10. Chapter: A/D and D/A converter principles

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1 Punčochář, Mohylová: TELO, Chapter 10: A/D and D/A converter principles Chapter: A/D and D/A converter principles Time of study: 6 hours Goals: the student should be able to define basic principles of D/A converters define basic principles of A/D converters describe aliasing use a multiply D/A converter for a control of circuit properties Text Basic considerations - ADC The devices which perform the interfacing function between analog and digital worlds are analog-to-digital (A/D or ADC) and digital-to-analog (D/A or DAC) converters, which together are known as data converters. The input to the system is a physical parameter such as temperature, pressure, flow, acceleration, and position, which are analog quantities. The parameter is first converted into an electrical signal by means of a transducer, once in electrical form, all further processing is done by electronic circuits x IN (t) Fig.1. An Analog to Digital converter is an electronic circuit which accepts an analog input signal (usually a voltage V(t)) and produces a corresponding digital number at the output see Fig.1. The resultant digital word goes to a computer data bus or to the input of a digital circuit. The analog-to-digital converter requires a small amount of time to perform the quantizing and coding operations. The time required to make the conversion depends on: the converter resolution, the conversion technique, and the speed of the components employed in the converter. The conversion speed required for a particular application depends on the time variation of the signal to be converted and on the accuracy desired. Aperture time - refers to the time uncertainty (or time window) in making a measurement and results in an amplitude uncertainty (or error) in the measurement if the signal is changing during this time Fig. 2.

2 Punčochář, Mohylová: TELO, Chapter 10: A/D and D/A converter principles 2 x IN (t) Antialiasing filtr LP x LP (t) Sampling x s (t) Quantizing x q (t) Coding (number) A/D converter (ADC) x(n) Digital signal y(n) y(t) Output filter processing DAC LP Fig. 1: Possible block diagram of digital signal processing Fig. 2: Aperture time T A and amplitude uncertainty For the specific case of a sinusoidal input signal, the maximum rate of change occurs at the zero crossing of the waveform, and the amplitude error is d V = T A ( Vm sin ωt) = TA ( Vmω cos ωt) = ω = 0 = TAVmω dt The resultant error as a fraction of the peak to peak full scale value is ε V 2V A m A m = = = = π m T V ω 2V m T V 2π f 2V m f T A From this result the aperture time required to digitize a 1 khz signal to 10 bits resolution can be found. The resolution required is one part in 2 10 or approximately 0.001, thus

3 Punčochář, Mohylová: TELO, Chapter 10: A/D and D/A converter principles 3 T A ε = π f 0,001 9 = ,14 10 The result is a required aperture time of just 320ns! It is evident that it is hard to find a 10-bit A/D converter to perform this conversion at any price! Fortunately, there is a simple and inexpensive way around this dilemma by using a sample-hold circuit Fig. 4. The aperture time of the A/D converter is therefore greatly reduced by the much shorter aperture time of the sample-hold circuit. In turn, the aperture time of the sample-hold is a function of its bandwidth and switching time. All signal sampling we can see in Fig. 3. SIGNAL SAMLING PULSES SAMLED SIGNAL SAMLED AND HELD SIGNAL Fig. 3: Signal sampling x(t) LP filter Sample and Hold Quantizing Coding The electrical signal is sampled (a sample-hold circuit acquires the signal voltage and then holds its value while an analog-to-digital converter converts the 2 B n value into digital form) Encoder x(n) 1 Quantizing is the process of transforming an analog signal into a set of discrete output states - there are 2 n -1 analog decision 1 points (or threshold levels) in the transfer function ( n number of the quantizer bits) Each threshold level corresponds to the number (code). Fig. 4: 3: Once more ADC The electrical signal is sampled (a sample-hold circuit acquires the signal voltage and then holds its value while an analog-to-digital converter converts the value into digital form)

4 Punčochář, Mohylová: TELO, Chapter 10: A/D and D/A converter principles 4 Quantizing is the process of transforming an analog signal into a set of discrete output states - there are 2 n -1 analog decision points (or threshold levels) in the transfer function ( n number of quantizer bits) Each threshold level corresponds to the number (code). LP filter - a low pass active filter which reduces high frequency signal components, unwanted electrical interference noise, or electronic noise from the signal; so-called antialiasing filter; its characteristics frequency must be ½ of sampling frequency f s. Sampling (sampling theorem, Nyquist theorem): If a continuous bandwidth-limited signal contains no frequency components higher than f c, then the original signal can be recovered without distortion if it is sampled at a rate of at least 2 f c samples per second, thus f s 2 f c Aliasing in the time domain Fig. 5 f c False signal alias frequency Fig. 5: Aliasing in the time domain; f s < f c 0 T 2T 3T 4T 5T 6T 7T

