slides electrical instrumentation lecture 6

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ET8.017El. Instr.Delft University of Technology

Electronic Instrumentation

[email protected]

Lecturer: Kofi Makinwa

015-27 86466Room. HB13.270 EWI building

Also involved:Saleh Heidary([email protected])


Introduction

Synonyms: coherent detection, synchronous demodulation, lock-in amplification, chopping??These are all modulation techniques that are used to improve the low frequency performance of measurement systemsWhen square-wave modulation is employed, the technique is referred to as chopping

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Coherent detection

TheoryWhy and when to use? Basic principle (time domain)Amplitude and phase measurementSignal-to-Noise Ratio (frequency domain)Switching detector (choppers)

Summary

Application example (DEMO 1)Strain gauges Wheatstone bridgeCoherent detector: design considerations


Coherent detection

When to use it?– When measuring low-bandwidth or quasi-static signals in the presence of high noise or disturbance levels.– If a high dynamic range is required

Therefore it is often used to readout sensors e.g.– (MEMS) Accelerometers (fF capacitance changes)– Optical (infrared) detectors (fA’s of current)– Magnetic sensors ( mV’s )– Strain gauges ( mV’s)

A coherent detector behaves like a bandpass filter

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Time domain analysis

( )ˆ ˆ1 cos 2

2r i

oM Amp mult iu uu G G tω= ⎡ − ⎤⎣ ⎦

( ) ( )

( )( ) ( )( )

ˆ ˆsin sin

ˆ ˆcos cos

2

oM Amp mult i i r r

r ioM Amp mult r i r i

u G G u t u t

u uu G G t t

ω ω

ω ω ω ω

= ⎡ ⎤⎣ ⎦

⎡ ⎤= − − +⎣ ⎦

Now if: (Coherent signals !) then:r iω ω=DC

GAMP


( )ˆ ˆ1 cos 2

2r i

oM Amp mult iu uu G G tω= ⎡ − ⎤⎣ ⎦

The low-pass filter removes the second harmonic:

r iω ω=

ˆ ˆ2r i

oF LPF Amp multu uu G G G=

This coherent detector detects the amplitude of the input signal.Note that in this case Ui and Uref are in-phase (Synchronous detection).What if they are not in-phase ????

Amplitude detection

GAMP

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Phase detection

( )ˆ ˆ[cos ]

2r i

oF LPF Amp multu uu G G G ϕ=

r iω ω=

This coherent detector is also phase sensitive.

Suppose Ui and Uref have a phase difference:

( ) ( )

( )( ) ( )( )

ˆ ˆsin sin

ˆ ˆcos cos

2

oM Amp mult i i r r

r ioM Amp mult r i r i

u G G u t u t

u uu G G t t

ω ϕ ω

ω ω ϕ ω ω ϕ

= ⎡ + ⎤⎣ ⎦

⎡ ⎤= − + − + +⎣ ⎦

=0

GAMP


Signal-to-noise ratio

What would be the S/N ratio of– an oscilloscope?– a coherent detector?

Some numbers:What would be the detection limit of the coherent detector if the white noise spectral density = 10 nV/√Hz and fLP = 1Hz ???

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Coherent detection in practice

Linear operation of an analog multiplier circuit, working over alarge range of input signals is difficult to realize in practice.

Chopper realization in CMOS:

Easy: just 4 transistors!Accurate: does not introduce offset But: switching spikes can cause problems (residual offset)

A much simpler realization is a switching detectora.k.a. a chopper


Summary

REMEMBER:An ideal coherent detector is only

sensitive to frequency components in a narrow band around a reference signal Uref.

The low-pass filter determines the signal bandwidth

So what happens if Uref is a square wave ??

