Simulation of an organ air pressure regulator

A Wurlitzer regulator
Fig 1. A small Wurlitzer regulator residing on its supply trunk. From the lid protrude three steering pins for the internal bars that sequentially push open the control valves when the lid lowers.		Fig 2. The trio of valves at the bottom of the regulator.

Analog circuit for simulation

For simulation of the regulator an analog network was developed as in fig. 3. Blue symbols indicate pressures at nodes, and red symbols for flows in branches. The analog diagram at right has symbols borrowed from electrical technology, though all its elements in our case are acoustical. The purpose of the diagram is to formalize which ways the different airflows U will take, and to visualize how you can compute partial pressure differences from the flows passing the various impedance elements, denoted in black. The basic principle is to apply Ohm's law in acoustical terms, that Pressure = Flow*Impedance. Further technical detail is developed in the trem article. All is driven from a fan by the pressure Pf  which is taken to be constant. Its airflow Uf is fed by a trunk where its length and diameter define the elements  Rf , Mf , and Cu.  Pu is the pressure entering the regulator input. Here the resistor Rr represents the valve that is controlled by the bellows lid elevation and allows the flow Ur to enter the bellows interior. Uv is a minor flow taken to compress the bellows volume of air in case its internal pressure Po should vary. The most important elements are Ml for the inertia of the bellows lid and Cl. Cl is a dual element that on one hand represents a next to infinite compliance that is determined by the bellows fold geometry. On the other hand its loading by springs and weights also tell the static pressure Pl , and to what degree this depends of lid lift and define what the regulator output pressure Po is supposed to be. Ideally Pl should be independent of lift.		Fig 3. Analog acoustical circuit for a regulator.
One purpose of the simulation is to see what happens to Po when the load flow Uo is varied. So Uo is an input to the simulation. It is not a computed result, but a prescribed load condition. This simulated load can be set in terms of average value Uoo and a superimposed component of amplitude Uoa , oscillating at freqency fo.

The parameters that are input with the left set of scrollbars need further explanation: Pf  is the driving fan pressure that must reasonably exceed the regulated output, required to nominally be Pn . Lf and df  are the length and the diameter of the feeding trunk.
ho is the maximum bellows lid lift and Ar is the maximum open area of the control valve(s). The default assumption is that the valve area is fully open until 40% lift, and then decreases linearly to zero at 90% lift. To reasonably approximate a Wurlitzer control board you can switch into a 'progressive' quadratic model using a checkbox at lower right. A sketch of the current model is shown at that place with a small insert figure. Fig 5 shows these models in more detail for Ar = 200 cm2 and ho = 100 mm, together with a measurement from actual hardware. For clarity at the low end, the area is shown on a square root scale. This explains the apparent contradiction between the red and blue model graphs of progressive vs linear area variation.		Fig. 5. Control valve open area vs. lid lift, linear and progressive model, and as measured from a small Wurlitzer regulator.  Square root vertical scale.
In a nearby checkbox you can select the regulator valve to be 'pressure relieved', meaning that the input pressure is not allowed to execute a force toward closure of the pallet. The lift at start of simulation is assumed to be h = 0.4ho . The bellows can normally not collapse more than this due to the thickness of the folding ribs. In Wurlitzer regulators such minimum lift is enforced by stop blocks. The precise value of this minimum lift relative to the nominal ho has too little influence on the simulation to warrant a separate setting option.
ml is the mechanical mass of the bellows lid, and Al is the lid area. These together with the desired nominal Pn indirectly define a required spring force working to compress the bellows.  Now there is an additional complication in that  the bellows geometry puts a requirement also on the spring constant. For an inward folded bellows the spring force should increase with lift in a controlled way. For constant pressure it is thus not enough with a certain force at a specific lift, but also the spring stiffness should be right. Details on this are elaborated in the separate bellows article. xCl is a special parameter to describe how well such spring tuning has been done, 0% is the ideal. It describes the percentage shift in pressure Pl as lid lift goes from zero to ho . In practice regulator springs are often too stiff, such that pressure increases with lift, this is then modeled with a positive xCl . In a reservoir one can fine adjust this parameter by balancing between spring force and ballast weights on the lid. But in a regulator any additional lid weights are dangerous to stability as a big lid inertia tends to belate and overshoot the regulating valve. The lowest parameters Uo and fo define the load to the regulator. The average load flow Uoo is stereotypically set into a pattern of zero flow for the first 0.4 sec, then in the next 0.1 sec increasing by a cosinusoidal arc smoothly to the value set by the scrollbar. After this time the prescribed load can oscillate at freqency fo around this average with the amplitude Uoa. With fo=0  the results display how the regulator settles when it is initially charged by a suddenly turned on input. This is not very relevant to a real situation since a turned on fan will deliver an input that grows very gradually, but it can serve as an indicator whether regulator stability is at issue. When the fo parameter is increased above zero the simulation shows what happens with a pulsating load, like when a tremulant is connected.