Instrumentation

Chris Carr

23rd February 2009

Instrumentation

• Today:

– 10:00 to 11:30Introduction to Instrumentation

• Chris Carr

– 14:00 to 15:30Magnetometers

• Patrick Brown

• Friday:

– 14:00 to 15:00Problem Sheet session

• Chris and Patrick

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The Scientific Method

Question:Which Greek philosopher is principally responsible for modern science?

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The Scientific Method

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A ‘Generic’ Instrument

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A Real instrument…

Fluxgate Magnetometer Instrument for the ESA/CNSA ‘Double Star’ Mission

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Input Transducers

• Our sensor is an input transducer:A device to convert a measureable (physical quantity) into an electrical signal

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Input Transducers

• Sensor often separate from the electronics:

– Because of InterferenceExample: Space Magnetometer

– Because of access or ease of useExample: Ultrasound (medical or industrial)

– Because the environment is hazardous (to people or electronics)Example: Nuclear/Particle Physics and Space

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Transducer Behaviour

• The ideal sensor…

• …does not exist!

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Transducer Behaviour

• A more realistic model…

• Examples: temperature sensors, pressure sensors, force/acceleration sensors, etc.

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Transducer Behaviour

• Real-world sensor characteristics

• Non-linearity is a major problem (as we shall see)

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Transducer Behaviour

• Real-world sensor characteristics

• We can define a range over which the sensor is approximately linear

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Transducer Behaviour

• We want a linear response because

– ‘Gain’ or ‘sensitivity’ is constant

– And because linear systems allow formalised mathematical analysis

• To achieve linear operation

– Limit operating range to the linear region

– Operate the sensor with feedback (this afternoon’s lecture)

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Static Response

• These graphs gives us the static response

– Output for a fixed, constant input

• Example: mechanical scales

– f(m) is the response the system reaches a long time (in theory, ) after theinput changes

t

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Frequency Domain Behaviour

• But for most physics applications the physical measureable is a time-varying quantity

• If the static response of the system is linear then we characterise the linear system with a Bode plot

– Gain and Phase of output(relative to input)

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Bode Plot

• Specifies the Transfer Function

• Gives the bandwidth (B)

• Applicable to sensor, filter, amplifier or the whole instrument

B

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Signals

• The system (sensor, amplifier, etc…) has frequency dependent behaviour

• Need to be aware of how the frequency content of our signal is affected

• We need a “Fourier Understanding” of signals

• Recall the Fourier Transform

• And the inverse transform

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Example: Square Wave

• An infinite signal (over all time)

• Has a spectrum consisting of an infinite series of sinusoids

Signal f(t) Spectrum F( )

AngularFrequency

Time Domain Frequency Domain

Time t

Amplitude Amplitude

0

11,3 5

/ . .( ), ....

n Sin n on

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Other Waveforms

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Some Further Examples

• Infinite Sinusoid has a finite spectrum

• Conversely, a finite sinusoid has an infinite spectrum

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Implications

• Signals with an infinite Fourier representation require an infinite bandwidth to support them

• This is unphysical

– Sensors and electronics have finite bandwidths

– Real signals are of finite duration and hence have wide spectrum

• So:

– The instrumentation messes with the spectral content of your signal

– Fidelity of recorded signal always compromised

– Example: see problem sheet

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Transmission Lines

• Sensor signal usually small and weak

• Transmission Line

– Preserves signal integrity

• Shape, amplitude, phase etc

– Protects from external interference

• Noise pickup

• Example:Co-ax cable whichconnects aerial to TV

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Impedance

– A complex quantity. We just need the magnitude.

• Longitudinal Compression Waves, e.g. sound

• EM Waves

• Electrical waves (signals) in a co-axial cable

– Where L is the inductance per metre, C the capacitance per metre

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Boundaries

• At the boundary between 2 media with different impedances, we get reflections

• Amplitude reflection coefficient KR

• The physics is the same:Mechanical Waves, Optical EM Waves, Electrical signals in a cable…

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Optics Example

• At air/glass interface we get 4% reflection (96% transmission)

• 10 interfaces gives 0.9610 = 66% light transmitted!