GAMP

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DEMO: Coherent detection for the readout of strain-gauges

R is typically 100-2000 Ω

ΔR is typically 10-1- 10-5 Ω

Very small changes in resistanceshould be measured


DEMO: Using strain gauges for force measurement

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DEMO: Using strain gauges for force measurement


Wheatstone ½ bridge

( )Δ Δ

Δ Δ++ +

=+ + +o b bR R R RU = U U

R R R 2R R

ΔΔ+ −= −

+ bd o oR=U U U U2R R

( )Δ Δ− =+ + +o b b

R RU = U UR R R 2R R

Non-linearity error is small if ΔR << 2R

DEMO: Wheatstone bridge with strain gauges

Construction enables only stressed and non-stressed strain gauges

Ub can be an AC source of well-known frequencyUo is a very small AC signal ( typically a few uV)

with a high noise level !

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Strain-gauge readout design

Design considerations:

What is the typical response time for this system used as a weighing scale ?

Frequency of uexc ?

What are the effects of offset in amplifier G


Design: Frequency domain

Demodulation causes a shift in the frequency domain

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DEMO: Coherent detection for the readout of strain-gauges

Some data:Strain gauges : 120 ohm, length 6mmSteel bar: area = 1cm2

1 kg gives: Δl =2,86 nmResistance change: ΔR = 120 μΩRelative change: ΔR/R = 10-6

Noise level:Force: 10N , or 1kgMax force: 15 kN, or 1500 kg ( Steel bar limit )Dynamic range: 1500/0.1 = 15*103 = 84 dB


Coherent detection:Overload and dynamic range

Important limitation of a coherent detector is the maximum allowable signal at the input of the multiplier.

( saturation and non-linearity will lead to unwanted effects such as harmonic sensitivity and self-detection as will be shown later )

Generally a maximum signal indicator is placed at the signal input of the multiplier to indicate overload situations

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Increasing the dynamic range

Use an AC amplifier

Offset suppression reduces overload on multiplier


Detection limits

Other detection limits are due to:o Offsets in both input- and output channel o Non-linearities in the reference channelo Non-linearities in the input channelo Non-linearities at the output of the multiplier

Signal-Noise ratio=

Green areaBlack area

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Coherent detection: Offset

Offset in the input AND reference channels are DC signals that are simply multiplied and pass through the low-pass filter, therefore:

oF mult LPF excDC bridgeDCU GG G U U=


Coherent detection: Non-linearities

o Non-linearities in the reference channelo Non-linearities in the input channelo Non-linearities at the output of the multiplier

In case of a second-order harmonic distortion only:

( ) ( )( ) ( )

( )( ) ( )( ) ( )( ) ( )( )

2

2 2

ˆ ˆ sin sin 2 sin

ˆ ˆ cos cos cos 2 cos 2

2

oM mult r r r i i

r imult r i r i r i r i

u G u t t u t

u uG t t t t

ω β ω ω

ω ω ω ω β ω ω β ω ω

⎡ ⎤= + × =⎣ ⎦

⎡ ⎤− − + + − − +⎣ ⎦

So input components at ωi = 2 ωr cause output!

Makes the coherent detector sensitive to SUPERHARMONICS in the input signal

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In case of a third-order harmonic distortion only:

( ) ( )( ) ( )

( )( ) ( )( ) ( )( ) ( )( )

1

1 1

ˆ ˆ sin sin 3 sin

ˆ ˆ cos cos cos 3 cos 3

2

oM mult i i i r r

r imult r i r i i r i r

u G u t t u t

u uG t t t t

ω β ω ω

ω ω ω ω β ω ω β ω ω

⎡ ⎤= + × =⎣ ⎦

⎡ ⎤− − + + − − +⎣ ⎦

This term generates output if ωi = ωr /3

Makes the coherent detector sensitive to SUBHARMONICS in the input signal (and also to noise in these bands !)




