this will be 312.5mV (1/16th) for a +5V input. Now lets see what happens to the output voltage if we connect input bit Vs HIGH to +5 volts.R-2R DAC with Input Vg, With input Vg HIGH and logic level “1” and all the other inputs grounded at logic level “O" the output voltage, Vour is calculated at 625m, and which is one-eighth (1/8th) the value of the +5V input (5/0.625 = 8) voltage. We can also see that itis double the output voltage when only input bit Vq was applied, and we would expect this asits the 2,, bit (input) so has double the weighting of the 1, bit. Now lets see what happens to the output voltage if we connect input bit Vc HIGH to +5 volts. R-2R DAC with Input Vc With input Vc HIGH and logic level “1” and the other input bits at logic level “O’, the output voltage, Vout is calculated at 1.25 volts, and which is one-quarter (1/4) the value of the +5V input (5/1.25 = 4) voltage. Again we can see that this voltage is double the output of input bit Vp but also 4 times the value of bit Va. This is because input Vcis the 3,qbit so has double the weighting of the 2,4 bit and four times the weighting of the 1. bit. Finally lets see what happens to the output voltage if we connect input Vp HIGH to +5 volts.R-2R DAC with Input Vp, With only input Vp HIGH and logic level "1” and the other inputs at logic level "0", the output voltage, Vouris calculated at 2.5 volts. This is on-half (1/2) the value of the +5V input (5/2.5 = 2) voltage. Again we can see that this voltage is double the output of input bit Vc,4 times the value of bit Vz and 8 times the value of input bit Va ast is the 4,, bit and therefore classed as the Most Significant Bit, (MSB). Then we can see that ifi put V represents the LSB and therefore controls the DAC’s resolution, and input Vg is double Va, input Vc is four times greater than Va, and input Vp is eight times greater than Va, we can obtain a relationship for the analogue output voltage of our 4-bit digital-to-analogue converter with the following equation:Digital-to-Analogue Output Voltage Equation vy. — Yat 2Vp +4Vo +8Vp our = 8 Where the denominator value of 16 corresponds to the 16 (24) possible combinations of inputs to the 4-bit R-2R ladder network of the DAC. We can expand this equation further to obtain a generalised R-2R DAC equation for any number of digital inputs for a R-2R D/A converter as the weighting of each input bit will always be referenced to the least significant bit (LSB), giving us a generalised equation of: Generalised R-2R DAC Equation + en ete + 4V, + 8Vp + 16V, + 32! F Where: “n’ represents the number of digital inputs within the R-2R resistive ladder network of the DAC producing a resolution of: Vise = Vin/2". Clearly then input bit Va when HIGH will cause the smallest change in the output voltage, while input bit Vp when HIGH will cause the greatest change in the output voltage. The expected output voltage is therefore calculated by summing the effect of all the in input bits which are connected HIGH. ualIdeally, the ladder network should produce linear relationship between the input voltages and the analogue output as each input will have a step increase equal to the LSB, we can create a table of expected output voltage values for all 16 combinations of the 4 inputs with +5V representing a logic"1” condition as shown. 4-bit R-2R D/A Converter Output pi[clela (8Vp+4°Ve+2°Va+ 1°Va)/24 inVolts Gilotlo lo (05 +0'5 +0°5 +0°5)/16 ° o}o}o|1 (05 +05 +0" + 1°5)/16 0.125 o}o}ajo (05 +05 +2°5+0°5/16 0.6250 ojo}als (os+orse2's+ 15/16 0.9375 o}1a}oljo (05+ 4°5+0°5+0°5/16 1.2500 ofa}o|. (sas 405+ 15/16 15625 Ota | 0 (05 +45 +2°5+0°5/16 1.8750 Oat a |e (ossarseasersyi6 2.1875 1}olojo (eS +05+0'5+0°5)/16 25000 afojo}s (e5+05+0r5+1°5/16 2.8125 1folajo (e5 +05 +2°5+0°5)/16 3.1250 afoja}a (505425 41°5/16 3.4975 1fajojo (85 +4°5 +0°5+0°5)/16 3.7500 afajo}a (85 +4°5 +0"5 + 1°5)/16 4.0625afajofa (erssarssorsersyi6 4.0625 afalafo (er544r5+2°5+0°5/16 43750 afajafa (ers +ar5 42541516 46875 Notice that the full-scale analogue output voltage for a binary code of 1111 never reaches the same value as the digital input voltage (+5V) but is less by the equivalent of one LSB bit, (312.5mV in this example). However, the higher the number of digital input bits (resolution) the nearer the analogue output voltage reaches full-scale when all the input bits are HIGH. Likewise when all the input bits are LOW, the resulting lower resolution of LSB makes Vout closer to zero volts. R-2R Digital-to-Analogue Converter Now that we understand what a R-2R resistive ladder network is and how it works, we can use it to produce a R-2R Digital-to-Analogue Converter. Again using our 4-bit R-2R ive ladder network from above and adding it to an inverting operational amplifier circuit, we can create a simple R-2R digital-to-analogue converter of:R-2R Digital-to-Analogue Converter Now that we understand what a R-2R resistive ladder network is and how it works, we can it R-2R resistive ladder network from above and adding it to an inverting operational amplifier circuit, we can create a simple R-2R digital-to-analogue converter of: use it to produce a R-2R Digital-to-Analogue Converter. Again using our 4- R-2R Digital-to-Analogue Converter (158) Va Ve Ve Vo (MSB) The digital logic circuit used to drive the D/A converter can be generated by combinational or sequential logic circuits, data registers, counters or simply switches. The interfacing of a R-2R D/A converter of “n’-bits will depend upon its application. Al boards such as the Arduino or Raspberry Pi have digital-to-analogue converters bui make interfacing and programming much easier. There are many popular DAC’s available such as the 8-bit DACO808. -one so