A throttle can hold the peak only in a small plenum

The plenum decides whether a compressor surges followed a compressor through the loss of its stability. The Moore–Greitzer model reduces the machine, its duct, its downstream plenum and its throttle to three variables — the flow through the machine, the plenum’s pressure rise, and the amplitude of a stall cell travelling round the annulus — and a single number, Greitzer’s B, the plenum’s compressibility against the duct’s inertia, decides what happens when the throttle is closed past the peak of the characteristic. A large B surges: the whole flow oscillates through the machine. A small B stalls: a cell of separated flow settles in and the pressure rise drops. And the stall comes with a hysteresis loop — it sets in at one throttle setting and does not clear until the throttle is opened well past it, to the tangency with the stalled characteristic.

The peak is where a compressor is most useful, the most pressure it can make, and it is where every one of those instabilities begins. The essay ended on active control. A fast actuator on the throttle — a bleed valve that opens when the flow starts to fall, or a throttle that responds to what a sensor reads — enters the same three equations, and the questions it posed were exact: the gain that stabilises the peak, how it scales with B, and whether the hysteresis loop can be closed, so that a stalled machine recovers without the throttle having to be opened past the tangency.

The answers divide along a line the essay would not have predicted. The two instabilities need two different feedbacks, one of the two feedbacks that looks natural does nothing at all, and the one that removes the stall’s jump works only when the plenum is small.

A throttle that listens #

The throttle in the Moore–Greitzer model sets how much flow the plenum lets out for a given pressure: $\Phi = \gamma\sqrt{\Psi}$. Here its setting is no longer fixed but moved by feedback,

$$\gamma = \gamma_0 + K_\Psi(\Psi - \Psi_0) + K_J J,$$

opening when the plenum’s pressure rises above its set point, and opening in proportion to the squared amplitude $J$ of a stall cell. A third possibility, a throttle that follows the flow, is examined alongside. The compressor’s characteristic is the previous essay’s cubic — shut-off pressure rise 0.3, height 0.18, half-width 0.25 — with its peak at a flow of 0.5 and a pressure rise of 0.66.

Surge: the pressure gain, and the flow gain that cannot help #

Surge is an oscillation of two of the three variables, the flow and the plenum pressure, with no stall cell. Linearised about an operating point, it is governed by a two-by-two matrix, and whether it grows is decided by that matrix’s trace — the sum of its two growth rates. The trace has two terms: the compressor’s slope, which pumps energy into the oscillation where the characteristic rises to the left of the peak, and the throttle’s own slope, divided by $4B^2$, which damps it. That is the previous essay’s surge criterion, and the feedback enters it in a revealing way.

A throttle that follows the plenum pressure adds directly to the damping term, $K_\Psi\sqrt\Psi/4B^2$. A throttle that follows the flow does not appear in the trace at all: it enters the matrix off the diagonal, where it changes the oscillation’s frequency and leaves the sum of its growth rates exactly where it was. The figure shows both at 90 per cent of the peak flow and B = 2. The pressure gain drives the growth rate steadily down and stabilises the mode at a gain of 7.3. The flow gain leaves the mean growth rate flat at every gain tried, to the last digit.

That is a structural result, not a tuning one. A bleed valve that opens as the measured flow falls — the instinctive design — cannot damp surge in this model however fast it is; it can only retune the oscillation. Damping comes from responding to what the plenum stores.

The gain grows as B squared #

The trace condition gives the least stabilising gain in closed form, $K_\Psi > (4B^2 S - \gamma/2\sqrt\Psi)/\sqrt\Psi$, with $S$ the compressor’s slope. At 90 per cent of the peak flow it is 0.053 for B = 0.5, 1.5 for B = 1 and 7.3 for B = 2 — close to the square of B, because the plenum’s compressibility, which is what B measures, is what the gain has to overcome. Nearer the peak, where the characteristic flattens and pumps less, a smaller gain does; further left, where it is steep, more. A high-speed compressor feeding a large combustor — large B — needs a pressure-feedback gain tens of times that of a low-speed rig, which is why surge control was first demonstrated on small machines.

Stall: what a throttle cannot do, and what it can #

Rotating stall is the third variable, and here the throttle’s power is limited from the start. A stall cell grows at a rate proportional to $1 - u^2 - J/4$, where $u$ measures the flow’s distance from the peak, and that rate depends only on where the flow sits on the characteristic. To the left of the peak it is positive whatever the throttle does: a one-dimensional actuator, which can only change the flow through the whole annulus, cannot stop a cell from starting. What it can change is what happens next.

