Control Authority (Elevator Power) Calculator

Pull back on the stick and the elevator deflects. Airflow over the horizontal tail changes, and a pitching moment builds around the aircraft's center of gravity. Control Authority (also called Elevator Power) puts a number on how much of that moment the pilot can actually command. Drop in elevator effectiveness and deflection angle to read off the change in pitching moment coefficient, or leave any one variable blank and solve for it instead.

What is Elevator Power?

Elevator Power is the control derivative $C_{M,\delta e}$. It tells you how much the pitching moment coefficient changes per unit of elevator deflection. Multiply that derivative by the deflection angle $\delta e$ and you have the change in the dimensionless pitching moment coefficient $\Delta C_M$:

The equation is short. The value of $C_{M,\delta e}$ is where the work hides. That single number is what designers spend months chasing when they're sizing a horizontal stabilizer or deciding how much authority a fly-by-wire system should hand the pilot.

How to use this calculator

Enter any two values; leave the third blank and it fills in. Elevator effectiveness is usually negative under the standard sign convention, and most deflection angles fall within ±25°. The output coefficient is dimensionless. To turn it into a real torque, multiply by dynamic pressure, wing area, and mean aerodynamic chord.

Understanding the formula

Suppose the elevator effectiveness is $C_{M,\delta e} = -0.02$ per degree. The pilot pulls the stick back, and the trailing edge of the elevator swings upward by 10°. In the standard convention, upward elevator is negative, so $\delta e = -10°$.

A positive pitching moment lifts the nose, which is exactly what the pilot wants during a rotation or a flare. The reason $C_{M,\delta e}$ itself is negative comes down to bookkeeping: a positive $\delta e$ points the trailing edge down, which lifts the tail, which pushes the nose down. So negative coefficient times negative deflection gives nose-up. The signs cancel and the airplane responds the way the stick asked.

Applications in aircraft design

Engineers size the elevator and the tail arm so that $C_{M,\delta e}$ stays big enough for the worst-case flight conditions. Takeoff rotation is usually the limiting case: low airspeed, low dynamic pressure, and the airplane still has to point its nose up. Landing flare in ground effect and recovery from a high angle-of-attack stall are the other two that come up regularly in handling-quality requirements. The same coefficient logic carries over to launch vehicles, drones, and submarine control planes. Only the geometry changes.

Tips for getting sensible results

Keep units consistent. Per-degree effectiveness needs degree deflections; per-radian needs radians. Mixing the two is the most common mistake here. For typical fixed-wing aircraft, $C_{M,\delta e}$ usually falls somewhere between -0.5 and -2.0 per radian, and elevator throw rarely exceeds ±25°. If the computed $\Delta C_M$ comes out larger than about ±0.5, look at the inputs again. You're probably outside the linear range the formula is built for.

Frequently asked questions

Why is elevator effectiveness usually negative?

It's just the sign convention. A positive $\delta e$ means the trailing edge goes down, which produces a nose-down (negative) pitching moment. A negative coefficient times a positive deflection gives a negative $\Delta C_M$. Every aerospace textbook uses this convention.

Does this formula account for airspeed?

No. $C_{M,\delta_e}$ is a geometric and aerodynamic property of the airframe alone. The actual torque scales with dynamic pressure ($\frac{1}{2}\rho V^2$), so the same deflection produces a much larger pitching moment at cruise than during takeoff.

What's the difference between control authority and stability?

Stability ($C_{M,\alpha}$) is how strongly the aircraft resists a change in pitch by itself. Control authority ($C_{M,\delta_e}$) is how strongly the pilot can force one. A well-designed aircraft has enough of both: enough stability to fly hands-off, and enough authority to maneuver when the pilot wants something different.

Can I use this for canard or V-tail configurations?

Yes. The form of the equation doesn't change; only the value and sign of $C_{M,\delta_e}$ change with the geometry. A canard typically has a positive $C_{M,\delta_e}$ because its control surface sits ahead of the center of gravity, not behind it.

Why does the formula break down at large deflections?

The linear assumption only holds while airflow stays attached to the elevator. Past roughly 20° to 25°, depending on the airfoil, the elevator stalls and effectiveness drops off non-linearly. After that, simple multiplication stops predicting the real moment.