Chapter 3 Fast Gas Chromatography: Theoretical considerations 3.1 Introduction

Chapter 3 Fast Gas Chromatography: Theoretical considerations 3.1 Introduction
Chapter 3: Fast GC Theory
Chapter 3
Fast Gas Chromatography:
Theoretical considerations
3.1 Introduction
Despite the tremendous decrease in analysis time that is possible with modern equipment,
limited useful applications have been found for fast gas chromatography. Most often
separation efficiency is sacrificed for a shorter run time. While shorter analysis times are
desirable, the concurrent increase in complexity of instrumentation is unfortunate. Special
low volume detectors and injectors1 that are capable of producing very short injection
bandwidths are required. Often the chromatographic run is a small part of the analysis
scheme where much larger time expenses are made in sample preparation and data
analysis. Thus it is not always wise to invest in new, complicated equipment when other
factors dominate the total analysis and reporting time.
In the case of comprehensive multidimensional chromatography it is, however,
paramount that the second separation should occur as fast as possible, since every 2nd
dimension chromatogram is repeated many times. Any increase in 2nd dimension runtime
is multiplied hundreds of times for each analysis. For a given 1st dimension runtime, the
analysis of the 2nd dimension should also be fast enough to allow a great number of these
to maintain the resolution of the 1st.
Normally fast GC alone is ineffective for detailed analysis of complex mixtures due to
the limited peak capacity that can be generated. Because of the first separation, each
transfer to the second column contains only a few compounds that can more readily be
separated with the limited separation efficiency attainable with fast analysis.
Chapter 3: Fast GC Theory
Comprehensive multidimensional chromatography is thus an ideal application for very
fast gas chromatography.
Retention mechanisms in gas chromatography are always dominated by volatility. There
is an approximate exponential relationship between retention time and solute boiling
point under isothermal GC conditions. When an intermediate temperature is chosen, the
chromatogram is characterized by poor separation of early eluting compounds, a long
analysis time and poor detectability of late eluting peaks due to band broadening. This is
commonly referred to as the general elution problem. For mixtures of analytes exceeding
a boiling point range of 100°C it is impossible to find a suitable analysis temperature. For
these samples only programmed modes can achieve complete separation in good time.
Both flow and temperature programming can be used. While linear temperature
programming is most often used due to simpler experimental implementation,
exponential flow programming offers many of the same advantages2. There is a small
chance that with modern electronic pressure control, flow programming may gain some
popularity. However, with these instruments flow rates are still calculated and the
accuracy depends on the precision to which column dimensions can be defined.
Efficiency is sacrificed, as columns are used at optimum flow rates only for a short period
during each run. Flow programming will never be able to cover the same wide spread in
volatility as temperature programming.
Negative thermal gradients in distance have been applied to chromatographic columns as
a form of moving focusing3. Cryogenic and retention gap focusing are usually single
events before or on the head of the column. With negative thermal gradients across the
length of the column the zones are continuously focused as they move towards the
column exit while separation takes place. Because the inlet side of the column is at a
higher temperature than the detector side, the rear of the band is at a higher temperature
than the front and thus moves at a slightly higher velocity. This counteracts some of the
chromatographic band spreading.
Unfortunately the same forces that make the zones narrower also move the zones closer
together reducing the separation because the trailing zone is at a higher temperature than
Chapter 3: Fast GC Theory
the leading zone4. While it can be helpful in reducing the effects of non-ideal
chromatographic conditions, gradient focusing can not increase resolution or speed of
analyses beyond what is theoretically achievable with conventional temperature
programmed analysis (PTGC) without gradient focusing5.
In comprehensive 2D gas chromatography, samples are first separated according to
boiling point. Since all analytes in a specific transfer have the same volatility, isothermal
chromatography can be used very successfully in the 2nd dimension. However, in our
proposed comprehensive 2D SFCxGC the chemical class separation precedes the boiling
point analysis and each subsequent transfer contains a wide boiling point range. It is
therefore required that a wide boiling point range be analyzed in a very short time.
To this end, the theory behind fast gas chromatography, especially that pertaining to
programmed temperature analysis, is explored in this chapter.
