PROCESS CONTROL
Characteristics by Frequency Response JOEL 0. HOUGEN Rensselaer Polyfechnic Institute, Troy, N. Y. SIDNEY LEES MassachusettsInstitute of Technology, Cambridge, Mass.
REQUENCY response methods for determining the dynamic characteristics of systems have been used for many years in analyzing electrical and electromechanical components and assemblies. Only rather recently, however, have similar methods been applied to processing apparatus and process control instruments in order to determine the dynamic properties of such systems. The objective of this work was t o demonstrate the applicability of the direct frequency response method in establishing the pertinent dynamic properties of a typical industrial controller. Specifically, the problem was to calibrate the dials of the reset and rate adjustments, relating these to the characteristic times appearing in the performance function describing the behavior of the controller and to determine the sensitivity of the instrument in terms of the proportional band settings. The direct method used in this work proved very time-consuming and necessitated the development of certain experimental skills and techniques of data reduction. Results, however, are considered reliable and will form the basis for comparison with more rapid methods of procuring the same information in future work. The instrument studied was a Leeds & Northrup Speedomax Model S, Series 49, pneumatic controller. Normally the instrument receives the output e.m.f. from a thermocouple as its input and produces a pneumatic pressure as its output. The input signal is detected and amplified electronically, the amplifier output positioning a slide wire through a two-phase a.c. servomotor. The motor also positions a baffle with respect to a small nozzle, this position being compared with a reference position. The difference betaeen actual position and reference position, hereafter referred to as t h r correction, forms the input t o the pneumatic systcm. The pneumatic system accepts the correction input, operates on it in a prescribed manner, and produces the output air pressure signal. When employed as a process controller this output normally is the input to the final control element directly or, more frequently, to intermediate components such as boosters or valve positioners. I n this work the output air line terminated in a pressure transducer. Accordingly the results obtained apply only t o the controller itself and do not account for the performance of subsequent output signal transmission and/or modifier components which would be present in any actual control application.
Description of Apparatus Figure 1 is a schematic diagram roughly indicating the mechanical elements of the Leeds & Northrup controller which converts mechanical motion t o air pressure. The essential components of the pilot unit are a circular baffle and nozzle; the relative separation between these parts determines the output pressure. The angular position of the nozzle may be changed, June 1956
but the locus of its end is a circle which parallels closely the arc of the circular baffle. At a given nozzle position the distance of separation between nozzle and baffle is determined by the baffle position. The lower end of the baffle is secured to the lower end of a slotted vertical lever by a flat-spring-steel flexure joint. This spring also keeps the upper end of the baffle always against the end of the screw, S. Thus the position of point P is determined by the position of the correction signal screw, 8. Motion of point P may. be accomplished by rotation of either the inside gear, GI,or the outer gear, Gz,or both. GI is positioned by the control point setter, while Gz is connected to the potentiometer slide-wire -positioning motor and thus takes a position directly proportional to the temperature signal to the controller.
20 PSI
i
OUTPUT PRESSURE
-
D E R I V A T I V E ACTION ADJUSTMENT
I N T E G R A L ACTION ADJUSTMENT
V.
V.
OP CEIRRECTIOR GEAR D R I V E N F ULAR
PILOT
Figure 1.
BbFFLE
UNIT
Essential mechanical features of pneumatic control unit
The vertical lever (baffle support arm), the half-section of which lies behind the baffle, is secured a t the top to a rigid frame by means of a flat-spring-steel flexure joint. The lower end is constrained in its motion by opposed springs. This entire lever may be rotated slightly around its upper flexural joint by horizontal forces developed by air pressure in the opposed bellows. If the net force of the bellows is to the right, the baffle will also move to the right, rotating around point P . The angular position of the nozzle, which is manually adjustable, determines the sensitivity or gain of the unit. When in its uppermost point, the sensitivity is greatest because for a given displacement of point P the change in the distance between nozzle and baffle is greatest.
INDUSTRIAL AND ENGINEERING CHEMISTRY
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ENGINEERING, DESIGN, AND PROCESS DEVELOPMENT
Figure 2.