5 Punčochář, Mohylová: TELO, Chapter 10: A/D and D/A converter principles 5 Aliasing spectrum Fig. 6 f s -f c (a) -2f s - f s 0 f c f s 2f s FREQUENCY FOLDING (b) - f s 0 f c f s 2f s Fig. 6: a) right sampling f s > 2f c ; practically at least 5x higher b) bad sampling f s < 2f c From the figure 6, if the sampling rate is increased such that f S - f C > f C, then the two spectra are separated and the original signal can be recovered without distortion. This demonstrates the result of the Sampling Theorem that f S >2f C. Frequency folding (aliasing) can be eliminated in two ways: first by using a high enough sampling rate, and second by filtering the signal before sampling to limit its bandwidth to f S /2 antialiasing filter. Quantizer, coding At any part of the input range of the quantizer, there is a small range of analog values within which the same output code word is produced. This small range is the voltage difference between any two adjacent decision points and is known as the analog quantization size, or quantum, q it is found in general by dividing the full scale analog range (FSR) by the number of output states. FSR is defined by the applied reference voltage V. Q is the smallest analog difference which can be resolved, or distinguished, by the quantizer (the quantization step; quantum - analog). q = V 2 n 1 V 2 n For a given analog input value to the quantizer, the output error will vary anywhere from 0 to ±q/2; the error is zero only at analog values corresponding to the code center points. This error is also frequently called quantization uncertainty or quantization noise. The quantizer output can be thought of as the analog input with quantization noise added to it. The noise has a peak-to-peak value of q but, as with other types of noise, the average value is zero. Its RMS value, however, is useful in analysis and can be computed from the triangular waveshape to be q/(2. 3). The most popular code is natural binary, or straight binary, which is used in its fractional form to represent a number

6 Punčochář, Mohylová: TELO, Chapter 10: A/D and D/A converter principles 6 N = a a2 2 + L a n n where each coefficient a i assumes a value of zero or one and N has a value between zero and one. The binary code word therefore represents the decimal fraction (1x0.5) + (1x0.25) + (0x0.125) + (1x0.0625) + (0x ) + (1x ) = or % of full scale for the converter. If full scale (V ) is +10V, then the code word represents V. The leftmost bit has the most weight, 0.5 of full scale, and is called the most significant bit, or MSB; the rightmost bit has the least weight, 2-n of full scale, and is therefore called the least significant bit, or LSB. The bits in a code word are numbered from left to right from 1 to n. The LSB has the same analog equivalent value as q discussed previously, namely LSB(analogvalue) = q = V 2 n 1 V 2 n Table 1 and Table 2 are useful summaries of the resolution, number of states, and LSB weights. Table 1: V n LSB (q) 1.00V mv 1.00V µv 2.00V mv 2.00V mv 2.00V µv 2.048V mv 2.048V µv 4.00V mv 4.00V mv 4.00V µv

7 Punčochář, Mohylová: TELO, Chapter 10: A/D and D/A converter principles 7 Table 2: Resolution = 2 n -1 [n = number of bites] error (FSR) n 2 n 1bit ppm [1x10-6] 8bits bits bits bits bits bits bits bits bits The dynamic range DR(dB) of a data converter in db is found as follows: n DR(dB) = 20log 2 = 20nlog 2 = 20n(0,3010) = 6, 02 n A 12-bit converter, for example, has a dynamic range of 72.2dB. Basic considerations - DAC A Digital to Analog converter is an electronic circuit which accepts a digital number at its input and produces a corresponding analog signal (usually a voltage) at the output. There are different DACs realizations. 1) Summation of binary weighted currents Fig. 7 MSB V LSB Fig. 7: DAC summation of weighted currents

