What Is Square-Root Extraction?
Square-root extraction, often abbreviated SRE, is a mathematical operation used to convert a differential pressure measurement into a value proportional to flow.
The reason is simple: differential pressure is proportional to the square of flow.
Therefore, to obtain flow from differential pressure, we take the square root:
This relationship is fundamental to many differential-pressure flow measurements.
Why Does Differential Pressure Measure Flow?
One common way to measure flow is to place a restriction in the pipe. Examples include:
- Orifice plates
- Venturi tubes
- Flow nozzles
- Other differential-pressure primary elements
As fluid moves through the restriction, a pressure difference develops between the upstream and downstream sides.
The amount of differential pressure depends on the velocity of the fluid. Since flow rate is related to velocity, the resulting relationship is approximately:
That means doubling the flow does not simply double the differential pressure.
If flow doubles, the differential pressure increases approximately four times.
If a system produces 25 units of DP at one flow condition, approximately 100 units of DP would correspond to twice the flow, assuming the same operating conditions and idealized relationship.
The Differential Pressure / Flow Relationship
The basic relationship can be expressed as:
where:
- Q = flow
- K = constant determined by the measurement system
- DP = differential pressure
The constant depends on the primary element, pipe, fluid, geometry, units, and other characteristics of the measurement installation.
For PLC and instrumentation troubleshooting, however, we often work with percentages rather than calculating the complete flow equation from the physical installation.
From Differential Pressure to 4–20 mA
A DP transmitter measures the pressure difference and converts it into an electrical output.
There are two important possibilities.
DP transmitter with linear output
The transmitter can produce a 4–20 mA signal proportional to differential pressure.
In this case, the PLC receives a signal representing DP. If the application requires flow, square-root extraction must happen somewhere downstream.
Transmitter configured for flow output
Some transmitters can perform the square-root operation internally.
In this configuration, the output can already be proportional to flow.
The Square-Root Extraction Formula
For a normalized signal, first determine the percentage of the 4–20 mA span:
Then take the square root:
If the flow range is 0 to 100%, the result can be expressed as:
For a general engineering range:
Worked Example: 0–100 GPM
Suppose a DP-based flow measurement has been configured for a flow range of:
- LRV = 0 GPM
- URV = 100 GPM
- Measured signal = 12 mA
Step 1: Calculate signal percentage
The signal is therefore at 50% of its electrical span.
Step 2: Take the square root
Therefore the flow is approximately 70.71% of its configured range.
Step 3: Convert to GPM
A 12 mA signal represents 50% signal span, but approximately 70.7% flow when square-root extraction is required.
This is one of the most important concepts to understand when troubleshooting DP flow measurements.
4–20 mA Square-Root Reference Table
For a 0–100% DP range, the following values illustrate the difference between signal percentage and flow percentage.
| Current | Signal % | Flow % After SRE |
|---|---|---|
| 4 mA | 0% | 0% |
| 8 mA | 25% | 50% |
| 12 mA | 50% | 70.7% |
| 16 mA | 75% | 86.6% |
| 20 mA | 100% | 100% |
Notice how the flow percentage increases rapidly at the low end and gradually approaches 100% at the high end.
Where Should Square-Root Extraction Happen?
There is no single location where SRE must always occur. It depends on the system design.
Inside the transmitter
The transmitter can measure DP, perform the square-root calculation, and produce a 4–20 mA output proportional to flow.
Inside the PLC
The transmitter can instead send a linear DP signal. The PLC then scales the signal and performs SRE.
Inside a flow computer or DCS
Some systems perform the calculation in a dedicated flow computer, DCS function block, or another control system component.
The important question is: "Where is SRE happening in this particular system?"
The Double Square-Root Problem
One of the easiest ways to create a bad flow reading is to perform square-root extraction twice.
For example, suppose the transmitter is already configured to output a signal proportional to flow. The PLC receives that signal and then applies another square root.
The error can be especially noticeable at lower flow rates.
If the transmitter already produces a signal representing 50% flow, treating that signal as though it were still a linear DP signal and taking another square root changes the result to:
The PLC would therefore report approximately 70.7% instead of the intended 50%.
When troubleshooting an unexpectedly high flow reading, always determine whether SRE has already been performed upstream.
Low-Flow Behavior
Square-root extraction has a very steep response near zero.
Mathematically:
So only 1% of the DP span corresponds to 10% of the calculated flow span.
This makes low-level noise and small measurement errors more significant after square-root extraction.
Low-flow cutoff
Many control systems use a low-flow cutoff or zero suppression function so that very small DP signals do not produce an unstable or unrealistic flow value.
The exact cutoff is application-specific and should not be assumed to be the same for every system.
Square-Root Extraction in the PLC
When SRE is performed in a PLC, the general sequence is usually:
- Read the analog input.
- Convert raw counts to a signal value or percentage.
- Determine the normalized signal fraction.
- Apply the square root.
- Scale the result into engineering units.
- Apply any required limits, filtering, or low-flow logic.
Conceptually:
The exact implementation depends on the PLC platform and analog module configuration.
Troubleshooting DP Flow Measurements
Flow is too high
Check:
- Whether SRE is being performed twice
- Transmitter configuration
- PLC SRE logic
- Transmitter LRV and URV
- Analog input scaling
Flow is too low
Check whether square-root extraction is missing. Also verify the transmitter range and analog scaling.
Flow is unstable at low rates
Investigate:
- DP signal noise
- Transmitter damping
- Impulse-line problems
- Low-flow cutoff configuration
- PLC filtering
- Square-root implementation
Flow suddenly changed after transmitter replacement
Compare the configuration of the old and new transmitters.
In particular, check whether the old transmitter was configured for square-root output while the replacement is configured for linear DP output, or vice versa.
PLC value doesn't agree with the transmitter display
Trace the complete signal path rather than immediately changing the PLC calculation.
Common Mistakes
- Applying SRE to a signal that is already linearized to flow.
- Forgetting SRE when the PLC receives linear DP.
- Applying square-root extraction twice.
- Taking the square root of the raw mA value instead of the normalized signal percentage.
- Assuming every flow transmitter uses differential pressure.
- Assuming 12 mA always means 50% flow.
- Ignoring transmitter configuration.
- Ignoring the analog input module's actual raw-count range.
- Treating low-flow noise as a scaling problem when the underlying issue is the measurement itself.
A Practical Field Troubleshooting Workflow
When a DP flow measurement doesn't look right, work through the signal chain one stage at a time.
- Identify the measurement technology. Determine whether the flow measurement is based on differential pressure.
- Check the transmitter configuration. Verify the pressure range, output range, units, and whether square-root extraction is enabled.
- Measure the actual current. Verify the 4–20 mA signal with appropriate test equipment and procedures.
- Verify the analog input. Determine what raw value the PLC is actually seeing.
- Verify scaling. Confirm the PLC converts the raw input to the expected signal percentage.
- Find the SRE. Determine whether the square root is being performed in the transmitter, PLC, DCS, flow computer, or another device.
- Check low-flow logic. Look for filtering, cutoff, limiting, or other signal conditioning.
- Compare the final value. Compare the PLC/HMI flow with the expected process condition and any independent measurement.
Flow & Scaling Calculators
Use the ControlsCalc calculators to work through the numerical side of the measurement.