In case of a quadratic distortion only:

So components at ωi = ωr cause outputSelf detection

( ) ( )( ) ( ) ( )( )

( )( ) ( )( ) ( )( )

( )( ) ( ) ( ) ( )( )( )

22

2

2

ˆ ˆ ˆ ˆcos cos cos cos

ˆ ˆ ˆ ˆcos 1 cos 2 1 cos 2

2 2ˆ ˆ ˆ ˆ 1cos 1 cos 2 cos 2 cos 22 2 2

oM mult r i r i r i r i

r i r iimult r i r i

r i r iimult r i r i r i

u G u u t t u u t t

u u u uG t t t

u u u uG t t t t

ω ω δ ω ω

δω ω ω ω

δω ω ω ω ω ω

⎡ ⎤= × + × =⎣ ⎦⎡ ⎤± + + + =⎢ ⎥⎣ ⎦

⎤⎡ ⎛ ⎞± + × + + + ±⎜ ⎟⎥⎢⎣ ⎝ ⎠⎦

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Lock-in amplifier

A lock-in amplifier is a coherent detector with all the extra features required of a general-purpose laboratory instrument.


DEMO: Coherent detection in an optical system

Source: LED modulated with a sine wave

Detector: Photodiode

Instrument: Analog lock-in amplifier (PAR 186)

Interference:

Ambient light, DC (offset), 50 Hz mains (lamp)

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Switching detector

Simplifying multiplier at the expense of super-harmonic sensitivity

Use Fourier series of square wave signal, S(t):( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

ˆ cos

cos 3 cos 54ˆ cos cos ...3 5

cos 2 cos 4ˆ2 cos cos 2 ...3 3

oM i

i

i

U u t S t

t tu t t

t tu t

ω ϕ

ω ωω ϕ ω

π

ω ϕ ω ϕϕ ω ϕ

π

= + ⎡ ⎤ =⎣ ⎦⎛ ⎞⎛ ⎞

+ − + − =⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠⎛ + + ⎞

+ + − − +⎜ ⎟⎝ ⎠

( )ˆ2 cosioF

uU θπ

= Similar transfer function as multiplier detector (apart from factor 2/π instead of ½).


Switching detectorSNR considerations

Simplifying multiplier at the expense of a slight decrease in SNR

Increase of noise due to super-harmonic sensitivity by a factor π2 /8.

( ) ( )2, 2 22

2 2 2 2 2

0

2 2 22 ....53 7

1 1 1 12 1 ... 2 .23 5 7 2 1 8

o LDF o LDF o LDFn SD o LPF

o LDF o LDF o LDFn

N f N f N fu N f

N f N f N fn

π∞

=

= + + + + =− −

⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞+ + + + = =⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟+⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎢ ⎥⎣ ⎦∑

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Phase detector

Further simplification of multiplier at the expense of super- and sub-harmonic sensitivity

Is often used in Phase Locked Loops (PLL).

Logic levels:No amplitude information


PLL for recovery of excitation voltage

The excitation frequency of remote sensors is not always directly available, but can be retrieved from the sensor’s output signal by a phase-locked loop (PLL).

The loop drives the VCO so as to minimize the error signal UoM

As a result, ωi ~ ωo with a small loop-gain-dependent phase error

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Digital lock-in amplifierAnalog functionality is replaced by digital signal processing (DSP)

⇒ Cost, Flexibility, User friendliness

ADC and DSP are critical (frequency limiting) components

Digital lock-in amplifiers are becoming more and more economical and common in test-setups in the lab and in industry


Coherent detection is a powerful technique to accurately measure low-level periodic signals in the presence of high levels of noise and interference.

Both amplitude and phase can be measured

The signal frequency should be known (reference channel) or it should be possible to regenerate it with a PLL.

The operating frequency should be in a frequency band where disturbing signals and noise are low.

Distortion in the input- or reference channel can lead to harmonic sensitivity

Modern (digital) lock-in amplifiers can also measure the amplitude and phase of harmonic signals (2f, 3f ,4f etc. ) e.g. for the measurement of signal distortion.

Summary

slides electrical instrumentation lecture 6

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