Control Authority (Elevator Power) Calculator

When the elevator is deflected, it changes the airflow over the horizontal stabilizer which in turn creates a pitching moment whose center of rotation is at the aircraft's center of gravity. The controllability, also called maneuverability, is a numerical representation of how much moments the pilot can actually control.

By entering the elevator efficiency and angle of attack you can see the change in the pitch moment coefficient. Also, you can leave one of the input fields blank so that the calculator will figure out the value for it.

What is the elevator efficiency?

The elevator efficiency is a control differential coefficient denoted by C_M,delta-e. It indicates how much the pitch moment coefficient changes when the elevator is deflected by a certain unit. If you multiply this differential coefficient with the angle of attack, you get the dimensionless change in the pitch moment coefficient, Delta C_M.

This equation is very simple but determining the values of the partial derivatives is a real challenge. Engineers often spend months trying to determine these values in order to set things like the size of the horizontal stabilizer or how much control authority the pilot has in a fly-by-wire flight control system.

How to use this calculator:

You can enter any two of the three sizes and the third size will be automatically calculated. The efficiency can be entered in degrees per unit or radians per unit, and for the pitch angle you can use either degrees or radians. Since the calculator ensures consistency between both units, common errors that occur with manual calculations are avoided. By convention, the efficiency is usually a negative value, and most elevators have a deflection range of typically plus/minus 25 degrees.

These three values are related by a single equation so they can be calculated from any one of them. You can determine the change in moment based on efficiency and angle of attack, calculate the required efficiency for a given design based on desired change in moment and angle of attack or find the required angle of attack to achieve a particular change in moment by inputting the efficiency and desired change in moment.

Understanding of formulas and symbols.

Assume the elevator effectiveness is negative and given in degrees (0.02). When the pilot pulls back on the control stick, the trailing edge of the elevator rotates up by 10 degrees. By convention, an upward movement of the trailing edge is a negative rotation so delta-e has a value of -10 degrees.

A positive pitch moment will lift the nose of the aircraft. This is the effect that a pilot wants when pulling up or intercepting. The negative value for the coefficient is just a result of sign conventions. A positive rotation means the trailing edge moves down, which results in the tail lifting and therefore the nose pointing down. Multiplying a negative coefficient by a negative rotation gives a pitch moment that pulls the aircraft up, thus responding to the stick input.

Converting coefficients to actual moments:

Delta C_M is a dimensionless quantity so it alone does not suffice to determine the moment in Newton-meters. To obtain the actual aerodynamic pitching moment this coefficient must be scaled according to flight conditions. The scaling factors include the dynamic pressure, wing area and mean chord length.

Even though they are the same elevators, they feel heavy during cruise flight and respond sluggishly to slow approaches. This is because while the coefficient does not depend on speed, the dynamic pressure q increases proportionally with the square of airspeed, which in turn increases the moment generated by elevator deflection. With a calculator you can open the "actual moment" option and enter values for q, S (wing area), and wing depth to calculate the moment in Newton meters.

Typical values

Actual values help validate the results. The following ranges are rough guidelines for typical fixed wing aircraft and are not design data.

If the calculated Delta C_M is significantly larger than about half, check your inputs again. You are probably outside of the linear range for which the formula was designed and the elevator has already started to stall.

Situations where controllability is important

Engineers determine the size of the elevator and the moment arm of the stabilizer so that efficiency is high enough under worst-case conditions. Nose-up pitching is often a limitation. At low speeds and low dynamic pressure, an aircraft must still be able to pitch up. Also, performing a flare on landing in ground effect situations as well as recovering from a stall at high angle of attack are other scenarios which need to be considered when determining the requirements for controllability. The same principle applies to control surfaces on rockets, drones and submarines. What changes is only the geometry.

Why the formula no longer works at large angles of attack:

The linear assumption only holds true when the airflow is attached to the elevator. At about 20 to 25 degrees, depending on the airfoil profile, the elevator becomes detached and its effectiveness drops in a way that cannot be predicted by a linear model. The moment vs angle of attack diagrams in the calculation tools show a linear model over the entire range so you can see where this model is reliable and when it starts to deviate from the actual curve.