Components of pneumatic system-pilot
Because the air flow capacity of the tiny restriction located in the amplifier unit is less than that of the nozzle, the nozzle back pressure is dependent on the proximity of the nozzle to the baffle. Changes in this pressure cause deflections of bellom BI in the amplifier unit which produces an opposite deflection a t the opposite end because of the flexure a t connecting link C. The latter motion adjusts the ball valve in the right end of the amplifier unit which regulates the flow of air through it and thus the output pressure from the unit. This pressure also exists in B2 and B,,the moments developed by these bellows thus being opposed and self-canceling. The restrictions used in such pneumatic systems are of the order 0.008 inch in diameter and are one third to one half the diameter of the nozzle. With a nozzle diameter of about 0.018 inch the nozzle back pressure nil1 vary from 2 t o 20 pounds/ square inch for a change in nozzle-baffle separation in the order of 0.002 inch. Thus in order to attain an accuracy of 1% in nozzle back pressure, the baffle must be positioned with an accuracy of a few millionths of an inch. Positioning of this exacting nature is impractical with simple mechanical linkages, and t o overcome this dilemma instrument makers have introduced a negative feed-back loop embodied in the proportional bellows (Figure 1). Through this arrangement a small displacement is produced as the diff erence between two larger displacements. Referring to Figure 1, if gear Gz rotates so as to move point P t o the right, a reduction in nozzle back pressure occurs. Accordingly, the output pressure from the direct acting relay or
1054
assembly
amplifier also decreases. If, for the moment, it is assumed that valves VI (and V2)are open and Va is closed, the proportional bellows will be collapsed. The rotation of the baffle around point P then brings t,he baffle toward the nozzle thus tending to reduce the result of the init,ial change. By proper design, t’he rotation of GBreqiiired to produce a full raiige of output air pressure may be considerably extended and smooth proportional posit’ioning of the final control element obtained. Since both lag and lead eharacteristice are desired in the control act,ion, other components are added. By introducing a throttling valve (VB)in the air line leading t o the proportional bellows, an adjustable lead act’ion is achieved. Adding the “droop correction” bellows and t’he throttling valve (V,) in its supply line gives a lagging action which may be changed depending on the extent of resistance offered by V,. Figure 2 is a cutaway view of the pneumatic pilot assembly, and Figure 3 indicates the nature of the throttling valves, V, and Va. Additional volume is also provided between throttle valves and bellows in each circuit. This is not indicated in either Figure 1 or Figure 2. As mentioned previously, the static sensitivity of the pneumatic system of this controller is determined by the angular position of the nozzle. The dynamic properties, hoir-ever, depend on the extent of throttling achieved by V2 and Va in addition t,o the position of the nozzle. Each valve is provided with a dial (Figure 3) which indicates the extent to which the valve is open. It was the calibration of bhese dials as well as the scale associated
INDUSTRIAL AND ENGINEERING CHEMISTRY
Vol. 48, No. 6
PROCESS CONTROL
Figure 3.
’[
Pictorial and cutaway views of throttling valves used in controller
db = -
R[ba-nld(cw)
$. R
b
(1 - R T )
R b
(3) with the angular position of the nozzle (proportional band scale) that formed the immediate objective of this work.
Mathematical Analysis The analysis of complicated systems is greatly facilitated by use of a fairly well defined and systematic procedure. The first step of such a procedure is to construct what may be called a functional diagram. These diagrams show schematically the action of all components and their relation to one another. With the relationships that follow from an analysis of each component and the inter-relations indicated by the functional diagrams, the next step is to prepare a mathematical diagram. I n this diagram each discrete component or subassembly of the system is shown as a block. The input signal (or cause) acting on the component is shown as a line directed toward the block, whereas the output signal (or effect) is shown as a line leaving the block. Within the block is inserted a mathematical expression such that the output is obtained by multiplying the input by the mathematical expression or operator. A signal may be directed to several destinations in which case a pick-off point exists, and all signals leaving the point have the same value. Signals may be added or subtracted; the operation is indicated by the signals entering a summation point, and the sum or difference is the signal leaving. Where the behavior of the components shown as blocks can be described by linear ordinary differential equations with constant coefficients, it is permissible to replace the signals themselves by the operational forms of the signals. This is a very important result since mathematical operations now become essentially algebraic. For the special situation where the input signal varies sinusoidally the operator ( p ) may be replaced by jw where w is the angular frequency of the signal. This further simplifies the mathematical treatment and permits relations between input and output terms t o be obtained quickly. I n this example, Figure 1 may be considered a functional diagram. However, to show more clearly the function of the various members in the pilot assembly Figure 4 has been constructed. I n this illustration the baffle support arm or force summing lever, which is normally in a very nearly vertical position, is shown displaced through an angle A[,,-II, rotation being ubout point 0. I n addition the angular relation of the baffle with respect to the force summing lever, A[c-bl, is greatly exaggerated. The analysis of the behavior of the subassembly shown in Figure 4 is as follows: The total displacement of the baffle a t the nozzle position may be assumed to be the result of two displacements-that of the correction signal screw, d(,,,), and that of the baffle axis, d ( b , ) . T h e total displacement of the baffle, d b , is therefore the sum of these two displacements each multiplied by the appropriate factor:
June 1956
The ratio, R r b a - n ] / R b , may approach but never exceed unity. The displacement of d(ba) from some equilibrium position occurs by virtue of a change in the net force developed by the reset and proportional bellows. Thus assuming displacements to the right as positive,
CORRECTION S I G N A L SCREW
IPROPORTIONA B A N D SCALE 1 I U
i
B A F F L E AXIS
(bo)
R E F E R E N C ~’ LINE
Figure 4.
Schematic functional diagram pilot assembly of controller
of pneumatic
If both throttle valves are open in the least, it follows that when equilibrium is established the pressure in both bellows is the same-that is p ( d & ) = &b) = p , a t equilibrium. Since i t is the function of the reset bellows t o effect a permanent shift in the position of the baffle axis as the pressure in both bellows changes i t follows that S ( d e b ) [ p , f l must differ from Scpb)[p,fl. For this particular instrument S ( d & ) [p,,l must exceed S(pb)[ P , l l since it is necessary t h a t the bellows axis be shifted to the left, (Figure 4) as the steady-state pressure in each bellow increases. Thus &a) becomes negative. The distance, Riba-n~, depends on the angular position of the nozzle. A position indicator in the form of the proportional band scale is associated with the mechanism by which the
I N D U S T R I A L A N D E N G I N E E R I N G CHEMISTRY
1055
ENGINEERING, DESIGN, AND P R O C E S DEVELOPMENT
1
-
I
I
p f b l O L - 5
II
I
I
I
I
I
Figure
5.