• In optics we use anti-reflective coatings

• In electronics we can directly change the impedances to minimisesignalloss

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Impedance Matching

• Z1 = Z2 means KR = 0

• Impedance matching is the condition for Minimum reflectionandMaximum Signal Power Transfer

• Example: see problem sheet

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Noise

• Have seen how sensors and electronics can change our signal characteristics

• And how signal power can be lost through impedance mis-matches

• A further consideration is the noise introduced into our measurements

• Generally, we need to worry about 3 sources

– Thermal Noise

– Shot Noise

– Flicker Noise

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Thermal Noise

• Arises from the random thermal movement of free-electrons in a conductor

• Is therefore a function of temperature

• The RMS noise voltage measured with an instrument bandwidth B is

• Where

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Shot Noise

• Arises from statistical fluctuations in charge carriers

• Wherever we have a current crossing a junction

• Typically in semiconductors and vacuum tubes

Photomultiplier Tube

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Flicker Noise (aka 1/f noise)

• Source not understood

• Fundamental property of all measurement systems

• Noise power is inversely proportional to frequency

• Contrast to Thermal & Shot noise

– Have power constant with frequency

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Composite Noise Spectrum

0

0.5

1

1.5

2

2.5

3

3.5

0 0.5 1 1.5 2 2.5 3 3.5

Log(Power)

log(f)

log (1/f)

log (shot)

log (thermal)

log (total)

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Consequences

• To reduce noise in our measurements

– Avoid low-frequencies

– Reduce temperature

• Of sensor and, often, electronics such as pre-amplifiers

– Reduce measurement bandwidth

• A filter is effectively a bandwidth reducing device

Pass-BandFilter

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Systems Analysis

• Mathematical framework to

– Model complex sensors and instruments

– Predict behaviour

– Extract measureable m from the output f(m)

– Test stability of systems

• Example of the “block diagram”approach used insystems analysis

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First Order System

• Simple Example: Mercury Thermometer

• A first order differential equation

• Characteristic of many sensors and electrical systems

Thermometer temperature

Initial: θ0

At time t: θ(t)

Heat Bath

temperature: θR

Input u(t)

Time t Time t

Output x(t)

t = 0 t = 0 t =

63.2%

Steady State

Initial Value

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Second Order System• Damped Harmonic Oscillator

• 2nd order D.E.

• Many mechanical andelectrical systems

• A physical representation:

Mass m

Damping Term

SpringConstant K

Scale

Pointer

Deflection x(t)

Time t

Time t

Time t

Time t

Input y(t)

Output x(t)

Output x(t)

Output x(t)

(b) Zero damping

Infinite oscillation

(c) Moderate damping

Oscillation decays to steady state

(d) Heavy Damping

No oscillation

Zero damping

Infinite oscillation

(a) Step Input

A mass is hung on the balance

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These are

Linear Time-Invariant Systems

• Frequency Preservation

– ω is not changed by passing through the system

• Linear Superposition

– For each frequency component in the signal only the amplitude and phase are modified

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We can build and analyse complex systems

• Where Gn is the Transfer Function for each part of the system

1. May be the sensor

2. Could be an amplifier

3. Could choose to be an integrator… etc…

• N.B. For Transfer Function, remember the Bode Plot

G1(s) G2(s) G3(s)Y(s) X(s)

1 2 3

G(s)

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Writing the Equations

• For each part we can write a DE of the form

• Then we would have a series of DE’s we could solve…

• But this rapidly becomes very hard

• Use the Laplace Transform to simplify the problem

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Laplace Transform very briefly…

• Transform all signals and transfer functions into the s-domain

• Differential equations become linear algebraic

G1(s) G2(s) G3(s)Y(s) X(s)

1 2 3

G(s)

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