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## Pneumatic Instrumentation - Analogy to OpAmp Circuits

In the following illustration, we see an opamp with no feedback (open loop), next to a baffle/nozzle mechanism with no feedback (open loop):

For each system there is an input and an output. For the opamp, input and output are both electrical (voltage) signals: Vin is the differential voltage between the two input terminals, and Vout is the single-ended voltage measured between the output terminal and ground. For the baffle/nozzle, the input is the physical gap between the baffle and nozzle (xin) while the output is the backpressure indicated by the pressure gauge (Pout).

Both systems have very large gains. Operational amplifier open-loop gains typically exceed 200,000 (over 100 dB), and we have already seen how just a few thousandths of an inch of baffle motion is enough to drive the backpressure of a nozzle nearly to its limits (supply pressure and atmospheric pressure, respectively).

Gain is always defined as the ratio between output and input for a system. Mathematically, it is the quotient of output change and input change, with “change” represented by the triangular Greek capital-letter delta:

Normally, gain is a unitless ratio. We can easily see this for the opamp circuit, since both output and input are voltages, any unit of measurement for voltage would cancel in the quotient, leaving a unitless quantity. This is not so evident in the baffle/nozzle system, with the output represented in units of pressure and the input represented in units of distance.

If we were to add a bellows to the baffle/nozzle mechanism, we would have a system that inputs and outputs fluid pressure, allowing us to more formally define the gain of the system as a unitless ratio of ΔPout/ΔPin

The general effect of negative feedback is to decrease the gain of a system, and also make that system’s response more linear over the operating range. This is not an easy concept to grasp, however, and so we will explore the effect of adding negative feedback in detail for both systems. The simplest expression of negative feedback is a condition of 100% negative feedback, where the whole strength of the output signal gets “fed back” to the amplification system in degenerative fashion. For an opamp, this simply means connecting the output terminal directly to the inverting input terminal:

We call this “negative” or “degenerative” feedback because its effect is counteractive in nature. If the output voltage rises too high, the effect of feeding this signal to the inverting input will be to bring the output voltage back down again. Likewise, if the output voltage is too low, the inverting input will sense this and act to bring it back up again. Self-correction is the hallmark of any negative-feedback system.

Having connected the inverting input directly to the output of the opamp leaves us with the non-inverting terminal as the sole remaining input. Thus, our input voltage signal is a ground-referenced voltage just like the output. The voltage gain of this circuit is unity (1), meaning that the output will assume whatever voltage level is present at the input, within the limits of the opamp’s power supply. If we were to send a voltage signal of 5 volts to the non-inverting terminal of this opamp circuit, it would output 5 volts, provided that the power supply exceeds 5 volts in potential from ground.

Let’s analyze exactly why this happens. First, we will start with the equation representing the open-loop output of an opamp, as a function of its differential input voltage:

As stated before, the open-loop voltage gain of an opamp is typically very large (AOL = 200,000 or more!). Connecting the opamp’s output to the inverting input terminal simplifies the equation:

Vout may be substituted for Vin(−), and Vin(+) simply becomes Vin since it is now the only remaining input. Reducing the equation to the two variables of Vout and Vin and a constant (AOL) allows us to solve for overall voltage gain (Vout- Vin) as a function of the opamp’s internal voltage gain (AOL). The following sequence of algebraic manipulations shows how this is done:

If we assume an internal opamp gain of 200,000, the overall gain will be very nearly equal to unity (0.999995). Moreover, this near-unity gain will remain quite stable despite large changes in the opamp’s internal (open-loop) gain. The following table shows the effect of major AOL changes on overall voltage gain (AV ):

AOL Internal gain |
AV Overall gain |

100,000 | 0.99999 |

200,000 | 0.999995 |

300,000 |
0.999997 |

500,000 | 0.999998 |

1,000,000 | 0.999999 |

Note how an order of magnitude change1 in AOL (from 100,000 to 1,000,000) results is a miniscule change in overall voltage gain (from 0.99999 to 0.999999). Negative feedback clearly has a stabilizing effect on the closed-loop gain of the opamp circuit, which is the primary reason it finds such wide application in engineered systems. It was this effect that led Harold Black in the late 1920’s to apply negative feedback to the design of very stable telephone amplifier circuits.

