Excess Power⚠ unverified

Understanding the Parameters

• Power available $P_a$ — for a jet, $P_a = T\,V$ (thrust times speed); for a propeller aircraft, $P_a = \eta_p\,P_{\text{shaft}}$. It decreases with altitude because engine thrust/power falls as air density drops. This altitude fade is what eventually erases the surplus and sets the ceiling.

• Power required $P_r$ — the power to overcome drag in level flight, $P_r = D\,V$. Because drag itself varies with speed (high at low speed from induced drag, high at high speed from parasite drag), $P_r$ traces a U-shaped curve with a minimum at the minimum-power speed. Its lowest point is where a propeller aircraft climbs and loiters best.

• The gap between the curves — plot $P_a$ and $P_r$ against speed on one chart (the classic "power curves"). The vertical gap between them is the excess power at each speed; the speed of maximum gap is the best-rate-of-climb speed. Where the curves just touch, excess power is zero and no climb is possible — the ceiling.

• Power vs thrust framing — excess power $(P_a - P_r)$ governs rate of climb and energy rate; excess thrust $(T - D)$ governs climb angle and acceleration. They are two views of the same surplus, related by a factor of $V$. Propeller analysis is usually done in power; jet analysis often in thrust.

Derivation (Approaching a Proof)

Excess power is a definition, but its meaning follows from the energy balance of an aircraft. Consider the total mechanical energy (kinetic plus potential):

$$E = \tfrac12 m V^2 + m g h.$$

Differentiate with respect to time — the rate at which the aircraft gains energy:

$$\frac{dE}{dt} = m V\frac{dV}{dt} + m g\frac{dh}{dt}.$$

By the work–energy theorem, the net rate of work done on the aircraft along its path equals thrust minus drag times speed (lift does no work, being perpendicular to the flight path):

$$\frac{dE}{dt} = (T - D)\,V = T V - D V = P_a - P_r = P_{\text{excess}}.$$

So excess power is the rate of total energy addition to the aircraft. Dividing by weight gives the rate of gain of energy per unit weight — the specific excess power $P_s = P_{\text{excess}}/W$, the central quantity of energy-manoeuvrability theory (see Energy Height):

$$P_s = \frac{P_{\text{excess}}}{W} = \frac{dh}{dt} + \frac{V}{g}\frac{dV}{dt}.$$

If the surplus is spent entirely on altitude ($dV/dt = 0$), $P_s$ is the Rate of Climb; if spent entirely on speed (level flight), it is the acceleration capability. The pilot chooses how to divide it.

Dimensional check. $P_a - P_r = \text{W} - \text{W} = \text{W}$ ✓ — subtraction of like quantities. And $(T-D)V = \text{N}\cdot\text{m}/\text{s} = \text{W}$, confirming the energy-rate interpretation.

History and Development

• The power curves (1910s–20s). Plotting power available against power required versus speed became the standard graphical method for predicting climb and ceiling in the first decades of aeronautics, taught in every performance course since.

• Energy manoeuvrability (1960s). The concept's modern significance comes from Colonel John Boyd and mathematician Thomas Christie, who in the early 1960s recast fighter performance in terms of specific excess power $P_s = P_{\text{excess}}/W$. Their Energy–Manoeuvrability (E–M) theory plotted $P_s$ across the speed–altitude–load-factor envelope, revolutionising fighter design and directly shaping the F-15 and F-16. A fighter that can hold positive $P_s$ where its opponent cannot dictates the fight.

• Why the surplus is the metric. Boyd's insight was that instantaneous turn rate matters less than the ability to regain energy — and that ability is exactly the excess power computed here. It links this humble subtraction to Energy Height and the whole apparatus of air-combat performance.

Related Concepts: Rate of Climb, Energy Height, Time To Climb, Best Climb Speed, Rate Of Climb Ft Min, Drag Force, Thrust, Power From Force Velocity

Notes: Exact definition (not an approximation). $P_a = TV$ (jet) or $\eta_p P_{\text{shaft}}$ (prop), falls with altitude; $P_r = DV$ is U-shaped in speed. Equals the aircraft's total energy-addition rate $(T-D)V = dE/dt$; divided by weight it is the specific excess power $P_s$ of Boyd's E–M theory. Zero surplus = ceiling. Defaults $1.0\ \text{W}$ placeholder.