Figure 6.
Mathematical diagram of pneumatic components
of controller giving proportional action
I
Mathematical diagram for controller with
both pneumatic components active
nozzle angle is adjusted. The loxer end of the proportional band scale corresponds to the larger values of Rl~,~-,,l. Thus a reciprocal relation is indicated. With the appropriate value of scale factor, S,, we may write
Using these relations it follows t h a t
p,=
(5) where
S.also depends upon the nozzle position. where Rb(PB)/S, is always greater than unity.
If
and
the following familiar form is found:
Figure 7,
Signal generation and measurement and recording system
Where only proportional action is desired the pressure in the proportional bellows is identical with the output pressure from the pneumatic relay. Also, Va is closed so t h a t the pressure in the droop correction bellows is held substantially constant at some reference pressure. Thus for actual pressure signals
If it is assumed t h a t both nozzle-baffle assembly and pneumatic amplifier are operating in a linear fashion the following additional relations may be written:
(7) 1056
It is important to note t h a t the proportional band adjuPtmeiit affects both the feed-forward and feed-back sensitivities. At low values of (PB)where &(PB)/S, approaches unity the relation indicates an increasing dependency on ( P B ) , whereas a t large values of (PB) where &(PB)/&>> 1 the product S f f S f bbeconies independent of ( P B ) . Experimental results bear out these deductions. The mathematical diagram shown as Figure 5 embodies the essential features of the analysis of Figures 1 and 2. When both sets of bellows and the throttling valves are active the mathematical diagram must be altered in order to account for the dynamic properties possessed by the two resistance-capacitance components in the feed-back path. Assuming t h a t the air behaves ideally a t a constant temperature, that flow through the throttling valves is always viscous, and that the displacement of the bellows is so small t h a t volume changee may be neglected, it is readiIy shown that the relation between force produced and pressure p , applied to the valve is described by a first-order differential equation:
Using the symbols shown in Figure 5 and noting that when the droop correction or reset bellows is active the output pressure replaces the reference pressure, the mathematical diagram shown in Figure 6 may be constructed.
INDUSTRIAL AND ENGINEERING CHEMISTRY
vat. 48,
NO.
6
I n establishing the signs associated with the force and displacement signals shown in this diagram it was assumed t h a t the force produced by the proportional bellows is positive, an increase in pressure giving a positive displacement which would occur also for a positive displacement of the correction signal screw. Thus, conversely, an increase in pressure in the droop correction bellows will produce a negative displacement. Following this convention the net displacement of the baffle will be the sum of t h a t caused by the correction signal screw and the net displacement caused by the forces developed by the opposed bellows. The latter produces t h e displacement of the force summing lever. This displacement, a t the nozzle position, is denoted as the feedback displacement, d(fb). With these conventions the complete performance function of the pneumatic assembly or ratio of outp u t t o input is expressed as Equation 10. On introducing the definitions of feed-forward and feed-back sensitivities as given previously, Equation 11 is obtained.
Experimental Apparatus and Methods
Rearrangement of Equation 1 1 yields
Figure 7 shows the front view of t h e instrument tested, the mechanism for producing the sinusoidal e.m.f. signal, and the two-channel recording oscillograph. The e.m.f. signal was produced by a sine-cosine potentiometer (Technology Instrument Corp., Acton, Mass., Type RVP3-359, 20,000 ohms &5%) which was part of a resistive circuit energized by a 1.3-volt mercury cell. The circuit was so arranged t h a t the output could be biased and changed in amplitude. The sine generator was driven by a 1/40th hp., 1800 r.p.m., synchronous motor. The speed reducer box and subsequent interchangeable gears permitted adjustments in frequency in discrete steps from 0.04 t o 18 cycles per minute. (Above 18 cycles per minute the roughness of the gears became noticeable and wave forms very poor.) Geared to the potentiometer shaft was a parallel shaft which rotated a Librascope (Librascope, Inc., Glendale, Calif.) mechanical sine-cosine resolving component. This mechanism produces a linear simple harmonic motion of a pin when the input shaft rotates uniformly. The pin deflected a flexible centilever near the base of which %-asattached a strain gage bridge. T h e signal from this follower mechanism, which bears a constant angular relation to the voltage signal, was re-
Besides these general forms several special forms are of interest. Performance Function for Proportional Action. I n this case V zand the by-pass are open, and Va is closed so t h a t
corded on one channel of the oscillograph. The other channel recorded t h e output pressure from the controller which waa sensed by a Statham (Statham Laboratories, Los Angeles, Calif., Model P6-206-120) strain gage pressure transducer.
(CT),, = 0 (CT),, = a
Experimental Data
From Equation 11 i t follows that
'
Data were collected with the objective of procuring information from which both static and frequency response performance
which agrees, as i t should, with Equation 9 as previously derived. Performance Function for Proportional Plus Rate Action. For this type of action, (CT),, = a and (CT),, has finite values so t h a t Equation 11becomes
Perfonnance Function for Proportional Plus Reset Action. I n this instance (CT),, = 0 and (CY),, assumes finite values. Equation 11 becomes
PROPORTIONAL
Figure 8.
