International Journal of Electronics Engineering, 1(1), 2009, pp. 1-6
Optimum Compensation and Stability of Potentiostat
P. M. Nawghare
Department of Electrical Engineering, Faculty of Engineering & Technology, University of Botswana, Gaborone, Botswana
Abstract: An attempt is made to investigate and eliminate the problems in a potentiostat employing positive feedback method
of solution resistance compensation which is widely used in electrochemical kinetic measurements.
A method of optimum adjustment of solution resistance compensation employs an adequate pilot signal obtained by using
selection criterion for minimum error.
An analytical investigation of the conditions to achieve stability along with ringing–free operation and the maximum flat
response is done. The effect of various factors is evaluated.
The illustration by means of numerical example and experimental results of the potentiostat with optimum compensation
and stable maximum flat response are given
Keywords: Potentiostat, Optimum compensation, Adjustment error, Frequency response.
1. INTRODUCTION
In the field of electrochemical kinetics, in order to set the
working electrode potential constant with respect to the
reference electrode, attempts have been made employing a
kind of feedback control loop. The name ‘potentiostat’ was
first given to a vacuum tube type constant voltage equipment
[1, 2, 3].
The major limitation of the potentiostat occurs due to
the presence of a non-fardaic resistance between the
reference and the working electrodes [4, 5, 6, 7]. This is
called ‘solution resistance’. Even in case of small value of
solution resistance, the electrode potential setting in a high
frequency region is very difficult because the solution
resistance drop becomes dominant over the true electrode
potential, as the latter is shunted by the double layer capacity.
The most generally applicable and hence most popular
method for eliminating solution resistance is the ‘positive
feedback’ or ‘series negative resistance’ method. It consists
of connecting a current detecting resistance in series with
the electrolytic cell and subtracting the voltage across it from
the voltage to be controlled. The method has been used by
many researchers and led to the improvement in
electrochemical kinetic results [8, 9, 10, 11, 12, 13, 14, 15,
16, 17, 18]
In using this method, the 100 % solution resistance
compensation is very difficult, if not impossible and the
solution resistance compensation tends to degradation of
potentiostat stability margin, or else reduction in bandwidth
[19, 20, 21, 22, 23, 24].
Many researchers adopted the trial and error adjustment
method in which the amount of compensating voltage is
increased till the ringing disappears in the system, presuming
*Corresponding author: pmnawghare@yahoo.com;
nawgharepm@mopipi.ub.bw
that the reason of the instability is the overcompensation
only [ 9, 10, 11, 12, 13, 14, 15, 16, 17 ]. This did not give
either a theoretical reasoning or the practical indication as
to whether actually an ideal compensation, under
compensation or overcompensation is obtained.
Thus, the study of the work done by the researchers until
now shows that the ‘potentiostat employing solution
resistance compensation’ suffers from the following
problems:
(1) No method of exact adjustment of solution
resistance compensation is available. The
incomplete adjustment results in the same
undesirable effects as those in ‘potentiostat without
solution resistance compensation.
(2) The positive feedback due to solution resistance
compensation usually results in oscillations or
instability of the system.
(3) The attempts to get rid of problem 2 above, results
in narrowing the bandwidth of the system so that
the setting of the true electrode potential cannot be
done at high frequency.
The research work presented here aims at eliminating these
problems.
2. THE POTENTIOSTAT
Fig.1 shows the potentiostat employing a solution resistance
compensation and a method of adjustment of compensation.
In this figure, a cell is replaced by its electrical equivalent
circuit consisting of a double layer capacity Cd shunted by
faradic resistance Rf.
First, a switch SW is placed in position 1, an adequate
high frequency signal (called pilot signal hereafter) is applied
at the input and the compensation is adjusted. Then the
feedback loop is closed by taking switch SW in position 2
International Journal of Electronics Engineering
Substituting for the transfer functions of amplifiers
A2 = K2/(1 + jwt2)
A3 = K3/(1 = jwt3) we get
K3 Rl
K 2 Rx
=0
1 + jwt3 1 + jwt2
get
Equating the real parts on both sides of equation, we
K2 = K3 Rl/Rx
(1)
Next, equating the imaginary parts on both sides of equation
and substituting from eqn.(1), we get
t2 * = t3
(2)
Equations (1) and (2) together form the condition for ideal
compensation of solution resistance drop [25].
3.2. Practical Optimum Adjustment
In practice, it is difficult to make the voltage Vs actually
zero, because Vs being a complex quantity, the adjustment
of the gain K2 only is not sufficient for its control. Therefore,
a phase detector is used to make the real part of Vs (i.e. the
component of Vs in phase with the current Ip) zero [25].