If we subject our negative feedback opamp circuit to a constant input voltage of exactly 5 volts, we may expand the table to show the effect of changing open-loop gain on the output voltage, and also the differential voltage appearing between the opamp’s two input terminals:

AOL Internal gain |
AV Overall gain |
Vout Output voltage |
Vin(+) − Vin(−) Differential input voltage |

100,000 | 0.99999 | 4.99995 | 0.00005 |

200,000 | 0.999995 | 4.999975 | 0.000025 |

300,000 | 0.999997 | 4.99998 | 0.00002 |

500,000 | 0.999998 | 4.99999 | 0.00001 |

1,000,000 | 0.999999 | 4.999995 | 0.000005 |

With such extremely high open-loop voltage gains, it hardly requires any difference in voltage between the two input terminals to generate the necessary output voltage to balance the input. Thus, Vout = Vin for all practical purposes.

One of the “simplifying assumptions” electronics technicians and engineers make when analyzing opamp circuits is that the differential input voltage in any negative feedback circuit is zero. As we see in the above table, this assumption is very nearly true. Following this assumption to its logical consequence allows us to predict the output voltage of any negative feedback opamp circuit quite simply. For example:

If we simply assume there will be no difference of voltage between the two input terminals of the opamp with negative feedback in effect, we may conclude that the output voltage is exactly equal to the input voltage, since that is what must happen in order for the two input terminals to see equal potentials.

Now let us apply similar techniques to the analysis of a pneumatic baffle/nozzle mechanism. Suppose we arrange a pair of identical bellows in opposition to one another on a force beam, so any difference in force output by the two bellows will push the baffle either closer to the nozzle or further away from it:

It should be clear that the left-hand bellows, which experiences the same pressure (Pout) as the pressure gauge, introduces negative feedback into the system. If the output pressure happens to rise too high, the baffle will be pushed away from the nozzle by the force of the feedback bellows, causing backpressure to decrease and stabilize. Likewise, if the output pressure happens to go too low, the baffle will move closer to the nozzle and cause the backpressure to rise again. Once again we see the defining characteristic of negative feedback in action: its self-correcting nature works to counteract any change in output conditions, such that the output pressure precisely tracks the input pressure. As we have seen already, the baffle/nozzle is exceptionally sensitive to motion. Only a few thousandths of an inch of motion is sufficient to saturate the nozzle backpressure at either extreme (supply air pressure or zero, depending on which direction the baffle moves). This is analogous to the differential inputs of an operational amplifier, which only need to see a few microvolts of potential difference to saturate the amplifier’s output.

Introducing negative feedback to the opamp led to a condition where the differential input voltage was held to (nearly) zero. In fact, this potential is so small that we safely considered it zero for the purpose of more easily analyzing the output response of the system. We may make the exact same “simplifying assumption” for the pneumatic mechanism: we will assume the baffle/nozzle gap remains constant in order to more easily determine the output pressure response to an input pressure. If we simply assume the baffle/nozzle gap cannot change so long as negative feedback is actively working, we may conclude that the output pressure is exactly equal to the input pressure for the pneumatic system shown, since that is what must happen in order for the two pressures to generate exactly opposing forces through two identical bellows so the baffle will not move from its original position.

The analytical technique of assuming perfect balance in a negative feedback system works just as well for more complicated systems. Consider the following opamp circuit:

Here, negative feedback occurs through a voltage divider from the output terminal to the inverting input terminal, such that only one-half of the output voltage gets “fed back” degeneratively. If we follow our simplifying assumption that perfect balance (zero difference of voltage) will be achieved between the two opamp input terminals due to the balancing action of negative feedback, we are led to the conclusion that Vout must be exactly twice the magnitude of Vin. In other words, the output voltage must increase to twice the value of the input voltage in order for the divided feedback signal to exactly equal the input signal. Thus, feeding back half the output voltage yields an overall voltage gain of two.

