Instrumental Peak Distortion 1. Relaxation Time Effects I. G . McWilliaml and H. C. Bolton2 Monash Unicersity, Clayton, Victoria 3168, Australia Distortion of a Gaussian curve due to the effect of a finite instrument time constant is first considered using the exact integral solution, and the expected changes in peak shape and peak separation for a concentration range of down to 1 ppm are shown. It is then established that Fourier analysis can be used to produce a relatively simple approximation to a Gaussian function, from which changes in peak width, height, and position can be readily obtained. Finally the convolution integral is introduced to enable the effect of two independent time constants to be evaluated.
THEDEGREE of separation of bands or peaks obtained from an analytical instrument is determined by two sets of factors which in general may be considered quite separately. The first set concerns only the basic principles which underlie the technique used, thereby determining the fundamental limitations of the method. One of the most clearcut examples is bandwidth and band separation in spectrophotometric techniques. In gas chromatography the limiting factors are again bandwidth and band separation, but in this case bandwidth is a complex function of column and operating factors and band separation is determined by stationary phase selectivity. The second set of factors is instrumental in origin, and leads to band broadening over and above that set by natural limitations of the technique. Frequently there is interaction between factors in the second set so that the results achieved are not determined solely by one particular factor. An example from spectrophotometry is finite slit width, which in turn is allied with the radiation source intensity and detector sensitivity. In gas chromatography one may include under this heading the effect of finite sample size (1) and, in common with other instrumental analytical techniques, the distortion caused by postcolumn factors arising in the detector, amplifier, and recording system. The evaluation and measurement of these postcolumn effects will be discussed. EFFECT OF A SINGLE RELAXATION TIME (OR TIME CONSTANT) The Integral Solution. Band broadening may occur as a result of slow response of either the detector, amplifier, or recorder. Since, however, recorder response is a nonlinear effect which is quite different from that due to a single relaxation time, it will be dealt with separately later (2). We shall here be concerned solely with the exponential form of response characterized by a single time constant or relaxation time for the system (these two terms being synonymous). 1
Department of Chemistry.
* Department of Physics.
(1) C. N. Reilley, G. P. Hildebrand, and J. W. Ashley, ANAL. CHEM., 34, 1198 (1962). (2) I. G. McWilliam and H. C. Bolton, ibid., 41, 1762 (1969).
In general, either the detector or the amplifier, not both, will be the dominant factor determining the speed of response of the system, and it will be possible to consider only one dominant time constant. However, the case of two noninteracting time constants of similar magnitude will be considered briefly in a later section. Furthermore, since we are mainly concerned with chromatographic peaks, we shall initially confine our attention to the Gaussian input function, but certain results will be generalized for application to any input function. Our system is defined by the linear differential equation
where D(t) is the response or output of the system, T is the relaxation time (or time constant), and the input function E(t) is in this case a Gaussian function with variance a2 centered at time t = 0. Equation 1 can be solved formally by use of an integrating factor, exp(t/r), and setting the boundary conditions D(- m ) = D(+ a ) = 0. The resulting equation is transformed to produce a solution containing the normalized error function integral which is available in tabulated form (3) or can be generated by computer techniques. The solution is D(t) = A
6
S exp
(i
S
- TS)
5 s-xmexp(
x2)
dx
where T = ria, S = U / T , and X = (T - S ) / d 2 . This is plotted in Figure 1 (solid lines) for several values of rja. Similar equations have been reported elsewhere ( 4 - 4 , but in each case a finite lower integration limit has been used in an effort to fit the equation more nearly to the case of a finite time scale. However, in any practical case this change would have no significant effect on the value of the integrand. Besides the evident increase in peak distortion as T / a increases, there are several further properties of this curve which are of immediate interest and which will be discussed in this and succeeding sections. It is evident from Figure 1 that, for the values chosen for T / U , the maximum of the output curve lies on the input curve. In fact, it is easily shown that this property holds for all values of rja. Furthermore, it is not restricted to the Gaussian input function but is common to all systems controlled by a single relaxation time. A simple example is that of a square wave input for which the maximum of the output curve always lies on the trailing edge of the input waveform. The (3) “Tables of the Error Function and its Derivative,” N.B.S. Applied Maths Series 41, U.S. Dept. of Commerce, 1954. (4) L. J. Schmauch, ANAL.CHEM., 31, 225 (1959). (5) H. W. Johnson and F. H. Stross, ibid., p 357. (6) J. C. Sternberg, in “Advances in Gas Chromatography,” Vol. 2, J. C . Giddings and R. A. Keller, Eds, Marcel Dekker, New York, 1966.
