Application Note #4 Potentiostat stability mystery explained I- Introduction As the vast majority o research instruments, potentiostats are seldom used in trivial experimental conditions. But potentiostats are not only acing the unusual nature o the research activity but also the great diversity o electrochemical systems and experiments. Even more, due to their nature, the electrochemical experiments evolve over extremely large ranges o values o the signiicant parameters. In corrosion applications, or example, recording the current over 5 or 6 current ranges in the same experiment is very common. It is not hard to imagine that, in such a demanding environment, potentiostats are oten pushed to their limits and used in situations that may compromise their perormance. There are always times when potentiostats are not unctioning as expected. Ringing or oscillations, or example, are signs that a potentiostat has diiculties to maintain or has even lost the control o the cell s potential. This document aims is to elucidate the origin o the stability problems o the actual potentiostats in close relation with the VMP multichannel potentiostat. Once you have better understanding o the instabilities, you will be more conident with playing with some o the experimental parameters like bandwidth or current range or choosing a resistor value in series with the working electrode to settle down your potentiostat without loss o accuracy. Although we will try to keep the text accessible, knowledge o potentiostats designs and terms like impedance, capacitance, and Bode representation is recommended as well as basic skills on complex number calculus. Note: VMP instrument illustrates this note, but the same speciications can be observed with VMP3, VSP, SP-50, SP-50 potentiostats. II- Potentiostats, basic principles Since 94, when Hickling built the irst three electrode potentiostat, a lot o progress has been made to improve the potentiostat s capabilities. Hickling had the genius idea to automatically control the cell potential by the means o a third electrode: the reerence electrode. His principle has remained the same until now. At a glance, a potentiostat measures the potential dierence between the working and the reerence electrode, applies the current through the counter electrode, and measures the current as an ir drop over a series resistor (R m in the Fig. ). Fig. : Basic potentiostat design. The control ampliier CA is responsible or keeping the voltage between the reerence and the working electrode as close as possible to the voltage o the input source E i. It adjusts its output to automatically control the cell s current so that this equality condition is satisied. To understand how it works, we must write down some equations very well known by electronics engineers. Fig. : An electrochemical cell and the current measuring resistor can be replaced by impedances. Bio-Logic Science Instruments, rue de l'europe, F-38640 Claix - tel: +33 476 98 68 3 Fax: +33 476 98 69 09
Beore going orward with the math, note that rom an electrical point o view the electrochemical cell and the current measuring resistor R m can be regarded as two impedances ( Fig. ). Z includes R m in series with the interacial impedance o the counter electrode and the solution resistance between the counter and the reerence. Z represents the interacial impedance o the working electrode in series with the solution resistance between the working and the reerence electrodes. The role o the control ampliier is to ampliy the potential dierence between the positive (or non-inverting) input and the negative (or inverting) input. This can be translated mathematically into the ollowing equation: E out =A. E + -E - =A.(E i -E r ) () where A is the ampliication actor o the CA. At this point, we should make the assumption that no or only insigniicant current is lowing through the reerence electrode. This corresponds to the real situation since the reerence electrode is connected to a high impedance electrometer. Thus the cell current can be written in two ways: I c or Eout = () Z + Z Er Ic = (3) Z Combining Equation and 3 yields Equation 4. E r Z = Eout = β E Z + Z out (4) where β is the raction o the output voltage o the control ampliier returned to its negative input; namely the eedback actor. Z = Z + Z β (5) Combining Equation and 4 yields Equation 6. E E r i β A = + β A (6) When the quantity βa becomes very large with respect to one, Equation 6 reduces to Equation 7, which is one o the negative eedback equations. E r = E i (7) Equation 7 proves that the control ampliier works to keep the voltage between the reerence and the working close to the input source voltage. III- Where are those oscillations coming rom? Let s have a closer look to the control ampliier. Equation 7 is true only when