3. The displacement of the peak center is equal to the sum of the two displacements which would result from the separate effect of each time constant. Result 3 follows directly from the earlier result that the shift (if small) of the maximum of a curve with a parabolic shape near its maximum is just the time constant, T . Both the first and second time constants leave the shape of the peak maximum substantially unchanged and, therefore, the shifts
must be additive. As stated earlier, the range of validity of this argument for 1% accuracy for each time constant is
TIU5 0.4. RECEIVED for review April 24, 1969. Accepted July 15, 1969. This work was made possible by the award of a Research Fellowship to one of the authors (I.G.M.) by the Shell Group of Companies in Australia.
Instrumental Peak Distortion II. Effect of Recorder Response Ti me I. G. McWilliam’ and H. C. Bolton2 Monash University, Clayton, Victoria 3168, Australia Distortion of square, sine, triangular, and Gaussian shaped peaks due to slow recorder response is considered. Frequency response can be used to determine both the maximum pen speed of a recorder and the recorder delay time. From the frequency response data, an effective response time can be established which determines the behavior of the recorder to a Gaussian input function. This time is generally about twice the manufacturer’s span step response time, and a simple relationship is derived between chromatographic column efficiency, the effective response time, and the minimum retention time for an undistorted peak. Finally the maximum acceptable recorder delay time is generally about one fifth of the effective response time.
IN STUDYING instrument performance, it is necessary to clearly separate those factors causing peak distortion which are introduced by the instrument, and those factors which are due to ancillary equipment. In a previous paper ( I ) , the effects of a single relaxation time (or time constant), and of two consecutive but independent relaxation times, have been discussed. Recorder response, however, must be considered separately because the recorder is a “non-linear” element, i.e., its behavior cannot be represented by a set of linear differential equations with constant coefficients. We shall be concerned primarily with the response of the recorder to a Gaussian input function, but the approach used is general and quite applicable to other waveforms. Our early proposals for studying recorder response were based on a direct approach requiring the generation of a Gaussian input function, and this led to the proposed use of Fourier synthesis as a possible means of generating this function. This has been described in some detail elsewhere (I) and early results using a 3-term Fourier expansion [ ( I ) , Equation 151 appeared to be quite promising. However, to generate pure first and second harmonics and keep these in phase at the low frequencies involved (of the order of several cycles/ sec) presents considerable problems. Initial attempts were made using a square wave generator and multivibrator, followed by filter circuits to isolate the required harmonics (2).
2
Department of Chemistry. Department of Physics.
However, peak distortion resulted because of the presence of unwanted harmonic components, and the performance of the filter circuits is, of course, frequency dependent, an undesirable property when it is required to alter the width of the Gaussian function. It is perhaps worth noting at this point that the response of a recorder to a Gaussian function cannot be deduced from its response to two frequencies,f and 2f, with subsequent application of Fourier analysis. This is because Fourier analysis, as distinct from methods based on the synthesis of a Gaussian function, is applicable only to linear systems. An alternative method which might be considered is the use of two cosine potentiometers driven by a common variablespeed motor, since this approach overcomes the filtering problem. However (particularly with the need to include an electronic gating system to isolate one section of the repetitive waveform) it is doubtful whether this offers any advantage over the use of a multi-tapped potentiometer loaded so as to reproduce a Gaussian function, or indeed over other possible methods such as waveform shaping (3, 4 ) or photoelectric curve-following techniques (5). An examination of recorder response to a step input function indicated that the problem might be solved by an indirect approach using digital approximation methods, and a description of the procedure used and results obtained follows. RECORDER RESPONSE TO A STEP INPUT
The response of a typical potentiometric recorder, with gain and sensitivity correctly adjusted, to a step input is shown in Figure l a . In Figure l b the derivative of Figure la, Le., the pen velocity, is shown. To a first approximation, the response can be represented by an instantaneous transition from zero to maximum pen velocity, the velocity falling sharply to zero when the impressed signal and the emf generated by the potentiometer are equal. As shown in Figure 1 , this ideal behavior (broken curve) appears to be met more closely at the start of the pen movement than at the settling point. However, in practice a slight delay may occur before the initial upward movement, even though the driving force is normally
(I) I. G . McWilliam and H. C. Bolton, ANAL. CHEM.,41, 1755
( 3 ) Chem. Eng. News, 51 (Nov. 15, 1965). (4) F. W. Noble, J. E. Hayes, and M. Eden, I . R. E . Proc., 47,
(1969). (2) H. Dyer and I. G . McWilliam, unpublished results.