ANALYTICAL CHEMISTRY, VOL. 41, NO. 13, NOVEMBER 1969
1755
-
1.
INTEGRAL SOLUTION
__--
1
rOURlER ANALYSIS
2
10
3
4
I(
5
10 6 ((0.6) (1.0) 0.3) ~~
,\c
'-
1
-1I
I
I
I
c
n
7 10
-3
-2
0 T I '/a
1
2
3
4
8
Figure 1. Distortion of Gaussian input curve due to a single time constant Id
same result is also obtained for a triangular input function. The proof that the output maximum lies on the input curve follows from Equation 1 ; the output maximum occurs at the point where
.t
IT1
-
IO
and on substituting in Equation 1 we get (4)
showing that the maximum of the output curve, D(tm),always lies on the input curve, E(t). This result has also been reported by Ashley and Reilley (7) whose work was based on Laplace transform analysis. In gas chromatography the peak profile near the base of the peak is often of greater importance than that of the major section of the peak shown in Figure 1. Whereas the separation of two materials may be deemed adequate when they are both present at comparable concentrations, one component may not be measurable in the other at low concentrations because of the swamping effect of the major peak. This effect is particularly pronounced when the minor peak is on the tail of the major peak. While peak tailing is usually attributed to adsorption effects, a substantial contribution can arise from relaxation time effects and this is shown in Figures 2 and 3. Figure 2 shows, for several fixed values of 1ti.l (or ITI), how the function D(r) varies with change in T / U . Negative values of T / U correspond to the leading edge of the peak. Superimposed on this figure is the undistorted original curve (broken line). As T / U increases there is a very large increase (7) J. W. Ashley and C . N. Reilley, ANAL.CHEM., 37,626 (1965). 1756
6
Figure 2. Change in Gaussian peak shape as a function of T / U . Positive values of T / U refer to the trailing edge; negative values of T / U refer to the leading edge. The T on the right hand side is to be interpreted as the modulus of T / U
in the trailing side of the peak, resulting in a pronounced tail, and a much smaller decrease at the leading edge. This decrease might lead one to expect that there might be an improvement in the separation of a minor component preceding a major component as T / U increases. However, the small shift in the leading edge of the main peak is more than offset by the increased tailing of the minor peak so that a net loss in resolution occurs. In Figure 3 the peak profile is shown on a linear scale for a minor component (a) preceding and (b) following a major component. In both cases the peak separation (distance between the major and minor peak centers) is 8a, and the figure shows the change in curve appearance corresponding to impurity levels of 1 %, 100 ppm, and 1 ppm for four different values of T / U . Where the minor component precedes the major component, it is evident that the loss in resolution with increased ria is very small. However, in the reverse case the importance of decreasing the value of T / U to obtain adequate resolution at lower impurity levels is clearly seen, although even in this case there is little to be gained by decreasing T / U below 0.1,
ANALYTICAL CHEMISTRY, VOL. 41, NO. 13, NOVEMBER 1969
(b)
(a) LEADING EDGE
TRAILING EDGE
I
0
0.2
