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Chemnitz – IWIS2012 – Tutorial 6, September 26, 2012Chemnitz – IWIS2012 – Tutorial 6, September 26, 2012
Electronics and Signals Electronics and Signals in Impedance Measurementsin Impedance Measurements
byby Mart Min Mart Min [email protected]@elin.ttu.ee
Thomas Johann Seebeck Department of Electronics, Thomas Johann Seebeck Department of Electronics, Tallinn University of Technology Tallinn University of Technology
Tallinn, EstoniaTallinn, Estonia
11
Old Hansestadt Reval – Today’s TallinnOld Hansestadt Reval – Today’s Tallinn
Tallinn / Reval was:
- a member of the Hanseatic League (since 1285)
- ruled under the Lübeck City Law (1248-1865)
- capital of the Soviet Socialist Republic of Estonia within the Soviet Union (1940-1991)
Tallinn is:
- capital of the Republic of Estonia, EU member state since 2004
- currency: EURO since Jan 2011
22
____________________________________________________________________________________________________________________________________________________________
The term was introduced by
Oliver Heaviside, mathematician, physicist, and self-taught engineer:
July 1886 - impedance
Dec 1887 – admittance
Ohm's law, published in 1826:
The concept of electrical impedance generalizes Ohm's law to AC circuit analysis. Unlike electrical resistance, the impedance of an electric circuit can be a complex number: Z = V/IZ = V/I, where Z = R + jXZ = R + jX, and R is a real part and X is an imaginary part.
Electrical impedance (or simply impedance) is a measure of opposition to sinusoidal electric current
In 1893, Arthur Edwin
Kennelly presented a paper “on impedance" to the American Institute of Electrical Engineers in which he discussed the first use of complex numbers as applied to Ohm's Law for AC
What is impedance ? What is impedance ?
33
Goal: making theGoal: making the identification faster and simpler! identification faster and simpler!
55
Both magnitude Both magnitude (amplitude) (amplitude) and phase are to be measuredand phase are to be measured
Excitation generator (sine wave)
Amplitude
meter
Phase meter
V/ I
Amplifier .
Z
.
Vz’ Vexc
Iexc
_
Vz
_
Φz
. phase lag Φz
.
Vz
Re Ż = R Re Ż
Im Ż
Im Ż = X
Ż
Ż = R + j X
Ż = R + j X
Magnitude and phase measurement
66
Synchronous or phase sensitive detection (demodulation) suppresses additive noise and disturbances and gives the results (Re or Im) in Cartesian coordinates
Synchronous or phase-sensitive detectionSynchronous or phase-sensitive detection
Excitation generator (sine wave)
LPF (filter)
V/ I
.
Z
.
Vz Vexc
Iexc
_
Vz · cos (Φ – φ )
Phase shifter
φ
Multiplyer
×
Phase lag Φz
- φ
Re Ż = R Re Ż
Im Ż
Im Ż = X
Ż
Ż = R + jX
phase lag Φphasephase lag Φ lag Φ
Synchronousdetection
77
Two-phase (inphase and quadrature, I & Q) synchronous detection (the simpliest
Fourier Transform) enables simultaneous measurement of Re and Im parts
Two-phase or quadrature synchronous detectionTwo-phase or quadrature synchronous detection
Excitation generator (sine wave)
V/ I
.
Z
Vexc
Iexc
.
Vz
Phase shifter
90º
LPF (filter)
Multiplyer
×
LPF (filter)
Multiplyer
×
_
Vz · cos (Φ ) = Re Ż = R _________
Re Ż
_________
Im Ż
_
Vz · sin (Φ ) = Im Ż = X
p h a se la g Φ
Vexc · sin (ω t)
Vexc · cos (ω t)
Fourier Transform
88
Impedance should be measured at several frequencies –a wide band spectral analysis is required. Impedance is dynamic - the spectra are time dependent.Examples: (a) cardiovascular system; (b) pulmonary system; (b) microfluidic device.
