Chemnitz – IWIS2012 – Tutorial 6, September 26, 2012
Electronics and Signals
in Impedance Measurements
by Mart Min
min@elin.ttu.ee
Thomas Johann Seebeck Department of Electronics,
Tallinn University of Technology
Tallinn, Estonia
1
Old 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
2
What is impedance ?
Ohm's law, published in 1826:
______________________________________________________________________________
The term was
introduced by
Oliver Heaviside,
mathematician,
physicist, and selftaught engineer:
July 1886 - impedance
Dec 1887 – admittance
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
Electrical impedance (or simply impedance) is a
measure of opposition to sinusoidal electric current
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/I, where Z = R + jX, and R is a real part and X is an imaginary part.
3
Dynamic system identification is the final aim!
Generation of Excitation
Digital
Synthesis of
Excitation
Signals
Processing of Response
M
u
x
ωexc ; Texc
waveform,
energy
Reference / Sampling
.
.
.
Impedance
Ż(ω,t)
Re{Z}; Im{Z}
speed of
changes
.
.
.
System under study
D
e
m
u
x
Time/frequency
domain
Processing
of Response
Signals
ω;t
resolution,
signal-to-noise
Sampling
Ż(ω,t)
Reference
4
Goal: making the identification faster and simpler!
Generation of Excitation
Digital
Synthesis of
Excitation
Signals
Reference
/ Sampling
Processing of Response
.
.
.
(Bio-)
Impedance
Ż(t)
.
.
.
System under study
Digital
Processing of
Response
Signals
Ż(ω,t)
Reference
5
Both magnitude (amplitude) and phase are to be measured
Magnitude and phase measurement
Im Ż
Ż = R +j X
Re Ż
Re Ż = R
F

