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ET8.017El. Instr.Delft University of Technology
Electronic Instrumentation
Lecturer: Kofi Makinwa
015-27 86466Room. HB13.270 EWI building
Also involved:Saleh Heidary([email protected])
ET8.017El. Instr.Delft University of Technology
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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ET8.017El. Instr.Delft University of Technology
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
ET8.017El. Instr.Delft University of Technology
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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ET8.017El. Instr.Delft University of Technology
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
ET8.017El. Instr.Delft University of Technology
( )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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ET8.017El. Instr.Delft University of Technology
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
ET8.017El. Instr.Delft University of Technology
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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ET8.017El. Instr.Delft University of Technology
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
ET8.017El. Instr.Delft University of Technology
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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ET8.017El. Instr.Delft University of Technology
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
ET8.017El. Instr.Delft University of Technology
DEMO: Using strain gauges for force measurement
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ET8.017El. Instr.Delft University of Technology
DEMO: Using strain gauges for force measurement
ET8.017El. Instr.Delft University of Technology
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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ET8.017El. Instr.Delft University of Technology
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
ET8.017El. Instr.Delft University of Technology
Design: Frequency domain
Demodulation causes a shift in the frequency domain
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ET8.017El. Instr.Delft University of Technology
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
ET8.017El. Instr.Delft University of Technology
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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ET8.017El. Instr.Delft University of Technology
Increasing the dynamic range
Use an AC amplifier
Offset suppression reduces overload on multiplier
ET8.017El. Instr.Delft University of Technology
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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ET8.017El. Instr.Delft University of Technology
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=
ET8.017El. Instr.Delft University of Technology
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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ET8.017El. Instr.Delft University of Technology
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 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 !)
ET8.017El. Instr.Delft University of Technology
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 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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ET8.017El. Instr.Delft University of Technology
Lock-in amplifier
A lock-in amplifier is a coherent detector with all the extra features required of a general-purpose laboratory instrument.
ET8.017El. Instr.Delft University of Technology
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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ET8.017El. Instr.Delft University of Technology
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).
ET8.017El. Instr.Delft University of Technology
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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ET8.017El. Instr.Delft University of Technology
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
ET8.017El. Instr.Delft University of Technology
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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ET8.017El. Instr.Delft University of Technology
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
ET8.017El. Instr.Delft University of Technology
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