June 1956
eAN0
S E T T I N G , "%"
Static sensitivity of pneumatic system
INDUSTRIAL AND ENGINEERING CHEMISTRY
1057
ENGINEERING, DESIGN, AND PROCESS DEVELOPMENT signal constant thioughout the entire aeries of tests; however, during a single test no change occurred. Total displacements of the pen or double amplitudes varied from 0.61 to 1.04 inches In most cases all information at a given frequency was obtained before proceeding to another setting of the controller adjustment. One dial a t a time m-as calibrated nith dial settings of 0, 2, 4, 6, 8, and 10. For the most part oscillograph chart speeds were used which gave records haviiig cycles from 20 to 700 mm. in length. Maxima and minima of the sinusoidal curves were found by bisecting the line segment drawn parallel t o the direction of the chart tiavel and intersecting the curve on either side of the point (Figure 9a). Where integration action was present some drift of the output signal frequently resulted. This was caused by failure to oscillate uniformly about the exact control point. In such cases the double amplitude was measured as ehos.n in Figure 9b, in which typical data are shown. Average dynamic response angles were calculated using measurements made at both top and bottom in order to compensate for any wave form deformation and drift.
Armeaiysio
Figure 9. Oscillograph record with derivative cofflponent active ( a ) a n d with integrating component active (b)
OB:
Data
Calibration of Rate Dial. Equation 11B describes the peiformance of the pneumatic components xhen proportional plu5 rate action is desired. As written, however, the perfornianw function is the ratio of the output pressure change to the di placement of the coriection signal sciew. The experimental dats give the ratio of output pressuie to di.placenieiit of the pen along the scale. Thus the information obtained is foi
po dm9, - P o x B l d g d c a s l [ P F ] ( ? , ? a ) / o b s . ) characteristics of the pneumatic components could be obtained. d(,,,, d pen) d w In addition, f i equency response data mere obtained on the entire controller so that the behavior of the electromechanical compoSimilarly the calibration data for puiely proportional action is nents could also be found in the frequency range of interest. given as The static data vere used to determine the sensitivity of the prieuinatic components and to calibrate the proportional band scale. Typical data are shown in Figure 8. These data Tyere obtained by observing the output pressure, as indicated by a merA nondimeneional performance function is olitained if the cury manometer, for various displaceinentx of the temperature observed performance function is divided by S [ d p p l . Thus indicator pen from t,he mid-point of the temperature scale and from Equation 11B there is obtained for the several proportional band settings. Pressure of 10.00 pounds/square inch mas maintained in the droop correction bellows during these tests. The sensitivity so obtained relates the displacement of the pcn to output air pressure, XIdn;gl. I t is an over-all sensitivity and may be considered as the product of For convenience tTvo auxiliary terms are defined such that t T o sensitivities-that for displacement of the pen to displacement of the correction signal screv-, S [ d p , d c i a l , and that for (1 5 ) displacement of the correction signal screw to pressure, S[dcss;pl. Thus and S [ d p ; p ] = S[dp,dersl .S [ d c s s ; p l (13) (C2'R)d = 1 XffSjb (16) ,S'ldp, dorsi was determined by observing that a pen displacement Thus of 8.24 inches v a s required to rotate the correction signal screnFirice the screw has 28 threads (CT),, = ( C T R ) d ( C T ) d (G2) through three revolutions. per inch the desired sensitivity >vas calculated as and ~
+
In all the frequency response tests the throttling range adjustment was set at 100% a t nhich setting the over-all static sensitivity, was -1.068 pounds lequare iiich/inch of pen displacement so that
When linear systems are forced sinusoidally the differential operator may be replaced by jw. Equation 17 then becomes
Three observations may be made by inspection of Equation 18: 1. As 20 0, [AvDPF1(p,+ra)rs,n) 1 2. As w Q , [-\7DPFI,,-.,,,,, n) (CTR)d = 1 3. S f f S f L 3. The magnitude or nondmlensional dgnarnic ainplitude ratio is given as
-f
pounds of output air pressure per inch of correction screw displacement. hToattempt was made to hold the amplitude of the input e.m.f. 1058
INDUSTRIAL A N D ENGINEERING CHEMISTRY
-
-f
Vol. 48, No. 6
PROCESS CONTROL
+ (CTR)a(CT)iwZ
of slope plus unity and intersecting the former a t a frequency equal t o the reciprocal of the denominator characteristic time. It follows that by plotting of experimental frequency response I n terms of logarithms data on appropriate logarithm coordinate paper 1 1 and by using standard profiles the desired inforlog [NDDARl(n+ra)(ain)= (CT)3wZ1 mation may be extracted very quickly. These 2 log [l f (CTR)2(CT)5w21- 3 log [I methods are not new, logarithmic plots having 1 = log [l (CTR)$(CT):Wq1’2 log 11 + ( c T ) ~ ~ z ] ” ~ been suggested many years ago (1). A somewhat detailed discussion is given here for the convenAt frequencies sufficiently high t o make (CTR);(CT);w2 ience of those not familiar with previous work. and (CT);w2> > 1 Figure 11 shows the results of the experimental amplitude data obtained for various settings of the rate dial and a sensitivity 1% [NDDARIw r a ) ( s i n ) G log (CTR)d(CT)dw - log ( C T ) ~ W corresponding t o 100% proportional band. The original plots wcre made to a scale such that one logarithm cycle occurs every 2’/2 inches. The shifting of the reference line to accommodate 20 all 4ata distinctly should be noted. By use of a standard pro13 file (2) the information shown in Table I is obtained. (CTR)d IC a (CT)d is calculated as 1/2~(bp)1,,,, and ( C T ) d as 6 Results are shown graphically in Figure 12.