From Fig. 1, we get
Re (Vs ) = 0 i.e.
K3 R f
Figure 1: Modified System with compensation
SW: Switch, PSD: Phase Sensitive Detector,
C, R, W Counter, Reference & Working Electrodes
Zc: Counter electrode impedance
Rf: Faradaid impedance
Cd: Double layer capacity of working electrode W
R1: Solution resistance between W & R
R1¢: Solution resistance between R & C electrodes
Rx: Series Resistance,
A1, A2, A3: Main Compensation and Buffer amplifers
.
2
1 + w2p t3
3.1 The Principle
When switch SW is placed in position 1 and an adequate
high frequency signal (called pilot signal hereafter) of
frequency wp is applied at the input Vi, the reactance of the
double layer capacity becomes very small so that the
impedance of the parallel combination of Rf and Cd is
negligible as compared to the solution resistance Rl. In this
situation, the potential drop between reference and working
electrodes equals the solution resistance drop IpRl where Ip
is a pilot signal current. Then the gain of amplifier A2 is
adjusted such that
Vs = VR – Vx = 0
i.e.
IpRl A3 – Ip Rx A2 = 0
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1 + w2p R 2f Cd2
+
Rl K3
RK
- x 22 2 = 0
2 2
1 + w p t3 1 + w p t2
from which we get
é R f 1 + w2p t22 1 - w2p R f Cd t3 R 1 + w2p t22 ù
+ l.
K 2¢ = K3 ê .
.
ú (3)
2 2
2 2 2
Rx 1 + w2p t32 úû
êë Rx 1 + w p t3 1 + w p R f Cd
3.3 Error in Adjustment
Let the error in adjustment be denoted by d. Then
and the system is ready for use in electrochemical kinetic
measurements [25]
3. OPTIMUM COMPENSATION
1 - w2p R f Cd t3
d=
K 2' - K 2*
K 2*
(4)
If d is positive, solution resistance will be over compensated
and if d is negative, it will be under compensated.
Substituting from equation (1) and equation (3) in equation
(4), and putting t¢2 = t3 +Dt, we get [25]
d=
é æ t f öæ Dt ö w2p t32 ù Dt w2p t32
.
ê1 + ç1 +
ú+ .
÷ç ÷ .
t3 ø è t3 ø 1 + w2p t32 ûú t3 1 + w2p t32
1 + w2p t2f ëê è
t f / tl
(5)
where tf = Rf..Cd and tl = Rl..Cd
As pilot signal frequency wp is quite high, w2pt2f >> 1 can be
assumed. Substituting this in equation (5), we get
é æ tf
d = d1 ê1 + ç1 +
tl
ë è
ö ù
÷ .d 2 ú + d 2
ø û
(6)
!
Optimum Compensation and Stability of Potentiostat
where
d1 »
1
(7)
w2p t f tl
2 2
æ Dt ö w p t3
d2 = ç ÷ .
2 2
è t3 ø 1 + w p t3
and
(8)
3.4 Minimization of Error
Using equation (7), d1 (in percent) is shown plotted [25]
against the normalized frequency wptf with tl /tf varied as a
parameter in Fig. 2.
12
10
3.5. Adequate Frequency of Pilot Signal
Thus, we define an adequate frequency of pilot signal wpo as
the frequency in the range where d1 and d2 and hence the
total error d lies within the permissible limit. Let the
permissible limit of error d be denoted by ‘a’. Then wpo is
given as [25]
wpl < wpo < wph
(9)
1/2
where wpl = (1/ a tf tl) from Fig. 2 and equation (7) And
wph = 1/t3 from Fig. 3.
Percent (%)
8
6
4
2
0 –1
10
The observation of Fig.3 shows that for a given value
of Dt/t3, if wt3 is increased , then d2 goes on increasing and
reaches a maximum value given by d2¢max = Dt/t3. However,
in practice, the useful range of operation of the amplifiers
A2 and A3 is wpt3 £ 1. The curves in Fig. 3 are symmetrical
with respect to the wpt3 axis. d2 is positive or negative
depending upon whether is positive or negative. Thus,
depending upon the value of Dt/t3 there exists a limit of
wpt3 below which d2 is negligibly small.
Now, as d1 is negligibly small above certain value of
wptf and d2 is negligibly small below certain value of wpt3,
the middle term of equation (6) which has the product of d1
and d 2 as a coefficient, is also negligibly small in the
frequency range where d1 or d2 or both are very small.
