Fluid signal square root extractor

A square root signal extractor generates an output signal which is a square root function of an input signal. First and second diaphragm amplifiers have cascaded output chambers connected to a high gain output amplifier. The first and second amplifiers are connected as the non-inverting input and the inverting input respectively of the operational amplifier. A high gain output diaphragm amplifier is connected to the inverting amplifier output. The output diaphragm amplifier is constructed such that only a low level input pressure change is required to create a full swing of the output pressure. A non-linear input restrictor and a linear feedback resistor are connected to the input chamber of the inverting amplifiers. The feedback resistor has a significantly greater resistance than the input resistor. The non-inverting diaphragm amplifier is connected to the transmitter by a fluid repeater. A reference signal source is connected to the input restrictor and thus to the inverting input. The input restrictor is selected such that the differential pressure across the restricter is the square root of the flow. An adjustable fluid resistor is connected between the non-inverting and the inverting inputs which creates a flow path between the transmitter and reference source. The fluid repeater prevents loading of the transmitter.

BACKGROUND OF THE INVENTION 
This invention relates to a fluid signal square root extractor and 
particularly to such an extractor operable to extract the square root of a 
flow-related pressure signal for producing an output pressure signal which 
is a linearized characteristic of pressure versus flow rate. 
Various pneumatic and other fluid flow systems develop signals which are 
the square of the measured quantity or condition. For example, in the 
heating ventilating and air conditioning art, variable air volume systems 
are finding wider usage in the control of the environmental air for 
enclosed spaces. In such systems, outside air and recirculated air are 
selectively mixed, conditioned and exhausted to maintain room temperature. 
The control responds to demand changes by varying of the air volume and 
maintaining the necessary temperature condition. The controls for such 
systems often are based on sensing the air velocity in the supply and/or 
exhaust ducts. Conventional velocity pressure sensors and transmittors 
such as pilot tubes are widely used. Such sensors generate a differential 
pressure signal which is the square function of the air velocity. 
Pneumatic differential pressure transmittors generally have a highly 
linear output over the transmitter span, and thus transmit the nonlinear 
velocity pressure signal versus velocity. The necessary summing, comparing 
and similar processing such non-linear pressure signals is difficult 
because of the nonlinear relationship. Therefore, it is desirable to 
linearize the velocity scale such that direct and simple signal processing 
of the velocity related signals with other signals is possible. Although 
various square root extractors have been suggested, the devices have 
normally relied on mechanical systems requiring careful processing and 
quality control, and are generally relatively expensive. There is 
therefore a need for a simple, reliable and relatively inexpensive square 
root extractor for processing of fluid signals such as pneumatic signals 
encountered in heating, ventilating and air control conditioning 
equipment, as well as other fluid control systems. 
SUMMARY OF THE PRESENT INVENTION 
The present invention is particularly directed to a fluid circuit for 
converting of an input signal to a corresponding related square root 
output signal. Generally in accordance with the present invention, a 
plurality of interconnected and interrelated fluid signal amplifying 
devices are interconnected into a cascaded circuit to define a high gain 
operational amplifier having a non-linear passive network which develops 
an output signal which is a square root function of an input signal. The 
system preferably employs diaphragm amplifiers with suitable input and 
feedback resistors to create an economical square root extractor while 
maintaining accurate translation of the input signal, such that the system 
is economically produced, and adapted to installation and maintenance 
based on standard skills in the fluid art. Generally, in accordance with 
the present invention, first and second diaphragm amplifiers having 
cascaded output chambers are connected to a high gain output amplifier to 
create an operational amplifier. The feedback resistor has a greater 
resistance than the input resistor. A suitable non-linear input restrictor 
means and a linear feedback resistor are connected to the input chambers 
of the cascaded amplifiers. The first non-inverting input diaphragm 
amplifier connected to the transmitter is constructed as a one-to-one 
fluid repeator such that the output signal is a duplicate of the input 
signal. The inverting input operational amplifier is a proportional high 
gain switch having its input chamber connected to reference pressure. The 
output chamber is connected in series with the output of the direct acting 
diaphragm amplifier, and is connected as the input to a high gain output 
stage which is preferably a diaphragm amplifier constructed to function as 
a repeater. The output gain stage is specifically selected and constructed 
such that only a low level input pressure is required to create a full 
pressure swing at the output of the high gain stage but with the output 
signal directly proportional to such low level pressure signal. 