3.2 Optimization of resolution for fast gas chromatography
The relation between the different chromatographic variables is demonstrated by the
well-known resolution equation:
k' α −1
4 k '+1 α
[eq 3-1]
where Rs is the resolution between two successive peaks, k’ is the capacity factor of the
most retained compound and α= k’2/k’1 is the relative retention (also known as the
The capacity factor is dependent on temperature through the thermodynamic partition
coefficient, K, and the phase ratio,β.
[eq 3-2]
β = Vm/Vs
[eq 3-3]
Chapter 3: Fast GC Theory
Vm as the volume of the mobile phase
Vs is the volume of the stationary phase
K ∝e
[eq 3-4]
− ∆µ
∆µ as the change in free energy.
Thus, according to equation 3-2 and 3-4 an increase in temperature will reduce the
capacity factor, resulting in a decrease in resolution (through equation 3-1) as well as
analysis time (equation 3-14).
For maximum resolution, conditions are chosen where the term [k’/(k’+1)] in equation 31 approaches the maximum value. For large retention values this term approach 1.
Figure 3- 1: The maximum of the [k’/(k’+1)] term.
Resolution is measured experimentally from a chromatogram as
tr2 − tr1
[eq 3-5]
A resolution of R>1 implies that the retention maxima of the two compounds differ more
than the band broadening of the zone in time units. This is normally expressed in terms of
Chapter 3: Fast GC Theory
the standard deviation (σ). For a peak showing a gausian profile, R=1 implies that the
difference in retention time is equal to the width of the peak at base (∆tr = 4σ).
The width of a peak is influenced by non-chromatographic factors such as the
introduction width and by band broadening caused during the chromatographic process.
The relative band broadening is expressed in terms of the theoretical plate height, H7.
H =L
t r2
The relationship between H and u (the average linear velocity) for open tubular columns
is given by the Gollay-Giddings equation6:
 2 k ' d 2f 
 1 + 6k '+11k ' 2 rc2 
 2 Dm 
H =
 f 2u
 f1 u + 
 f1 + 
Dm 
 u 
 96(1 + k ')
 3 (1 + k ') Ds 
Dm= diffusion of solutes in the mobile phase and
Ds= diffusion of solutes into the stationary phase.
f1 =
9 P4 −1 P2 −1
P3 −1
f2 =
3 P2 −1
2 P3 −1
P = pi / po (the ratio between the column inlet and outlet pressures)
This can be simplified to
+ Cmu + Csu
B = 2Dm
longitudinal diffusion
rc 2
C m = F (k )
resistance to mass transfer in mobile phase
Chapter 3: Fast GC Theory
2  k'  Df
Cs = 
f 2u
3  (k '+1)2  Ds
resistance to mass transfer in stationary phase
(1 + 6k '+11k ' )
F (k ' ) =
96(1 + k ')
A graphical presentation of plate height against linear flow rate is known as the Van
Deemter curve. Figure 3-2 is a calculated curve for hydrogen and a thin film 0.25mm
capillary column. It graphically demonstrates the additive effect of B, Cm and Cs as
described by equation 3-8. For thin film columns the effect of diffusion into the stationary
phase is usually negligible. The value of Ds is assumed to be 3.3x10-6cm-2.s-1 at 85°C
and the value of Dm is assumed to be 0.2cm2/s for hydrogen .
Figure 3-2: Calculated Van Deemter curve
Ds=3.3x10-6cm-2.s-1 (at 85°C), df=1x10-4cm dimethyl silicone
Dm=0.2cm2/s (hydrogen), k’=10, rc=0.0125cm.
linear flowrate (cm/s)
Chapter 3: Fast GC Theory
Differentiation of equation 3-8 with respect to mobile phase velocity, followed by setting
the result to zero, leads to an optimum mobile phase velocity with a corresponding
minimum in plate height6.
u opt =
(C m + C s )
≈ Dm/rc
H min = 2 B(C m + C s ) ≈ dc
The highest column efficiency will be obtained at uopt.
After separation between compounds is effected, minimization of analysis time of the
separation problem is of interest.
