A differential pressure transmitter is configured for:
0...100 mbar
and provides:
4...20 mA
to a PLC.
At 25 mbar differential pressure, the PLC indicates 25% flow.
Is that correct?
For a conventional differential-pressure flow measurement using an orifice plate, nozzle or Venturi tube, normally not.
The reason is the physical relationship between differential pressure and flow. In simplified form:
Q ∝ √Δp
.
Flow therefore does not increase linearly with differential pressure. 25% of the maximum differential pressure already corresponds to 50% of the maximum flow.
For a 4…20 mA signal to be directly proportional to flow, a square-root function must therefore be applied at exactly one point in the signal chain.
This can be done:
- in the differential pressure transmitter,
- in the PLC,
- in the process control system,
- in a separate evaluation device.
The decisive point is:
The square-root function must not be omitted – and it must not be applied twice.
This is why commissioning errors can occur even though the differential pressure transmitter, PLC input and 4…20 mA current loop are each functioning correctly on their own.
Why Flow and Differential Pressure Are Not Linearly Related
In differential-pressure flow measurement, a primary element such as an:
- orifice plate,
- flow nozzle,
- Venturi nozzle,
- Venturi tube
is installed in the pipeline.
The reduction in cross-section locally increases the flow velocity. At the same time, the static pressure changes.
The differential pressure between the high-pressure and low-pressure sides is measured by the differential pressure transmitter.
Under constant process conditions, the simplified relationship is:
Δp ∝ Q²
.
Therefore:
Q ∝ √Δp
.
For normalized flow, this can be written in simplified form as:
Q / Qmax = √(Δp / Δpmax)
.
This relationship is the reason why the electrical output signal must not simply be scaled linearly from the measured differential pressure to flow.
What Does Square-Root Extraction Mean?
In this context, square-root extraction means:
calculating the square root
.
The measured differential pressure is mathematically processed so that the result becomes proportional to flow.
For example:
Δp = 25%
.
Then:
Q = √0.25 = 0.5
therefore:
Q = 50%
.
At:
Δp = 50%
the result is:
Q = √0.50 ≈ 0.7071
therefore:
Q ≈ 70.7%
.
Only at:
Δp = 100%
do the differential-pressure and flow percentages coincide again:
Q = 100%
.
Option 1: Transmitter Provides Linear Differential Pressure
The differential pressure transmitter can be configured so that its 4…20 mA output is linear to differential pressure.
For example:
0 mbar = 4 mA
100 mbar = 20 mA
.
Then:
25 mbar = 8 mA
.
These 8 mA represent:
25% differential pressure
but:
50% flow
.
In this case, the PLC must not simply scale the signal linearly to the flow range.
It must first normalize the differential-pressure value and then calculate the square root.
The signal chain is:
differential pressure → linear 4...20 mA signal → square-root extraction in PLC → flow
.
This configuration is useful when the unprocessed differential pressure is also required as a process value or when the complete flow calculation is deliberately performed centrally in the control system.
Option 2: Transmitter Provides Square-Rooted Flow
Many intelligent differential pressure transmitters can perform the square-root function internally.
The transmitter still measures the actual differential pressure, but processes it internally using the square-root function.
The 4…20 mA output is then linear to flow.
For example:
0% flow = 4 mA
50% flow = 12 mA
100% flow = 20 mA
.
For a measuring range of:
0...1,000 m³/h
this means, for example:
12 mA = 500 m³/h
.
The PLC then only needs to scale this signal linearly.
The signal chain is:
differential pressure → square-root extraction in transmitter → linear 4...20 mA flow signal → PLC
.
In this configuration, no additional square-root function may be active in the PLC.
Comparing Linear and Square-Rooted 4–20 mA Signals
| Differential Pressure | Flow | 4…20 mA Linear to Δp | 4…20 mA Square-Rooted / Linear to Q |
|---|---|---|---|
| 0% | 0% | 4.00 mA | 4.00 mA |
| 6.25% | 25% | 5.00 mA | 8.00 mA |
| 25% | 50% | 8.00 mA | 12.00 mA |
| 50% | 70.71% | 12.00 mA | 15.31 mA |
| 100% | 100% | 20.00 mA | 20.00 mA |
The table illustrates an important point:
Only at 0% and 100% are the linear differential-pressure signal and the square-rooted flow signal identical.