The inverting switch amplifier remains closed and prevents transmission of 
an output until the level of the input signal rises above that of the 
reference signal. Thereafter a proportionally related output signal is 
transmitted which is a square root function of the input signal as the 
result of the nonlinear input-feedback network. In a practical 
implementation of the invention, the input signal is preferably coupled to 
the circuit through a fluid repeater. An adjustable fluid resistor is also 
connected between the non-inverting and the inverting inputs. The latter 
resistor provides a flow path between the input signal input and the 
reference source. A fluid repeater is preferably interposed between the 
input signal source and the operational amplifier input unit to prevent 
loading of the transmitter and the differential pressure signal. 
The reference source preferably includes a fluid diaphragm regulator and 
diaphragm amplifier system such as shown in U.S. Pat. No. 4,199,101, 
MULTIPLE LOAD INTEGRATED FLUID CONTROL UNITS, filed on Jan. 26, 1979 by 
Scott B. Bramow et al and assigned to the same assignee as this 
application. The regulator is adjustable for establishing zero offset and 
provides a stable reference signal. 
The input side of the output gain stage is also connected by a restrictor 
to ground to return the output to zero pressure in response to closing of 
the non-inverting or reverse acting diaphragm amplifier. 
The present invention, particularly in the preferred construction, produces 
a square root output signal function using known fluid devices in an 
economical apparatus while maintaining a high degree of accuracy such as 
required for usage in the commercial practice.

DESCRIPTION OF THE ILLUSTRATED EMBODIMENT 
Referring to the drawing and particularly to FIG. 1, a schematic circuit of 
a preferred construction of a square root extractor constructed in 
accordance with the teaching of the present invention is schematically 
shown at 1. The extractor 1 is shown as part of a variable air volume 
control system 2 for monitoring and controlling of air supplied to an 
enclosed area or space 3. The variable air volume control system includes 
a supply duct 4 adapted to supply tempered or conditioned air 5 to the 
conditioned space. A corresponding volume or air is exhausted through 
exhaust duct 6. The air 5 supplied to the conditioned space 3 via the 
supply duct 4 is drawn from the exterior via an outdoor air input duct 7 
and/or a recirculation of the exhaust air from the exhaust duct 8. Exhaust 
air not recirculated is discharged via an output duct 9. The control 2 
includes various means responsive to the velocity of the air in the supply 
and/or the exhaust duct. In the variable air volume systems, for example, 
the air flow may be readjusted on the basis of the supply air demand, the 
flow in supply duct 4 and/or the exhaust air flow. For purposes of 
describing the present invention, a velocity responsive pressure 
transducer 10 and a transmitter 11 is shown associated with the exhaust 
duct 6 to develop a pressure signal related to the velocity of the supply 
air. The transmitter 11 is connected by the square root extractor 1 to a 
controller 12, the output of which is connected in any suitable manner to 
control the temperature and volume of air supplied to the enclosed space 
3. The conditioned air supplied to the space 3 may be similarly monitored 
by a similar sensor 10a associated with the supply duct 4 and connected 
via transmitter 11a and extractor 1a to the controller 12. The present 
invention is particularly directed to the square root extractor. The other 
components will be readily understood by the ordinary worker in the art 
and are therefore only described in such detail as necessary to the full 
and clear description of the square root extractor. 
The transducer 10 may be a well known Pitot tube connected to a suitable 
pressure signal transmitter 11. The output is a differential pressure 
signal which is a square function of the fluid velocity in the duct 7. 
Over the operating range of the transmitter 11, the output is linear. For 
example, in a practical heating, ventilating and air conditioning system, 
the output pressure signal will vary with flow over a range of three to 
fifteen PSI (pounds per square inch). The output pressure signal, however, 
changes as a nonlinear square function with velocity such as shown at 13 
in FIG. 2. The sensed transmitted signal to the control unit 11 is 
desirably modified to produce a linear straight line relationship with 
velocity as shown at 14 in FIG. 2. The square root extractor 1 of the 
present invention serves to convert the output signal 13 of the 
transmitter 11 to a related square root output signal 14 and thus a 
linearized transform of the curve 13, as presently described. 
In the illustrated embodiment of the invention, proportional fluid 
switching and relay devices are interconnected with passive resistance 
devices to produce an operational amplifier 15, the output of which is the 
square root function of the input. Generally, in accordance with a 
teaching of the present invention, a direct acting non-inverting diaphragm 
amplifier 16 is cascaded with an inverting or reversed acting diaphragm 
amplifier 17 to define the noninverting and the inverting inputs of the 
fluid operational amplifier 15. The output of the inverting amplifier 17 
is connected to an output gain stage, also shown as a diaphragm amplifier 
18. Amplifier 18 is constructed to respond to a very low level input 
pressure change and establish a full pressure change in output which is 
directly proportional to the applied signal input signal. A fluid feedback 
resistor 19, which is larger than the input resistance as subsequently 
discussed, is interconnected between the output of the high gain stage 18 
and the input to the inverting amplifier 17. A reference signal source 20 
is connected to the inverting input in series with a non-linear input 
restrictor 21. The non-linear input restrictor 21 is selected to produce a 
pressure signal which is the square root of the flow in the restrictor, as 
more fully developed hereinafter. Signal source 20 is preferably 
adjustable to supply a regulated reference and which permits adjusting the 
system for a zero offset. The reference pressure source 20 is shown 
connected to the input of the inverting amplifier 17 in series with the 
input restrictor 21 and may be set for adjusting the offset of the 
operating response characteristic with a zero input signal from the 
transmitter 11. This permits maintaining a predetermined positive output 
signal in the presence of a zero input signal. As shown in FIG. 2, the 
system has been set for an offset of 3 PSI for a practical installation. 
