# Sensor Systems

### Sidebar

theory:sensor_technology:st3_measurement_domains

# 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.

## Introduction

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.

Fig. 1: Three measurement tools: oscilloscope, bode analyzer and multimeter (from the National Instruments ELVIS toolbox)

## Amplitude Domain

### Amplitude domain: examples

• Measure battery voltage, cable resistance, etc. in a car
• Current clamp in Irms
• Measure temperature with a resistive sensor

### Amplitude domain: characteristics

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

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

### Amplitude domain: models

• Assume things are (quasi) static
• Apply Kirchhoffs laws ($\Sigma$Inode=0, $\Sigma$Uloop=0)
• Notice voltage dividers:

$$U_{out}=U_{in} \frac{R_{out}}{R_{series}+R_{out}} \label{eq:VoltageDivider}$$

Fig. 2: The voltage divider

### Amplitude domain: equipment

• 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 ($50 \Omega$)

Fig. 3: A function generator and a Thevenin equivalent circuit

## Time Domain

### Time domain: examples

Fig. 4: Time domain evaluation of a circuit

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

### Time domain: characteristics

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

### Time domain: models

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

Fig. 5: An RC network as a low pass filter

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. $$\begin{matrix} I_{C}=C\frac{\partial U_{C}}{\partial t} \\ U_{R}=I_{R}R \\ I_{C}=I_{R} \\ U_{out}=U_{C}=U_{in}-U_{R} \end{matrix} \label{eq:RC_Network}$$ After combining the equations, we find $$U_{out}+RC\frac{\partial U_{out}}{\partial t}=U_{in} \label{eq:RC_DiffEquation}$$

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

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

### Time domain: equipment

• 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

## Frequency Domain

### Frequency domain: examples

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

### Frequency domain: characteristics

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

### Frequency domain: models

• 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 $$\begin{matrix} I_{C}=C\frac{\partial U_{C}}{\partial t} U_{R}=I_{R}R \end{matrix} \label{eq:RC_NetworkImpedanceComponentEq}$$ 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 $$U_{out}\left ( j\omega \right ) = \frac{Z_{C}\left ( j\omega \right )}{Z_{C}\left ( j\omega \right )+Z_{R}\left ( j\omega \right )}U_{in}\left ( j\omega \right ) \label{eq:RC_ImpedanceVoltageDivider}$$

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

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

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.

Fig. 6: Frequency transfer function of the RC filter

### Frequency domain: equipment

• 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

## Summary

Domain Characteristics Equipment Mathematics
Amplitude DC gain, hysteresis, RMS values, offset, $V_{bias}$, $V_{diode}$, $h_{fe}$, $\beta$ Multimeter Kirchhoff
Time rise time, peaks, events, period, delay Counter, interval analyzer, oscilloscope differential equations
Frequency phase, gain, transfer functions, bandwidth Oscilloscope, spectrum analyzer Fourier, Laplace, FFT
Tab. 1: Every measurement domain has its own tools, mathematics and vocabulary