Classical excitation – a sine wave – enables slow measurements. Excitation must be:
1) as short as possible to avoid significant changes during the spectrum analysis;
2) as long as possible to enlarge the excitation energy for achieving max signal-to-noise ratio.
Which waveform is the best one?A unique property of chirp waveforms – scalability – enables to match the above expressed contradictory requirements (1) and (2) and the needs for spectrum bandwidth (BW), excitation time (Texc), and signal-to-noise ratio (S/N).
The questions to be answered: a. A chirp wave excitation contains typically hundreds and thousands of cycles, if the impedance changes slowly. What could be the lowest number of cycles applicable when fast changes take place?
b. Are there any simpler rectangular waveforms to replace the sine waves and chirps in practical spectroscopy?
Excitation current
Response voltage
Excit. Z(t)
Problems to be solvedProblems to be solved
99
. response Vz
1) excitation amplitude is strictly limited ! 2) excitation time is limited !
|Ż (f )|
(f )
Short-time DFT or FFT: directly or via binary transforms (Walsh, Hadamard)
Frequency: f 1 to f n
Timing/synchro: t 1 to t 2 Measurement time, Tm
. Response Vz(t)
Generator of excitation current Iexc(t)
|Z (ω,t)| Φ(ω,t) ReŻ (ω,t) ImŻ (ω,t)
Ż(t)
internal noise, a time-variant
nonlinear system
Results as
Spectrograms
outer noise
Focus: finding the best excitation waveforms for the fast and wideband time dependent spectral analysis: intensity (Re & Im or M & φ)
versus frequency ω and time t
10
freq: f1 to f2
Generation ofexcitationwaveform
Cross
correlation
CVz(t),Vr(t,τ)
Fourier Transform
(DFT, FFT)
gz(t) Sz (jω,t) time: t1 to t2
excitation, Vexc response, Vz
Impedance
spectrogram
Ż
Focus:Focus: finding the bestfinding the best excitation waveforms excitation waveforms and signal processing methods and signal processing methods ffor the fast or the fast and wideband, scalable, and and wideband, scalable, and time dependent spectral analysistime dependent spectral analysis: :
intensity ( intensity (ReRe & & ImIm oror M M & & ΦΦ) versus frequency ) versus frequency ωω andand time time tt
Signals and signal processing Signals and signal processing in wideband impedance spectroscopyin wideband impedance spectroscopy
A
a – short rectangular pulse
Crest factor CF = Peak / RMS
A
t1t1t2
b – chirp pulse (t1 to t2)
covers BW (f1 to f2), scalable, acceptable CF=1.414
t1 t2
c – binary sequence (chirp pulse) from t1 to t2 covers BW from f1 to f2 , scalable, ideal CF=1.0
A
Δt
- very high CF (10 to 1000) - BW = 0 to 0.44(1/Δt), - low signal energy, - not scalable
Excitation
control
reference, Vr
1111
Fast simultaneous measurementat the specific frequencies of interest!+ Simultaneous measurement/analysis;+ Frequencies can be chosen freely;+/- Signal-to-noise level is low but acceptable;− Both limited excitation energy and complicated signal processing restrict the number of different frequency components.