3.3 Optimization of separation speed 6,7
Retention in chromatography is described by6:
tr =
(1 + k ') = N (1 + k ') H
or, since the number of theoretical plates (N) attainable with a column is defined as the
column length (L) divided by the theoretical plate height (H)
N= L/H
Retention can thus also be described as
tr = N(1+k’)H/u
where the flow rate is defined as
u = L/tm
Retention times can be reduced at will, by reducing L or k’ or increasing u. However,
changing any of these parameters has a pronounced effect on the resolution between the
compounds of interest. The possibilities for increasing speed of analysis for a given
separation problem is limited by the relationship between retention time, column
resolution and plate number.
Chapter 3: Fast GC Theory
When equation 3-16 is combined with the resolution equation (equation 3-1) an equation
is obtained that relates retention time with resolution. If tr is chosen as the last eluting
compound then the well known equation 3-18 gives an indication of the total analysis
time of a sample:
tr =
  α  2 (1 + k ')3 2 
Rs 
k '2
  α − 1 
[eq 3-18]
While column length does not feature directly in equation 3-18, it is indirectly defined by
the relationship between R and N in equation 3-1. An excess of resolution should be
avoided as this leads to an increase in analysis time (tr ∝Rs2). Resolution tends to increase
proportionally to the square root of the number of theoretical plates. Analysis time,
however, is directly proportional to the column length. Thus, increasing column length to
improve resolution is time-expensive.
3.3.1 The influence of capacity factor
Figure 3-3: The minimum retention time is obtained when k’=2.
The influence on analysis time with variation in k’ is graphically represented by Figure 33. It can be seen that the (1+k’)3/k’2 term (equation 3-18) reaches a minimum at k’=2.
However, it should be remembered from Figure 3-1 that the maximum resolution is
Chapter 3: Fast GC Theory
obtained for large values of k’ where the k’/(k’+1) term in the resolution equation
(equation 3-1) approaches 1 (Figure 3-2). At k’=2 only 2/(2+1) i.e. 66% of the
maximum resolution (at higher k’ values) can be obtained. The column length could be
increased to counteract the loss in resolution.
16 R 2  α   k '+1 
 ×
 α − 1  k' 
This equation is obtained by combining equation 3-1 and 3-15 and rearranging to solve
for column length. Working at k’=2 an increase in column length of (2+1)2/ 22 i.e. 9/4 or
2.25 is required to achieve the same resolution as would be obtained when working with
higher retention factors.
3.3.2 The influence of selectivity
When a column is used with high selectivity between the compounds of interest,
resolution is easier to obtain. This generally leads to a faster analysis as shorter columns
or smaller k values can be used to obtain the required resolution. In gas chromatography
high values of selectivity are generally only obtained for compounds that differ widely in
boiling point. However, it can be calculated from equation 3-1, that for a conventional
capillary chromatographic column that provides about N=100 000 plates, a selectivity of
only α = 1.02 is required for Rs=1 with k’=2.
3.3.3 Influence of carrier gas flow rate and pressure drop
Replacing H in equation 3-18 by equation 3-7 the influence of column diameter (dc) and
carrier gas on tr becomes apparent. H can be simplified to B/u+Cmu when working with
thin film open tubular columns and low pressure drop conditions (|pi-po| < 0.8pi ). At high
flow rates, where the B term reaches a minimum, H≈Cmu and the ratio H/u stays
constant. Under these conditions an increase in column length can be compensated for
with a proportional increase in flow rate to maintain a constant R and retention time8.
That means doubling the column length will not lead to an increased retention time,
provided the flow rate is also doubled, ensuring constant N and therefore R2.
Chapter 3: Fast GC Theory
 (1 + k ')  rc
2 α
tr = F (k ' ) 16 Rs 
k '2  Dm
 α − 1
When the pressure drop is high (pi-po > 0.8pi ) , f1 in equation 3-7 approximately equals
9/8 and f2 approaches 3/(2P) . The retention time is then better described as6
α  (1 + k ')4
tr = F (k ' ) 64 R 
 (α − 1)  k '
3η  rc
pa  Dm
where η is the dynamic viscosity
and po is the outlet pressure (normally atmospheric).
