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## Square-Root Characterizations

This quadratic relationship between flow and pressure drop due to fluid acceleration requires us to mathematically “condition” or “characterize” the pressure signal sensed by the differential pressure instrument in order to arrive at an expressed value for flow rate. The traditional solution to this problem was to incorporate a “square root” function relay between the transmitter and the flow indicator, as shown in the following diagram:

The modern solution to this problem is to incorporate square-root signal characterization either inside the transmitter or inside the receiving instrument (e.g. indicator, recorder, or controller).

In the days of pneumatic instrumentation, this square-root function was performed in a separate device called a square root extractor. The Foxboro corporation model 557 pneumatic square root extractor was a classic example of this technology1:

Pneumatic square root extraction relays approximated the square-root function by means of triangulated force or motion. In essence, they were trigonometric function relays, not square-root relays. However, for small angular motions, certain trigonometric functions were close enough to a square-root function that the relays were able to serve their purpose in characterizing the output signal of a pressure sensor to yield a signal representing flow rate.

The following table shows the ideal response of a pneumatic square root relay:

Input signal |
Input % |
Output % |
Output signal |

3 PSI | 0% | 0% | 3 PSI |

4 PSI | 8.33% | 28.87% | 6.464 PSI |

5 PSI | 16.67% | 40.82% | 7.899 PSI |

6 PSI | 25% | 50% | 9 PSI |

7 PSI | 33.33% | 57.74% | 9.928 PSI |

8 PSI | 41.67% | 64.55% | 10.75 PSI |

9 PSI | 50% | 70.71% | 11.49 PSI |

10 PSI | 58.33% | 76.38% | 12.17 PSI |

11 PSI | 66.67% | 81.65% | 12.80 PSI |

12 PSI | 75% | 86.60% | 13.39 PSI |

13 PSI | 83.33% | 91.29% | 13.95 PSI |

14 PSI | 91.67% | 95.74% | 14.49 PSI |

15 PSI | 100% | 100% | 15 PSI |

As you can see from the table, the square-root relationship is most evident in comparing the input and output percentage values. For example, at an input signal pressure of 6 PSI (25%), the output signal percentage will be the square root of 25%, which is 50% (0.5 = √0.25) or 9 PSI as a pneumatic signal. At an input signal pressure of 10 PSI (58.33%), the output signal percentage will be 76.38%, because 0.7638 = √0.5833, yielding an output signal pressure of 12.17 PSI.

Although analog electronic square-root relays have been built and used in industry for characterizing the output of 4-20 mA electronic transmitters, a far more common application of electronic square-root characterization is found in DP transmitters with the square-root function built in. This way, an external relay device is not necessary to characterize the DP transmitter’s signal into a flow rate signal:

Using a characterized DP transmitter, any 4-20 mA sensing instrument connected to the transmitter’s output wires will directly interpret the signal as flow rate rather than as pressure. A calibration table for such a DP transmitter (with an input range of 0 to 150 inches water column) is shown here:

Differential pressure |
% of input span |
Output % |
Output signal |

0 ”W.C. | 0% | 0% | 4 mA |

37.5 ”W.C. | 25% | 50% | 12 mA |

75 ”W.C. | 50% | 70.71% | 15.31 mA |

112.5 ”W.C. | 75% | 86.60% | 17.86 mA |

150 ”W.C. | 100% | 100% | 20 mA |

Once again, we see how the square-root relationship is most evident in comparing the input and output percentages. Note how the four sets of percentages in this table precisely match the same four percentage sets in the pneumatic relay table: 0% input gives 0% output; 25% input gives 50% output, 50% input gives 70.71% output, etc.

An ingenious solution to the problem of square-root characterization, more commonly applied before the advent of DP transmitters with built-in characterization, is to use an indicating device with a square-root indicating scale. For example, the following photograph shows a 3-15 PSI “receiver gauge” designed to directly sense the output of a pneumatic DP transmitter:

Note how the gauge mechanism responds directly and linearly to a 3-15 PSI input signal range (note the “3 PSI” and “15 PSI” labels in small print at the extremes of the scale, and the linearly-spaced marks around the outside of the scale arc representing 1 PSI each), but how the flow markings (0 through 10 on the inside of the scale arc) are spaced in a non-linear fashion.

An electronic variation on this theme is to draw a square-root scale on the face of a meter movement driven by the 4-20 mA output signal of an electronic DP transmitter:

As with the square-root receiver gauge, the meter movement’s response to the transmitter signal is linear (note the evenly-spaced, triangular marks on the bottom of the scale arc representing increments of 4 mA each), but the markings drawn on the top of the scale are spaced in a non-linear (square-root) fashion. This makes it possible for a human operator to read the scale in terms of (characterized) flow units. Instead of using complicated mechanisms or circuitry to characterize the transmitter’s signal, a non-linear scale “performs the math” necessary to interpret flow.