Several sine waves simultaneously – Multisine excitation Several sine waves simultaneously – Multisine excitation
1212
Sine wave signals and synchronous sampling: Sine wave signals and synchronous sampling: multisite and multifrequency measurementmultisite and multifrequency measurement
MultisiteMultisite (frequency distinction method, (frequency distinction method, slightly different slightly different ff11 and and ff22))
Multifrequency Multifrequency (sum of very different frequency sine waves)(sum of very different frequency sine waves)
1313
Multisine excitation: optimization (a sum of 4 equal level sine wave components – 1, 3, 5, 7f)
Sum of 4 sine waves Ai = 1, Φi = 0, CF=2.08
Sum of 4 sine waves Ai =1, Φi = opt, CF=1.45(the best possible case)
Normalized to ∑Ai = 1, Φi = opt: Vrmsi = 0.344, CF=1.45Normalized to ∑Ai = 1, Φi = 0: Vrmsi = 0.241, CF= 2.08
Sum of 4 sine waves Ai =1, Φi = 900, CF=2.83 (the worst possible case)
RMS levels of sine wave components in the multisine signal
the best caseΦi = opt; 0.344 Φi = 0; 0.241
Sine waves: A=1, RMS = 0.707
Φi = 90; 0.177 the worst case
14
Waveforms of wideband excitation signalsCrest Factor CF = (max level) / RMS value
Multisine waveform: Σsin
Chirp, chirplet / titlet: ch(t)
Binary sequences (BS)
A
Tch
+V
-V
0
T s
+ V
-V
0
Σsin = Σ Ai sin(ωi t), discrete spectrum
Ai V/ n, i = 2 to n max P 0.5 V2 max RMS 0.707 V CF = 1.38 to (2n)½
max Pexc = max P 0.5V2
Ideal signal-to-disturb. ratio S/ D (no disturbances)
ch(t) = A sin [ ω(t)dt], continuous spectrum
A = V, ω(t) = var, ω1 to ω2
P = 0.5 V2 RMS = 0.707 V CF 1.41
Pexc P = (0.4…0.5) V2
Good S/ D, complicated signal processing
- multifrequency binary sequence (MFBS), discrete spectrum, as signΣAisin(ωit); - binary chirp, cont. spectrum, as sign ch (t); - binary random, cont. spectrum, as MLS
P =1.0 V2 ( the highest possible level ! ) RMS =1.0 V
CF = 1 (the best possible value!)
Pexc < P = (0.6…0.9)V2
Lower S/ D, complicated processing, plenty of disturbing components
A1
T2 T1
+ V
- V
0
A2
∫
≈
≥
≤
≈
≈
≈
15
A. Scalability in frequency domain: bandwidth BW changes, Texc = const = 250 μs 1.0
-1.0
-0.8
-0.5
-0.2
0.0
0.2
0.5
0.8
250u0 25u 50u 75u 100u 125u 150u 175u 200u 225u
Texc = 250 μs
t
100m
1u
10u
100u
1m
10m
10M1k 10k 100k 1M
2.24 mV/Hz1/2
1.12 mV/Hz1/2
1 mV / Hz1/2
BW = 100 kHz
BW = 400 kHz
Texc = 1000 μs
Excitation energy Eexc = 0.5V2 ∙250 μs = 125 V2∙μs
Voltage Spectral Density @ 100 kHz = 2.24 mV/Hz1/2
Voltage Spectral Density @ 400 kHz = 1.12 mV/Hz1/2
Changes in the frequency span BW reflect in spectral density
48 cycles 12 cycles
Excitation time Texc = 250 μs = const
Scalable chirp signals:Scalable chirp signals: two chirplets 1 two chirplets 1
1616
B. Scalability in time domain: duration Texc changes, BW = const = 100 kHz 1.0
-1.0
-0.8
-0.5
-0.2
0.0
0.2
0.5
0.8
1m0 100u 200u 300u 400u 500u 600u 700u 800u 900u
Texc = 250 μs
Texc = 1000 μs
100m
1u
10u
100u
1m
10m
10M1k 10k 100k 1M
2.24 mV/Hz1/2
4.48 mV/Hz1/2
1 mV / Hz1/2
BW = 100 kHz
Energy E250μs = 125 V2∙μs Energy E1000μs = 500 V2∙μs
Voltage Spectral Density @ 250μs = 2.24 mV/ Hz1/2 Voltage Spectral Density @ 1000μs = 4.48 mV/ Hz1/2
Changes in the pulse duration Texc
reflect in spectral density
Bandwidth BW = 100 kHz = const
48 cycles 12 cycles
Scalable chirp signals:Scalable chirp signals: two chirplets 2 two chirplets 2
1717
10
100u
1m
10m
100m
1
10M1k 10k 100k 1M
-40 dB/dec
RMS spectral density (relative)
10
1
10-1
10-2
10-3
10-4
1k 10k 100k 1M f, Hz
2.26 mV/Hz1/2
BW = 100 kHz
Instant frequency , , rad/s - a linear frequency growth chfin Ttfdt
tdt /2
)(
Current phase , rad; chfin Ttfdttt 2/2)( 2
100kHzTch = 10 μs
Texc = Tch = 10 μs,
A very short Chirplet 3 - Half-cycle linear titlet A very short Chirplet 3 - Half-cycle linear titlet
chfin Ttfdttt 2/2)(sinsin 2Generated chirplet 1818
Rectangular (binary) wave based impedance measurementRectangular (binary) wave based impedance measurement
Clock
The current switch operates as a multiplier!