βa is very large. Since the β raction is always inerior to one, this is equivalent to saying that the ampliication actor A must be very big. In practice, the control ampliier ampliies about,000,000 times the input dierence voltage. In act, this is true only or low requency signals. A real control ampliier is made o real, hence imperect, components. Thereore, it does not ampliy in the same way as a low and a high requency signal. It is natural to think that a slowly varying signal is ampliied better than a high-speed signal. The control ampliier is more and more embarrassed as the requency increases because it cannot catch-up with high-speed variation signals. So the ampliication decreases as the requency increases. Furthermore, the output signal is somehow shited with regard to the input signal. Obviously the ampliication is a unction o requency, which can be expressed by a simpliied mathematical model described by the complex Equation 8. A = a +j a (8) where is the requency, a the low requency ampliication, a is called the break down requency, and j =. As any complex number, the amplitude can be expressed in polar orm in terms o magnitude and phase: ϕ j A A = A e (9) Bio-Logic Science Instruments, rue de l'europe, F-38640 Claix - tel: +33 476 98 68 3 Fax: +33 476 98 69 09
According to Equation 8, the magnitude is calculated as: A = a (0) + a and the phase as: ϕ = A arctg () a Fig. 3 shows the control ampliication magnitude and phase plotted versus requency or some common values o a and a. This graphical representation is very intuitive and very close to the real behaviour o the control ampliier. The ampliication actor goes down or requencies bigger than the break down requency. When the ampliication reaches unity, the control ampliier no longer ampliies; it becomes an attenuator. The requency at which the ampliication reaches unity is called the unitygain bandwidth. Fig. 3: Bode plot o the ampliication magnitude and phase or a = 0 6 and a = 0 Hz. Now, let s go back to the Equation 6 and note that both the raction β and ampliication A are complex numbers. What is happening when the quantity βa approaches minus one? β A = () Well, it is not diicult to see that the Equation 6 approaches /0, which heads to minus ininity. In this case, the control ampliier output heads to the power supply limit as ast as it can. When the limit is approached, the control ampliier enters a nonlinear zone. At this point, it can either stay orever or head to the other power supply limit and so on until the power supply is disconnected. The second state is named oscillatory. In both states, the potentiostat has lost the control o the cell, and the system has become unstable. Note that the stability is determined only by the βa actor according to Equation. Thus a stability problem is exclusively due to the control ampliier characteristics, the current measuring resistor (included in Z ), and the cell. It has nothing to do with the excitation signal! Replacing the polar orm o both β and A in the Equation yields Equation 3: β. A.e j( + ) =- (3) which is equivalent to: β A = (4) and: ϕ β + = ± 80 (5) ϕ A We have seen that the phase shit associated with the control ampliier can reach 90 or requencies over the break requency (see Fig. 3). I phase shit associated with the eedback is important, then the total phase shit may reach 80. I this occurs at requencies where Equation 4 is satisied, then the system becomes unstable. A very simple graphical method (also known as the Bode method) can be developed rom Equations 4 and 5 to determine the stability o a potentiostat. Both IAI and /IβI are plotted as a unction o requency on log-log coordinates as shown in Fig. 3 and Fig. 5. Equation 4 is ulilled at the interception o the two curves. The total phase shit at the intercept can be determined by relating the phase shit to the slopes o the IAI and /IβI curves. As shown in Fig. 3, the magnitude rolls-o with a actor 0 within one decade o requencies and the phase shit reaches 90 or requencies over the break requency. Generally a negative magnitude slope o -0/decade corresponds to 90 phase shit while a positive 0/decade to +90 phase shit. Thus, i at the intercept point the IAI slope alls with -0/decade and the /IβI slope rises with +0/decade, then the total phase shit Bio-Logic Science Instruments, rue de l'europe, F-38640 Claix - tel: +33 476 98 68 3 Fax: +33 476 98 69 09 3