A test limited to:
4 mA
and:
20 mA
can therefore completely miss an incorrectly configured signal chain.
At least one intermediate point is required for a functional test.
Why 25% Differential Pressure Means 50% Flow
This relationship is particularly useful for a quick plausibility check.
At:
Q = 50%
the following applies:
Δp = Q²
therefore:
Δp = 0.5² = 0.25
.
Only:
25% of the maximum differential pressure
is therefore required for 50% flow.
With a measuring range of:
0...100 mbar
this means:
25 mbar = 50% flow
.
If the transmitter is already configured with square-root extraction, its current output at 25 mbar should therefore be approximately:
12 mA
.
If the output is linear to differential pressure instead, the correct value would be:
8 mA
.
This makes it possible to identify the configured characteristic very effectively using just one suitable intermediate point.
Programming Square-Root Extraction Correctly in the PLC
If the transmitter provides a linear differential-pressure signal, the PLC must first convert the 4…20 mA signal into a normalized value.
For a standard 4…20 mA range:
x = (I - 4 mA) / 16 mA
.
Where:
x = 0
corresponds to the minimum differential pressure and:
x = 1
to the maximum differential pressure.
Then:
Q = Qmax × √x
is calculated.
Example with 8 mA
First:
x = (8 - 4) / 16 = 0.25
.
Then:
√0.25 = 0.5
.
With:
Qmax = 1,000 m³/h
the result is:
Q = 500 m³/h
.
The sequence is important:
normalize current signal → calculate square root → scale to flow
.
For non-zero measurement starts, bidirectional applications or manufacturer-specific characteristics, this simplified equation must not be used without verification.
Detecting Double Square-Root Extraction
One of the most common configuration errors occurs when:
the transmitter applies square-root extraction
and at the same time:
the PLC applies square-root extraction
.
The signal is then processed twice with a square-root function.
Example
The actual flow is:
25%
.
The correctly configured square-rooting transmitter therefore outputs:
8 mA
.
The PLC interprets the 8 mA as a normalized value of:
25%
and incorrectly calculates the square root again:
√0.25 = 0.5
.
The displayed value is therefore:
50% flow
even though the actual flow is only:
25%
.
At 50% actual flow, the second square-root extraction would indicate approximately:
70.7%
.
The error only becomes zero again at the upper range limit.
This explains why a system can appear to be scaled correctly at 100% while the intermediate values are significantly incorrect.
What Happens If No Square-Root Extraction Is Applied?
The opposite problem occurs when the transmitter operates linearly to differential pressure and the PLC nevertheless scales the signal directly as flow.
Example
Actual flow:
50%
.
This produces:
25% differential pressure
.
The linear transmitter outputs:
8 mA
.
A PLC without square-root extraction interprets 8 mA as:
25% flow
.
The actual flow is therefore indicated far too low.
This error also disappears at 0% and 100% and may therefore remain undetected during a simple end-point test.
Correctly Testing and Calibrating a Square-Rooted Transmitter
When testing a differential pressure transmitter, the configured output characteristic must first be known.
Assume the transmitter has:
0...100 mbar differential pressure
and:
4...20 mA
.
With a linear characteristic
| Test Pressure | Range | Expected Output |
|---|---|---|
| 0 mbar | 0% | 4.00 mA |
| 25 mbar | 25% | 8.00 mA |
| 50 mbar | 50% | 12.00 mA |
| 75 mbar | 75% | 16.00 mA |
| 100 mbar | 100% | 20.00 mA |
With a square-root characteristic
| Test Pressure | Differential Pressure | Equivalent Flow | Expected Output |
|---|---|---|---|
| 0 mbar | 0% | 0% | 4.00 mA |
| 6.25 mbar | 6.25% | 25% | 8.00 mA |
| 25 mbar | 25% | 50% | 12.00 mA |
| 50 mbar | 50% | 70.71% | approx. 15.31 mA |
| 100 mbar | 100% | 100% | 20.00 mA |
A technician who automatically expects:
50 mbar → 12 mA
from a square-rooted transmitter would therefore incorrectly judge a properly functioning transmitter as defective.