1952 (1959). (5) R. D. Johnson, J . Gas Chromatogr.,6, 43 (1968)
1762
ANALYTICAL CHEMISTRY, VOL. 41, NO. 13, NOVEMBER 1969
$1
I
HI I
I
PEN TRAVEL
I
Figure 1. Recorder response to step input
Figure 2. Recorder response (idealized) to sine wave input
much greater at this initial point than at the settling point where the recorder driving emf is approaching zero.
I
“c
RESPONSE TO A SINE WAVE INPUT Because of the availability of sine wave generators, it is generally much easier to check the response of a recorder to a continuous sine wave input than to a Gaussian input function. Moreover, this offers advantages over the measurement of step response time, particularly when the recorder response is fast but the chart speed is relatively slow, i.e., when the slope of the step input response (Figure 1) is too great to accurately measure on the chart. This approach, frequently used in process control investigations, has been reported for recorder testing (6, 7) but care is necessary in interpreting the results obtained. Unlike normal frequency response techniques, the recorder results do not have the same significance as those produced by a linear element, nor has the response behavior previously been explained. Since the sine wave input is a recurring function, the method of treatment can be much simpler than that required for the Gaussian input function. Let us first consider a recorder whose maximum pen speed, u, is too low to accurately follow the input sine wave starting at t = 0 shown in Figure 2. Provided that u is less than u t (see below), and the recorder response is ideal (Figure l), the response will settle out over a few cycles to an equilibrium position as shown. Since the resultant response must be symmetrical, hilt1
=
-hhz/tz =
(1)
L‘
-+
Figure 3. input
Critical velocities for sine wave
make the maximum traverse of the recording equal to a, the amplitude of the input sine wave at any time, t , is given by y =
(4)
sin 2xft
- hd/a = ( t l + t&/a
= u/2fa
- d dtr =
-xfa cos 2xft = v t
(6) R. B. Bonsall, J . Gas Chromatogr., 2, 211 (1964). (7) H. L. Daneman and G . S. Talbot, “Application of Recorders to Gas Chromatography,” k e d s and Northrup Reprint ND4691(7), Philadelphia, Pa., 1961.