expressed by the Equation 5 gets close to -80 and the potentiostat is unstable. IV- Practical situations Connecting a highly capacitive cell to a potentiostat can be a troublesome experience especially when the application requires a sensitive current range. Generally things get worse on more sensitive current ranges. The reason is that this type o cell, along with the current measuring resistor, introduces important phase shits in the eedback signal. Let s take a simple cell equivalent circuit or a nonaradaic system (Fig. 4). Fig. 4: Dummy cell or a nonaradaic system. In this equivalent circuit, the uncompensated solution resistance between the reerence and the working electrodes is represented by the resistor R u, C d is the double-layer capacitance o the working electrode, and R m is the current measuring resistor. The impedance o the counter electrode and the solution resistance between the counter and the reerence electrodes have been neglected or the sake o simplicity (these impedances can be added to the series with R m or a more sophisticated analysis). For the Fig. 4 circuit, the previously deined Z and Z impedances are expressed by Equations 6 and 7. Z = R m (6) Z = R (7) j πc u + d Replacing terms in Equation 5 yields the eedback actor: + j + j β = (8) where = (9) π R m +R u C d and = (0) πr u C d Now let s perorm the stability analysis by the Bode method, or some particular values o the dummy cell circuit (Fig. 5). The ampliication magnitude A corresponds to the VMP control ampliier with the bandwidth actor set to 5. The /IβI quantity is calculated or C d = µf, R m = 00 kω (0 µa current range), R u = kω (curve b ), and R u = 0 Ω (curve a ). The requencies and deined by the Equations 9 and 0 correspond to the /IβI break requencies. Fig. 5: Bode plots or Fig. 4 dummy cell. C d = µf, R m = 00 kω, R u = kω (b), R u = 0 Ω (a). According to the Bode method, the phase shit can be correlated to the slope o the IAI and /IβI curves at the critical interception point. When R u is set to zero, the /IβI a curve has a slope o 0 by one decade o requency and the IAI curve has a 0/decade slope or about 80 total eedback phase shit at the interception point requency (307 Hz). This Bio-Logic Science Instruments, rue de l'europe, F-38640 Claix - tel: +33 476 98 68 3 Fax: +33 476 98 69 09 4
situation will cause oscillations. When the R u = kω, the intercept point moves to a higher requency where the /IβI b curve has a slope very close to zero. Under these circumstances the oscillation condition is not met, thus the system should be stable. This stability analysis is in perect agreement with the true behaviour o the VMP connected to this type o cell. Fig. 6 shows a voltage step response o the system recorded with the EC-Lab sotware. Counter, counter sense, and reerence leads were stucktogether (CA, REF3 and REF) as well as the working with the sense lead (CA and REF). In this test, the cell potential and current are recorded on the 0 µa current range ollowing a 00 mv voltage step. oscillation period o 3.3 ms thus a requency o 300 Hz. As a summary, potentiostats generally provide dierent ir compensation techniques to reduce R u solution resistance. Normally the ir compensation cannot completely remove the uncompensated resistance and oten leads to instability problems. This behaviour can now be perectly understood by the stability analysis prescribed in this note. V- The bandwidth parameter To adapt to most o the practical situations, the VMP was designed with the ability to change the control ampliier bandwidth. By changing the bandwidth, one can move the system rom an unstable state to a stable one. Seven stability actors (also called compensation poles) are proposed which correspond to the same number o bandwidths o the control ampliier. As a reerence, the highest value (7) corresponds to the highest bandwidth o 680 khz and the lowest () to the lowest bandwidth o 3 Hz. Intermediate values are shown Table. Fig. 6: Step response o the VMP or the Fig. 5 dummy cell values. Fig. 5 predicts a stable state when R u = kω. Indeed, Fig. 6 shows that the cell potential quickly reaches the 00 mv level with a small overshot ollowing the voltage step made at.0 seconds. Conversely, when R u is set to zero, the system oscillates as expected. Although, the oscillation does not last orever. The oscillation amplitude is attenuated in time, and the system inally converges to the 00 mv voltage level. Accurate calculation at the intercept point shows that the phase shit misses about 0.7 rom the perect 80 oscillation condition. It is interesting to note that the requency o the oscillation matches the intercept point requency. One can count about 6 periods in 0 ms, which yields an Fig. 7: VMP control ampliier bandwidths. Generally, the narrower the bandwidth (i.e. the lower the value), the more stable it gets, but this is not compulsory as can be shown in Fig. 7. Sometimes the system may become stable when the bandwidth is increased, so i decreasing does not render the potentiostat stable, try to increase it. Fig. 7 shows, along with the VMP gain magnitude or the dierent bandwidth actors, the /IβI quantity or the previously deined dummy cell. As can be quickly seen, the Bio-Logic Science Instruments, rue de l'europe, F-38640 Claix - tel: +33 476 98 68 3 Fax: +33 476 98 69 09 5
