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AUTO-EXHAUST PPOPOftTIONAL SAMPLER
STABILITY ANALYSIS
Arthur J. Kline
Submitted 10/26/64
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Discussion
This report presents the results of a stability analysis
performed on the auto-exhaust proportional sampler control
system. The system functions to control the flow rate of a
sample of auto-exhaust gas proportional to the flow rate of
air into the engine carburetor. The exhaufit gas sample is
collected in a bag for later analysis in' the laboratory.
To control the sample flow rate a pressure transducer
senses the pressure across a laminar flow element attached
to the inlet of the carburetor and produces a d.c. voltage
directly proportional i.o the pressure drop. A second pressure
transducer is used to produce a d.c. voltage proportional
to the pressure across a laminar flow element in the sample
line. These two voltages are compared in the input circuit
of an electronic amplifier to produce an error voltage. The
error voltage is amplified and used to drive a motor which
positions a throttling valve in series with the laminar flow
element in the sample line. The throttling valve is positioned
in a direction to reduce the error to zero, thereby, main-
taining the pressure across the laminar flow element in the
sample line equal to that across the laminar flow element at
the inlet to the carburetor. The flow rate in the sample'line
is thus directly proportional to the.flow rate into the car-
buretor, the proportion depending on the relative size of the
two laminar flow elements.
The x>erformance characteristics of prime importance for
the sv stein include} steady state accuracy, speed of response
to transients, and steady state stability. For a clearer
und-arstanding of the factors which determine these character-
istics it is convenient, to represent the system in terms of
a functional block diagram as shown on figure 1.
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From this diagram it can be seen that the feedback loop
contains the amplifier, motor and gear train, throttling valve,
and the sample line laminar element and pressure transducer.
The carburetor laminar element and pressure transducer are
external to the loop and serve to form the reference input.
In general the steady state accuracy of the system depends
on the relative linearity of the two laminar flow elements
and their pressure transducers and on the magnitude of error
required to overcome friction in the motor and valve. To
minimize error due to friction the amplifier gain should be
high and the motor should have high stall torque characteris-
tics. For this system errors due to friction are negligible
since friction is low, amplifier gain is extremely high and
the stall torque characteristics of the motor are adequate.
In addition the friction is the same for both directions so
*•
that any small error wo.ild tend to average out.
The speed of response depends on the gain and dynamic
characteristics of. all the components and on the maximum rate
at which the flow area of the throttling valve can be changed.
The rate at which the flow area can be changed*depends on the
maximum speed of the motor, the gear ratio, and the relation
between flow area and angular rotation of the valve. For
fast response the characteristic.delays and lags of the com-
ponents should be small and their gains high, especially the
gain from motor speed to rate of change of valve area. Thus
for both fast response and good accuracy the gain of the com-
ponents should be high, however, this can lead to instability.
In general instability in a system is caused by the trans-
mission of a signal around a loop in such a manner and mag-
nitude so a? to reinforce the original signal and cause it to
build in magnitude. Although as shown on the diagram, the
feedback is negative and «=icts to reduce the original signal,
the characteristics of tht components within the loop cause
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the signal to be shifted in time. For a sine wave or oscilla-
tion this results in a phase shift. A phase whift of 180° is
equivalent to a reversal of sign so that for this condition
the feedback becomes positive and acts to reinforce the
original signal, If the loop gain is equal to or greater
than one the feedback signal will be equal to or Creator than
the original signal and oscillations will build up and be
sustained. This leads to the Simplified Nyquist stability
criteria, which states in effect, a negative feedback system
will be unstable and will oscillate at a frequency whe.ro the
phase ehift around th? loop is 180° if' the loop gain, at that
frequency is equal to or greater than ohe. Conversely a .
system will be stable if the loop gain is less than one at
the frequency whore the pha.de shift around the loop is 180°.
To provide adequate stability margin, a rule of thumb that is
frequently used is to design for a loop gain of one or slightly
less at a .frequency where the phase shift is 135 « This
corresponds to a stability phase margin of 4|> .
From the foregoing it is evident that a high loop gain
is desirable for fast response and . good accuracy however the
maximum valve that can be used Is limited by ln.e rtiquirouionts
i'or stability. The analysis of' the proportional sampler thu*
resolves into four general steps:
1. Determination of the magnitude and phase characteristics
of the components within the feedback loop as a function
of frequency.