[NDDAR](p+ra)(ain) =
1/1
4\/1+(CTW
+
+
+
I
4
-
2
Table 1.
, 1.0
5
.e : z
Summary of Results for Various Settings of Rate Valve (Proportional band Break Points,
.6
I
=
100%; reset valve closed)
Dial (CTR)d(CT)d, Setting Lower Upper Minutes
2
2 I
.COP .GO4
.01
.C2
.04 .C6,06.1 FREQUENCY,
,2
RADIANS
.4
PER
,6 .6 1.0 UNIT TIME
2
4
6
Figure 10. Logarithmic plots of dynamic amplitude ratios for typical functions On a logarithmic plot of [NDDAR](,+ra)(sin) versus frequency, the two right-hand terms may be represented as two straight lines with slopes of one and minus one, respectively. Since 1 is larger than unity, i t follows that the value of the first term always exceeds t h a t of the second. As zu .-, a the difference in logarithms approaches log (CTR)d = log(1
+
4 6 8 10
0.057 0.79 0.23 2.75
0.50 1.03 1.80
6.50
... ...
2.795 0.694 0.318 0.154
0.0885
(CTR)d-
(CT)d,
Minutes Calcd. 0.202 0.058 0.0246
... ...
13.8 12.0 12.9
... 1 . .
Obsvd. 13.2 13.3
14.0
... ...
+
A value of (CTR)d = 1 S f f S f b of 13 is taken. Superposition of Figure 10 on the upper three curves of Figure 11 will show t h a t the experimental data agree well with the profile constructed for the example where the coefficient in the numerator term is 13 times as great as that in the denominator term.
+
s f / S f b*) On the other hand as w -.,0 each right-hand term within brackets approaches unity, the frequency a t which the term containing w ceases to become significant being dependent on the relative magnitude of the coefficients of w. The situation is illustrated in Figure 10, where (cTR)d and ( C T ) d are arbitrarily taken as 65 and 5, respectively. The solid lines represent the individual terms and the quotient of the two, and the dotted lines represent the approximations z (CT)iw2 >> 1. These are extrawhere ( C T R ) ~ ( C T ) z wand polations of the straight-line portion of each curve. Each extrapolation intersects the unity nondimensional dynamic amplitude ratio line at a frequency equal t o the reciprocal of the coefficient of frequency in the term in which i t appears. This frequency is called the break point frequency. Each curve can be approximated by two straight lines-one horizontal, located on the unity 0 ordinate and the other of unity slope passing through the FREOUEWCY. CYCLES PER M I N U T E break point. I n the example cited the break points are 0.0154 Figure 1 1 . Experimental data for various settings of rate and 0.20 radian per unit time so that (CTR)d(CT)d = 65 and valve (CT)d = 5, as assumed. The curve representing [ N D D A R ] is found by subtracting (CTR)~(CT)”,w]l’2 Calibration of Reset Dial. If [PF](D+,,),written as Equafrom the curve representing log [I the value for log [1 ( C T ) ~ ~ W Zas] ~ measured ‘~ above the tion 11C, is divided by Sff/l f SffSfb a nondimensional perordinate of unity. The complete curve is shown; it becomes formance function for proportional plus reset action is found: coincident with the value of (CTR)d at high frequencies. Also, the upper right-hand portion of the complete curve can be approximated by two straight lines-one of zero slope representing [NDDAR](p+,a)(sin) = (CTR)d(in this case 13) and the other
+
June 1956
+
I N D U S T R I A L A N D E N G I N E E R I N G CHEMISTRY
1059
ENGINEERING, DESIGN, AND PROCESS DEVELOPMENT Convenient auxiliary terms in this case are
(cT)i SZ (1 f
s//s/b)( C T ) , ,
and
(CT),, 3(CTR),(CT)% so t h a t
Figure 13 shows logarithmic plots of the experimental amplitudes for various settings of the reset valve; observations at frequencies sufficiently low t o find the low frequency break point were not made. This prevents the direct experimental determination of (CT)d. The break points corresponding to the numcrstor coefficient, (CTR)i(CT)i,ho.;r-ever,are clearly discernible. Measurements n-ere not sufficiently accurate t o indicate that [+vDDAR](++.,)(sin) approaches the ratio of 1/[1 S//S,fb (1 (SR)a)]rather than unitmy. This is because the former differs from unity by about 2%, as will be shown later, and amplitude measurements were not made with precision greater than this. The values of (CTR)d(CT)idetermined by fitting a standard profile to the plots in Figure 13 are shon-n in Table 11 and Figure 12. Also shown in Figure 12 is the line representing ( C T ) ; . This is calculated assuming the characteristic time ratio ( C T R ) ; = 1/(1 S / / S f a )has a value of '/Is under the conditions of the tests.
+
Equation 19 then becomes, for sinusoidal forcing,
e I00
1l
1 1
---hO '3R
CXPERIMEN'AL TH1S :UF\'E
+
OhT& E X I S I S I T I S LOCA'EO
H E DILiL S E T T I U O S
Table II.