Im Ż = X
Ż 
Ż = R +j X
6
Synchronous or phase-sensitive detection
_
Synchronous
detection
V z · cos (Φ – φ )
Im Ż
phasephase
phase
lag Φ lag Φ
lag Φ
Re Ż = R
Re Ż
F -φ
Ż
Im Ż = X
Ż = R + jX
Synchronous or phase sensitive detection (demodulation) suppresses additive
noise and disturbances and gives the results (Re or Im) in Cartesian coordinates
7
Two-phase or quadrature synchronous detection
Fourier Transform
sign [sin]
D
C
QI
FF I
302
____
QI
C lock
f clk = 4·f
D
C
"1"
QQ
FF Q
____
QQ
sign [cos]
335
Two-phase (inphase and quadrature, I & Q)
synchronous detection (the simpliest
Fourier Transform) enables simultaneous
measurement of Re and Im parts
8
Problems to be solved
Excitation
current
Response
voltage
Impedance should be measured at several frequencies –
Excit.
Z(t)
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-tonoise 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?
9
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
Signals and signal processing
in wideband impedance spectroscopy
Focus: finding the best excitation waveforms and signal processing methods
for the fast and wideband, scalable, and time dependent spectral analysis:
intensity (Re & Im or M & Φ) versus frequency ω and time t
excitation, Vexc
Excitation
control
time: t1 to t2
freq: f1 to f2
Generation of
excitation
waveform
response, Vz
Ż
Cross
correlation
reference, Vr
gz(t)
C{Vz(t),Vr(t,τ)}
Fourier
Transform
Impedance
spectrogram
Sz (jω,t)
(DFT, FFT)
A
A
A
a – short rectangular pulse
- very high CF (10 to 1000)
- BW = 0 to 0.44(1/Δt),
- low signal energy,
- not scalable
t1
Δt
Crest factor CF = Peak / RMS
t2
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
11
Several sine waves simultaneously – Multisine excitation
Fast simultaneous measurement
at 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.
12
Sine wave signals and synchronous sampling:
multisite and multifrequency measurement
Multifrequency (sum of very different frequency sine waves)
Multisite (frequency distinction method,
slightly different f1 and f2)
13
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 = 900, CF=2.83
(the worst possible case)
Sum of 4 sine waves Ai =1,
Φi = opt, CF=1.45
(the best possible case)
RMS levels of sine wave components in the multisine signal
Sine waves: A=1,
RMS = 0.707
the best case
Φi = opt; 0.344
Φi = 0; 0.241
Φi = 90; 0.177
the worst case
Normalized to ∑Ai = 1, Φi = opt: Vrmsi = 0.344, CF=1.45
Normalized to ∑Ai = 1, Φi = 0: Vrmsi = 0.241, CF= 2.08
14
Waveforms of wideband excitation signals
Crest Factor CF = (max level) / RMS value
M ultisine w avefo rm : Σsin
+V
+V
B inary sequ ence s (B S )
C hirp , chirplet / titlet : ch(t)
+V
A2
A
A1
0
0
0
T ch
Ts
T2
T1
-V
-V
Σsin = Σ A i sin(ω i t),
-V
ch(t) = A sin [ ∫ ω (t)dt ],
- m u ltifreq u en cy b in ary seq u en ce (M F B S ),
d iscrete sp ectru m
co n tin u o u s sp ectru m
d iscrete sp ectru m , as sign { ΣA isin(ω it)};
A i ≈ V / n, i = 2 to n
A = V , ω (t) = var, ω 1 to ω 2
- b in ary ch irp , co n t. sp ectru m , as sign{ ch (t)};
- b in ary ran d o m , co n t. sp ectru m , as M L S
m ax P ≈ 0.5 V 2
C F = 1.38 to (2n) ½
P = 0.5 V 2
R M S = 0.707 V
C F ≤ 1.41
m ax P exc = m ax P ≈ 0.5V 2
P exc ≥ P = (0.4… 0.5) V 2
m ax R M S ≈ 0.707 V
Ideal signal-to-disturb.
ratio S / D (no distu rbance s)
G ood S / D , com plicated
signal processing
P =1.0 V 2 ( th e h ig h est p o ssib le level ! )
R M S =1.0 V
C F = 1 ( th e b est p o ssib le valu e! )
P exc < P = (0.6… 0.9)V 2
Low er S / D , com plicated processing,
p lenty of disturbing com ponents
15
Scalable chirp signals: two chirplets 1
A. Scalability in frequency domain: bandwidth BW changes, Texc = const = 250 μs
1.0
0.8
48 cycles
12 cycles
0.5
t
0.2
0.0
-0.2
-0.5
-0.8
-1.0
0
25u
50u
75u
100u
125u
150u
175u
Texc = 250 μs
200u
225u
250u
Texc = 1000 μs
100m
Excitation time Texc = 250 μs = const
Excitation energy Eexc =
0.5V2 ∙250
μs = 125
2.24 mV/Hz1/2
V2∙μs
10m
1 mV / Hz1/2
1.12 mV/Hz1/2
Voltage Spectral Density @ 100 kHz = 2.24 mV/
Hz1/2
BW = 100 kHz
1m
Voltage Spectral Density @ 400 kHz = 1.12 mV/Hz1/2
BW = 400 kHz
100u
Changes in the frequency span BW
reflect in spectral density
10u
1u
1k
10k
100k
1M
10M
16
Scalable chirp signals: two chirplets 2
B. Scalability in time domain: duration Texc changes, BW = const = 100 kHz
1.0
0.8
12 cycles
0.5
48 cycles
0.2
0.0
-0.2
-0.5
-0.8
-1.0
0
100u
200u
300u
400u
500u
Texc = 250 μs
600u
700u
800u
900u
1m
Texc = 1000 μs
100m
Bandwidth BW = 100 kHz = const
4.48 mV/Hz1/2
2.24 mV/Hz1/2
Energy E250μs = 125 V2∙μs
Energy E1000μs = 500 V2∙μs
Voltage Spectral Density @ 250μs = 2.24 mV/
10m
Hz1/2
Voltage Spectral Density @ 1000μs = 4.48 mV/ Hz1/2
Changes in the pulse duration Texc
reflect in spectral density
1 mV / Hz1/2
1m
100u
10u
BW = 100 kHz
1u
1k
10k
100k
1M
10M
17
A very short Chirplet 3 - Half-cycle linear titlet
sin θ
RMS spectral density (relative)
sin θ(t)
10
10
1
-40 dB/dec
1
θi= θ(ti)
θfin = π
θ
1
θ0= 0
cos θ
0
t =0
t
ti
tfin= Tch = 10 μs
1
2.26 mV/
10
100m
-1
f
ffin= 100 kHz
10m
10-2
Tch = 10 μs
100kHz
f(ti)
f0= 0
t=0
t
ti
d  (t )
dt
10-3
1m
tfin= Tch = 10 μs
100u
10-4
Texc = Tch = 10 μs, BW = 100 kHz
Instant frequency t  
Hz1/2
1k
1k
10k
10k
100k
100k
1M
1M
f, 10M
Hz
 2   f fin  t / T ch, , rad/s - a linear frequency growth
Current phase  t     ( t )  dt  2   f fin  t 2 / 2 T ch , rad;
Generated chirplet sin  t   sin
2

(
t
)