3.3.4 The influence of column radius
Following equation 3-20, the retention times increase at a rate equal to the square of the
column radius (tr ∝ rc2). Thus when a column with 50 µm is used as opposed to a 250 µm
column, a 25 times faster analysis can be effected. However, when a high pressure drop is
present across the column, the retention time is proportional to the column radius ( tr ∝ rc ,
eq3-21) and the increase in speed is reduced to 5 times that of the wider bore.
3.3.5 The influence of diffusion coefficients
Using the kinetic model of gases9 the diffusion coefficient can be described as
D = λc
where λ is the mean free path without collision of gas molecules:
σ is the collision cross section of the molecule
and c is the average speed derived from the Maxwell distribution of speeds:
Chapter 3: Fast GC Theory
These equations imply that the diffusion coefficients of various gasses depend on their
molecular mass and the collision cross section (σ).
Table 3-1: Collision cross sections/ nm2
Carbon Dioxide
A comparison of the molecular mass and the collision cross-sections reveals that when
hydrogen is used, diffusion coefficients 9 times larger than for carbon dioxide will be
In the low pressure drop case (eq 3-20) analysis times are 9 times faster when hydrogen is
used as opposed to carbon dioxide. When a high pressure drop (eq3-21) exists the
influence of diffusion coefficients (tr ∝ 1/ (Dm)-2) are less and retention times are reduced
by a third.
3.3.6 The relative contributions of column diameter v/s carrier gas identity
With the proposed two-dimensional SFCxGCftp the first separation will be effected with
high pressure CO2. The possibility exists that it could also serve, after depressurization,
as carrier gas for the GC analysis. When the influence of the carrier gas identity is
compared to the decrease in analysis time through reduction of the column inner
diameter, it can be concluded that for a narrow bore capillary (50µm) with CO2, faster
results will be obtained than using a wide bore capillary (250µm) with H2. However the
combined effect of using a narrow bore capillary together with H2 will produce the best
resolution in the shortest time.
In this way, resolution between a critical pair of analytes can be optimized with
isothermal GC. When resolution between the critical pair is obtained it may happen that
resolution between the other interesting compounds of interest is also obtained.
Chapter 3: Fast GC Theory
3.4 Temperature programmed analysis
While temperature programmed analysis does not improve the resolution that can be
obtained from a specific set of chromatographic conditions, speed and the detection limits
of such chromatograms are considerably increased. It has been shown that the
dependence of the analysis time on the column inner diameter for a capillary column is
the same in both isothermal and temperature programmed analysis10.
Diffusion coefficients are a function of temperature. However, it can be seen from
equation 3-22 to 3-24, that the respective dependencies on temperature is the same for
each of the gases. It can thus be assumed that hydrogen should also be the fastest gas for
temperature programmed GC analysis. The difference in the change of viscosities for the
different carrier gasses with temperature is small. This difference has been calculated to
be roughly equal to 1°C over the entire temperature-programming range of several
hundred degrees centigrade11. Thus, it seems safe to assume that the influence on runtime
of diffusion coefficients should also be the same for isothermal and temperature
programmed chromatographic runs.
3.4.1 Heating rates
The reduction in analysis time in temperature programmed GC analysis depends on the
heating rate - the higher the rate the shorter the analysis time. Unfortunately, an increase
in heating rate causes a reduction in column peak capacity. The selection of the best
heating rate requires a compromise between maintaining a minimum acceptable
resolution for the sample while obtaining the shortest separation time.
The argument is the exact parallel to the role of temperature and capacity ratio (k’) on the
separation in the case of isothermal GC. Too high heating rates imply elution of
compounds at too low k’ values with consequent reduction in R (see figure 3-1). Too low
heating rate implies final elution of compounds at too high k’ values with resultant loss in
separation speed (see Figure 3-3).