Figure 3. Effect of time constant on peak separation Elsewhere it has been stated that asymmetry which takes the form of sharpened fronts and extended tails roughly cancels (8). Figure 3 supports this view for a minor peak preceding a major one, but also shows that it is unacceptable for the reverse case. Figures 2 and 3 also show that the concept of exact separation (8) (corresponding to a peak separation of 6a) may be useful for peaks of similar magnitude but becomes quite meaningless where components are present at, or below, the 1 level. The results shown in Figures 2 and 3 apply to either of two possible situations. The first is when the detector time constant is the dominant response factor, and the importance of reducing this for analyses at the ppm level and below is evident. The second case is where the amplifier time constant is dominant and the amplifier sensitivity at the input is fixed during a scan. This applies to logarithmic amplifiers, and to linear amplifiers which are attenuated at the output or which have been preset to record only the minor peaks on scale. It also applies to integrators which are designed to accept a very wide range of input signals, particularly the more recent digital integrators which are claimed to have a linear dynamic range of up to lo6to 1. When the amplifier time constant is the limiting factor, and the detector output is attenuated at the input to the amplifier (which may be regarded as the normal recording mode), the curves of Figures 2 and 3 no longer apply. E(r) in Equation 1 is then not the complete Gaussian function but a series of small segments of it, each being selected to keep the recorder on scale and each having a different value of A . The construction of the curves shown in Figures 2 and 3 can be facilitated by the use of approximate asymptotic relationships (9) for the normalized error function integral in Equation 2. Thus, for the leading edge of the curve (subscript f), and X 5 - 3, we can use the asymptotic expression (8) J. H. Purnell, J . Chem. SOC.,1268 (1960). (9) E. Jahnke and F. Emde, “Tables of Functions,” 4th Ed., Dover Publications, New York, 1945, p 24.
and Equation 2 then reduces to
For the trailing edge of the curve (subscript tr), and X 2 2, we can use 2 +lXI exp(-x2)dxN 2 (7)
so that
The computer program used to calculate the curves shown in Figures 2 and 3 employed an error function integral subroutine based on Chebyshev polynomials for the range -4 < X < 4 (10, 11). Outside this range the above approximations (then accurate to better than 0.3 %) were used. In the past, typical time constants for gas chromatography detectors and amplifiers have been about 1 second which implies a minimum peak “base-width” (corresponding to 4a) of about 40 seconds if significant peak distortion is not to occur. The development of ionization detectors with inherent time constants nearer to 1 msec, and associated low time constant amplifiers, has made capillary column systems a practical possibility. However, even a 10-msec time constant implies a minimum peak basewidth of 0.4 sec. If we are (10) C. W. Clenshaw, “Chebyshev Series for Mathematical Functions,” 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),” Commonwealth Scientific and Industrial Research Organisation, Canberra, 1964.
ANALYTICAL CHEMISTRY, VOL. 41, NO. 13, NOVEMBER 1969
1757
1.0
Table I. Fourier Coefficients for Equations 18 and 27 as a Function of ria 0.8
riff
Bi 0.4839 0.4833 0.4815 0.4785 0.4745 0.4695 0.4635 0.4493 0.4328 0.4149
0 0.05 0.10 0.15 0.20 0.25 0.30 0.40 0.50 0.60
91
0 0.0500 0.0997 0.1489 0.1974 0.2450 0.2915 0.3805 0.4636 0.5404
Bz 0.1080 0.1075 0.1059 0.1034 0.1003 0.0966 0.0926 0.0843 0.0764 0.0691
+z
Ba
0 0.0997 0.1974 0.2915 0.3805 0.4636 0.5404 0.6747 0.7854 0.8761
0.0089 0.0088 0.0085 0.0081 0.0076 0.0071 0.0066 0.0057 0.0049 0.0043