The stalled equilibria form a branch: for each flow left of the peak there is a cell amplitude $J = 4(1 - u^2)$ and a pressure on the stalled characteristic, and a throttle setting that holds them, $\gamma_0(\Phi) = \Phi/\sqrt{\psi_s(\Phi)} - 4K_J(1 - u^2)$. With no feedback the branch leaves the peak towards a more open throttle, overshooting the peak’s setting by 0.025 before turning back. That is a subcritical bifurcation, and it is the hysteresis: closing the throttle past the peak there is no nearby stalled state, so the machine jumps to a large cell, and opening it again the large-cell state persists until the branch’s turning point. The slope of the branch at the peak is, in closed form,

$$\frac{d\gamma_0}{d\Phi} = \frac{1}{\sqrt{\psi_p}} - \frac{6H}{\psi_p^{3/2}} + \frac{8K_J}{W},$$

and it changes sign at the critical gain $K_J^* = (W/8)(6H/\psi_p^{3/2} - 1/\sqrt{\psi_p})$, 0.0245 for this compressor. Above it the branch leaves the peak towards a more closed throttle, and the cell grows from nothing as the throttle closes. The jump is gone — statically.

With a small plenum the loop closes #

The dynamics confirm it where the plenum is small. At B = 0.2 the throttle is closed over twenty thousand rotor radii from just right of the peak to well past it, and opened again over the same. Without feedback the cell appears suddenly at a setting of 0.609, jumps to an amplitude near three, and vanishes on the way back only at 0.643: a loop between the two legs with an area of 0.078. With stall feedback at twice the critical gain the cell grows gradually as the throttle closes and shrinks along the same curve as it opens, and the loop’s area is 0.0014 — closed, to the resolution of a sweep. A stalled machine recovers where it stalled, without the throttle having to go past the tangency. That is the previous essay’s question answered yes, for this B.

Twice the critical gain rather than once is not an accident of the choice. Near a bifurcation that has just been made supercritical the cell grows and decays slowly, and a throttle moving at a finite rate outruns it; at 1.1 times the critical gain a loop of 0.0055 remains, four times the loop at twice the gain, from that lag alone. A real controller needs margin above the static number.

With a larger plenum it cycles #

At B = 0.5 the same feedback does something different. The cell starts gradually, as it should. Then the machine begins to cycle. The cell grows; the throttle opens in response; with the throttle open the flow recovers and the cell collapses; with no cell the throttle closes again; and the cell grows again. Through the sweep the cell’s amplitude swings between nearly zero and 3.3 over and over. The static picture — a smooth branch of small cells — is still there, but none of its states is stable.

The plenum is what does it. Opening the throttle in response to a growing cell lets the plenum’s pressure fall, and the stored pressure’s fall overshoots: by the time the flow has recovered and the cell has gone, the plenum has emptied past the point that state needs, the throttle closes on a low pressure, and the cycle repeats. The feedback that removes the jump has coupled the stall to the surge mode, and the surge mode, which B controls, is now driven by the stall controller.

The window #

The cycle can be predicted without integrating anything. Linearised about each controlled stalled equilibrium, the full three-variable system has a cubic characteristic equation, and the Routh–Hurwitz test says whether all three of its roots decay. The largest B at which they do is the window the feedback works in. Near the peak, at 90 per cent of the peak flow, it closes at B of 0.36 for 1.5 times the critical gain and 0.30 for four times. More gain makes the window smaller: the feedback that removes the jump is the same feedback that feeds the surge mode, and more of it feeds more. Deep in stall, below about 70 per cent of the peak flow, the equilibria are stable at every B drawn — but deep stall is not where anyone wants to hold a compressor.

Adding the surge controller does not recover the window in this model. With a plenum-pressure gain up to ten on top of the stall feedback, the controlled stalled equilibria near the peak stay unstable at B of 0.5, 1 and 2. The two feedbacks act on the same throttle, and the throttle has one degree of freedom for two modes that now share it.

A controller that makes its own oscillation #

The cycle is a familiar character in this collection, met before in machines with no blades at all. A slow sensor makes the bearing cycle found a journal bearing’s temperature controller turning a steady state into a hunt, because the sensor’s lag put the correction a quarter-cycle late; a warm bush stops the bearing hunting found the physical coupling that removed it. The compressor’s version has no sensor lag: the throttle responds at once. The lag is in the plenum, which takes time to empty and fill, and it plays the same part — a correction applied to a stored quantity arrives after the quantity has moved on.