As opposed to the standard goal of achieving:
Good separation of a critical pair of solutes in the shortest time;
Chapter 3: Fast GC Theory
The optimization criteria of achieving:
An adequate separation of a required number of analytes in the shortest time,
is particularly useful for the general optimization of the proposed multidimensional
application of fast GC. The required number of analytes can be expressed through the
peak capacity (n). The analysis time of the sample (ta) is taken as the elution time of the
last eluting sample component. Blumberg12 defined certain constraints relating to
different pressure drop scenarios and obtained an optimum ramp rate (Rt) for each of the
different conditions. These optima were expressed in unit temperature increase per void
time (tm). Void time is the time it takes for an unretained compound to elute from a
chromatographic column.
r = Rt × t m
3.4.2 Normalized heating rates
The concept of normalized heating rate (r) substantially simplifies the optimization of the
heating rate by reducing the range of possible values that represent the heating rate. Once
an optimum heating rate has been experimentally found for a particular method, there is
no need to make another set of experiments to find an optimum heating rate for each set
of column dimensions, carrier gas, gas flow rate, outlet pressure or any other combination
of translatable changes. Translatable variations allow one to change the heating rate
without moving from the optimal normalized heating rate.
3.4.3 Default Optimum Heating Rate
Based on experimental data, Blumberg recommended that for columns with silicone
stationary phases with β≈250 (the thin film case, Cs≈0) the optimum normalized heating
rate is 10°C/tm. Low pressure drop conditions require a factor 2 higher and an increase of
12% is suggested for every factor 2 increase in film thickness.
Chapter 3: Fast GC Theory
Table 3-2: Heating rate (°C/min) vs. column dimensions for H2 at 10°C/tm13
Diameter/ µm
Length /m
As can be seen from Table 3-2 for shorter columns, the heating rate (RT) is impossible
with standard commercial gas chromatographs. For these high-speed separations,
alternative methods of column heating are required. The major limitation on heating rates
attainable with conventional stirred bath ovens is the huge thermal mass of the oven that
needs to be heated together with the column. Modern methods provide heat directly to the
small thermal mass of the capillary column. Very high heating rates up to 1200°C/min or
more can be achieved. Some of the methods for obtaining these fast heating-rates will
now be discussed.
3.5 Achieving fast heating rates
3.5.1 Resistive heating
Resistive heating has been applied successfully to direct heating of capillary columns. It
is achieved by applying a voltage drop across a capillary that has been made electrically
conductive. The increase in column temperature depends on the amount of power
dissipated. The dissipated power (W) depends on the current (I) through and the voltage
(V) across the column.
W= V I
[eq 3-15]
The current is dependent on the voltage drop and the resistance (R) of the column.
[eq 3-16]
Chapter 3: Fast GC Theory
The electrical resistance is dependent on the length, diameter wall thickness and
composition of the electrically conductive column or conductive layer.
The amount of heat required for increasing the temperature of any substance by an
increment ∆T is given by14
Q = mC∆T
[eq 3-17]
Where Q is the heat (Joules or Watt⋅seconds), m is the mass of the material (gram), C is
the heat capacity of the material( J/g.C°) and ∆T is the change in temperature of the
material. Eq 3-17 demonstrates that objects with large mass require more heat or power to
reach the same temperature in a given time period than an object of smaller mass. The
low thermal mass of a capillary column allows it to be rapidly heated while using much
less power than is normally required with a GC oven. Even more important, low thermal
mass allows for faster cooling and thus cycle times.
Methods of making columns electrically conductive
Resistive heating of flexible fused silica columns with metal cladding was first suggested
by Lee in 198415. The first practical demonstration of this technology for GC analysis
was by Hail and Yost16 who used a short section of a commercial aluminum clad fused
silica column. A programmable DC power supply was used to regulate the voltage across
the column. The power supply output was regulated through a 0-5V signal derived from a
digital to analog board.
Philips and Jain painted fused silica columns with a thin layer of electrically conductive
paint and regulated the output from a programmable DC power supply in a similar way17.
Mechanical instability due to differences between the thermal expansion coefficients of
the fused silica and the coatings caused rupture of the conductive layer. This was further
accelerated by local hot and cold spots caused by uneven coating. Chromatographic
efficiency was degraded and the analytical column was damaged.
