Engine Control and Transients

Spool dynamics, what a fuel step does to surge margin, acceleration and deceleration schedules, FADEC architecture and limits, and a digital PI speed loop

The PW2000, 1984

In 1984 the PW2000, designed for the Boeing 757, was certified as the first civil engine controlled by a full-authority digital engine control — a FADEC (the PW2000). NASA and Pratt & Whitney had first flown an experimental FADEC in the 1970s, on an F-111 with a highly modified TF30 (FADEC) — the engine that, in the F-14, was prone to compressor stalls at high angle of attack when the pilot moved the throttles aggressively (the TF30). The certification rules leave no room for that. Move the power lever from minimum to maximum in one second, and the engine must reach take-off thrust without surge, stall or overtemperature. From flight idle it must reach 95 % of take-off thrust within 5 seconds (14 CFR 33.73). Fast, but never too fast: that is the control problem.

A rotor changes speed only when its turbine and compressor powers differ. Newton's second law for rotation, with power equal to torque times angular speed, gives the spool equation below. Everything in a transient follows from it and from one fact: fuel changes

T_{t4}

within milliseconds, while the rotor needs seconds to change speed.

J\,\omega\,\frac{d\omega}{dt} = P_t - P_c \qquad\Longrightarrow\qquad \frac{dN}{dt} = \frac{P_t - P_c}{J\,\omega_d^2\,N}

Spool dynamics: N = ω/ω_d is the speed as a fraction of design, J the rotor's moment of inertia, Pt and Pc the turbine and compressor powers.

Derivation

The spool equation

A net torque

Q

accelerates the rotor,

Q = J\,d\omega/dt

, and the net torque is the net power divided by the angular speed,

Q = (P_t - P_c)/\omega

. With

\omega = N\omega_d

J\,\omega_d^2\,N\,\frac{dN}{dt} = P_t - P_c.

The excess power pays for the rotor's kinetic energy

\tfrac12 J\omega^2

: for this unit's turbojet,

J = 8

kg·m² at 12,000 rpm stores about 6.3 MJ.

💡What fuel does first

At constant speed, a hotter turbine throat passes less corrected flow (Lesson 8-1). So the instant fuel is added, the operating point jumps up its speed line, towards surge, onto the line of the new

T_{t4}/T_{t2}

. Only then does the excess turbine power accelerate the rotor, and the point slides back towards the running line as the speed rises. Taking fuel away does the opposite: the point drops towards choke, and the danger moves to the flame.

📐Fuel steps from idle

Flight idle:

N = 0.60

T_{t4} = 825

K, 10.4 kN (24 % of the rated 44.0 kN), surge margin 9.8 % (Lab 1).

Step to 1000 K: the margin drops to 4.3 % at once; the spool settles near $N = 0.69$ , 95 % of the way there after 2.2 s.
Step to 1100 K: the margin drops to 2.2 %.
Step to 1500 K: the margin is −5.7 % — the compressor surges before the rotor can move. That is what slamming the fuel to full would do.

💡Acceleration and deceleration schedules

The control therefore limits fuel as a function of spool speed (and of pressure, in a real engine):

an acceleration schedule that keeps a chosen surge margin;
a deceleration schedule that keeps enough fuel for the flame. Cut fuel too fast and the mixture falls below the combustor's weak-extinction limit (Lesson 7-2), and the flame blows out.

Between the two the engine may move as fast as the rotor allows.

📐Margin against the 5-second rule

Idle to 95 % of rated thrust with the fuel held at the acceleration schedule (Lab 2):

Keep 5 % surge margin: 2.91 s. Keep 7 %: 3.92 s.
Keep 8 %: 5.13 s — the engine fails the certification rule.
Keep 10 %: impossible. At idle the running line itself has only 9.8 %, so the schedule allows no extra fuel at all — which is why compressors open bleed valves at low speed.

💡The FADEC

A FADEC is a digital computer with full authority over fuel flow, variable stators, bleed valves and nozzle. It reads spool speeds, pressures and temperatures many times a second and runs the schedules above around a loop that holds spool speed or pressure ratio at the value the power lever asks for. Redundancy comes from two or more separate, identical channels, each able to run all engine functions without restriction (FADEC). Between the limits a simple loop does the work — typically proportional plus integral control of spool speed.

📐A digital speed loop

A PI loop at 50 Hz commands

T_{t4}

between the two schedules, for a throttle step from

N = 0.70

to 0.97 and back to 0.75 (Lab 3):

With anti-windup — the integrator pauses while the fuel is held at a schedule: 90 % of the step in 1.90 s, no overshoot, settled 0.98 s after the step down while riding the weak-extinction limit at $f = 0.0061$ .
Without: the integrator keeps winding up against the schedule, the engine runs on to $N = 0.999$ — almost full speed — and takes 2.06 s to settle after the step down.

Make haste slowly.

– Augustus, as reported by Suetonius: festina lente (Wikiquote)

⚠Always inside the limits

A FADEC keeps the engine within its limits all the time — even in a crisis such as imminent terrain contact, when an engine without one could be pushed well beyond its rated thrust (FADEC). Whether that is the right trade is a design decision, not a law of nature.

💡Where you will use this

The same spool equation, schedules and limit logic run in every modern engine, in industrial gas turbines and in rocket-engine controllers. Unit 10's adaptive-cycle and electric-assist engines add more actuators — and more limits — to the same loop.

📚Further reading

Link C. Jaw and Jack D. Mattingly, Aircraft Engine Controls: Design, System Analysis, and Health Monitoring (AIAA, 2009).
H. I. H. Saravanamuttoo et al., Gas Turbine Theory (7th ed., Pearson, 2017), on transient behaviour.
14 CFR 33.73, the power and thrust response rule, and FADEC on Wikipedia.

↓But there is a problem

The engine now matches its parts, holds its rating on any day and changes speed on command. Inside it, rotors spin at thousands of revolutions per minute and their blades pull outwards on the disks with forces the next unit will calculate. Bearings, disks, casings, mounts: how an engine holds together.