system should be stable with the bandwidths actors 7, 6, and 5; it will probably maniest an important overshot with 4 and go into strong ringing or even oscillations or 3,, and. VI- Stability criterion or a capacitive cell A straightorward stability criterion can be deduced when the cell is a simple capacitance: BW Imax < () 4π C where BW is the unity-gain bandwidth in Hz (see Table ), C is the capacitance in F, and I max is the maximum current o a current range in A. Table : Bandwidth poles Bandwidth actor Pole requency ( BW ) 3 Hz 38 Hz 3 3. khz 4 khz 5 6 khz 6 7 khz 7 680 khz Equation yields to a simple abacus shown in Fig. 8. To ind the bandwidth actor or a stable system, locate the intercept point o the capacitance with the desired current range. All the bandwidths on the right side o this point will provide stability. Fig. 8: VMP stability abacus; current range vs. capacitance: ma/µf, µa/nf, na/pf. Examples:. C = nf, I max = 0 µa the stability can be acquired or BW5 - BW. C = µf (000 nf), I max = 00 µa the stability can be acquired or BW - BW 3. C = 0 µf (0000 nf), I max = 0 µa the stability cannot be acquired I the stability cannot be acquired with one o the bandwidth actors, a resistor should be added in series with the capacitance. A series resistor will have the same eect as the uncompensated solution resistance: it will stabilize the system but it will introduce an ir drop error. The resistor should have a minimum potential drop across it in order to have minimum inluence on the working electrode potential. A good compromise is to admit a maximum ir drop o mv. The minimum resistor in series with a capacitance or a given current range and a given bandwidth actor is given by Equation. R min = π BW I max C () As an example, or C = 000 µf, I max = 0 µa, and bandwidth 7 ( BW = 680 khz), the stabilizing resistor would be about 0 Ω. Note that higher the bandwidth the smaller the series resistor value, thus the smaller the ir drop error. VII- Settle down the potentiostat The irst thing to do when your potentiostat gets mad is to admit that the cell might have its part o the responsibility. Ater all, the cell is part o the eedback element o the control ampliier. The rigorous way to ind out what is happening is to draw a circuit model o the cell, compute the eedback actor β, and use the Bode method or the stability analysis. This may be a diicult task since the electrochemical cells are seldom made o just simple capacitors and resistors. I you want a quick solution to your problem without going into detailed stability analysis, you may ollow these steps: Bio-Logic Science Instruments, rue de l'europe, F-38640 Claix - tel: +33 476 98 68 3 Fax: +33 476 98 69 09 6
Check your reerence electrode. Make sure that the inside solution o the reerence electrode has good contact with the bulk electrolyte o the cell. I the porous junction is not wet, then the electrode may have enormous impedance and together with the electrometer input capacitance may introduce a supplementary phase shit on the eedback. Change the Bandwidth actor. Start with a lower value. I decreasing does not work, try to increase it. Choose a higher current range. Since the current measuring resistor is part o the eedback, the lower it is the more stable the system gets. But there is a limit on how small a measuring resistor can be. I it is too small, you won t be able to detect the low currents. I ater the previous steps, the system is still unstable, then you have to think about adding a resistor in series with the working electrode. When the cell is highly capacitive and you have an idea about the double layer capacitance, then use Equation to determine the resistor value. Reduce, i possible, the surace o the working electrode. Since the double layer capacitance is proportional to the electrode area lowering the surace will reduce the capacitance, which is generally responsible or the instabilities. Reduce also, i possible, the impedance between the counter and the reerence electrode. This includes the interacial impedance o the counter electrode and the solution resistance between the two electrodes. Reerences: [] Ronald R. Schroeder, Irving Shain, The application o eedback principles to the instrumentation or potentiostatic Studies, Chemical Instrumentation, (3), pp.33-59, Jan. 969 [] Allen J. Bard, Larry R. Faulkner, Electrochemical Methods Fundamentals and Applications 00 [3] Jerald G. Graeme, Feedback plots deine op amp ac perormance, Burr Brown Applications Handbook 94-06, 994 [4] Ron Mancini, Op Amps For Everyone Texas Instruments, SLOD006B Bio-Logic Science Instruments, rue de l'europe, F-38640 Claix - tel: +33 476 98 68 3 Fax: +33 476 98 69 09 7