8 Punčochář, Mohylová: TELO, Chapter 10: A/D and D/A converter principles 8 - It is a summing connection (inverting) of an operational amplifier. - For given ratios is output voltage V DAC : V DAC = = 8, V - For example, in a 12-bit converter you would need a range of resistor values of 2000:1, with corresponding precision of the small resistor values an elegant solution is R 2R ladder. - The switch resistance must be smaller than 1/2 n of the smallest resistor. - This principle is used only in fast, low-precision DACs. 2) R 2R ladder Fig. 8 C B A VIRTUAL GROUND I VG LSB MSB Fig. 8: DAC R 2R ladder - Only two resistor values are needed (R and 2R). - The resistor must be precisely matched, though the actual resistor values are not critical. - Electronic switches connect resistors to either ground or the V line. The result is binary weighted current I VG flowing down VIRTUAL GROUND (The Thevenin resistance of an R/2R ladder is always R regardless of the number of bits in the ladder. Since an R/2R ladder is a linear circuit, we can apply the principle of superposition to calculate I VG.). - Operational amplifier current to voltage converter (inverting). - The circuit shown generates an output of zero to 10 V (the maximum input count is 15, with output voltage -10x15/16). - The operational amplifier tends to be the slowest part of the DAC we can use a converter with current output.

9 Punčochář, Mohylová: TELO, Chapter 10: A/D and D/A converter principles 9 3) DAC with current output (current switched DAC) Fig. 9 U I R VIRT. GROUND OZ T T 1 T 2 T 3 T 4 16A E 8A E 4A E 2A E 1A E 4A E for example I I 1 I 2 R 2R R 2R I 3 2R R R 2R I /2 I /4 I /8 Fig. 9: DAC R 2R ladder; current output - The current sources are ON all the time, and their output current is switched to the output terminal or to ground (very fast and cheap). - The transistor areas are scaled (16A E : 8A E : 4A E : 2A E :1A E ), thereby ensuring equal current densities in all the transistors for optimum VBE matching (the emitter areas of the BJT devices must be proportional to the emitter current). - OZ + T create current I = U /R (T is an inverting structure, thus feedback is negative, virtual ground). - I 1 = (( I R + U BE ) U BE )/ 2R = I / 2 - I 2 = (( I R + U BE ) U BE R I / 2) / 2R = I / 4 - I 3 = (( I R + U BE ) U BE R I / 2 R I / 4) / 2R = I / 8 - etc. There are a few ways to generate an output voltage from a current DAC. To generate large swings, or to buffer into small load resistances, a current to voltage amplifier (with an opamp) can be used. 4) Multiplying DACs (MDAC) - can be made from DACs that have no internal reference by using the reference input for the analog input signal. A DAC with good multiplying properties (wide analog input range, high speed, etc.) will be called a multiplying DAC. When used like this, MDAC behaves as a digitally controlled audio attenuator because the output V 0 is a fraction of the voltage representing the input digital code and the attenuator setting can be controlled by digital logic. If followed by an op-amp integrator, the MDAC provides digitally programmable integration which can be used in the design of digitally programmable oscillators, filters.

10 Punčochář, Mohylová: TELO, Chapter 10: A/D and D/A converter principles 10 5) Generally, the output voltage V DAC 1 2 n ( a 2 + a 2 + a ) = ± V N = ± V 2 Sign + means a noninverting DAC. Sign - means an inverting DAC. a 1 MSB; a n LSB. Basic principles of ADCs There are different ADCs realizations. 1 2 L 1) Flash conversion (parallel encoder) Of all conversion techniques, one of the fastest is direct conversion, better known as "flash" conversion. ADCs based on this architecture are extremely fast and perform their multibit conversion directly, but they require intensive analog design to manage the large n Fig. 10: ADCs based on the direct-conversion architecture (better known as flash converters) include 2 N -1 comparator banks and a reference resistor-divider network number of comparators and reference voltages required. As shown in Fig. 10, a converter with N-bit resolution has 2 N -1 comparators connected in parallel, with reference voltages set by a

11 Punčochář, Mohylová: TELO, Chapter 10: A/D and D/A converter principles 11 resistor network and spaced V /2 N (~1 least significant bit, or LSB) apart. Input voltage offset of operational amplifiers must be less than the "LSB / 2". A change of input voltage usually causes a change of state in more than one comparator output. These output changes are combined in a decoder-logic unit that produces a parallel N- bit output from the converter. 2) Dual slope ADCs The dual-slope ADC architecture was truly a breakthrough in ADCs for high resolution applications such as digital voltmeters (DVMs), etc. A simplified diagram is shown in Fig. 11, and the integrator output waveforms are shown in Fig. 12. Fig. 11: Dual slope ADC Fig. 12: Dual slope ADC output waveforms

