Controller Equations
THE BASIC PNEUMATIC CONTROLLER EQUATION
In the basic controller equation, the actual value of the controlled variable measured by a sensor is related to the setpoint value for the controlled variable, which is the midpoint of the controller pressure range. That difference in value between measured value and the set-point or midset-point value is multiplied by the controller sensitivity rate to give a pressure difference. This difference is added to or subtracted from the midpoint pressure to give the predicted output pressure value for the measured value. The same equations are used for the calculation of set-up parameters for other variables, such as relative humidity and static pressure.
The basic pneumatic controller equation is:
Pout = Psp± T1± SP1 °F TR1
× PR (4-2)
Where:
Pout = Output or branch pressure from the controller, psig.
Psp = Pressure at setpoint or the pressure corresponding to set-point temperature, 8 psig for 3 to 13 psig system or 9 psig for 3 to 15 psig system.
SP1 = Setpoint temperature, °F.
T1 = Measured temperature of the controlled medium, °F TR1 = Throttling range, °F, of the controller.
PR = Pressure Range of the controller as temperature changes through throttling range, 10 psig on 3 to 13 psig system or 12 psig on 3 to 15 psig system.
± = Sign for pressure change due to action; additive (+) for direct action and subtractive (–) for reverse action.
Assigning Throttling Ranges
Throttling ranges are generally assigned rather than calculated, because they are subject to adjustment during the control system set-up process to obtain a stable control system. The change in the controlled variable which will take place during the control process must be con-sidered in the assignment of a numerical value of throttling range to be set up on a controller.
If a space controller is to position a cooling only VAV at 75°F plus or minus 2°F, the throttling range might be assigned as 4°F. With a 4°F throttling range, a change in space conditions of 2°F warmer will cause the VAV actuator to be at full output at 77°F and a change in space conditions of 2°F cooler will cause the actuator to be at zero output at 73°F.
That throttling range will probably give a fairly stable and comfort-able system. If a space temperature controller is to position both cooling and heating control devices through their full ranges and the space
tem-perature setpoints have been established as 75°F on cooling down to 70°F on heating, a desired throttling range can be worked out as follows.
Because there is a 5°F difference between cooling and heating setpoints, which should be the midpoints of the respective cooling and heating control actuators, it can be seen that a throttling range of 10°F, or twice the difference between cooling and heating space temperature setpoints, will cause the cooling actuator to be at full output at 77.5°F and the heating actuator to be at full output at 67.5°F. That may turn out to be too wide a throttling range for occupant comfort but will give a very stable system.
Throttling Range and Proportional Band
A pneumatic controller with integral sensor has a throttling range adjustment. A pneumatic controller with a remote sensor does not have a throttling range adjustment, but is set up by use of a proportional band adjustment. Throttling range and proportional band are closely related.
The term “throttling range” is used with sensors having variable sensing ranges of sensed medium, while the term “proportional band” is used with sensors having fixed sensing ranges of sensed medium, which must be related to sensor pressure span.
The basic pneumatic controller equation used on controllers with integral sensors is modified for use on controllers with remote sensors to use PB as follows:
Pout = Psp± T1± SP1
PB × Span × PR (4-3)
Where:
PB = Proportional Band, % of sensor span.
Span = Sensor span, expressed in °F, % RH, or “ wc for output variation from 3 to 15 psig for 12 psig span system or from 3 to 13 psig for 10 psig span system.
Calculating Proportional Bands
The value of the PB is calculated for a specific sensor which is to be used on a controller with an assigned throttling range as follows:
PB = TR
Span × 100
(4-4)
Where:
TR = Throttling Range of Controller, expressed in °F, % RH, or psig.
Example of proportional band calculation. Consider a controller with an assigned throttling range of 4°F. Use Equation 4-4 to determine the PB in % when using a temperature sensor with a 100°F span.
PB = 4°F
100°F× 100 = 4%
The proportional band setting for a controller varies for sensors with different spans but the throttling range stays the same. When a 50°F sensor is used with the controller, the proportional band setting is:
PB = 4°F
50°F× 100 = 8%
If the sensor is changed to one having a span of 200°F, the proportional band setting becomes:
PB = 4°F
200°F× 100 = 2%
Calculating Controller Sensitivity
The sensitivity of the controller in this set-up on a 3 to 13 psig system is determined as follows:
Sensitivity =PR span
TR =10 psig
4°F = 2.5 psig/°F Output Pressure and Throttling Range
Take a controller with 3 to 13 psig or 10 psig output or branch pressure range, 78°F setpoint, 6°F throttling range, and a 3 to 13 psig range system. With the controller set at and sensing 78°F, the output pressure of a properly calibrated controller will be the midpoint value of 8 psig.