43 0 0.1489 0.2915 0.4229 0.5404 0.6435 0.7328 0.8761 0.9828 1.0637
0‘6
0’4
0 ‘2
D
prepared to accept a minimum separation between adjacent peaks of 6a (which corresponds to an overlap of 1 % and would therefore be inadequate for trace analyses), the permissible rate of peak development would be less than 2 peaks/ sec. Moreover, this applies only to the initial peaks in the chromatogram. The permissible rate falls off with increasing peak width because of the need to maintain at least a 6u separation between the peaks. This can be avoided only by the use of temperature or flow programming in which case the peak widths then remain substantially constant. Fourier Analysis. An alternative approach to the problem lies in the use of approximation methods. One example is the replacement of the Gaussian input function by a triangular waveform which allows a relatively simple analytical solution (12, 13). A more valuable approach proves to be the use of Fourier analysis. Not only does Fourier analysis provide a straightforward analytical solution to the problem, but it also offers a method for the generation of Gaussian peaks. With certain exceptions, which need not concern us here, any function can be represented in the interval from --?r to r by the Fourier series expansion
.0’2
I
+ 2
a0
= --
+ bl sin x ) + (a, cos 2x + 62 sin 2x) +.. . . + (a, cos nx + b, sin nx) +.. . . . (9)
(a1
I
-2
7r
The coefficients ao, . . . .a, are given, in the general case, by the expression a, =
f JI, F(x)
COS
nx dx
(12)
a, =
SI,
1 lr
exp
(-
xz) cos nx dx.
The value of ao/2can be obtained from relevant tables (15) and is equal to 0.3983. However, to the authors’ knowledge no tables of definite integrals exist for the expression given in (12) I. G. McWilliam, J. Appl. Chem., 9, 382 (1959). (13) A. E. Banner, J. Sci. Instr., 43, 138 (1966). (14) H. S. Carslaw, “Introduction to the Theory of Fourier’s Series and Integrals,” Macmillan and Co. Ltd., London, 1921. (1 5 ) “Tables of Normal Probability Functions,” N.B.S. Applied Maths Series 23, US. Govt. Printing Office, Washington, 1953.
1758
(-0.02
5.2)
7r
Successive coefficients in the Fourier expansion are then approximated by 4 2 = 0.3989 a1 = 0.4839 CPZ = 0.1080 a3 = 0.0089 a4 = 0.0002. The fourth and higher terms can clearly be neglected, and the first three terms, together with their sum (Equation 15) are
H-3 1
plotted in Figure 4. The error exp
-x2 -F(x)
is also
plotted in Figure 4 for the two approximations
+ 0.4839 cos x + 0.1080 cos 2x (ii) F ~ ( x=) Fz(x)+ 0.0089 3 ~ . COS
(13)
’
((- 1
(i) F2(x) = 0.3989
and for the specific case of the Gaussian function by
I
2
xz) cosnxdx
-m
= -1 6 e x p
(10) (11)
exp
a, cv 1 Jm
and therefore (14)
...= 0.
I
1
Equation 13 for n 2 1, nor is there an indefinite integral form. A sufficiently close approximation can be obtained by changing the integration limits to =k m so that (16)
In the case of the Gaussian function, F(x) = exp (-+z)
b 1 = . , . = b,=
1
0
Figure 4. Fourier approximation to a Gaussian function
cos x
F ( - x ) = F(x)
I
-1
w t (radians)
+
F(x)
I
-3
(15) (16)
The first expression, Equation 15, reproduces the desired curve to within f1 Z of the maximum value throughout most of the range. The second expression, Equation 16, is accurate to within =k0.03% for -2.1 < x < 2.1 [F(x) > 0.11 and within 1 0 . 1 for -2.5 < x < 2.5 [F(x)> 0.051. The increasing error as x approaches T is due to the fact that Equation 9 represents a periodically repeated curve, the (16) D. Bierens de Haan, “Nouvelles Tables D’Integrales DBfinies,” G. E. Stechert and Co., New York, 1939, p 385.