100
101
wptf
102
C=
tf
103
t1
Figure 2: Error in Adjustment of Compensation in Low
Frequency Range
The observation of Fig.2 reveals that for a given tl /tf ,
d1 goes on increasing with the decrease of wptf and then
reaches saturation at its maximum value given by d1 max =.
tl /tf . If tl /tf is increased by two orders, the curve is shifted
to the left by about one order of wptf. Thus, depending upon
the value of tl /tf, a limit of wptf exists above which d1 is
negligibly small.
Next, using equation (8), d2 (in percent) is shown plotted
[25] against the normalised frequency wpt3 with Dt/t3 varied
as a parameter in Fig. 3.
Figure 3: Error in Adjustment of Compensation in High
Frequency Range
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4. FREQUENCY RESPONSE CONSIDERATION
Stability is the necessary condition for any system. Moreover,
as the electrochemical system has electrolytic cell as a load,
it needs to operate without occurrence of ringing, because
ringing creates undesirable long period oscillations in the
cell current and increases the settling time too much. This
requires that the frequency response of the system should
not have any peaking. The system also requires a wide
bandwidth in order to be useful for the electrochemical
experiments at high frequencies.
Therefore, a condition for the non-occurrence of
peaking in the frequency response is required to be
investigated. Such a condition will not be necessary but will
be sufficient for stability [26]
Consider the system in Fig.1. Let the adjustment of
compensation of solution resistance be made with switch
SW in position 1. The value of gain K2 obtained by practical
method of adjusting output of PSD to zero is given by
K¢2= K3. Rl /Rx (1 + d)
(10)
Now, let the feedback loop be closed by taking switch SW
to position 2. Then the closed loop transfer function of the
system can be evaluated and is of third order.
Here, the following two reasonable assumptions are
made
(1) w << 1/dtl i.e. the highest angular frequency of
interest is much less than 1/dtl.
(2) t1 >> t3 i.e. the bandwidth of amplifier A3 is much
wider than that of amplifier A1.
(3) t3 » 0
"
International Journal of Electronics Engineering
These assumptions simplify the transfer function to
second order and it is given as [26]
Vs( jw)
k
=
Vi( jw) 1 + jwt f b - w2 t f 2 c
(11)
where
k =
b =
c =
K1 (1 - x)
+
1 K1 (1 - x )
-DK1 (1 - x) + x + (t1 / t f )
1 + K1 (1 - x)
x(t1 / t f )
1 + K1 (1 - x)
It can be seen from equation (11) that for a given cell
with time constant tf and a constant x, the factors affecting
the frequency response are time constant t1, compensation
error d and main control amplifier gain K1.
Using the theory of control systems, the damping factor
for the system is given as
Damping factor
z=
b
2 c
@
-DK1 (1 - x) + x + (t1 / t f )
2 x (t1 / t f )[1 + K1 (1 - x)]
(12)
And the bandwidth is given as
wc =
1
tf c
=
1
tf
1 + K1 (1 - x)
x(t1 / t f )
(13)
For getting maximum flat response without peaking putting
d = 0 and damping factor = 1/ 2 in Eq. (12), we get
x tf
1
K1 = .
.
2 (1 - x) t1
(14)
Figure 4: Effect of Variation of t 1 on Frequency Response
k = 0.99, d = 0, Curve 1: t1/tf =2.50×10–3, Curve 2: t1/tf
= 1.75 × 10–3, Curve 3: t1/tf = 1.25 × 10–3, Curve 4:
t1/tf = 0.75 × 10–3 Curve 5: t1/tf = 0.25 × 10–3
absence of peaking. If t1 is decreased the peaking is reduced.
The value of t1 for optimum frequency response can be
obtained. Curve 3 is nearly optimum response.
It can be seen from Fig. 5 that curves 1 and 2 illustrate
the peaking. Curves 4 and 5 illustrate no peaking but are
under damped. If d is increased (i.e. over compensation),
the peaking is increased. Curve 3 is nearly optimum response.
In Fig. 6, Curve 1 shows peaking which is an effect of
over compensation. Curves 2 and 3 also are over
compensated but have reduced gain i.e. increased steady state
error. Thus, the effect of lesser K1 which gives greater steady
state error is to increase damping and reduce the chances of
occurrence of peaking even in presence of a little over
compensation, but at the same time the bandwidth of the
response is reduced.
for K1(1 – x) >> 1 and x >> t1/tf
Substituting Eq. (14) in Eq. (13) the bandwidth of the
maximum flat response of the system is given as
5.