2. Determination of the variation in loop gain as a
function of environmental conditions.
3. Definition of maximum loop gain that will satisfy sta-
bility characteristics.
4. Distribution of loop gain among components to provide
minimum error and maximum speed of response.
The analysis, procedures used, arid results obtained are pre-
sented in the appendix.
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Conclusions and Recommendations
1. The proportional sampler can be stabilized and still
provide adequate accuracy and speed of response with-
out the addition of stabilization circuitry.
2. The amplifier gain is somewhat high, however, its
range of adjustment is adequate. ;
3. The 10 to 1 gear ratio should be used.
4. The relief valve on the tank whould be set to maintain
tank pressure at 3 in. of Hg below atmospheric. The
pump should have adequate capacity to hold the relief
valve open at maximum sample flow conditions.
5. The flow area of the throttling valve as a function
of angular displacement should be such that, when tested
with the sample line laminar flow element and with
tank pressure set at -3 in. of Hg, the slope of the
curve of pressure across the laminar flow element
plotted verses angular 'displacement of the valve is
.006 -10% ini of.HpO per degree at a prensurft 1 in.
of HpO across the .laminar flow element.
6. Transient testing should be performed on the final
system to demonstrate adequate recovery time following
a throttle chop.
?. Work still needs to be ,dcne to minimize the loading
effect", of the integrator or counter circuits on the
input circuit to the amplifier.
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Appendix
General equations representing the dynamic character-
istics of the components within the feedback loop can be
written as follows.
Amplifier Motor and Gear Train
The speed of the output shaft of the gear train is
proportional to the voltage at the input to the amplifier,
however, the speed will not change instantaneously with a
step change in voltage but will build up at a rate determined
by the viscous friction and electrical damping. This can
be represented by:
» - V - Tm t! <1>
N = output shaft speed, rev/s«c
K-, = amplifier and iiiotor steady state gain,
rev/sec/volt.
e = amplifier input volgage, volts.
T » motor time constant, sec.
t = time, sec.
Output shaft angular position can be represented by:
e = 360/*Ndt (2)
where 0 » output sh^ft angular position, degrees.
Throttling Valve arid LaiuinaT' Element in Rampl e Line
The flow through the sample line and thus the pressure
across the laminar element is proportional to the position
of the throttling valve. Because of the compressibility of
the gas and the volume in the syptern the pressure across the
laminar element will not change instantaneously with a change
in valve position.
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Thus:
P8 - K2e - TV s_ (3)
where P « pressure across the laminar flow element
in the sample line, in. fl^O
Kp a throttling valve and laminar element steady
state gain, in. HpO/ Degree.
T » gas-volume time constant, sec.
Sample Line Pressure Transducer
The output voltage of the pressure transducer is pro-
portions! to the pressure across the laminar flow element.-'
Thus :
where V * sample line laminar element pressure
transducer vpltage, volts
K* = pressure transducer steady state gain,
volts /in. HpO
Amplifier Input Circuit
The input voltage to the amplifier is the difference
between the voltages of the two pressure transducers.
Thus:
e . VR - Vs (5)
where Vr, s Carburetor laminar element pressure
K
transducer voltage, volts.
Equations 1 through 5 can be linearized and represented
in operational form to yield:
= Kl _ Ae (6)
V*1
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Tvp
(8)
AVS = K? APB (9)
Ae = AVR - AVS (10)
where A represents a email change in a variable
Equations 6 through JO can be, represented in block
diagram form as shown on figure 2,
The value of K-, was measured by disconnecting the valve
from the gear train output shaft and recording shaft speed
for various values of voltage (V^-V.,). These results are
n 8
plotted on figure 3 in terms of motor shaft speed versus
error voltage* K-, is the slope of this line divided by the
gear ratio. From figure 3 it can be seen that the value of
KI will depend on th^ magnitude of the input signal . For
stability purposes the effective gain can be approximated by
the slope of the dashed Line. .
Kp was determined by taking steady state data of valve
position, pressure across the valve, and pressure across the
laminar flow element. The valve was set at various angles,
at each an&le the flow in the sample line was restricted
various amounts by clamping the tube. At each angle, readings
of pressure across the valve and pressure across the laminar
element, wore taken for the various amounts of restriction.