ANALYTICAL CHEMISTRY, VOL. 41, NO. 13, NOVEMBER 1969
repetition interval being 2 a . Thus at the point x = a , the error due to overlap by the repeated curve centered at x = 2 s is exp { -0.5(2x - T ) ~ ] 0.0072. A closer fit can be obtained in this region, if required, by slight adjustment of the coefficients but only by simultaneously introducing errors at other points. We now return to Equation 1, and substitute Equation 15 for E(t)i.e.,
7
where x has been replaced by ut. The solution to Equation 17 is
+
D(t) = A { ~ o / 2 Bi
COS (ut
- 41)+ Bz COS ( 2 ~ t - 4 2 ) )
(18)
where
Figure 5. Peak displacement and height as a function of
+
B1 = a1 ( 1 UT)^}-''^ 41 = tan-' ( o r ) Bz = a2(l $z =
simple to calculate the peak displacement and change in peak height as a function of r/u. At the peak maximum dD(t)/dt = 0 and hence, from Equation 18,
+ (2~0r)*)-"~
tan-' (2wr)
and u = 2af, wherefis the repetition frequency. The physical meaning of Equations 17 and 18 is that each Fourier component is reduced in amplitude and suffers a phase lag on going through the linear system. The amplitude of each Fourier component decreases and the phase lag increases as the frequency of the Fourier component increases, meaning that the system cannot follow the more rapid oscillations faithfully. To relate the results obtained from Equation 2 with those obtained from Equation 18, we note that in Figure 1 time, t , is given as a function of u , If the results from Equation 18 are also to be plotted as a function of u, the limits for the Fourier representation will then be fTU. Since f is the repetition frequency,
f = l/(2xu)
(20)
2afr
(21)
and UT
=
=
TJU.
T/U
Selecting appropriate values for r J u , we can now calculate values of D(t) from Equation 18 and these are shown as the broken lines in Figure 1. The results obtained by the Fourier technique agree very well with those obtained by the integral solution over the center of the range. The divergence at the extremities results from the fact that the Gaussian curve has been approximated by a repeating function. A closer approximation could be obtained by, in effect, selecting a wider repetition intervale.g., i 2 a u . However since we are limited to an interval of ia for the Fourier expansion, we would then have to replace x in the Gaussian function in Equation 13 by a new variable u = x/2. When this is done, we find that the resulting Fourier series has twice as many significant terms because successive coefficients no longer fall off so rapidly. Provided that the error introduced by the approximation is acceptable, which it is for many applications, the Fourier solution given in Equation 18 can be calculated readily, either by hand or by a digital computer if a large number of points is required. In the former case it may be advantageous to expand the cosine terms rather than to repeatedly recalculate them for the small differences dl and 42. After the Fourier representation is obtained, it is relatively
Bl sin (ut - 4J
+ 2B2 sin ( 2 w t - 42) = 0 .
(22)
If we now assume that 42 = 291 which is a reasonable approximation (see Table I), then Equation 22 can be expanded to give B1 sin (ut - 4')
+ 4B2 sin (ut - 4') cos (ut - 4J = 0
(23)
and the solution to this is sin (ut
- 41) = 0 or ut
=
4I
where 4' is given by Equation 19, and for small displacements this reduces to
41 == o r
=
TJU.
In other words, since the term containing B1is the dominant frequency-dependent term in Equation 18, the displacement is closely approximated by 41,which in turn is approximately equal to T / U . This is shown in Figure 5 (solid curves), together with peak displacement values calculated from the exact integral solution given in Equation 2. Alternative methods of approach, which produce the same result, are to solve Equation 22 for ut small, i.e. sin(ut - 4) 'v (ut - +), or to differentiate Equation 6 which applies for T / U small (Slarge). In gas chromatography we are more interested in the shift in retention time caused by peak distortion than in the shift from the peak center. It is readily shown that the relative peak shift due solely to the finite relaxation time, 7, is given by
where Vis the uncorrected retention volume or retention time (17) and N is the number of theoretical plates. Thus the relative accuracy increases with increased retention volume, but at the same time the number of compounds likely to have similar retention times also tends to increase. While the increased relative accuracy may therefore be of little consolation, the error due to the finite time constant will eventually become (17) ASTM Committee E-19 on Gas Chromatography, J . Gas Chromatogr., 6, 1 (1968).