V-to-I
Vin
S1
I –
I+ I
Driving Flip-Flop
transor
S2
+
–
Vout
I-to-V
1 3 5 7 9 11 13 15 17 19 21 23 25 ht h = 1, 3, 5, 7, 9, 11,A1
9 11 13 15 17 1
A1 t
A1 = (4/π)A > A
h = 1, 3, 5, 7, 9, 11, ...
A3 = (4/3π)A
tA
A problem: sensitivity to all the odd higher
harmonics !contains the products of all odd higher harmonics in addition to the response to signal component A1
A5 = (4/5π)A
1919
Ż
Ternary SD
reference signal excitation signal
response
+1 -1
0 +1 -1
0
1.0
-1.0
-0.5
0.0
0.5
1.00.0 0.2 0.4 0.6 0.8
1.0
0.0
0.2
0.4
0.6
0.8
260 5 10 15 20FIG. 2B
Ternary signals – waveforms and spectraTernary signals – waveforms and spectra
-111 reference
excitation
7th 11th
13th
17th
19th
23rd
3rd 5th
1st
9th 25th
- coinciding spectral lines
2020
+1
-1
Ternary SD
reference
0 -1
+1
0 response
Ternary signal processing Ternary signal processing – – 3-positional synchronous switching 3-positional synchronous switching
2121
Generator of binary and ternary signals
22
C ____
CE QQ
CE QI
C ____
QI
RG
CLOCK
FFQ
FFI
OR
XOR
"1"
"0"
D1 Q 1
Q 2
Q 3
Q 4
Q 5
RG Q 6
Q 7 C
Q 8
Q 9
Q10
Q11
Q12
Q13
D14 Q14
0°
6°
12°
18°
24°
30°
36°
42°
48°
54°
60°
66°
72°
78°
84°
90°
NAND
Binary 2-level signals
Ternary3-level signals
Different rectangular waveforms (binary and ternary) of excitation signals
(b)
(c)
(a)
(a) – binary (2-state) chirp, scalable; (b) – binary pseudorandom (MLS), not scalable, waveform is quite similar to the multifrequency binary signal, see next slide
(c) – ternary (3-state) chirp, scalable.23
0
18
30
Spectra and power of binary/ternary chirpsSpectra and power of binary/ternary chirps
Binary(0): Pexc= 0.85P Ternary(18): Pexc 0.93P
Ternary(30): Pexc 0.92P
Binary(0)
Trinary(30)
Trinary (21.2): Pexc= 0.94P – max. possible!
Pexc – excitation powerwithin (BW)exc=100kHz
100kHz
2424
Synthesized multifrequency binary sequences
(4 components – 1, 3, 5, 7f)
Equal-level components
Growing-level components !
Decreasing levels: usual case!A simple rectangular waveform
25
The spectrum contains 14 components at 1, 2, 4, 8f,..., until 8192 f with mean RMS value of 0.22 each. Max level deviation is +/- 3.5 %; 67 % of the total energy is concentrated onto desired frequencies
A section of one binary wave sequence: 14 frequency components and 81920 samples
While multisine signals concentrate all the energy into wanted spectral lines, the binary ones only about 60 to 85%
Despite of losses (15 to 40%), the energy of the desired frequency components in binary sequences have greater value than the comparable components in multisine signals !