A major disadvantage to the use of these non-linear indicator scales is that the transmitter signal itself remains un-characterized. Any other instrument receiving this un-characterized signal will either require its own square-root characterization or simply not interpret the signal in terms of flow at all. An un-characterized flow signal input to a process controller can cause loop instability at high flow rates, where small changes in actual flow rate result in huge changes in differential pressure sensed by the transmitter. A fair number of flow control loops operating without characterization have been installed in industrial applications (usually with square-root scales drawn on the face of the indicators, and square-root paper installed in chart recorders), but these loops are notorious for achieving good flow control at only one setpoint value. If the operator raises or lowers the setpoint value, the “gain” of the control loop changes thanks to the nonlinearities of the flow element, resulting in either under-responsive or over-responsive action from the controller.

Despite the limited practicality of non-linear indicating scales, they hold significant value as teaching tools. Closely examine the scales of both the receiver gauge and the 4-20 mA indicating meter, comparing the linear marks (one mark for every 1 PSI increment on the gauge, and one mark for every 4 mA increment on the meter), then compare what you see on the scales against figures in the tables provided earlier for characterized instruments (the square-root extractor and the characterized DP transmitter). Do you see the correspondence? Note how the points marked by the linear divisions match with points on the square-root scale in the exact same manner as the input percentage values in the characterizer tables correspond with the output (square-root) percentage values. At an input value of 25% (6 PSI on the receiver gauge, and the first non-zero linear mark on the meter) match precisely with the 50% point on the square-root scale. Note also how a linear value of 50% (the half-way point on the needle’s sweep for both the receiver gauge and the meter movement) points to just under 71% on each indicator’s square-root scale. A few checks like this verify the fact that the square-root function is encoded in the spacing of the numbers on each instrument’s non-linear scale.

Another valuable lesson we may take from the faces of these indicating instruments is how uncertain the flow measurement becomes at the low end of the scale. Note how for each indicating instrument (both the receiver gauge and the meter movement), the square-root scale is “compressed” at the low end, to the point where it becomes impossible to interpret fine increments of flow at that end of the scale. At the high end of each scale, it’s a different situation entirely: the numbers are spaced so far apart that it’s easy to read fine distinctions in flow values (e.g. 94% flow versus 95% flow). However, the scale is so crowded at the low end that it’s really impossible to clearly distinguish two different flow values such as 4% from 5%.

This “crowding” is not just an artifact of a visual scale; it is a reflection of a fundamental limitation in measurement certainty with this type of flow measurement. The amount of differential pressure separating different low-range values of flow for a flow element is so little, even small amounts of pressure-measurement error equate to large amounts of flow-measurement error. In other words, it becomes more and more difficult to precisely resolve flow rate as the flow rate decreases toward the low end of the scale. The “crowding” that we see on an indicator’s square-root scale is a visual reflection of this fundamental problem: even a small error in interpreting the pointer’s position at the low end of the scale can yield major errors in flow interpretation.

A technical term used to quantify this problem is turndown. “Turndown” refers to the ratio of high-range measurement to low-range measurement possible for an instrument while maintaining reasonable accuracy. For pressure-based flowmeters, which must deal with the non-linearities of Bernoulli’s Equation, the practical turndown is often no more than 3 to 1 (3:1). This means a flowmeter ranged for 0 to 300 GPM might only read accurately down to a flow of 100 GPM. Below that, the accuracy becomes so poor that the measurement is almost useless. Advances in DP transmitter technology have pushed this ratio further, perhaps as far as 10:1 for certain installations. However, the fundamental problem is not transmitter resolution, but rather the nonlinearity of the flow element itself. This means any source of pressure-measurement error – whether originating in the transmitter’s pressure sensor or not – compromises our ability to accurately measure flow at low rates. Even with a perfectly calibrated transmitter, errors resulting from wear of the flow element (e.g. a dulled edge on an orifice plate) or from uneven liquid columns in the impulse tubes connecting the transmitter to the element, will cause large flow-measurement errors at the low end of the instrument’s range where the flow element hardly produces a differential pressure at all. Everyone involved with the technical details of flow measurement needs to understand this fact: the fundamental problem of limited turndown is grounded in the physics of turbulent flow and potential/kinetic energy exchange for these flow elements. Technological improvements will help, but they cannot overcome the limitations imposed by physics. If better turndown is required for a particular flow-measurement application, a wholly different type of flowmeter should be considered.

1Despite the impressive craftsmanship and engineering that went into the design of pneumatic square root extractors, their obsolescence is mourned by no one. These devices were notoriously difficult to set up and calibrate accurately, especially as they aged.

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written by N RAMANAIAH, August 14, 2016