ANALYTICAL CHEMISTRY, VOL. 41, NO. 13, NOVEMBER 1969
1759
/- I
I 1.1
I
TWO (OR MORE) TIME CONSTANTS-THE CONVOLUTION INTEGRAL
An alternative method for obtaining the response of a system to an input function E(t) is to divide the function into small time intervals, at, and to sum the resultant response of the system to each increment. Provided that the intervals are sufficiently small, the output response to each incremental excitation 6E(t)is simply the response to a step input function of magnitude 6E(t). It can then be shown (18) that the total response at any time, t‘, is given by D(t’)
E(t)h(t’
m=: J
- i)dt
(31)
where h(t) is the impulse response function, defined as the output of the system to a Dirac delta function input at t = 0. Alternatively, the function h(t) can be obtained from 0.8
0.2
0
0.4
0.6
VU
Figure 6. Increase in peak width, and decrease in chromatographic column efficiency, as a function of T / U
small compared with errors arising from other sources, e.g., flow and temperature variations. The decrease in the peak maximum is readily obtained from the peak displacement and the knowledge that the maximum of the displaced peak lies on the input curve. For completeness, the peak maximum is also plotted as a function of ria in Figure 5 (broken curve). The remaining parameter of interest is the peak width. It is common to use the width at half height to obtain a measure of column efficiency, although the base-line intercept method is “recommended practice” (17). Because the threeterm Fourier expression, Equation 15, overestimates the Gaussian function by almost 1 at half-height (see Figure 4), it is necessary to use the more accurate four-term expression, Equation 16, to determine the peak width. Substituting for E(t) in Equation 1, we now obtain &(t) = A {ao/2
+ Bi
COS
(ut
- 41) + B2 COS ( 2 ~ -t 42) + B3 COS (3 ut (27) $3))
where B1,Bz,q5i, and 42are given by Equation 19, and
where g(t) is the response of the system to a unit step function input (which has the value zero for t < 0 and unity for t > 0). The integral in Equation 31 is known as the convolution integral, and its use has been reported for the correction of spectroscopic line shapes for the effects of scanning speed, field and frequency modulation widths, and circuit time constants (19). In the case of a system controlled by a single relaxation time, the function h(t) is the solution of Equation 1 with E(t) = S(t), the Dirac delta function; this is h(t) = (117) exp (-ti.).
For the Gaussian input function centered at t = 0, Equation 31 then becomes D(T’) =
S’h
0 2
- UO’ ‘v
72
- T ) ) di’ (34)
which is in fact equivalent to Equation 2. It can for instance be used as an alternative to the argument in the first section to show that the maximum of the output curve falls on the input curve. Equation 31 can also be used to obtain the position of the maximum of the output when the displacement is small. To do this we transform Equation 31 by putting u = t ’ - t and we get
som
E(t’ - u)h(u)du.
(35)
For small displacements we can approximate the Gaussian function by a parabola which, if we insert into Equation 35
allows us to define the maximum of the output by dD(t’) = 0 = dt
som
(2’
- u)h(u)du.
(37)
(18) M. Strain, “Mathematical Methods for Technologists,” EUP,
London, 1961, pp 393-5.
or u/uo -?i { 1
1760
1
A exp (- - T Z ) S exp (-S(T’ 2
D(t’) =
Values of the coefficients B3 and 43are given in Table I. In Figure 6 the increase in peak width, based on half-width calculations, is plotted as a function of rja. These results also correspond closely to the increase in base-line intercept. In addition, the corresponding reduction in the effective number of theoretical plates is shown in Figure 6, again based on the half-width measurements. It is also seen from Figure 6 (broken curve) that, for r / u small, the increase in peak is given by (6, 7)
(33)
+ (r/u)2)
1’2.