12 Punčochář, Mohylová: TELO, Chapter 10: A/D and D/A converter principles 12 The input signal is applied to an integrator; at the same time a counter is started, counting clock pulses. After a predetermined amount of time (T), a reference voltage having opposite polarity is applied to the integrator. At that instant, the accumulated charge on the integrating capacitor is proportional to the average value of the input over the interval T. By choosing that time interval to be a multiple of the power-line period, the converter becomes insensitive to 50 Hz hum (and its harmonics) Fig. 13. The integral of the reference is an oppositegoing ramp having a slope of V/RC. At the same time, the counter is again counting from zero. When the integrator output reaches zero, the count is stopped, and the analog circuitry is reset. Since the charge gained is proportional to VIN T, and the equal amount of charge lost is proportional to V tx, then the number of counts relative to the full scale count is proportional to tx/t, or VIN/V. If the output of the counter is a binary number, it will therefore be a binary representation of the input voltage. Fig. 13: Frequency response of integrating ADC Dual-slope integration has many advantages. Conversion accuracy is independent of both the capacitance and the clock frequency, because they affect both the up-slope and the downslope by the same ratio. 3) SAR ADCs Although there are many variations in the implementation of a SAR (successive approximation register) ADC, the basic architecture is quite simple (see Fig. 14). The analog input voltage (V IN ) is held on a track/hold. To implement the binary search algorithm, the N- bit register is first set to midscale (that is, , where the MSB is set to '1'). This forces the DAC output (VDAC) to be V /2, where V is the reference voltage provided to the ADC. A comparison is then performed to determine if V IN is less than or greater than VDAC.

13 Punčochář, Mohylová: TELO, Chapter 10: A/D and D/A converter principles 13 If V IN is greater than VDAC, the comparator output is a logic high or '1' and the MSB of the N-bit register remains at '1'. Conversely, if V IN is less than VDAC, the comparator output is a logic low and the MSB of the register is cleared to logic '0'. The SAR control logic then moves to the next bit down, forces that bit high, and does another comparison. The sequence continues all the way down to the LSB. Once this is done, the conversion is complete, and the N-bit digital word is available in the register. Fig. 14: Simplified N-bit SAR architecture Fig. 15: SAR operation 4 bit example

14 Punčochář, Mohylová: TELO, Chapter 10: A/D and D/A converter principles 14 Fig. 15 shows an example of a 4-bit conversion. The y-axis (and the bold line in the figure) represents the DAC output voltage. In the example, the first comparison shows that V IN < VDAC. Thus, bit 3 is set to '0'. The DAC is then set to and the second comparison is performed. As V IN > VDAC, bit 2 remains at '1'. The DAC is then set to , and the third comparison is performed. Bit 1 is set to '0', and the DAC is then set to for the final comparison. Finally, bit 0 remains at '1' because VIN > VDAC. Notice that four comparison periods are required for a 4-bit ADC. Generally speaking, an N-bit SAR ADC will require N comparison periods and will not be ready for the next conversion until the current one is complete. This explains why these types of ADCs are power- and space-efficient, yet are rarely seen in speed-and resolution combinations beyond a few Msps at 14 to 16 bits. The two critical components are the comparator and the DAC. 4) Voltage to frequency ADC Voltage-to-frequency ADCs convert the analog input voltage to a pulse train with the frequency proportional to the amplitude of the input (see Fig. 16). This can be done simply by charging a capacitor with a current proportional to the input level and discharging it when the ramp reaches a preset threshold. The pulses are counted over a fixed period to determine the frequency, and the pulse counter output, in turn, represents the digital voltage. Fig. 16: Principle of voltage to frequency DACs Voltage-to-frequency converters inherently have a high noise rejection characteristic, because the input signal is effectively integrated over the counting interval. Voltage-tofrequency conversion is commonly used to convert slow and noisy signals. Voltage-tofrequency ADCs are also widely used for remote sensing in noisy environments. The input voltage is converted to a frequency at the remote location and the digital pulse train is transmitted over a pair of wires to the counter. This eliminates noise that can be introduced in the transmission lines of an analog signal over a relatively long distance. 5) Delta sigma converter see the 9. Chapter