(4)
and the pen speed is also equal to Ut =
(2)
wherefis the impressed sine wave frequency and a is its peakto-peak amplitude. The maximum pen speed at which a pure triangular output would result, u t (Figure 3), is an important limiting value for Equation 1, since this Equation is valid only when the maximum pen speed is less than, or equal to, the slope of the sine curve at the point of intersection. Since we have chosen to
(3)
When the slope of the sine curve is equal to the pen speed, we know that
and therefore (hl
TlME,I
a sin 2xft 1I2f
so that we have -tan 2xft = x / 2 or 2xft
=
2.138 radians
Substituting for 2xjr in Equation 4 gives ut N 1.7 fa
or
fi= 0.59 u/a
ANALYTICAL CHEMISTRY, VOL. 41, NO. 13, NOVEMBER 1969
(7) 1763
lor
I
I
f
RIKADENKI
r ------l
1
B-O11(91%RESPONSE,REEB)
/
.il
L a N MODEL G (OLD MODEL,86%RESPONSE)
/
+TIME
-NORMAL
RESPONSE
----MAXIMUM
0
0.2
0.4
0.8
0.6
1.0
Figure 5. Recorder response (idealized) to square, sine, and triangular inputs
DAMPING
1.2
1.6
1.4
1.8
l/f (,e,)
Figure 4. Frequency response to sine wave input for several recorders
The behavior at the higher frequencies is readily explained by the fact that any initial delay (Figure 1) has been neglected in deriving Equation 2. Referring to Figure 2, if a delay time d is introduced at point D,we have ti
and from Equation 2 we get (hl
- h2),/a N 0.85
(8)
We also know that when u is equal to, or greater than, the maximum curve slope, urnax, the curve will be faithfully reproduced. From Equation 4 we thus have urnax= . f a
+ + d = 1/2f tz
(11)
so that Equation 2 now becomes
and, at the 85 % response point, the full scale response time is now given by
or fmax
N
0.32 uja
(9)
A third critical velocity, uc, can be defined at which the peak maximum is just attained; it is also shown in Figure 3. It can be seen by inspection that 2rft, N - 914 where the recorder line departs from the curve, and this makes
ve ‘v 1 ( a f a sin T),/(+ 2 4
&)
N
2.28 fa
or
fc ‘V 0.44 via
(10)
This means that an ideal recorder rated at 0.25 sec full scale response time ( u = a/0.25 = 4a) would just record a sine wave of 1.8 cycles/sec without attenuation or, from Equation 9, a frequency of 1.3 cycles/sec without distortion. According to Equation 2 the plot of recorder response us. l/f for a sine wave input should be linear up to, from Equation 8, approximately 85 % of the amplitude of the input sine wave. Such a plot is shown for several commercial recorders in Figure 4, which also includes some data obtained from previously published frequency response tests (6,8). It will be noted that, although a linear relationship is obtained, the lines do not pass through the origin, and in one case the linear relationship is not maintained up to the expected 85 % of the input sine wave amplitude. (8) Rikadenki Kogyo Co. Ltd., Tokyo, Japan, Recorder Catalogue CNO.R674. 1764
In Table I the full scale response times calculated from Equation 1 3 (using linear extrapolation if necessary), and the delay times obtained from Equation 12 are given for those recorders included in Figure 4. The “span step response time” is generally used as a measure of recorder performance, and this is the time required for full scale response to a full scale step input. To obtain the span step response time, we must also include the “settling time” (see Figure 1) which is caused by the recorder pen slowing down as the emf generated by the recorder potentiometer approaches the input signal. The departure from linearity at the lower frequencies is due mainly to the settling time effect. It is more pronounced with some recorders than with others, and is markedly increased by increasing the recorder damping (see Figure 4). The effect is particularly evident when comparing the response to triangular, sine, and square wave input functions. As shown in Figure 5 , the difference between the input signal and the recorder emf is smaller in the case of the triangular function than in the case of the sine or square wave. The corresponding frequency response plots for one particular recorder are shown in Figure 6. At the higher frequencies, the difference between the input signal and the recorder emf is generally greater, and the effect becomes negligible. Under these conditions it can be shown simply that Equation 12 also applies to both the triangular and square waves (and in fact to any symmetrical repetitive function) so that the response at the higher frequencies falls on a common line (see Figure 6).
ANALYTICAL CHEMISTRY, VOL. 41, NO. 13, NOVEMBER 1969
Table I. Recorder Response Times (sec) from Figure 4 Full scale Sine wave response time Delay time Total from Eq. 12 generator usedn from Eq. 13
Chart width (in.)