If we make the same (analogous) change to the pneumatic system, we see the same effect:

Here, the feedback bellows has been made smaller (exactly half the surface area of the input bellows). This results in half the amount of force applied to the force beam for the same amount of pressure. If we follow our simplifying assumption that perfect balance (zero baffle motion) will be achieved due to the balancing action of negative feedback, we are led to the conclusion that Pout must be exactly twice the magnitude of Pin. In other words, the output pressure must increase to twice the value of the input pressure in order for the divided feedback force to exactly equal the input force and prevent the baffle from moving. Thus, our pneumatic mechanism has a pressure gain of two, just like the opamp circuit with divided feedback had a voltage gain of two. We could have achieved the same effect by moving the feedback bellows to a lower position on the force beam instead of changing its surface area:

This arrangement effectively reduces the feedback force by placing the feedback bellows at a mechanical disadvantage to the input bellows. If the distance between the feedback bellows tip and the force beam pivot is exactly half the distance between the input bellows tip and the force beam pivot, the effective force ratio will be one-half. The result of this “divided” feedback force is that the output pressure must rise to twice the value of the input pressure, since the output pressure is at a mechanical disadvantage to the input. Once again, we see a balancing mechanism with a gain of two.

It should be noted that this mechanism, which achieves different gains by applying different lever ratios, is more properly classified as a moment-balance system rather than a pure force-balance system. In all the previous pneumatic examples where one bellows directly opposed another bellows (both bellows sharing the same centerline), the balancing action was direct force against direct force. Here, in this leverage mechanism, the balance is not a matter of one bellows force countered directly by another bellows force, but rather one bellows’ moment4 countered by another bellows’ moment. Pneumatic instruments built such that bellows’ forces directly oppose one another in the same line of action to constrain the motion of a beam are known as “force balance” systems. Instruments built such that bellows’ forces oppose one another through different lever lengths (such as in the last system) are technically known as “moment balance” systems, referencing the moment arm lengths through which the bellows’ forces act to balance each other. However, one will often find that “moment balance” instruments are commonly referred to as “force balance” because the two principles are so similar.

4In physics, the word moment refers to the product of force times lever length. By the same token, we could classify this pneumatic mechanism as a torque-balance system.

An entirely different classification of pneumatic instrument is known as motion balance. The same “simplifying assumption” of zero baffle/nozzle gap motion holds true for the analysis of these mechanisms as well:

In this mechanism there is no fixed pivot for the beam. Instead, the beam hangs between the ends of two bellows units, affixed by pivoting links. As input pressure increases, the input bellows expands outward, attempting to push the beam closer to the nozzle. However, if we follow our assumption that negative feedback holds the nozzle gap constant, we see that the feedback bellows must expand the same amount, and thus (if it has the same area and spring characteristics as the input bellows) the output pressure must equal the input pressure:

We call this a motion balance system instead of a force balance system because we see two motions canceling each other out to maintain a constant nozzle gap instead of two forces canceling each other out to maintain a constant nozzle gap.

The gain of a motion-balance pneumatic instrument may be changed by altering the bellows-to-nozzle distance such that one of the two bellows has more effect than the other. For instance, this system has a gain of 2, since the feedback bellows must move twice as far as the input bellows in order to maintain a constant nozzle gap:

Force-balance (and moment-balance) instruments are generally considered more accurate than motion-balance instruments because motion-balance instruments rely on the pressure elements (bellows, diaphragms, or bourdon tubes) possessing predictable spring characteristics. Since pressure must accurately translate to motion in a motion-balance system, there must be a predictable relationship between pressure and motion in order for the instrument to maintain accuracy. If anything happens to affect this pressure/motion relationship such as metal fatigue or temperature change, the instrument’s calibration will drift. Since there is negligible motion in a force-balance system, pressure element spring characteristics are irrelevant to the operation of these devices, and their calibrations remain more stable over time.

Thus, adding a relay to a self-balancing pneumatic system is analogous to increasing the openloop voltage gain of an opamp (AOL) by several-fold: it makes the overall gain closer to ideal. The overall gain of the system, though, is dictated by the ratio of bellows leverage on the force beam, just like the overall gain of a negative-feedback opamp circuit is dictated by the feedback network and not by the opamp’s internal (open-loop) voltage gain.

1An “order of magnitude” is nothing more than a ten-fold change. Do you want to sound like you’re really smart and impress those around you? Just start comparing ordinary differences in terms of orders of magnitude. “Hey dude, that last snowboarder’s jump was an order of magnitude higher than the one before!” “Whoa, that’s some big air . . .” Just don’t make the mistake of using decibels in the same way (“Whoa dude, that last jump was at least 10 dB higher than the one before!”) – you don’t want people to think you’re a nerd.

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