15 Punčochář, Mohylová: TELO, Chapter 10: A/D and D/A converter principles 15 6) Pipelined analog-to-digital converters The pipelined analog-to-digital converter (ADC) has become the most popular ADC architecture for sampling rates from a few megasamples per second (Msps) up to 100Msps. Resolutions range from eight bits at the faster sample rates up to 16 bits at the lower rates. These resolutions and sampling rates cover a wide range of applications, including CCD imaging, ultrasonic medical imaging, digital receivers, base stations, digital video, etc. Fig. 17: Pipelined ADC with four 3-bit stages (each stage resolves two bits) Fig. 17 shows a block diagram of a 12-bit pipelined ADC. In this schematic, the analog input, V IN, is first sampled and held steady by a sample-and-hold (S&H), while the flash ADC in stage one quantizes it to three bits. The 3-bit output is then fed to a 3-bit DAC (accurate to about 12 bits), and the analog output is subtracted from the input. This "residue" is then gained up by a factor of four and fed to the next stage (Stage 2). This gained-up residue continues through the pipeline, providing three bits per stage until it reaches the 4-bit flash ADC, which resolves the last 4LSB bits. Because the bits from each stage are determined at different points in time, all the bits corresponding to the same sample are time-aligned with shift registers before being fed to the digital-error-correction logic. Note when a stage finishes processing a sample, determining the bits, and passing the residue to the next stage, it can then start processing the next sample received from the sample-and-hold embedded within each stage. This pipelining action is the reason for the high throughput. Although each stage generates three raw bits in the Figure 1 example, because the interstage gain is only 4, each stage (Stages 1 to 4) effectively resolves only two bits. The extra bit is simply to reduce the size of the residue by one half, allowing extra range in the next 3-bit ADC for digital error correction, as mentioned above. This process is called "1-bit overlap" between adjacent stages. The effective number of bits of the entire ADC is therefore = 12 bits.

16 Punčochář, Mohylová: TELO, Chapter 10: A/D and D/A converter principles 16 7) Comparison of ADCs As ADCs can consume a large percentage of power in a device, it is of vital interest to minimize ADC power consumption. Optimal power consumption for different sampling rates, and resolutions we can see in Fig. 18 (qualitative description only). Fig. 18: A/D Converter technologies, resolution and bandwidth When selecting an ADC, some of the factors to consider are - Precision - Speed - Accuracy (external trimming required? monotonicity?) - Required supply voltages and power dissipation - Reference (internal or external? if internal, is it accessible externally?) - Input impedance and analog voltage range (unipolar, bipolar, or both?) - etc.

17 Punčochář, Mohylová: TELO, Chapter 10: A/D and D/A converter principles 17 [1] [2] Basic texts Other text [3] Horowitz, P-Hill, W.: The art of electronics. Cambridge University press 2001, str.612 až 636 Questions Answers you find in this text 1. Explain basic operating principles of DACs. 2. Explain basic operating principles of ADCs. 3. Can we use DAC as a variable controlled attenuator? 4. Explain the operating principle of an ADC, usually referred as a SAR. 5. Explain what aliasing is, how it happens, and what may be done to prevent it from happening to an ADC circuit. Problems 1. Determine the required sampling frequency f s if the maximum signal frequency is 16 khz. 2. Suppose an analog-digital converter inputs a voltage ranging from 0 to 5 volts DC and converts the magnitude of that voltage into an 8-bit binary number. How many discrete "steps" are there in the output as the converter circuit resolves the input voltage from one end of its range (0 volts) to the other (5 volts)? How much voltage does each of these steps represent? 3. Determine the output voltage of a multiplying DAC (inverting), who s V = 10 V, a binary word is Determine the maximum input voltage offset of comparators in parallel converter according to problem 3. Problems key Ad1) Use the equation f s 2 f c.

18 Punčochář, Mohylová: TELO, Chapter 10: A/D and D/A converter principles 18 Ad2) This ADC (Analog-to-Digital Converter) circuit has 256 steps in its output range, each step representing mv. 1 2 n Ad3) Use the equation V = ± V N = ± V ( a 2 + a 2 + a 2 ) DAC 1 2 L n ; n = 8. Ad4) We know n = 8, LSB(analogvalue) voltage offset must be less than q/2. V V = q = ; thus comparators input n n Recommendation If you can solve and answer more than circa 60 % of the problems and questions, you may continue your study.