Mechanical stability was improved by Ehrmann et al18. They compared the use of a
coaxial metal tube or collinear heater wire as an at-column heating element. Both
approaches proved to be satisfactory, but the coaxial heater provided better retention time
Chapter 3: Fast GC Theory
reproducibility. The tubular design allowed the use of an auxiliary sheath gas to even out
heat distribution along the column. While this gas had a statistically significant
advantageous effect, the benefit was too small to justify the additional instrumental
complexity. A pulse width modulator operating at 100Hz was used to control the
An instrument using a coaxial heater is commercially available. The Flash-GC19
embodies a conventional 0.25mm id column, either 6 or 12 meters long, placed inside a
precision-engineered metal tube. The tube can be heated up to 1200°C/min but was sold
in 2001 for approx. £20 000 pounds.
3.5.2 Microwave Heating
A recent commercial development uses a modified exterior polyimide coating that
absorbs microwave radiation. The column is placed in a cell that can be installed into a
traditional GC. With the microwave generator turned on, the column can be heated at
rates up to 600°C/min. It takes about 60 seconds for the column to cool down to starting
temperature, resulting in a typical cycle time of 180 seconds. Resolution and repeatability
is claimed to rival conventional GC20.
Induction or infrared heating could potentially be used for heating of the column.
3.5.3 Methods of sensing temperature
In the beginning, resistive heating was calibrated through an iterative process where the
heating profile was changed until the desired normal paraffin separation was obtained17.
This time consuming process was not very flexible and discouraged the changing of ramp
rates. The actual column temperature and heating rates were also unknown.
Thermocouples could be considered impractical for this application due to their relatively
large thermal mass in comparison to the capillary column wall onto which they are to be
connected. Even if small sensors were placed on the column, local cold spots would be
caused and this may lead to inaccurate temperature measurements, possibly producing
peak tailing due to local cooling.
Chapter 3: Fast GC Theory Resistance measurements
It was opted instead to use the resistance of the conductive capillary column as an
indication of the average temperature17. The resistance of any metallic conductor is
linearly related to its temperature over a large temperature range and is given by:
RT = Ro (1 + αT )
Where RT is the resistance at T, Ro is the resistance at 0°C and α is the temperature
coefficient of resistivity of the metal. The simplest way to measure the resistance is to
calculate it from the current through, and the voltage drop across, the column:
Since the current through every point of a circuit is the same, the current through a high
Wattage, low resistance, resistor in series with the column can be measured by measuring
the voltage drop across this known and constant resistor. This was the approach followed
by Hail and Yost16.
The resistance of the column was calibrated against known temperatures and used as a
direct measure of temperature. Resistance measurement with superimposed AC signal
Philips measured the resistance of the column by superimposing a supposedly constant
current 10kHz square wave on top of the DC heating current. The square wave was
sampled at 20kHz and the amplitude of this voltage measurement was proportional to the
resistance and hence the temperature of the column17.
This circuit (Figure 4-2) was somewhat delicate and required empirical tuning to obtain
good results. Very small signals of about 10 to 20mV had to be measured at high
frequency. Electronic noise is definitely a problem in a laboratory environment with lots
of electronic equipment. Despite the incorporated band pass filters, noise equal to 10°C
can be discerned from the published graphic results.
Chapter 3: Fast GC Theory Separate sensing wire
With the column-in-a-sleeve design by Ermann18 it was also possible to incorporate a
separate sensor wire. A small constant sensing current was passed through the wire. The
resultant voltage drop across the wire is proportional to the resistance of the wire (eq 319) and this resistance is proportional through equation 3-18 to the temperature. Infrared temperature sensing
Infra red temperature sensors are excellent non-contact sensors and thus do not cause cold
spots. They are available in models that offer much the same temperature range and
linearity as type K thermocouples. However, even with close focusing optics the smallest
measurement spot size of commercial models are 2.5mm2. This is many times bigger than
the surface of a capillary column. A couple of column windings could potentially be
coiled tightly together but each coil would have to be electrically insulated from its
neighbors and this will increase thermal mass and cool down times.