Summary of Results for Various Settings of Reset Valve (Proportional band = 100%; rate andby-pass valves open) (CTR)i(CT)i = Break Point (B.P.), 1 (CTh Dial Setting Cylces/Min. 27r ( B . P . ) (Calcd.") 2 0.023 6.96 90.30 4 6 8 10 (9.8)
0.10 0.23 0.48 0.69
1.59 0.695 0.332 0.231
19.75 9.03 4.13 2.8'9
Figure 12. Characteristic times and t i m e ratios associated with rate and reset dial settings
0 C-
Results using dynamic response angle data Results using dynamic amplitude ratio data
Equation 20 does not have the sinip!e form of Equation 18, differing both by the coefficient l/(CYE); and denominator term, S/,S,b(l Logarithmic plot of the nondimensional dynamic amplitude ratio corresponding to [-VDPF (sin) is composed of three parts:
(sn),).
1. A horizontal line corresponding to log l / ( c T E ) i = log (1 Sf/S/b) 2 . The curve for log [l (CTB)i(CT),jzo] S,jSjs(l - (&?)a) (CT)ijwl 3. The curve for log [l
+
++
+
Since ( C T X ) iis less than unity it follows that (CY')(is alrvays greater than (CTEi,(CT)d. Thus the denon2.inator frequency coefficient exceeds that of the numerator, and the break point for the denominator term on a log-log p!ot lies t o the left of t'hat for the numerator. Since on such a plot the denominator magnitude is to be subtracted from a constnnt, log(1/(c~n)i)= log (1 SffSfb), it follows that a t low frequencies [ArDDAE1(,,+ re)(sin) approaches (1 X f f S f 6 ) / [ l S / j S f b ( l (SR)b)].At'frequencies sufficient,ly great to make only t h e frequency dependent term in the denominator significant but Tvith t h a t of the numerator term still small, the log plot of [NDDAR](p+,,,(sin) has a slope of minus unity. When the frequency increases to the point where the numerator term is no longer insignificant, the curve flat,tens out and approaches the value 1/[1 S f / S / b ( l- (8R)b)] at high frequencies.
+
+
+
+
1060
Figure 13.
Experimental data for various settings of reset valve
Information is available from the Leeds R. Northrup Co. for estimation of both S Z [ / ~ ~ ~ S ( and & ~ )SgI,,dl [ ~ , J S(pb) I [p,~i which are the over-all sensitivities for pressure in, to displacement of baffle axis out of the droop correction and proportional bellows, respecis made by using the fact that tively. Evaluation of Si[f;:d]S(pb)[P,fl an increase of pressure in the proportional bellow of from 3 00 t o 15.00 pounds/square inch will move the baffle axis 0.0383 inch when the pressure in the reset bellows is 9 pounds/ square inch. Thus
Sl[/;,$(pb)[p,f] = 0.0383 = 0.0032 inches/(lb./sq. inch)
IN D U S T R I A L A N D E N G I N E E R I N G C H E M I S T R Y
Vol. 48, No. 6
PROCESS CONTROL SO
with the system being forced. This is conveniently done, in a mannei similar t o that employing dynamic amplitude ratios, by a curve fitting technique. If a system whose performance function is 1 ( C T ) , is forced sinusoidally the operator may be replaced by jw so that
I
w
, , / j 1 ,1
I ,
70
,
,
,
! ,
v \I. A I , ,
, ~
, , u!! !
I
!
+
[PFI(,,,, = 1
+ (CT)jw
This may be written as a magnitude and an associated angle: Magnitude
E
[ D A R ]=
dl + ( C T ) 2 ~ 2
Angle 3 [DRA] = tan-' ( C T ) w Angles will vary from 0 to 90" as the frequency varies from 0 to infinity. When (CT)w equals unity the angle is 45', hence ( C T ) = 1/w(460)where W(r5O) is the frequency a t which [DRA] is 45'. This suggests the method of finding characteristic times by analysis of experimental data.
E -30 y
'40
2 -45 -10 0
'M1 -70.001
,002
,004 ,006
01
02
ANGULAR
Figure 14.
.04 .06 0 .2 .4 .e F R E Q U E N C Y . RbOlANS P E R UNlT
1.0
2
6
6
10
TlHE
Semilogarithmic plots of dynamic response angles for typical functions
The manufacturer also states that for this assembly an increase of pressure in both proportional and droop correction (reset) bellows from 3.00 to 15.00 pounds/square inch will displace the baffle axis to the left (Figure 4) a distance of 0.0008 inch. Thus
0.000066 inches/(lb./sq. inch)
A performance function of the form written as
f (cT)dw may be 1
+ (CT*)jW
[ D AR [DAR]ze3P or more conveniently as
[DAR]e i ( a - 0 ) = [DAR]e m where CY = tan-l(CT)lw and p = tan-'(Cl')*w and y = CY - @. Thus the dynamic response angle associated with the denominator term is subtracted from that of the numerator term to arrive a t the angle associated with the quotient. Figure 14 illustrates the graphical combustion of two-angle loci to yield the dynamic 1 65 jw rebponse angle for the performance functions and 1 5jw 1 + 565 jw ~jw. For maximum utility it has been found convenient
+ +
+
Verification Using Dynamic Response Angle Measurements The output signal of a linear system being forced sinusoidally bears a fixed angular relation to the input at a given frequency and after all transients disappear. Thus dynamic response angles, determined experimentally, may also be used to find the characteristic times in the performance functions associated
Figure 15. Experimental dynamic response angle data for various settings of rate dial June 1956
to plot such data on semilogarithmic coordinates with frequency on the abscissa logarithm scale (I,!?). When the coefficient of the frequency term in the numerator exceeds that of the denominator the angles are always positive, whereas when the opposite is true the angles are negative. Also, the extremity of each component curve may be fitted by a dynamic response angle profile for a performance function of the form 1 (CT)jw with the appropriate value of ( C T ) . Figure 15 shows the results with proportional plus rate action for various settings of the rate dial. The standard profile has bem used to construct both ends of the curves for which sufficient data appear. Despite the lack of data in the intermediate frequency range smooth curves can be drawn which fit the data remarkably well and from which the characteristic times may be
+
Figure 16. Experimental dynamic response angle data for various settings of reset dial
INDUSTRIAL AND ENGINEERING CHEMISTRY
1061
ENGINEERING, DESIGN, AND PROCESS DEVELOPMENT
Table 111.
