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Digital Systems

Digital Systems. General characteristics. Digital systems - General. Most modern control and measurement systems are micro-controller based and include both analog and digital signals. Analog vs. Digital signals. Digital signals – discrete in magnitude Transfer function:

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Digital Systems

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  1. BAE 3023 Digital Systems General characteristics

  2. Digital systems - General BAE 3023 • Most modern control and measurement systems are micro-controller based and include both analog and digital signals

  3. Analog vs. Digital signals BAE 3023 • Digital signals – discrete in magnitude • Transfer function: • Many input voltagesmap to each outputcode at any quantizationlevel • Resolution of codecontrols the digitalquantization error • Digital code resolution= 1/2nwhere n = no. of bits • Examples: • 8 bits = 1 in 28 • or 1 in 256 • 12 bits = 1 in 4096 • 16 bits = 1 in 65536

  4. Analog to digital resolution BAE 3023 • For an analog to digital converter (A/D) • Voltage resolution is the voltage range divided by the digital range: 0-5V range and 8 bit resolution • Voltage resolution = 5V / 256 = 0.02 V /bit • Assumes linear mapping • Consider a thermocouple used with an 8 bit converter and a 0.1 V range. (the data acquisition systems at the lab. could be set this way.) • Type “T” thermocouple data: • Voltage resolution = 0.1 / 256 = 0.00039 V / bit • From the table above Temperature resolution = 100 / 0.00428 = 23400 C / V • Temperature resolution per bit = (0.0039)(23400) C / bit = 9.12 degrees C per bit ! • Must use a higher resolution converter or smaller voltage range.

  5. Binary representation of numerical data BAE 3023 • Binary integers may beused to represent decimalnumbers. See counting right: • Consider addition: • Normally signed data are represented as 8, 16, 32, 64 bit numbers with the MSB = 1 for negative. Uses 2’s complement representation • Hexidecimal is convenient because each digit represents 4 binary bits

  6. Discrete in time BAE 3023 • Within computer based systems, data are normally sampled in time. These systems are also known as sampled data systems • In sampled data systems, nothing is known of the value between samples

  7. Relationship to control systems BAE 3023 • Sampling is normally fast enough and resolution high enough that • The system can be considered analog • For analysis, a dead-time term should be added with a dead-time of ½ the sample period • Usually acceptable for sample period < 1/10 smallest time constant • For slower sampling rates • Apply digital analysis of the system • Z-transform approach • Analogous frequency domain and root locus analysis

  8. Differential equations as difference equations BAE 3023 • Simple methods may be used to convert differential equations into difference equations(Analog in time to discrete in time) • See development on digital controllers in earlier lecture • Example for converting a first order differential equation into a backward difference equation:

  9. Calculation of coefficients for difference equations BAE 3023 • Solve for p2

  10. Comparison of differenc equation to theoretical BAE 3023 • Here, p^ is the digital approximation (p2) and p is the theoretical solution

  11. Feed-forward application BAE 3023 • Numerical modeling can be used within controllers to implement controllers or predict system output to allow feed-forward control. (See Chapter 12 in S&C)

  12. Feed-forward TF BAE 3023 • The feed-forward transfer function can be calculated easily from the load and process response transfer functions. The feed-forward transfer function can then be converted to a difference equation and used to directly compensate for the load. The load must be measured.

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