Manufacturer's span step response time
Recorder Hitachi 0.04 0.90 vc v = v,
Ladder
yl
Stepwise
TL = 0.0 Tu = 1.0
Spiral
Ti = 0.0
Ladder
W
A
- c)/V = a exp(- Tn2/2)
yB
a 0 . 5 (TL Tu) yo u exp(-Tn2/2) YE= vTn+C T L = Tn, ya > YE Tu Tn, YO < YE
T,
+
= VTi c Ti = 1.0 yi = 0.6065~ ~i
C
*
(~n-1
TB = 0.0
None
Ladder
Tn y,
TI Yl
0.0
+
=
Tn = [21n(~/y,-~)]1’2 Y n = VTn c Tn = v/yn-i y, = a exp(-Tn*/2)
+
Tn = V/.~n-i yn = a exp(-Tn*/2)
=a
Tc
None
=
TB
YC = YB
Stepwise
TL = 1.0 Tu = Tc Yc/V C’ = VTc YC
++
ANALYTICAL CHEMISTRY, VOL. 41, NO. 13, NOVEMBER 1969
1767
0
0.1
0.2
0.3
0.4
0.5
~~
0.6
(
PEN SPEED(V)
Figure 12. Peak parameters as a function of pen speed, V, for ideal recorder response
required is relatively small, and for an accuracy limit of 0.00001 generally varies between 3 and 20. However a larger number is required for point B when V is between 0.56 and 0.60. Using a CDC 3200 computer, all 4 points, together with width and area calculations (see below), can be computed for 35 values of V in about 5 seconds. If the accuracy requirement is reduced to 0.001, the number of steps required generally varies between 2 and 12. The two parameters, half-width (width at half height, w ) and area are now readily obtained using the following expressions
Figure 12 shows peak height OBfor V 6 V,), peak displacement (TB), half-width and area plotted on a relative basis as a function of pen speed, V. Effect of Delay Time. The introduction of a delay time into the above calculations would have no effect on the peak height but would increase the other three parameters-peak width, peak area, and displacement of the center of the peak. The inclusion of a delay time would be relatively simple for the situation shown in Figure 8a, when it would occur at point B, and this implies an increase in peak width, area, and displacement (mid-point) for pen speeds below V t . However, consideration of the experimentally determined values of the delay time, Table I, shows that for the faster recorders it is generally the predominant term for all pen speeds less than Vmax. In Figure 13 we consider an example where the peak basewidth is 0.4 sec (a = 0.1 sec). V,,, then corresponds to a response time of 0.165 sec. If we specify that the maximum allowable delay is to be such that the recorder pen passes through the true inflection point of the curve, so that there is no increase in peak width, we find that the error in peak area (shaded) is 0.7%, and this corresponds to a delay of 0.035 sec. (Had the delay time been increased by a further 10% to 0.0385 sec, the error in area would then have been approximately 1.5% and this would have been accompanied by a similar increase in half-width.) For comparison, the broken lines show the case where the pen speed corresponds to the critical speed V , (which in the present example means a span step response time of 0.222 sec) but with the same delay time, and the net increase in area is now 2.8%:. It is also seen from Figure 13 that, once the delay time is set, even a large increase in response speed beyond V,,, will have little effect on the resulting error. The maximum acceptable ratio of delay time to pen response time is thus about 0.21. Effect of Settling Time. The case where the settling time is pronounced (Figure 1 ) can be treated in the following manner. We wish to define the conditions under which the recorder will accurately reproduce a Gaussian input function, and to do this we approximate it by a sine curve. This is shown in Figure 14 in which both the curves have the same peak-topeak amplitude and the same width at half-height. This particular approximation has been selected from several alternatives to ensure that the pen speed required to reproduce the Gaussian curve shall be less than that required for the corresponding sine curve over the major part of the curvei.e.,it is a conservative approximation. The two curves can be plotted as shown in Figure 14 by introducing a frequency factor, a , which is equivalent to changing the scale of the Gaussian curve. Thus we can write for the sine curve
and for the Gaussian curve
The integrals required for the area calculations were evaluated using a subroutine based on Chebyshev polynomials (I, IO, 11). (10) C. W. Clenshaw, “Chebyshev Series for Mathematical Func-