Example: optimized multifrequency binary sequence
(14 binary rated components – 1, 2, 4, 8 f,...,8192 f)
2626
Based on diamond transistor
How to make a current sourcesHow to make a current sources
Cparasitic
Based on current feedback
is a problemCparasitic
2727
Simple resistive V-to-I converter
How to make passive current sourcesHow to make passive current sources
Cparasitic
Compensated resistive V-to-I converter
is a problem
2828
Tends to be unstable (both negative and positive feed-backs)
Howland current sourceHowland current source
Cparasitic is a problem
2929
We designed a current source using differential difference amplifier.We got the output impedance: 250 kΩ. At higher frequencies a part of excitation current is flowing down through a parasitic capacitance 40pF. We added a voltage follower (more exactly, an amplifier with a gain 0.9) and reduced the parasitic capacitance about 10 times !
Dual differential amplifier
AC
0.9XZ
Instru-mentationamplifier
0.9X U:Uz
+
-
+
-
Shunt
How to make the current excitation better and to couple the How to make the current excitation better and to couple the excitation signal with the impedance to be measuredexcitation signal with the impedance to be measured
Cparasitic
3030
Dual differential amplifier
AC
0.9X
Trans- impedance
amplifier
ZInstru-
mentationamplifier
0.9
X U:Uz
U:Iz
+
-
+
-
+
-
Shunt
Programmableshunt
We added a trans-impedance amplifier for the measurement of excitation current. Result – degradation of the current source at higher frequencies can be taken into account
How to make the excitation more accurate?How to make the excitation more accurate?
Cparasitic
An alternative: voltage measurement on a shunt
3131
How to measure voltage drop across the impedanceHow to measure voltage drop across the impedance
Instrumentation amplifier (IA)SolutionGood BW can be reached when the IA is constructed from separate high performance op-amps.
Magnitude
Phase
Voltage aquisition amplifier
3232
1) 1) Frequency stepping or sweeping Frequency stepping or sweeping together withtogether with multiplexing of multiplexing of traditional traditional sine wave sine wave excitation is excitation is tootoo time consuming, especially when time consuming, especially when the the dynamic impedances dynamic impedances are to be are to be measured.measured.
2) 2) Simultaneous applying of several sine wave Simultaneous applying of several sine wave excitationsexcitations with different frequencies with different frequencies ((multisinemultisine) ) is a better, but more complicatedis a better, but more complicated solution. solution.
3) 3) We propose specific chirp based excitation signals as We propose specific chirp based excitation signals as chirpletschirplets and and titletstitlets, also , also binary and ternary chirpsbinary and ternary chirps and chirplets and chirplets for carrying out the fast and wide band for carrying out the fast and wide band scalable scalable spectroscopyspectroscopy of dynamic objects. of dynamic objects.
4) 4) Also Also multi-sine binary and ternary multi-sine binary and ternary (trinary) (trinary) signalssignals are proposed for excitationsare proposed for excitationsin impedance spectroscopy and tomography.in impedance spectroscopy and tomography.
5) 5) Synthesis of the above mentioned excitation signals enables to provide independentSynthesis of the above mentioned excitation signals enables to provide independent,, time and frequency domain scalable spectroscopytime and frequency domain scalable spectroscopy, , which is which is adaptadaptableable to given to given measurement situation (speed of impedance variations, frequency range, measurement situation (speed of impedance variations, frequency range, S/NS/N level level).).
6) Use discrete and digital signal generation/processing methods as much as possible, 6) Use discrete and digital signal generation/processing methods as much as possible, but you can never avoid analog part of the measuring system.but you can never avoid analog part of the measuring system.
7) Be careful with current sources, avoid if possible.7) Be careful with current sources, avoid if possible.
8) Using of field programmable gate arrays (FPGA) is challencing. Both, microcontrollers 8) Using of field programmable gate arrays (FPGA) is challencing. Both, microcontrollers and signal processors, make troubles with synchronising and throughput speed.and signal processors, make troubles with synchronising and throughput speed.
SummarySummary
3333