A/D and D/A Conversion

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A/D and D/A Conversion

• Shuvra Das
• University of Detroit Mercy

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Flowchart of Mechatronic Systems
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Objective

• The objective of this section is to describe how
  the D/A or the A/D converter blocks function.
• And highlight important issues an user should be
  aware of.

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Analog to Digital Conversion
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A/D conversion

• Sensors usually receive analog data, e.g. sensor
  signals from thermocouples or strain gages are
  voltages.
• These signals are converted to digital form by
  A/D converters.
• Modern A/D converters are available as single IC
  chips (integrated circuit).

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Types of A/D converters

• The tracking ADCs (Analog to Digital converters)
• The integrating ADCs
• The flash ADCs
• successive approximation ADCs

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Quantization

• The process of A/D converting an analog voltage
  (or current) to digital form requires that the
  analog signal be quantized and encoded into
  binary form.
• Quantization consists of subdividing the range of
  the signal into a finite number of intervals.
• Each quantized level is then assigned a binary
  word.
• Usually if n bits are available in the binary
  word one employs (2n -1) intervals. (e.g. if n
  4, no. of intervals 15, if n 8, no. of
  intervals 255, etc).

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QuantizationExample

• Let va represent analog voltage and vd its
  quantized counterpart
• Let the analog voltage range be 0-16 V
• Then for all va lying between 0 and 1 V, vd 0.
• For all va between 0and 1 vd0, 1 and 2, vd 1,
  etc, for 15-16 vd 15.

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QuantizationExample
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Quantization

• As the table shows each quantized voltage level
  may be represented by a unique binary
  counterpart.
• There is always some error associated with
  quantization. As the number of bits increases
  the error decreases.

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Quantization Error

• The difference between actual analog voltage and
  the level to which the quantization assigns the
  value.
• Example If 5 bits are used to represent a 50 V
  range, what is the quantization error for
  representing 10 V?

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Quantization Error

• Example If 5 bits are used to represent a 50 V
  range, what is the quantization error for
  representing 10 V?
• Repeat for 12 V , 31 V, etc for practice.

• Q 50V/(25-1) 1.6129V per level.
• kQ lt 10 V
• k needs to be an integer number of levels
• k lt 10/1.61296.2
• k 6
• quantized value for 10 61.6129 9.6774
• error 10-9.6774 0.3226V

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Quantization Error

• Example If 8 bits are used to represent a 50 V
  range, what is the quantization error for
  representing 10 V?
• Repeat for 12 V , 31 V, etc for practice.

• Q 50V/(28-1) .19607 V per level.
• kQ lt 10 V
• k needs to be an integer number of levels
• k lt 10/0.19607 51.002
• k 51
• quantized value for 10 510.19607 9.99957
• error 10-9.99957 0.00043V

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Quantization Error

• Quantization error may be reduced by using higher
  number of bits in the A/D converter
• Typical A/D converters have 8, 12 or 16 bits.
• In the scheme described earlier Maximum
  quantization error thus is the length of the
  quantized level. (Max error voltage
  range/(2n-1))

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Saturation Error

• A/D converters have specific upper and lower
  voltage ranges. Typical values used are 0-10V or
  -10 to 10 V.
• If the input signal is higher than the upper
  limit or lower than the lower limit the converter
  saturates. This can be prevented by appropriate
  signal conditioning.

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Conversion Time

• The time required for A/D converter to provide
  the digital equivalent of the analog output.
• What happens if analog value changes (time
  varying signal) before conversion is complete?
• Sample and hold-maintains constant input to A/D
  while conversion is taking place.

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Conversion Time sampling rate
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Conversion time Sampling Rate

• Sampling rate must be at least at twice the
  frequency of the maximum frequency of interest in
  the analog signal. This critical sampling rate
  is called the Nyquist frequency (2fmax). If
  sampling is not done properly aliasing will
  occur.
• Aliasing Form of signal distortion as a result
  of improper sampling.
• In practice about 5-10 times frequency of the
  signal is used.

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Sampling Rate Example

• Datasheet for ADC 574 (a particular ADC IC) says
  that the maximum conversion time is 35micro-sec.
  What is the highest signal frequency that can be
  converted with this ADC?

• Highest conversion frequency for this ADC
  fmax 1/(35 10-6) 28.57kHz
• Therefore the maximum sampling frequency can be
  28.57khz.
• Therefore the maximum frequency of the signal
  should not be more than 1/2 of this, i.e. 14 kHz.
• In reality it should be even less.

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Digital to Analog Conversion
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D/A conversion

• Takes a binary word and converts it into an
  analog signal.
• Binary word is represented by 1s and 0s where
  typically 0 corresponds to 0 V and 1 corresponds
  to 5 V.

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Example

• Consider 4-bit binary word representing a
  positive number
• Binary number (b3b2b1b0)2 (b323b222b121
  b020)10

• The analog voltage corresponding to the binary
  word B is
• va (b38b24b12 b01)dv
• where dv is the smallest step size by which va
  can increment.

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D/A conversion

• The step size is determined by the number of bits
  in the digital word to be converted.
• Extending the previous example
• vamax (2n-1) dv gt vamax/(2n-1) dv
• This is the smallest change in voltage when the
  least significant bit changes from 0 to 1.

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D/A conversion process

• A summing amplifier is used. Each bit is
  represented by a 5 V source and a switch.
• If bit is 0 switch is off and if bit is 1 switch
  is on.
• The output va is proportional to the binary word.
  
• Va -(S(RF/Ri)biVin)
• RiR0/2i

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D/A conversion process

• Va -(S(RF/Ri)biVin)
• RiR0/2i
• Va -RF/R0(2 n-1 bn-1 2n- 2 bn-2 .. 20b0)Vin

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Example

• Consider a 4-bit DAC with 0-15V range (I.e. va
  max 15)
• Rs chosen are 10kohm, 5, 2.5 and 1.25 kohms, RF
  2kohm., Vin5V
• dv 15/(24-1)1V
• 0101 gt -(02k/1.25k 12k/2.5k02k/5k12k/10k)
  5 -5V
• 1001gt -(12k/1.25k 02k/2.5k02k/5k12k/10k)
  5 -9V

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Example

• Determine the smallest step size (or resolution)
  of an 8 bit DAC for a range of 0-12 V.

• Resolution is dv, smallest non-zero voltage
  value.
• dv Voltage range/(2n-1) 12/(28-1) 12/255
  47.1milli Volt

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Example

• Find the minimum number of bits required in a DAC
  if the range is 10 V and a resolution of 10mV is
  required.

• dv (voltage range)/(2n-1)
• 10(10-3) 10/ (2n-1)
• (2n-1) 10/0.01
• n (log2) log(10001)
• n log(1001)/log2 9.97
• minimum bits req. 10