dt

2


f

t
/ 2 T ch
fin

18
Rectangular (binary) wave based impedance measurement
A problem: sensitivity to all
The current switch operates as a multiplier!
the odd higher harmonics !
V-to-I
I+
I
+
Vout contains the products of all odd
I-to-V
higher harmonics in addition to the
response to signal component A1
I– –
Vin
S1
S2
Driving
Flip-Flop
transor
Clock
A1 = (4/π)A > A
t
A1
A
t
sign [sin]
D
C
QI
FFI
A3 = (4/3π)A
302
A5 = (4/5π)A
____
QI
C lo c k
h = 1, 3, 5, 7, 9, 11, ...
f clk = 4·f
D
C
QQ
FFQ
_ __ _
QQ
sign [cos]
1
3
"1 "
5 7
t
h = 1, 3, 5, 7, 9, 11,
A1
9 11 13 15 17 1
9 11 13 15 17 19 21 23 25
h
335
19
Ternary signals – waveforms and spectra
1.0
excitation
0.5
0.0
-111
reference
-0.5
Ż
response
-1.0
0.0
Ternary
SD
1.0
0.2
0.4
0.6
0.8
1.0
1st
0.8
+1
0
-1
excitation signal
+1
0
-1
reference signal
0.6
- coinciding spectral lines
0.4
7th
0.2
0.0
3rd
5th
11th
13th
9th
17th 19th
23rd 25th
FIG. 2B
0
5
10
15
20
26
20
Ternary signal processing – 3-positional synchronous switching
Ternary SD
+1
response
0
-1
reference
+1
0
-1
21
Generator of binary and ternary signals
N AND
CE
C
331
QI
FFI
__ _ _
QI
Binary
2-level signals
0°
"1 "
D1
Q 1
Q 2
Q 3
Q 4
Q 5
RG
RG Q
XOR
C LO C K
18°
30°
42°
Q 8
48°
Q 11
Q 12
Q 13
Q 14
332
(3 3
333
(3 3
24°
Q 7
Q 10
D 14
12°
6
Q 9
"0 "
6°
36°
C
Ternary
3-level signals
54°
60°
334
66°
72°
334
78°
84°
90°
QQ
C
OR
CE
FFQ
_ __ _
QQ
335
22
Different rectangular waveforms (binary and ternary)
of excitation signals
(a)
(b)
(c)
(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
Spectra and power of binary/ternary chirps
0
Binary(0)
18
100kHz
Trinary(30)
30
Pexc – excitation power
within (BW)exc=100kHz
Binary(0): Pexc= 0.85P
Ternary(18): Pexc  0.93P
Ternary(30): Pexc  0.92P
Trinary (21.2): Pexc= 0.94P – max. possible!
24
Synthesized multifrequency binary sequences
(4 components – 1, 3, 5, 7f)
A simple
rectangular
waveform
Decreasing levels: usual case!
Equal-level components
Growing-level components !
25
Example: optimized multifrequency binary sequence
(14 binary rated components – 1, 2, 4, 8f,...,8192f)
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%
The spectrum contains 14 components at 1, 2, 4,
8f,..., until 8192f with mean RMS value of 0.22 each.
Max level deviation is +/- 3.5 %; 67 % of the total
energy is concentrated onto desired frequencies
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 !
26
26
How to make a current sources
Cparasitic
Cparasitic is a problem
Based on diamond transistor
Based on current feedback
27
How to make passive current sources
Cparasitic is a problem
Simple resistive V-to-I converter
Compensated resistive V-to-I converter
28
Howland current source
Cparasitic is a problem
Tends to be unstable (both negative and positive feed-backs)
29
How to make the current excitation better and to couple the
excitation signal with the impedance to be measured
+
-
Dual
differential
+ amplifier
Cparasitic
-
0.9X
Z
0.9X
AC
Shunt
Instrumentation
amplifier
U:Uz
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 !
30
How to make the excitation more accurate?
+
Shunt
Cparasitic
-
Dual
differential
+ amplifier
0.9X
Programmable
shunt
Z
0.9X
-
AC
Instrumentation
amplifier
U:Uz
U:Iz
Transimpedance
amplifier
+
We added a trans-impedance amplifier for
the measurement of excitation current.
An alternative:
Result – degradation of the current source
voltage measurement on a shunt
at higher frequencies can be taken into account
31
How to measure voltage drop across the impedance
Instrumentation amplifier (IA)
Solution
Good BW can be reached when the IA is
constructed from separate high performance
op-amps.
Magnitude
Phase
Voltage aquisition amplifier
32
Summary
1) Frequency stepping or sweeping together with multiplexing of traditional sine wave
excitation is too time consuming, especially when the dynamic impedances are to be
measured.
2) Simultaneous applying of several sine wave excitations with different frequencies
(multisine) is a better, but more complicated solution.
3) We propose specific chirp based excitation signals as chirplets and titlets, also
binary and ternary chirps and chirplets for carrying out the fast and wide band
scalable spectroscopy of dynamic objects.
4) Also multi-sine binary and ternary (trinary) signals are proposed for excitations
in impedance spectroscopy and tomography.
5) Synthesis of the above mentioned excitation signals enables to provide independent,
time and frequency domain scalable spectroscopy, which is adaptable to given
measurement situation (speed of impedance variations, frequency range, S/N level).
6) Use discrete and digital signal generation/processing methods as much as possible,
but you can never avoid analog part of the measuring system.
7) Be careful with current sources, avoid if possible.
8) Using of field programmable gate arrays (FPGA) is challencing. Both, microcontrollers
and signal processors, make troubles with synchronising and throughput speed.
33