With a direct-acting (DA) controller, as the ambient temperature at the controller sensor drops through one-half the throttling range, or 78°F minus 3°F to 75°F, the controller output pressure will drop through one-half of its 10 psig pressure range to 3 psig. On warm-up, when the ambient temperature rises through one-half of the throttling range, or 3°F, to 81°F, the output pressure will rise through one-half of its pressure range to 13 psig.
Thus, as the temperature changes through 6°F from the bottom to the top of the throttling range, the output pressure of the controller will change through a 10 psig range from minimum to maximum. When a reverse-acting (RA) controller is used in the same example, the output pressure will be observed to be 13 psig at the lower end of the throttling range, and at the upper end of the throttling range the output pressure will be 3 psig.
For example, with the set-up described above, to determine the output pressure of a DA controller at 77°F, using Equation 4-2:
Pout= 8 psig +(77 ± 78)°F
6°F × 10 psig = 6.3 psig.
Pressure Response Beyond Throttling Range
If the measured value of the controlled variable increases or de-creases beyond the limits of the throttling range, then the output pres-sure will also change beyond the system prespres-sure range PR to drop to zero or to increase to the main supply air pressure.
For example, to predict the output pressure of the controller when measuring 85°F with the previous set-up parameters, using Equation 4-2:
Pout= 8 psig +(85 ± 78)°F
6°F × 10 psig = 19.7 psig.
Although the calculated output value is within the normal main air supply pressure range, if the controller air supply pressure is on a changeover system, with a daytime air supply of 13 or 18 psig, then the controller output cannot exceed the main air supply pressure. When the sensed medium value is outside the assigned throttling range limits, the control system has positioned the cooling or heating controlled devices to the limits of their operating ranges and the system is out of control.
Unless the control components are selected and the system is setup for higher pressures, any output pressure less than 3 psig or greater than 13 psig can do no controlling because the spring-compensated actuators move between fully retracted and fully extended as the branch pressure varies between 3 and 13 psig. From Equation 4-2, it can be predicted that, at 81°F, a 13 psig controller branch output pressure will occur, which will fully extend an actuator with a spring having an upper limit value of 13 psig.
Examples of Calculations
The following examples illustrate how to use the controller equa-tions and how to calculate the other variables needed for calibration of a controller.
PROBLEM 1
Take a single-input direct-acting controller used to control space temperature and set-up for 72°F setpoint with an 8°F throttling range assigned.
Using the controller equations predict values for the following parameters:
(a) Controller PB setting when remote sensors with spans of 50°F, 150°F, and 200°F are used.
(b) Controller pressure range.
(c) Controller setpoint.
(d) Controller output pressure at T = 72°F using the same sensors as in (a).
(e) Controller output pressure at T = 76°F.
(f) Predict the output pressure of the controller at T = 80°F when the main air supply pressure to the controller is 18 psig.
(g) Predict the output pressure of a direct-acting controller at T = 71°F on a 3 to 15 psig system.
(h) Predict the output pressure of a reverse-acting controller at T =
75°F on a 3 to 15 psig system with the same throttling range and setpoint.
PROBLEM 2
Take a single-input reverse-acting control used to control space relative humidity and set-up for 30% RH setpoint with a 10% throttling range assigned. The system pressure range is from 3 to 13 psig. Predict the following:
(a) Controller PB setting of the controller with humidity sensor having 15% to 75% range.
(b) Controller output pressure, when space relative humidity is 35%.
(c) Space relative humidity when controller output pressure is mea-sured to be 12 psig.
Solutions to Problems PROBLEM 1
(a) Proportional Band for various sensor spans:
For sensor with 50°F span , PB = 8°F TR50°F = 16%.
For sensor with 150°F span , PB = 8°F TR
150°F = 5.3%.
For sensor with 200°F span , PB = 8°F TR 200°F = 4%.
(b) Range is 13 psig minus 3 psig = 10 psig.
(c) The controller setpoint pressure is the midpoint pressure. Add 3 psig and 13 psig, or 16 psig, which, when divided by 2, equals 8 psig.
(d) The controller output pressure remains the same for all the
differ-ent sensor spans.
Pout= 8 +(72°F ± 72°F)
8°F × 10 = 8 psig.
(e) Pout= 8 +(76°F ± 72°F)
8°F × 10 = 13 psig.
(f) Pout= 8 +(80°F ± 72°F)
8°F × 10 = 18 psig.
(g) Pout= 9 ±(71°F ± 72°F)
8°F × 12 = 7.5 psig.
(h) Pout= 8 ±(75°F ± 72°F)
8°F × 10 = 4.25, say 4.3 psig.
PROBLEM 2
(a) PB = 10%
(75% ± 15%)= 16.7%
(b) Pout = 8 ±(35°F ± 30°F)
10° TR × 10 psig = 13 psig.
(c) Space relative humidity is the unknown in this problem. Solve the equation for % RH.
Pout = 8 ±RH% ± 30%
10% TR × 10 psig = 12 psig.
Solving the above equation, we find that RH = 34%.