Rate Dial Setting
Characteristic Time and Time Ratios Determined by Analysis o f Dynamic Response Angle Data
Numerator Term B.P., cyles/min. (CTR)d(CT)d
...
0 2 4
6 8
10
Denominator Term B.P., cgcles/min.
(CT)d
(CTR)d
0.088 0.73 2.50 5.30 11.5
1.82 0.22 0.064 0.030 0.014
10.9 11.4 11.1 9.4
...
0.067 0.22 0.50 1.23 2.00
2.38 0.72 0.32 0.13 0.08
Table IV.
...
Reset Dial Setting
...
2 4 6 8 10
...
...
Denominator Term B.p., cycles/min. (CTR)i(CT)i 0.021 0.095 0.235 0.51 0.80
7.60 1.68 0.68 0.32 0.20
Results of Experimental Determination of Feed-Forward and Feed-Back Sensitivities
Sensitivitv. -4
1 5 10
-37.95 -19.27 -10.39 -7.38 -5.59 -4.36 -3.17 -2.28 -1.63 - 1.068 -0.800 -0.534 -0.186
15 20 25 35
50
75 (70) 100 150 (135) 200
500
2915 1480 798 566 430 335 243 175 125 82 61.5 41 14.3
... ...
... ...
*.. ...
...
10.9 12.0
9.9 11.1
8700 6800
1.14 1.62
13.0 13. 13.0 12.7 12.8 I3 I3 13
12 12 12
4340 3160 2270 1590 1050 800 534 186
2.78 3.80 5.28 7.35 11.22 15.00 22.50 64.5
...
derived with considerable confidence. Table I11 and Figure 12 show the results and the excellent agreement vc.ith characteristic times determined from dynamic amplitude ratio data. Figure 16 is a plot of dynamic response angle data obtained when calibrating the reset dial. Frequencies sufficiently low t o enable detcrrnination of ( C T ) ,r e r e impractical t o attain, hence only the right-hand portion of the curves are useful. Fitting the standard profile to these curves permits the determination of (CTR),(CT)i.
...
11.7 11.8 12 12 12
...
...
...
I/(PB)
.....
.. 8.7 10.2
0.10 0.067
10.8 11.1 11.4 11.9(11.1) 10.5 12.0(10.8) 10.7 9.30
0.040 0.0286 0.020 0.0133 0.010 0.0067 0.005 0.002
...
e . .
These are also tabulated in Table I11 and plotted in Figuic 12 for comparison with previous results. Agreement is considered good although the value of (CTR)dappears to be about 11 as compared with 13 determined from amplitude data. The smoothed curves in Figure 12, however, give a ratio of 13 at the lower dial settings falling to 11 at the highest reading. This might indicate deviations from viscous flow as the throttle valve approaches a full-open position. Determination of Feed-Forward and Feed-Back Sensitivities
(-----) +
100
ao 60 40
+
eo IO
E 6 4
a c
+
+
0
cx
Static determinations of the term S , d p dessl - '// wcrc 1 Si/S/b made in connection with the calibration of the proportional band scale (Figure 8). Furthermore S l d p d c s s ] has been found to be 1 0.0130 inch/inch of pen displacement Hence values of -Si,/ s//S/bmay be calculated as shown in Table It-. The valuc of [1 S//S,?b]was found for the 100% proportional band setting from high frequency rate dial calibrations. A value of 13 proved to fit the data quite well. By determining the nondimensional amplitude ratio for high frequencies for other proportional band settings the corresponding values of [ 1 3. S//S/b]may likewise be found. A rate dial setting of 4 was used for these tests. TTith values of -8/,/l 4- S//AS'S/~ and 1. SffSfbboth AS'// niid S/b may be calculated. These are shonn in Table IV. Foi S/,SJI proportional bands in excess of 25%, i t appears that 1 remains constant a t about 13; hence this value is assumed for the settings above 25% which were not obtained experimentally Figure 17 shows that S,fb varies approximately as the square of the proportional band throughout the range 10 to 200%, while S J fvaries inversely as the square of (PB)in the region 35 to 200% but varies less rapidly in the low range and more rapidly in the high range of ( P B ) settings. Figure 18 indicates an approximate linear relation betmeen S J , and 1 / ( P B ) a t high values of ( P B ) (low values of l / ( P B ) ) . I n addition a plot of S f f ( P B )versus l / ( P B ) is given. This plot is very sensitive to the values of (PB) assumed for the given setting. Since the scale is crude and unlabeled a t several intermediate points and since fine adjust-
2
+
v)
I 8
6
I I I
I 5
I
1
IO
1
15
PROPORTIONAL
Figure 17.