3.5.4 Temperature Control21,22
In order to do useful gas chromatography it is necessary to accurately and reproducibly
control the temperature. Isothermal temperatures should be well maintained and
temperature ramps promptly and precisely followed. This implies that the temperature
should be constantly monitored and control variables need to be continuously altered as
the set-point changes or when the measured temperature is different from the set point.
The process where a measurement is compared with a set point before corrective action is
taken is called feedback control. The difference between the set point (SP) and the
measured signal, also called the process variable, (PV) is the error(e)
e= SP-PV
[eq 3-20] The proportional controller
A proportional controller attempts to apply power, W, to the heater in proportion to the
size of the error, where P is known as the proportional gain of the controller:
W=P e
[eq 3-21]
Chapter 3: Fast GC Theory
As the gain is increased, the system responds faster to changes in set point and may
eventually start to oscillate as the controller becomes unstable. When a small
proportionality constant is chosen (Figure 3-2 no.2) the final oven temperature after a
step function disturbance (Figure3-2 no.1) lies below the set point because the product of
the error and proportionality constant is too low to request adequate power from the
heater. Increasing the gain alleviates this problem but at very high gain the process
variable may overshoot and start to oscillate around the set point (Figure 3-2 no.4).
Figure 3-2: Proportional control
1. A step increase in temperature of set point.
2,3. Increasing the gain (P) causes faster response to set point changes
4. At very high gain, temperature (PV) oscillate around control value (SP). Proportional+Derivative Control
Adding the time-derivative of the error signal to the control output can improve the
stability and overshoot problems that arise when a proportional controller is used at high
d 
W = P e + D e 
dt 
[eq 3-22]
Chapter 3: Fast GC Theory
This technique is known as PD control. The value of the damping constant, D, can be
adjusted to achieve a critically damped response to changes in the set-point temperature,
as shown in Figure 3-3. Too little damping results in overshoot and ringing (Figure 3-3
no.2), too much cause an unnecessarily slow response (Figure 3-3 no5).
Figure 3-3: PD control
1. Set point with step increase in temperature
2. High gain causes ‘ringing’ of process variable
3-5. Increasing damping improves oscillations Proportional+Integral+Derivative Control
Although PD control corrects the overshoot and ringing problems associated with
proportional control, it does not cure the offset problem encountered when a small gain is
used. Fortunately, it is possible to eliminate this steady-state error while using relatively
low gain by adding an integral term to the control function, which becomes:
W = P ×  e + D e + I ∫ edt 
[eq 3-23]
Here, I, the integral gain parameter is sometimes known as the controller reset level. This
form of function is known as proportional-integral-differential, or PID, control. The
Chapter 3: Fast GC Theory
effect of the integral term is to change the heater power until the time-averaged value of
the temperature error is zero. The method works quite well but complicates the
mathematical analysis slightly because the system is now third-order.
Figure 3-4: PID Control
1. Step increase in set point.
2. Process variable (temperature) follows set point.
3. Controller output.
Figure 3-4 shows that, as expected, adding the integral term has eliminated the steadystate error. Proportional+Integral Control
Sometimes, particularly when the sensor measuring the oven temperature is susceptible to
noise or other electrical interference, derivative action can cause the heater power to
fluctuate wildly. In these circumstances it is better to use a PI controller or set the
derivative action of a PID controller to zero. When a ramp as opposed to a step function
is used to set the temperature, derivative action is often not required.
Chapter 3: Fast GC Theory
3.5.5 The Control variable
For resistive heating there are two control possibilities:
A continuous current can be increased or decreased depending on the size of the
error signal
or a fixed current output can be turned on and off for various lengths of time in
response to the error signal. The latter case is called pulse width modulation
3.6 Chapter conclusion
Temperature programming of the chromatographic column is required for the separation
of the wide boiling point range of samples with the 2nd dimension of the SFCxGCftp. Fast
heating rates are required, because of the limited time available for the 2nd dimension
analysis. Resistive heating is a well-established technique for fast heating of capillaries.
While commercial resistive heating instrumentation is available, when attempting to
duplicate such a system, the various methods of temperature sensing should be compared.
Chapter 3: Fast GC Theory
Chapter 3
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