Before every calibration, it must therefore be clarified whether the analog output represents differential pressure or already represents flow.
Why the Lower Flow Range Is Critical
The square-root function is particularly sensitive to small differential-pressure changes close to zero.
For example:
1% differential pressure
already corresponds to:
10% flow
.
And:
0.25% differential pressure
theoretically corresponds to:
5% flow
.
This means that even small influences such as:
- zero-point drift,
- measurement noise,
- pressure pulsations,
- different liquid columns in the impulse lines,
- temperature changes,
- small hydrostatic differences
can produce an apparent flow indication.
The problem does not necessarily indicate a poor measuring cell. It also results from the mathematical amplification of very small differential-pressure values by the square-root function.
Setting the Low-Flow Cutoff Correctly
To avoid an unstable or physically implausible flow indication close to zero, many flow calculations include a:
Low-Flow Cutoff
or:
low-flow suppression
.
Below a defined threshold, the flow is set, for example, to:
0
.
This prevents a very small differential-pressure offset from being displayed or totalized as actual flow.
However, the threshold must not be selected arbitrarily high.
A cutoff that is too high suppresses actual low flow.
A cutoff that is too low may allow:
- zero-point noise,
- pulsations,
- thermal drift
to appear as apparent flow.
The correct value therefore depends on:
- the primary flow element,
- maximum differential pressure,
- turndown,
- transmitter accuracy,
- process dynamics,
- required minimum flow.
Zero-Point Errors and Incorrect Flow Indication
A small zero-point error is often easy to identify with a linear differential-pressure indication.
With square-rooted flow, its effect can appear significantly larger.
Example
With the process actually at standstill, the transmitter measures an offset of:
1% Δp
.
After applying the square-root function:
√0.01 = 0.10
therefore:
10% flow
.
This is exactly why the following are important in differential-pressure flow measurement:
- correct zero adjustment,
- correct mounting position,
- symmetrical or correctly designed impulse lines,
- fully filled impulse lines for liquid measurements,
- defined condensate pots for steam applications,
- appropriate low-flow suppression.
Before changing the low-flow cutoff, it should therefore first be verified that the zero point is actually stable.
Accounting for Density Changes in Gases and Steam
The simplified relationship:
Q ∝ √Δp
assumes that the other relevant process variables are sufficiently constant or have already been accounted for in the design.
For a differential-pressure primary element, the simplified relationship can be written as:
Q ∝ √(Δp / ρ)
.
This means the density:
ρ
plays an important role.
For liquids with largely constant density, a simple differential-pressure measurement is often sufficient.
For gases and steam, however, density can change significantly with:
- pressure,
- temperature,
- composition.
If accurate standard volumetric flow or mass flow is required, additional:
- absolute pressure measurement,
- temperature measurement,
- density or process compensation
may therefore be necessary.
Square-root extraction alone does not compensate for changes in gas density.
Considering Bidirectional Flow Separately
A simple square-root function is formulated for positive differential pressure.
In systems where the flow direction can reverse, the differential pressure can also change sign.
In that case:
Q = √Δp
is not sufficient.
Suitable bidirectional measurement concepts fundamentally require signed processing, for example:
Q ∝ sign(Δp) × √|Δp|
.
Whether this is supported by the transmitter, control system and, in particular, the primary flow element used must be checked for the specific measuring point.
A conventional 0…100% flow configuration must not simply be transferred to a bidirectional application.
Systematically Troubleshooting the Signal Chain
If the indicated flow is implausible, the entire signal chain should be considered:
primary flow element → impulse lines → differential pressure transmitter → 4...20 mA → analog input → PLC scaling → display
.
Step 1: Check the differential pressure
First determine what differential pressure the transmitter is actually measuring.
Step 2: Check the transmitter configuration
Is the output:
linear to Δp
or:
square-rooted and linear to Q
?
Step 3: Measure the current signal
Check the actual 4…20 mA current using a suitable measuring instrument.
Step 4: Check the PLC raw value
The analog input must detect the same current or the corresponding raw value.
Step 5: Check PLC scaling
Now verify whether:
- only linear scaling is applied,
- a square-root function is active,
- a second square-root extraction is being performed,
- a low-flow cutoff is active.