1062
1
1
25 35 50 BAND
7 5 100 150200
SETTING
500
"%'
Feed-forward and feed-back sensitivities vs. proportional band setting
INDUSTRIAL AND ENGINEERING CHEMISTRY
Vol. 48, No. 6
PROCESS CONTROL 12
d indicates derivative action i indicates integral action
12
II
10
(CTR)
= characteristic time ratio; subscripts d and i defined
db
= displacement of baffle a t nozzle position
above
11
9 B
10
$,tx 106
9
d(aa)
df b
= displacement of baffle axis = feed-back displacement
dff
= feed-forward displacement
d(eas)
[DAR] [DRA] F
= = = =
j
=
5 4
8
displacement of correction signal screw dynamic amplitude ratio dynamic response angle force produced by the bellows component; subscript denotes particular bellows
47
[ N D D A R ] = nondimensional dynamic amplitude ratio :
3
Subscripts p ra refers to proportional lus rate action p re refers to proportionafplus reset action sin denotes that forcing is sinusoidal
+ +
2 I
0 02
01
03
RECIPROCAL
OF
04
05
06
07
PROPORTIONAL
08 BAND
09
IO
I!
SETTING,
,&
12
13
14
= reference nondimensional performance function;
subscripts same as [ N D D A R ]
= differential operator, d/dt (in special cases may
Figure 18.
Dependence of feed-forward sensitivity and Sfi ( P B ) on reciprocal proportional band setting
ments are impossible, a smooth curve is not expected. However, if the setting assumed t o correspond to 75% were 70% and that for 150% taken as 135%, a fairly smooth curve is found which probably represents the true variation of S f f ( P B ) as a function of l / ( P B ) with moderate accuracy. Figures 12, 17, and 18 along with the static sensitivities of the feed-back bellows give all information needed to predict both the static and dynamic performance of the pneumatic components of this controller for any setting of the adjustments in the frequency range of practical interest and when operating so that no pneumatic load is imposed on the output. Where the dynamic performance of the process to be controlled is known as well as the influence of loading on the controller it is possible to select the controller settings which mill confer upon the entire closed loop system the most desirable over-all dynamic behavior using a controller of this type.
Acknowledgment The procurement of the experimental data was made possible by virtue of a grant in aid from the National Science Foundation for sustenance of one of the authors (Joel 0. Hougen) during the period June 15 to Sept. 15, 1955. The recording oscilloscope was purchased from funds supplied by the Rensselaer Polytechnic Institute Research Fund. The authors wish to express their thanks for this assistance.
Nomenclature The notation used follows the general philosophy described in reference ( 2 ) . I t is functional in character, the objective being to permit the reader to associate a self-defining symbol with the concept without the need of memorizing particular relationships. Although this type notation may appear a t first sight t o be somewhat cumbersome, its use has been well received by students and those studying the subject of systems engineering for the first time.
A [ c a - ~ ~ = angle between force summing lever and instrument case
A[i-b]
(bp)
(CT)
= angle between baffle and force summing lever = break point angular frequency = characteristic time:
Subscripts b indicates bellows ra indicates rate bellows re indicates reset bellows
June 1956
also represent the complex Laplacian operator) = pressure in droop correction bellows = nozzle back pressure
= output pressure = pressure in proportional bellows = proportional band setting, %
= performance function or ratio of output to input; = = =
= =
= =
= =
= = = =
= = =
=
= = =
subscripts same as [ N D D A R ] ; p c denotes pneumatic components radius of motion of point P around baffle axis (length of baffle) radius of motion of baffle at nozzle position around the baffle axis radius of motion of force summing lever a t point of contact with bellows around point 0 radius of motion of baffle axis around point 0 (length of force summing lever) radius of motion of baffle a t nozzle position around point P sensitivity of pneumatic amplifier for pressure in, to pressure out sensitivity of droop correction bellows for pressure in, to force out static sensitivity of pneumatic components for displacement of correction signal screw in, to prissure out static Sensitivity of mechanical system for displacement of pen in, to displacement of correction signal screw out static sensitivity of instrument for pen displacement in, to pressure out (an over-all sensitivity) sensitivity of the feed-back path for displacement of correction signal screw in, to pressure out sensitivity of the feed-forward path for displacement of correction signal screw in, to pressure out sensitivity of force summing lever for force in, to displacement of baffle axis out sensitivity of force summing lever for force in, to displacement out, a t point of force application by bellows sensitivity of nozzle-baffle system for displacement in, to pressure out sensitivity of proportional bellows for pressure in, to force out sensitivity ratio; sensitivity of droop correction bellows for pressure in, to force out, divided by sensitivity of proportional bellows for pressure in, to force out proportional band scale factor angular frequency, radians per unit time angles associated with complex numbers
Literature Cited (1) Bode, H. W., “Network Analysis and Feed-Back Amplifier Design,” Van Nostrand, New York, 1945.
(2) Draper, C. S.,McKay, Walter, Lees, S.,“Instrument Engineering,” vol. 11, chapt. 24, McGraw-Hill, New York, 1953. RECEIVED for review January 21, 1956.
INDUSTRIAL AND ENGINEERING CHEMISTRY
ACCEPTEDApril 16, 1956.
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