And as with the shake that is not resonance, what grows is not a forced response but a coupling of two modes that were each stable alone. The stalled equilibrium with fixed throttle is stable deep in stall; the surge mode with no stall is stable at a small enough B. The feedback joins them, and the joined system has a root in the right half-plane that neither had. That is why adding more of either gain cannot fix it: the instability lives in the coupling, and the throttle’s one degree of freedom is the coupling.

Underneath all of it is the blade row’s own work, which work out of a change of swirl prices and which is what the characteristic’s peak is the most of: the controller is fighting to keep the machine at the point where the swirl it puts in is greatest, and the stall cell is what happens when the blades are asked for more.

Where this leaves a compressor #

The result reads as a design rule with a number in it. A compressor whose B is below about a third — a low-speed machine, or one whose plenum is small against its duct — can be held at its peak by a single fast throttle: stall feedback above the critical gain removes the jump and the hysteresis, and the surge mode it might excite is too weak to matter. A compressor whose B is larger cannot be held there by a throttle alone. Its surge can be damped by pressure feedback, but its stall can only be traded for a cycle, and the cure for that needs actuators that act on the stall cell itself — air injected at the rotor tips at the cell’s own angular position, which the one-dimensional model here cannot represent. That division is the same one matched at one speed and at no other found for the machine’s design point: a single parameter of the installation, not of the blades, decides what can be done.

The plenum that makes surge possible is also what defeats the stall controller, and the analogy with a tank that turns a hammer into a swing is close: a compliance placed to absorb one transient becomes the spring of a new oscillation.

Checks on the controlled machine #

The surge gain from the trace condition is checked by perturbing it: a tenth of a per cent below it the surge mode grows and a tenth above it decays. A flow gain of fifty moves the trace by exactly zero. The closed-form slope of the stalled branch at the peak agrees with a finite difference of the branch to $8\cdot10^{-6}$ for three stall gains, and the stalled branch meets the unstalled characteristic at the peak’s throttle setting exactly, with no cell. The critical gain found from the closed form, 0.0245, agrees with the gain at which the whole branch stops overshooting the peak’s setting, found numerically. The tests also refuse a non-positive B.

What a three-variable compressor leaves out #

More than one harmonic. The model keeps one circumferential harmonic of the stall cell. Real cells have shapes, and higher harmonics can be unstable where the first is controlled.

An instantaneous actuator. The throttle responds at once. A real valve has a bandwidth, and a lag adds phase that narrows the window further.

Measurement. The controller reads the stall cell’s amplitude perfectly. In a machine it is inferred from wall-pressure sensors round the annulus, with noise, at the instant the cell is smallest and hardest to see.

A cubic characteristic. The numbers are for one characteristic, a smooth stand-in for what the blade rows’ velocity triangles and the relative-frame Bernoulli balance actually give. The critical gain depends on its height and width through the closed form, and the window’s B on the whole system.

The convention: Greitzer’s B and the throttle coefficient #

Flow and pressure rise are the model’s dimensionless coefficients; time is in rotor radii travelled. Greitzer’s B is half the blade speed over the speed of sound times the square root of the plenum’s volume over the duct’s area times its length. The throttle coefficient γ is the flow over the square root of the plenum pressure rise, so a larger γ is a more open throttle, and gains are throttle coefficient per unit of the quantity fed back.

Epstein, Ffowcs Williams and Greitzer #

Active control of compressor instabilities was proposed by Epstein, Ffowcs Williams and Greitzer in 1989, and surge suppression with a plenum-pressure feedback was demonstrated on a small centrifugal compressor by Ffowcs Williams and Huang the same year; Simon, Valavani, Epstein and Greitzer compared actuator and sensor pairs systematically in 1993. The stall-amplitude feedback that makes the bifurcation supercritical is Liaw and Abed’s, 1996, and its sensitivity to B and to actuator rate was analysed by Wang and Krstić in the years after. The coupled window computed here, and the cycle it ends in, are what this calculation draws from the same model.

Still open: the tip injector that acts on the cell #

A throttle acts on the whole annulus; a stall cell lives at one angle. The next calculation adds the actuator that can reach it — air injected at the rotor tips with an amplitude and a phase set by the cell’s own measured position, entering the Moore–Greitzer equations as a forcing of the first harmonic itself — and asks whether that actuator, combined with the throttle’s pressure feedback, holds the peak at B of one and two, where the throttle alone cannot, and how much injected air, as a fraction of the machine’s flow, it takes.