1 1
. for x >> t1 / tf
2 t1
1
(15)
ILLUSTRATION
The effect of t1 , d and K1 on frequency response is illustrated
by numerical example [26].
Let the gain K1 be selected so that the system error at
w = 0 may be 1%. i.e. k = 0.99. Also, assume x = 0.25 so
that K1 = 132.
Substituting these values in Eq. (11), the frequency
response curves are shown plotted in Fig. 4, Fig. 5 and Fig. 6
for a series of values of t1 , d and K1 respectively.
It can be seen from Fig. 4 that curves 1 and 2 illustrate
the occurrence of peaking. The curves 4 and 5 illustrate the
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1.2
|VsV1|
wc max @
1.4
0.8
0.6
0.4
0.2
0
100
101
102
wtf
103
104
Figure 5: Effect of Variation of d on Frequency Response k = 0.99,
t1/ tf. = 1.25 × 10–3, Curve 1: d = 1% Curve 2: d = 0.5%,
Curve 3: d = 0, Curve 4: d = –0.5% Curve 5: d =– 1%
#
Optimum Compensation and Stability of Potentiostat
Figure 6: Effect of Variation of K1 on Frequency Response
d = 0.5%, t1/ tf. = 1.25 × 10–3, Curve 1: k = 0.99 Curve 2:
k = 0.97 Curve 3: k = 0.95
6.
EXPERIMENTAL RESULTS
Using the principles analytically obtained above, a
potentiostat was constructed. A multiplier with C preamplifier and a low pass filter was used as a phase sensitive
detector. The specifications of the constructed system were
measured and found as follows:
A1: K1= 20 dB to 40 dB adjustable, t1 = 0.16 × 10–4 sec.
A2: K2= 26 dB to 0 dB adjustable, t2 = t3 within permissible
limit
A3: K3= 0 dB, t3 = 0.16x10-6sec, CMRR = 98 dB at 1 KHz
The chemical cell was constructed using gold (Au) wires
as counter and working electrodes and a silver/silver chloride
(Ag/AgCl) wire as reference electrode. An electrolytic
solution of 0.1 Mol/litre potassium chloride (KCl) was used
without reacting species in it. This cell was put to
experimentation using the potentiostat constructed as above.
For this cell tf = RfCd is of the order of several seconds
and t1 = 0.16 × 10–4 sec, so that from equation (7), lower
limit of pilot signal frequency is about 400 Hz. The upper
limit is the cut off frequency of amplifiers A2 and A3 from
Fig.3 i.e. 10 MHz. Therefore, using equation (9), 10 KHz
was selected as a pilot signal frequency. The value of Rx
was decided by trial and error to be 3 KW. With switch SW
in position 1 and with pilot signal applied, the suitable value
of K2 for getting optimum compensation was found to be
0.3, revealing that the solution resistance of the cell is about
1 KW.
Next, switch SW was taken to position 2 and the gain
of main control amplifier A1 was adjusted. The frequency
responses plotted experimentally are shown in Fig. 7. Curve
1 is a case of 1 % over compensation with optimum value
of K1. It shows a peaking due to over compensation. Curve
2 is a case of optimum compensation with optimum value
of K1. It shows a maximum flat response. Curve 3 is a case
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101
102
103
104
Frequency (Hz)
105
Figure 7: Experimentally Drawn Frequency Responses for Cell
without Reacting Species
of 1% under compensation with optimum value of 1. It shows
over damping due to under compensation. Curve 4 is a case
of 1 % over compensation with K1 lesser than optimum value.
It shows the over damping due to under compensation and
also increased steady state error due to reduced K1. Curve 5
is a case of optimum compensation with value of K1 greater
than optimum. It shows the effect of incorrect adjustment
of K1.
7.
CONCLUSION
In a potentiostat, the true electrode potential setting by the
adjustment of solution resistance compensation within
permissible error can be achieved by the adjustment of the
gain of compensation amplifier done under the conditions
of opening the loop and applying an adequate high frequency
pilot signal (i.e. lying in the range with lower limit depending
upon the cell parameters and the upper limit depending upon
the time constants of the buffer and compensation amplifiers)
to its input and detecting its zero output voltage by the phase
sensitive detector. The effect of compensation adjustment
error and that of main control amplifier gain and time
constant are illustrated by numerical example. The maximum
wide frequency band operation of the potentiostat employing
solution resistance compensation can be obtained to be free
from ringing if the gain and time constant of the main control
amplifier in the system and the degree of solution resistance
compensation satisfy the critical damping condition. The
experimental results obtained are in agreement with the
theoretical principles.
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