(19) C. P. Flynn and E. F. W. Seymour, J. Sei. Instr., 39, 352
(1962).
ANALYTICAL CHEMISTRY, VOL. 41, NO. 13, NOVEMBER 1969
The position of the maximum tm’ is then given from Equation 37 as
and inserting h(u) from Equation 33 we get tm’ =
(39)
T.
We can see how accurate this result is from Figure 5; to 1 accuracy it agrees with the numerical calculations up to T / U N 0.4. However the simple result, Equation 39, holds only for the parabolic curve, Equation 36; if we chose to include quartic terms in E(t), then Equation 39 would not be valid. An important property of the system defined by Equation 1 is the fact that the area under the output curve is independent of TICS.This can be shown using the convolution integral as follows. Using Equation 35, the area is D(t’)df’ =
s:”
E(t’
- u)dt’
h(u)du
(40)
Because the limits of the t’ integral are infinite, this integral is independent of the value of u. We therefore get D(t’)dt’
=
s:m
E(t’)dt
h(u)du
(41)
-,I 10 -6
showing that the area under the output curve is proportional to that under the input curve. We will now prove, without using the explicit form for h(u), that the constant of proportionality, namely the integral
1
h(u)du
is equal to unity. We know that h(u) is the solution of
dh(u) + 1- h(u) = 1- 6(u) du
i-
i-
and if we integrate this over all values of u we get
The first integral on the left hand side vanishes since h( - ) = h(+ w ) = 0. The integral on the right hand side is unity from the property of the Dirac delta function. Also, h(u) = 0 for u < 0, which is an expression of the causal nature of the system. Thus Equation 43 reduces to
Lm
h(u) du
=
1
(44)
and we see that the area under the output curve is equal to that under the input curve and is thus independent of T / U . The main reason for introducing the convolution integral is to enable systems involving two independent time constants to be evaluated. Putting Equation 34 into incremental form so that it can be handled on a digital computer, we have
where the value of X (which should be - 0) to be strictly correct) is chosen to give the desired degree of accuracy, and
/’-0,6 -0.3 k 0 . 6 -0.1
0
I
-4
-2
0
2
4
6
t/6 Figure 7. Distortion of Gaussian input curve due to two independent time constants
E1(T)is the function resulting from the effects of the first time constant, S1. In practice a value for AT of about 0.001 is necessary to give an overall relative accuracy of better than 1 throughout the range for D(T’). The large number of calculations thus required can be reduced in two ways. First, the values of B ( T ) , using the smaller of the two time constants (this gives better overall accuracy), are calculated from Equation 2-not by using the convolution integral. Second, the number of exponential calculations, which are most time consuming, can be considerably reduced by using a larger step length for these calculations followed by linear interpolation for the smaller step length required for the summation. In this way a reduction in computing time by a factor of about 4 can be achieved with a simultaneous reduction in computer store requirements by a factor of 10. Alternatively, or in addition, simpler forms for the exponential calculation might well be employed (20). Some results obtained for a number of two-time constant systems are shown in Figure 7. In addition, the behavior of the curve over the range t/u = 0 to 1 and (r1 T * ) / u 5 0.6 was investigated in more detail. The numerical results can be summarized as follows. 1. The peak height is the product of the heights of the two peaks which would be obtained if each of the time constants acted separately on the original peak. 2. The increase in peak width is the sum of the increases which would result from each of the time constants acting separately.
+
(20) C. Hastings, “Approximations for Digital Computers,” Princeton IJniversity Press, Princeton, N. J., 1955, pp 181-4.
ANALYTICAL CHEMISTRY, VOL. 41, NO. 13, NOVEMBER 1969
1761
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)
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