Analogy to OPAMP circuits

Self-balancing pneumatic instrument mechanisms are very similar to negative-feedback operational
amplifier circuits, in that negative feedback is used to generate an output signal in precise proportion
to an input signal. This section compares simple operational amplifier (“opamp”) circuits with
analogous pneumatic mechanisms for the purpose of illustrating how negative feedback works, and
learning how to generally analyze pneumatic mechanisms.
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

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:

This “negative” or “degenerative” feedback is 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 typifies the very nature of
negative feedback.