tions,’’ Mathematical Tables 5, National Physical Laboratory, Her Majesty’s Stationery Office, London, 1962. (11) J. J. Russell, “Error Function and Normal Distribution Function (3600 Fortran),” C.S.I.R.O., Canberra, 1964. 1768
from which we obtain
a
N
0.75
(30)
(Had we chosen our two curves so that they both had the same maximum slope, we would then have found that the above factor was equal to 0.83.) Using the frequency response results (Figure 4), we know that we can determine the critical frequency, fc,with reasonable accuracy since, by definition, this is the frequency at
ANALYTICAL CHEMISTRY, VOL. 41, NO. 13, NOVEMBER 1969
,
-3
-2
-1
1
0
3
2
t
Figure 13. Peak distortion due to recorder delay time
Figure 14. Sine curve approximation to a Gaussian function
which full response is just attained (Figure 3). From Equations 9 and 10 we have fmax N
0.32fJO.44
(31)
We can now define a new frequency, fG, for the Gaussian function in Figure 14 which corresponds to the speed Vmax (Figure 9), required for an undistorted curve, but which is determined in terms of the measurable frequency, fc. The latter, and hence also fG, now includes the settling time contribution. Thus we have fG
=
a! fmnx
1 : 0.55
fc
(32)
The significance of fa can be seen by considering the relationship between recorder speed and the minimum retention time, tm, for an undistorted full scale chromatographic peak. The peak shape approximates a Gaussian distribution MINIMUM RETENTION T I M E ( S ~ C )
y = a exp(-P/2u2)
(33)
and, by definition, the number of theoretical plates in the column is given by
N
from which we obtain (34)
= (tm/u)2
Since we are now interested in absolute times, rather than the function t/a (= T ) which has been used up till this point we must now introduce an absolute speed, V', and we have VImax=
Figure 15. Minimum chromatograph retention times for an undistorted, symmetrical, full-scale peak
e)
= 0.6065 a/o
max
(35)
For an ideal recorder whose span step response time is s, it is evident that the limiting value V',,, must correspond to VIrnax=
a/s
(36)
tm N
0.6 s
~
N
(37)
We see from Figure 12 that a constant of 0.5 should be acceptable and in Figure 15 the expression tm = 0.5 s d ; is shown plotted for various recorder step response times, but it must be emphasized that these results so far apply only to the ideal recorder. If we now consider the sine curve approximation to a Gaussian function shown in Figure 14, we have already derived a characteristic frequency, fG, for which we would expect accurate recording to be obtained. From Equation 9 we know that this corresponds to a speed uG, given by
Table 111. Effective Response Times, sec Recorder Leeds and Northrup Rikaden ki Varian G-1000 Varian G-14A-1 Moseley 80A Brown Approximately 90
Response time from Eq. 13 0.76 0.21 0.20 0.45
lifc 4.OU 1 .oa 0.9 1.6
S.
2.3 0.6 0.5 0.9
Mfr's span step response time (ssrf) 1.0 0.25 0.35 0.6 0.25 1 .o
SJlssrf 2.3 2.4 1.4 1.5 2.3 1.6
full scale.
ANALYTICAL CHEMISTRY, VOL. 41, NO. 13, NOVEMBER 1969
e
1769
fG 1 :0.32
vola
(38)
We can now define an effective response speed, S,, derived from Equation 38 and for which Equation 37 will apply, thus S, = a/uo 1:0.32/fa
(39)
and in terms of the measurable critical frequency, A, this gives us S,
‘v
0.32/0.55fc N 0.58/f,
This means that Figure 15 and Equation 37 can be used for any recorder provided that the effective full scale response time defined by Equation 40 is employed. In Table 111, values of S, are compared with the full scale response times previously obtained from Equation 13, and they are seen to be about 2 to 3 times the latter. Also the values of S, are about twice the manufacturers’ span step response times, and it is suggested that a minimum factor of 2 should be adopted in the absence of relevant frequency response data. The last two sets of figures in the Table were obtained from published chromatographic results (12) for which we know that d&/21: 16. The relevant minimum retention times found were 11 and 30 sec, respectively, giving the ratio figures shown. These results are also in reasonable agreement with the suggested factor of 2.0. (12) S. Dal Nogare and J. Chiu, ANAL.CHEM., 34, 890 (1962).
