theory:sensor_technology:st3_measurement_domains

*An oscilloscope is good to measure delays in time, but is bad for high precision amplitude measurements. A Bode analyzer good for phase shifts, but bad for measuring delays. A multimeter good for high precision voltage measurements, but bad if they change faster than 10 seconds. In this way, every domain has its own vocabulary, toolset, mathematical models and application areas. *

In figure 1 we see three screenshots of typical tools that are specific for three measurement domains. To make a convincing measurement that captures a problem or solution in a convincing way, the rigth tool must be selected. The same is true for the mathematics. In the next sections, the three most important domains are discussed.

- Measure battery voltage, cable resistance, etc. in a car
- Current clamp in I
_{rms} - Measure temperature with a resistive sensor

- We talk about absolute values like Volts
- Gain to relate an output simply to an input
- Use RMS for powers and noise

\begin{equation} U_{RMS}=\sqrt{\frac{1}{T}\int_{0}^{T} U^{2} \left ( t \right )} \label{eq:RMS} \end{equation}

- Assume things are (quasi) static
- Apply Kirchhoffs laws ($\Sigma$I
_{node}=0, $\Sigma$U_{loop}=0) - Notice voltage dividers:

\begin{equation} U_{out}=U_{in} \frac{R_{out}}{R_{series}+R_{out}} \label{eq:VoltageDivider} \end{equation}

- The multimeter is our high precision friend
- The RMS-mode is normally not “true-RMS”
- A (handheld) multimeter is battery operated and is therefore perfectly differential: there are no ground problems
- The internal resistance may change per range (systematic errors)

- A function generator has a Thevenin equivalent with an internal resistance ($50 \Omega$)

In the time domain, we have to deal with transients: rise times and delays.

- Design a controller board: connect a memory chip to the microcontroller
- Optimize a PID controller
- Work on echo cancellation

- Signals are electrical voltages that change in time
- Low bandwidth signals change slowly
- We talk about rise-times, overshoot, delays

- Differential equations make good models
- Example: RC Network (remember?)

To capture such a system in a mathematical equation, we first write down the component equations. In this case, that is Ohm's law for the resistor and the voltage-current relation for the capacitor. Next, we need the network equations. These follow from Kirchhoff's current and voltage laws. \begin{equation} \begin{matrix} I_{C}=C\frac{\partial U_{C}}{\partial t} \\ U_{R}=I_{R}R \\ I_{C}=I_{R} \\ U_{out}=U_{C}=U_{in}-U_{R} \end{matrix} \label{eq:RC_Network} \end{equation} After combining the equations, we find \begin{equation} U_{out}+RC\frac{\partial U_{out}}{\partial t}=U_{in} \label{eq:RC_DiffEquation} \end{equation}

which can be solved analytically for a step-input at t = 0 sec as

\begin{equation} U_{out}\left ( t \right ) = U_{in} \left ( 1-e^{-\frac{t}{RC}} \right ). \label{eq:RC_RC_AnSolution} \end{equation}

- Oscilloscope for analog signals
- Logic analyzers for digital signals
- Function generator or AWG to generate a signal
- Counter to count events
- Two options:
- Event based signals
- Need a trigger to locate the effect

- Periodic signals
- Need a trigger to synchronize signals

- AM modulation as used in radio
- Mechanical vibrations (Tacoma Narrows Bridge 1940)
- Acoustics: Wah-wah guitar pedal

- Phase, gain (bode plot)
- Cut-off frequencies
- Spectrum
- “Impedance” and not “resistance”

- Laplace transforms (or Fourrier) give easier mathematics than differential equations

As an example we look at the same RC network as used in figure 5. The component equations become \begin{equation} \begin{matrix} I_{C}=C\frac{\partial U_{C}}{\partial t} U_{R}=I_{R}R \end{matrix} \label{eq:RC_NetworkImpedanceComponentEq} \end{equation} We could combine the component equations with the network equations (from Kirchhoff), but because all components have become simple impedances, we can also work with simple voltage dividers. Using a voltage divider, we can see that \begin{equation} U_{out}\left ( j\omega \right ) = \frac{Z_{C}\left ( j\omega \right )}{Z_{C}\left ( j\omega \right )+Z_{R}\left ( j\omega \right )}U_{in}\left ( j\omega \right ) \label{eq:RC_ImpedanceVoltageDivider} \end{equation}

which can be substituted easily with the component equations in impedance form

\begin{equation} U_{out}\left ( j\omega \right ) = \frac{1}{1+j\omega RC}U_{in}\left ( j\omega \right ). \label{eq:RC_AnZSolution} \end{equation}

The result is an expression of the frequency transfer function of the RC circuit as a filter. It has a shape as shown in figure 6.

- Oscilloscope
- Not a good instrument for precision (U, t) measurements: good for signal shapes
- Digital scopes (and so the scope function of MyDAQ) have discrete phenomena like aliasing

- Bode diagram
- shows phase and magnitude information as a function of frequency
- Log-log plot shows R, L and C components as asymptotes easily

- Chapter 1: Measurement Theory
- Chapter 2: Measurement Errors
- Chapter 3: Measurement Domains
- Chapter 4: Circuits, Graphs, Tables, Pictures and Code ← Next
- Chapter 5: Basic Sensor Theory
- Chapter 6: Sensor-Actuator Systems
- Chapter 7: Modelling
- Chapter 8: Modelling: The Accelerometer - example of a second order system
- Chapter 9: Modelling: Scaling - why small things appear to be stiffer
- Chapter 10: Modelling: Lumped Element Models
- Chapter 11: Modelling: Finite Element Models
- Chapter 13: Modelling: Systems Theory
- Chapter 14: Modelling: Numerical Integration
- Chapter 15: Signal Conditioning and Sensor Read-out
- Chapter 16: Resistive Sensors
- Chapter 17: Capacitive Sensors
- Chapter 18: Magnetic Sensors
- Chapter 19: Optical Sensors
- Chapter 20: Actuators - an example of an electrodynamic motor
- Chapter 21: Actuator principles for small speakers
- Chapter 22: ADC and DAC
- Chapter 23: Bus Interfaces - SPI, I
^{2}C, IO-Link, Ethernet based - Appendix A: Systematic unit conversion
- Appendix B: Common Mode Rejection Ratio (CMRR)
- Appendix C: A Schmitt Trigger for sensor level detection

theory/sensor_technology/st3_measurement_domains.txt · Last modified: 2017/10/10 18:37 by glangereis