Step 6: Test intermediate values
Particularly informative points include:
25% differential pressure
and:
50% differential pressure
.
Testing only 0% and 100% is not sufficient for this type of troubleshooting.
Practical Example: 0…100 mbar and 0…1,000 m³/h
An orifice plate has been designed for the following operating point:
Δpmax = 100 mbar
at:
Qmax = 1,000 m³/h
.
The differential pressure transmitter has an output of:
4...20 mA
.
Case A: Square-root extraction in the transmitter
The transmitter is configured for flow or a square-root characteristic.
At:
Δp = 25 mbar
it calculates:
Q = 1,000 × √(25 / 100)
.
Therefore:
Q = 500 m³/h
.
The output is:
12 mA
.
The PLC is configured linearly:
4 mA = 0 m³/h
20 mA = 1,000 m³/h
.
Case B: Square-root extraction in the PLC
The transmitter outputs differential pressure linearly.
At:
25 mbar
its output is:
8 mA
.
The PLC interprets this as:
25% Δp
and calculates:
√0.25 = 0.5
.
The displayed value is again correctly:
500 m³/h
.
Case C: Double square-root extraction
The transmitter from Case A already outputs:
12 mA = 50% flow
.
The PLC incorrectly applies the square-root function to the 50% value again:
√0.5 ≈ 0.707
.
The displayed value is therefore incorrectly:
707 m³/h
instead of:
500 m³/h
.
At this operating point, the error is more than 40% of the actual flow value.
Commissioning Step by Step
- Check the design data of the primary flow element.
- Determine the maximum differential pressure at maximum flow.
- Check the measuring range of the differential pressure transmitter.
- Define where in the signal chain square-root extraction is to be performed.
- Configure the transmitter either for linear differential pressure or square-rooted flow.
- Check which process variable the 4…20 mA output actually represents.
- Configure PLC scaling to match the transmitter configuration.
- Ensure that only one square-root function is active.
- Check the zero point of the differential pressure transmitter.
- Set the low-flow cutoff according to the measurement task.
- Check the 4 mA and 20 mA end points.
- Check at least one characteristic intermediate point.
- For a square-rooted output, for example, verify that 25% Δp corresponds to 50% Q.
- For gas or steam measurements, check whether pressure and temperature compensation is required.
- Document measuring range, unit and scaling in the transmitter, PLC and visualization system.
Common Mistakes
- Scaling differential pressure directly as flow: The required square-root function is missing.
- Both transmitter and PLC apply square-root extraction: The signal is processed twice with the square-root function.
- Testing only 4 and 20 mA: An incorrect characteristic remains undetected at both end points.
- Equating 25% Δp with 25% Q: In reality, 25% differential pressure corresponds to 50% flow.
- Equating 50% Δp with 50% Q: In reality, 50% differential pressure corresponds to approximately 70.7% flow.
- Calibrating a square-rooted transmitter linearly against pressure: The expected intermediate values are calculated incorrectly.
- Setting the low-flow cutoff too high: Actual low flow is suppressed.
- Setting the low-flow cutoff too low: Zero-point drift and noise appear as flow.
- Ignoring zero-point errors: Because of the square-root function, even a small Δp offset can produce a significant flow indication.
- Ignoring gas density: Square-root extraction alone does not compensate for larger pressure and temperature changes.
- Using the wrong maximum differential pressure: The transmitter range and primary-element design do not match.
- Not documenting the process variable: Later it is unclear whether 4…20 mA represents differential pressure or already represents flow.
Suitable Differential Pressure Transmitter
For differential pressure and differential-pressure flow measurements, the Siemens SITRANS P320 is one suitable option.
The digital process pressure transmitter is suitable, among other things, for:
- differential pressure measurement,
- volumetric flow measurement,
- mass flow measurement,
- level measurement using differential pressure.
For flow applications, a square-root characteristic can be used so that the measured differential pressure is converted into an output value proportional to flow.
Depending on the configuration, functions for the lower flow range and low-flow suppression are also available.
HART communication additionally allows configuration and diagnostics using suitable communication or engineering tools.
For particularly demanding applications, the SITRANS P420 is also available within the product family.