The results are plotted on figure 4. Lines of constant
pressure across the s^mpl* line are superimposed on this greph.
Figure 5 is a cross plot of figure 4. The value of K0
can be determined from the slope of these lines at the
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operating point. It can be seen that the value of Kp will
depend on the pressure in the tank as well as the pressure
across the laminar flow element. Figure 6 shows the variation
in KO as a function of tank pressure at a flow corresponding
to a pressure' across the laminar element of 1 inch of water.
This figure shows that the gain of the valve increases as
tank pressure is reduced. Thus the tank relief valve setting
will have an effect on the stability of the system tending to
decrease stability for lower tank pressure settings.
The value of K^ was determinod by taking readings of
pressure across the laminar flow element and corresponding
readings of pressure transducer voltages. These results are
plotted on figure 7«
To determine the value of T a frequency response of
V versus valve position was run. This was accomplished by
oscillating the velve through an angle of about - 30\ degrees
by means of an external drive attached to the valve. \ The
valve was oscillated at various frequencies. At each ' frequency
a photograph of valve position and transducer voltage, V ,
\
displayed on a dual beam oscilloscope was taken. Thesq
1 \
photographs are shown on figure 8 along with the reduction
of the data. The value of T can be determined from this data
in the following 'manner. For steady state response to a
sinusoidal signal p can be replaced ty Jw where = 2nt
and $ ^ V-l . Thus equation 6 combined with equation 7
becomes
Only the last term in equation 9 need be considered to
determine T • The test data was plotted on figure 9- Using
straight line approximations to curve fit the data, T. was
found to be approximately .025 sec.
8
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The product of equations 6 through 10 yields the open
loop transfer function and describes the characteristics of
the components within the loop.
360 K, KP K,
The time constant of the motor T is the only, value
that was not determined by test, however, representative
values for a motor of this size and loading, range from about
.01 to .02 seconds.
Figure 10 shows a plot of the open loop transfer function,
(equation 11) with p replaced by ja) , for assumptions of T = 0,
.01, and .02 seconds. This plot is for the 10:1 gear train,
a tank pressure of -3 in» of Hg, and low gain throttling valve.
This curve indicates that the pjiase margin is from about 4-7°
to ^7° depending on the time constaht of the motor.
It should be noted that the value of K-, is a function of
the magnitude of the input signal. The value near null is
about 1/4 the effective gain for large signals. This hns
th$ effept of modifying the rule of thumb for 4-5° of phase
margin to a value of 15° to 20° provided the plot is based on
the maximum gain condition as is the case on figure 30. As
a consequence the gain of the present system can be increased
by a factor of 2 to nearly 3 without causing instability.
Figure 11 shows the response of the system to throttle
bursts and chops. These traces show that the recovery from
a chop is very poor. This can be explained as follows:
When the throttle is chopped V-o decreases about 1.6 volts in
K
.1 second. The error needt; to be only about .18 volts to be
sufficiently large to cause maximum motor speed. This takes
slightly less than .015 seconds. From figure 3 it can be seen
/
that the maximum speed the motor can attain. is, 2?. 3 rev. per
second even for error signals much larger than .18 volts.
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The maxirmm rate that V can be reduced is
(2?.3)(1/]0)(360)(.00205X1.6) = 3-22 volts /sec.
This is in clo?e agreement with the slope of the trace of V_
*""* S
foil owing the initial dip caused by the decrease in exhaust'
pressure immediately following the chop.
This can be improved considerably by increasing, the
gain of the valve sc that a given motor speed wi] 1 cause a
higher rate of change of flow area. If the gain of the valve
is increased by about 3, the recovery time will be reduced
to nearly 1/3 that indicated by the trace. According to
figure 10 this nuch increase in gnin might cause instability
if the motor time constant is as high as .02 seconds. However,
if the system is unstable or marginal, the gain of the amplifier
can be reduced the appropriate amount by adjustment without
degrading the speed of response.
10
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.5/3* mpL.ee.
P/ee-ssu&e
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FN-156 (8-50)
GENERAL ELECTRIC COMPANY. SCHENECTADY. N. Y.. U.S.A.
180x250 1mm DlvUiom
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180x250 Issm Division?
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KN-156 (8-50)
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3 eyc.x 170 1mm Dmnona
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