# What is the shape of the power required curve?

I understand that when flying level, some power is required to maintain altitude, this power amount being:

• Normally larger when the speed is larger, this is flying on the forward side of the required power curve.

• But below the minimum required power, this amount of power needed to maintain altitude goes down when speed increases. This is flying the backside of the curve.

This can be represented like this:

Being on the backside of the curve means flying in an area of strong instability, possibly reaching the stall point if not managed correctly.

Question: What is the shape of this power required curve for a small GA airplane, and for a large commercial airplane?

I mean is the backside a small portion where it is unlikely to be without obvious pilot mistake? (left hand side below) or is it a large area where it is easy to fall? (right hand side)

• No, the left part is instable: if there is a random decrease in speed in the red portion without changing the power setting (power available), the speed will continue to decrease by itself instead of self-correcting as this is the case on the other side. See "low speed" section here.
– mins
Commented Feb 10, 2018 at 15:40
• The shape of these curves (even qualitatively) depends on the aerodynamic characteristics of the aircraft in question, I believe. As such, this will be hard if not impossible to answer generically. If you were asking about a specific aircraft type, one could try to find data for it. Commented Feb 10, 2018 at 16:28
• is it not true that the left side of the curve represents a condition where the angle of attack is large enough that part of the lift required to maintain altitude is being generated by the vertical component of the propellor's thrust vector? Commented Feb 10, 2018 at 18:20
• @nielsnielsen Not necessarily. The left side of the curve “simply” represents a condition where induced drag (drag due to high lift coefficient = high angle of attack) increases disproportionately with decreasing speed, provided level flight is maintained. For some aircraft, it is possible that the thrust vector is angled upward enough in these conditions to generate a meaningful upward force, but that’s not intrinsic to what the curve represents. Commented Feb 10, 2018 at 23:21
• got it, thanks. it's the "backwards" component of the lift vector then, yes? Commented Feb 11, 2018 at 0:43

What is the shape of this power required curve for a small GA airplane, and for a large commercial airplane?

The answer depends on the span loading and flight altitude of the aircraft. For most practical cases, the left hand side power curve is more realistic, but it still can be entered if the pilot is unaware of what he/she is doing. Just witness AF447

The power curve is the sum of two drag curves, times flight speed:

1. The induced drag is dominant at low speed and decreases with increasing dynamic pressure, and
2. the friction (and parasitic and interference) drag grows with increasing dynamic pressure.

Below I have plotted the theoretical power curves for a small GA airplane at sea level. The dashed lines show the power curve if no flow separation would happen.

Aircraft with low span loading will exhibit low induced drag, so the friction drag (zero lift component) will become dominant soon. The higher the airplane flies, the more the available power is reduced and the induced drag is increased, so the power curve will shift to something resembling your right plot. The point splitting the left side of the curve from the right side is the point of minimum power. Where is this point for actual configurations?

We can calculate the lift coefficient at minimum power in order to know where the speed range of the backside of the curve starts. In this answer, I derived this already, so here is only the result: $$c_{L_{min.\:power}} = \sqrt{\frac{2-n_v}{n_v+2}\cdot c_{D0}\cdot\pi\cdot AR\cdot\epsilon}$$ The symbols are:
$\kern{5mm} c_L \:\:\:$ lift coefficient
$\kern{5mm} n_v \:\:\:$ thrust exponent, as in $T = T_0\cdot v^{n_v}$
$\kern{5mm} c_{D0} \:$ zero-lift drag coefficient
$\kern{5mm} \pi \:\:\:\:\:$ 3.14159$\dots$
$\kern{5mm} AR \:\:$ aspect ratio of the wing
$\kern{5mm} \epsilon \:\:\:\:\:\:$ the wing's Oswald factor, normally between 0.8 and 1.0

Now let's input the values for a glider ($n_v$ = -1): $$c_{L_{min.\:power}} = \sqrt{3\cdot 0.00935\cdot\pi\cdot 21.43\cdot 0.98} = 1.36$$ So the minimum power speed for the ASW-20A is very close to its stall speed. Falling into the backside is practically equivalent with stalling.

Now the values for a jet fighter ($n_v$ = 0): $$c_{L_{min.\:power}} = \sqrt{0.0172\cdot\pi\cdot 2.45\cdot 0.76} = 0.317$$ So the Starfighter pilot is on the backside of the curve until he accelerates beyond 191 m/s or 371 knots (at maximum take-off mass). That is already Mach 0.54 at sea level (and more the higher you go), and explains why the Starfighter prefers to be flown fast. Its best climb speed is at Mach 0.9!

Since you asked about a commercial airplane, here we go: $$c_{L_{min.\:power}} = \sqrt{1.67\cdot 0.01277\cdot\pi\cdot 9.58\cdot 0.74} = 0.687$$ This is for the clean configuration of course and falls between the glider and fighter values. The numbers for a GA aircraft are a little higher (around $c_L$ = 1) due to the piston engine, with a higher Oswald factor due to an unswept wing and a somewhat lower aspect ratio. Commercial aircraft prefer to cruise close to the minimum drag point, which for this example is at $c_{L_{min.\:L/D}}$ = 0.532, so at altitude their operating point is already quite close to the backside range. In contrast to that, a GA aircraft in cruise will fly low (no pressurisation) and much faster that at its minimum power, so its cruise point will be far to the right of the backside range, making the power curve look more like your left hand side example.