The transmitter 11 is coupled to the direct acting input in series with a 
flow isolating repeater 22 and a resistor 23. A span adjusting resistor 24 
is preferably connected between the input resistor and the input of the 
non-inverting amplifier 16. The resistor 24 inserts a dead band within 
which the signal pressure may vary without change in the output. The 
resister 24 also establishes a flow path to the reference source 20, which 
is however isolated from transmitter 11 by the repeater 22 to prevent 
undesirable loading of the transmitter because such loading could 
adversely effect the linear output of the transmitter. 
The direct acting amplifier 16 thus operates to transmit the level of the 
input signal to the inverting amplifier 17 which acts as a proportional 
high gain switch. The output of the inverting amplifier 17 is positive as 
long as the direct acting or non-inverting input signal exceeds the 
reference signal level applied to the inverting input amplifier 17. If the 
positive input drops below that of the inverting input, the amplifier 17 
closes to prevent any further transmission of an output signal. 
If the cascaded signal from the amplifier 16 is larger than the reference 
signal applied to the input chamber of the inverting amplifier 17, a 
corresponding proportional output signal is applied to the high gain 
output stage 18, which in turn provides a proportional output signal. The 
linear resistor 19, and the non-linear input 21, with the relative higher 
resistance of the linear feedback resistor 19, results in an output signal 
which is the square root of the input signal. 
The pressure characteristic of the system is generally shown in FIG. 2. The 
output signal of the transmitter is a squared curve 13 of the transmitter 
pressure input signal versus the velocity which varies in the range or 
span between a positive 3 PSI and 15 PSI. Thus, the output pressure varies 
as the square of the velocity. For system pressure in the controller 12, a 
linearized output pressure vereus velocity characteristic such as shown at 
14 is desired. The linearized curve 14 for the offset of 3 PSI, is defined 
by the equation: 
##EQU1## 
The input flow to the reference source is the square root of the 
differential input pressure; or 
##EQU2## 
while the feedback flow is directly proportional to the feedback pressure 
or Q.sub.2 =K.sub.2 P.sub.2. These two flows are equal and of an opposite 
sense relative to the inverting input to the operational amplifier. The 
differential output pressure .DELTA.P.sub.T, equals the sum of the two 
differential pressures which establishes the final equation; 
EQU .DELTA.P.sub.T =P.sub.1 +K.sub.1 /K.sub.2 .DELTA.P.sub.1, 
which reduces to the above equation. Thus, with the illustrated operational 
amplifier which has a high gain and wherein the feedback resistance is 
greater than the input resistance, the characteristic is essentially and 
for practical application a function of only the input resistances and the 
feedback resistance. For example, in a practical system, the input 
resistance changer over the 3 to 15 PSI signal, the input resistance 
changed from approximately 0.2 to 0.45 resistance which in relation to the 
feedback resistance of 1.18 gives a ratio of approximately 2 to 6. The 
ratio may of course vary and suitable resistors and orifices can be 
readily determined. The constant K.sub.1 and K.sub.2 are slightly 
different but with the feedback resistance larger than the input 
resistance, a single constant produces a practical output characteristic 
or response. The output is therefore a linearized translation of the 
transmitter signal. 
Further, because of the flow through the feedback system to the reference 
signal source, the resistance of restrictor 21 should be substantially 
larger than the system input line resistance in order to avoid any 
possibility that signal line resistance will effect the set point or 
closed loop gain of the operational amplifier. 