It should perhaps be emphasized at this point that our discussion has been concerned with a chromatographic peak of height equal to the full scale deflection of the recorder. For smaller peaks, a smaller value of tmwould be obtained, but not in direct relationship to the peak heights. This is because the settling time effect becomes more pronounced for the smaller peaks, as can be seen from the two sets of frequency response results (50% and 10% full scale) for the Honeywell recorder in Figure 4. Finally, we must again consider the effect of the delay time. As outlined in the previous Section (Figure 13), provided that the delay time does not exceed one fifth of the response time, the error introduced is negligible. It might be expected that the introduction of the effective response time, which makes allowance for the settling time effect, would also compensate to some extent for delay time effects. By redefining the allowable maximum delay time in terms of S,, thus d,,,
‘v
0.2 S,
(41)
we retain the correct factor for use when the delay time dominates and s = S, but relax the specification for most practical cases for which the values of S, will be about twice the manufacturers’ span step response times. On this basis, the delay time figures given in Table I are therefore acceptable. RECEIVED for review May 19, 1969. Accepted July 15, 1969. This work was made possible by the award of a Research Fellowship to one of the authors (1.G.M.) by the Shell Group of Companies in Australia.
Detection and Resolution of Overlapped Peaks for An On-Line Computer System for Gas Chromatographs A. W. Westerberg’ Control Data Corp., 4455 Eastgate Mall, LaJolla, Calif. 92037 Peak detection and resolution methods are discussed for an existing computer system which handles several concurrently operating on-line gas chromatographs. The use of digital filtering to smooth the input data and then of heuristic criteria to distinguish peaks from noise solves the detection problem; the design criteria for the required digital filter are given together with a minimum sampling rate. After a detailed study of two commonly used resolution techniques of triangulation and perpendicular drop, the paper concludes these methods are too inaccurate and are poorly reproducible. It then presents an alternate method of resolving peaks which uses model curve fitting and covers the details for a Gaussian, a modified Gaussian, and a general tabular data model which may be used with this approach. The paper concludes with a sample trace analyzed by the system.
THISPAPER is the second of a series presenting a software system developed on the CDC 1700 computer for on-line gas chromatographic data input and analysis. An earlier paper (1) presented a real-time periodic sampling algorithm. This paper will discuss the detection and resolution of peaks in the nonbaseline segments of a trace. (1) A. W. Westerberg, ANAL..CHEM., 40, 1595 (1969). 1770
COMPONENT DETECTION
One problem for the computer in gas chromatography data analysis is the detection of the number of components contributing peaks within a nonbaseline segment of a trace. A human interpreter can manage to detect peaks from noise quite successfully because he can view the whole trace segment and make a judgement based on this perspective. Unfortunately this problem is much more difficult for the computer. Two approaches will be discussed here. The first technique, used successfully for NMR (2, 3), assumes a precise peak shape model. One first guesses the number of components by finding only the obvious major peaks in the trace. Using this number of components, one easily least squares fits the data to the model (see Peak Resolution, Curve Fitting Section). If the fit is poor, as evidenced by the sum of squares of differences between the data and fitted model, one assumes added components in those local Present address, Dept. of Chemical Engineering, University of Florida, Gainesville, Fla. 32601
(2) W. D. Gwinn,A. C. Luntz, C. H. Sederholm, and R. Millikan, J . Cornp~ifafional Phys., 2, 439 (1968). (3) W. Keller, T. Lusebrink, and C. H. Sederholm, J. Chem. Phys., 44, 782 (1966).
ANALYTICAL CHEMISTRY, VOL. 41, NO. 13, NOVEMBER 1969