Further differential pressure transmitters, pressure transmitters and accessories for industrial process measurement can be found under Pressure Measurement Technology at ICS Schneider.
Further fundamentals on the linear assignment of a 4…20 mA signal can be found in the article Scaling a 4–20 mA Digital Display: Correctly Displaying the Process Value.
Conclusion
In differential-pressure flow measurement, differential pressure is not linear to flow.
In simplified form:
Δp ∝ Q²
and therefore:
Q ∝ √Δp
.
For this reason, square-root extraction must be applied exactly once in the signal chain.
If the differential pressure transmitter provides a linear Δp output, the square-root function must be performed in the PLC or process control system.
If the square-root function is already active in the transmitter, the 4…20 mA signal directly represents flow and must then be scaled linearly.
Checking intermediate points is particularly important. At 25% differential pressure, the flow is already 50%. At 50% differential pressure, it is approximately 70.7%.
In the lower measuring range, the square-root function also amplifies the influence of small differential-pressure errors. Correct zero adjustment and appropriately configured low-flow suppression are therefore particularly important for a stable flow indication.
For gases and steam, it must also be taken into account that density changes caused by pressure and temperature can influence the flow calculation.
The most important question during commissioning is therefore: Do the current 4…20 mA represent the measured differential pressure or already the calculated flow – and at which exact point in the signal chain is the square root applied once and only once?
FAQ: Square-Root Extraction in Differential-Pressure Flow Measurement
Why must a differential pressure signal be square-rooted for flow measurement?
With conventional differential-pressure primary elements, differential pressure is approximately proportional to the square of the flow. To obtain a value proportional to flow from differential pressure, the square root must therefore be calculated.
How much flow corresponds to 25% differential pressure?
50%. The square root of 0.25 is 0.5. With a maximum flow of 1,000 m³/h, 25% differential pressure therefore corresponds to 500 m³/h.
How much flow corresponds to 50% differential pressure?
The square root of 0.5 is approximately 0.7071. 50% differential pressure therefore corresponds to approximately 70.7% of the maximum flow.
Should square-root extraction be performed in the transmitter or in the PLC?
Both are generally possible. What matters is that the square-root function is applied at only one point. If the transmitter provides a linear differential-pressure signal, the PLC must calculate the square root. If the transmitter already provides a square-rooted flow value, the PLC must scale the signal linearly.
How can I tell whether my differential pressure transmitter is already applying square-root extraction?
A simple test uses an intermediate point. With a range of 0…100 mbar, apply 25 mbar. A linear differential-pressure output will provide 8 mA. A square-rooted flow output will instead provide 12 mA.
What happens with double square-root extraction?
The indicated flow becomes significantly too high. At 25% actual flow, for example, a second square-root function would indicate 50%. Only at zero and at the upper range limit do the values coincide again.
What happens if no square-root extraction is applied?
The flow is indicated significantly too low in the partial-load range. At 50% actual flow, only 25% of the maximum differential pressure is generated. Linear scaling would therefore incorrectly indicate only 25% flow.
What is a low-flow cutoff?
The low-flow cutoff or low-flow suppression sets very small flow values below a defined threshold to zero. This prevents zero-point drift or differential-pressure noise from being interpreted as actual flow.
Why is the flow indication often unstable near zero?
The square-root function is very sensitive in the lower differential-pressure range. Even 1% differential pressure theoretically corresponds to 10% flow. Small zero shifts or pressure fluctuations can therefore become clearly visible.
How do I test a square-rooted 4…20 mA transmitter?
The test pressure points must be evaluated according to the square-root function. With a 0…100 mbar range and a square-rooted output, for example, 25 mbar corresponds to an expected output of 12 mA and 50 mbar to approximately 15.31 mA.
Is square-root extraction sufficient for accurate gas flow measurement?
Not always. With gases, differential-pressure flow also depends on density. If process pressure or temperature changes significantly, additional pressure and temperature compensation may be required for accurate volumetric or mass flow measurement.
Can bidirectional flow also be measured using differential pressure?
In principle, yes, provided a suitable primary flow element, transmitter and evaluation concept are used. However, a simple positive square-root function is not sufficient because the sign of the differential pressure and therefore the flow direction must also be taken into account.