More particularly, in the illustrated embodiment of the invention the input 
amplifiers 16 and 17 are preferably constructed as convoluted diaphragm 
devices such as more fully disclosed in U.S. Pat. No. 3,662,779. Referring 
particularly to the direct acting input amplifier 16, a convoluted 
diaphragm 27 is secured between opposing body members 28 and 29 to define 
a closed input chamber 30 and an output chamber 31. The input chamber is 
connected to the fluid signal transmitter 11 in series with the dropping 
resister 23 and the fluid repeater 22. The output chamber 31 includes an 
orifice 32 which is connected to ground or reference atmosphere, as shown 
at 33. The orifice 32 is located in slightly spaced and opposed relation 
to the central portion of the diaphragm 27. As the diaphragm 27 moves 
toward the orifice 32, it serves to effectively close the orifice and 
reduce the flow to atmosphere, thereby increasing the pressure in chamber 
31 to balance the pressure in the input chamber 30. If the diaphragm 27 
moves into complete engagement with the orifice 32, it of course closes 
the orifice and transmits maximum or full supply pressure to the output 
line. As the diaphragm 27 moves away from the orifice, it opens the 
orifice and in the opposite extreme position, fully opens the orifice and 
connects the chamber 31 essentially to ground bypassing of the signal to 
ground and establishing a zero output. The intermediate positioning of the 
diaphragm provides for a corresponding proportionate output signal. The 
diaphragm amplifier 16 is constructed to function as a fluid repeater with 
the output signal an accurate one-to-one transform of the pressure at the 
input chamber 30. The input or supply port to chamber 31 is connected to a 
suitable air supply such as a common manifold 34 having a dropping 
resistor 35. The output port to chamber 31 is connected by a series 
resistor 36 to the output chamber of the inverting amplifier 17. 
The inverting amplifier 17 and the output stage 18 are constructed as 
similar diaphragm amplifiers. Amplifier 17 functions as a proportional 
high gain switch which produces an output which is proportional to the 
difference in reference pressure in input chamber 38 and the 
condition-related pressure in the output chamber 39 as received from the 
direct acting amplifier 16. The diaphragm amplifier 18 is specially 
constructed to respond to a low level input pressure signal and to a 
established full pressure swing of its output in direct proportion to the 
output signal. 
A midpoint bleed or by-pass resistor 40 is connected from ground 33 to the 
connection between the inverting amplifier 17 and the high gain amplifier 
18. When the inverting amplifier 17 closes, the pressure in the input 
chamber of the high gain stage 18, which is also closed chamber, should be 
removed. Resistor 40 is an adjustable pin valve or the like which relieves 
the input chamber pressure and thereby returns the output signal to a zero 
output pressure. 
The illustrated reference source 20 is similar to that disclosed in the 
previously identified United States Patent application and includes a 
spring loaded pressure regulator 41 having an output connected to the 
input chamber of a fluidic repeater 42 to provide a corresponding output 
reference signal connected by the input restrictor 21 to the inverting 
amplifier 17. The regulator 41 is a spring-loaded leak-port diaphragm unit 
41 having a diaphragm 43 which is preloaded by an adjustable spring 44 to 
close an orifice in an output chamber 45. A supply port is connected to 
the supply through a suitable manifold unit 45a and an output port is 
connected to the input chamber of fluid repeater 42. The supply pressure 
builds in the output chamber 45 to the level necessary to balance the 
spring force at which point the leakage or bleed to ground through the 
orifice reduces the pressure to the level of the spring. The diaphragm 
regulator 41 produces a closely regulated pressure. The output pressure is 
applied to the input side of the fluid repeater through a pressure 
dividing resistance network. In the illustrated embodiment of the 
invention, a pair of resistors 46 and 47 of relatively large resistance 
are selected to drop the regulated 9 PSI to 3 PSI which is transmitted by 
the fluid repeater 42 to the operational amplifier 15. The resistor 
network not only serves to drop the pressure to the desired reference 
level but also serves to increase the sensitivity of the regulator 
adjustment to provide an accurate reference signal input to the 
operational amplifier 15. 
As noted previously, fluid flow exists between the input network including 
the reference source of the transmitter 11. The regulator as such is 
generally somewhat more sensitive to loading than the repeater. The 
repeater thus isolates the sources such that slight changes in the 
reference flow does not result in a practical detectable change in the 
regulated pressure supplied to the input of amplifier 17. 
The present invention, by use of appropriate diaphragm units interconnected 
into an operational amplifier configuration and with the suitable 
non-linear passive regulator network, generates an output signal which is 
a square root function of the input signal. The input signal which is the 
square of the flow and is thereby converted to a corresponding straight 
line function 14. 
The active components employed are known devices which are commercially 
available with linear and accurate characteristics. The passive elements 
are also known commercial elements having the desired linear and 
non-linear characteristics which can be readily constructed to produce the 
necessary accuracy. The device therefore provides a particularly practical 
implementation and application for developing a square root conversion of 
an input signal in an inexpensive and commercially producible product. 
Various modes in carrying out the invention are contemplated as being 
within the scope of the following claims, particularly pointing out and 
distinctly claiming the subject matter which is regarded as the invention.