Two different reference locations are used to summarize how an airfoil's distributed aerodynamic loading produces lift and pitching moment:

• the center of pressure, where the resultant aerodynamic force has zero pitching moment;
• the aerodynamic center, where the pitching moment is locally independent of angle of attack.

They answer different questions and generally do not coincide.

Center of pressure

Let $x$ be measured aft from the leading edge and let positive pitching moment be nose up. If the lift-dominated resultant is represented as a force $L'$ acting at $x_{cp}$, then its moment about a reference point $x_{ref}$ is

$$M'{ref}=-L'(x{cp}-x_{ref}).$$

Solving,

$$\boxed{x_{cp}=x_{ref}-\frac{M'_{ref}}{L'}}.$$

Using section coefficients,

$$\boxed{ \frac{x_{cp}}{c} =\frac{x_{ref}}{c}-\frac{c_{m,ref}}{c_l} }.$$

The center of pressure is therefore the line-of-action location of the resultant lift in the thin-section approximation.

Why the center of pressure can behave badly

If

$$c_l\to0$$

while

$$c_{m,ref}\ne0,$$

then

$$\left|\frac{x_{cp}}c\right|\to\infty.$$

A finite aerodynamic couple cannot be represented by an almost-zero force acting at a nearby point; the equivalent line of action must move arbitrarily far away.

This is why the center of pressure can change strongly with angle of attack and can even move outside the physical airfoil.

Aerodynamic center

The aerodynamic center $x_{ac}$ is defined by the condition that the pitching-moment coefficient about that point does not change, to first order, when angle of attack changes:

$$\boxed{ \frac{dc_{m,ac}}{d\alpha}=0 }.$$

Shift the moment coefficient from a known reference $x_{ref}$ to $x_{ac}$:

$$c_{m,ac} =c_{m,ref}+c_l\frac{x_{ac}-x_{ref}}c.$$

Differentiate with respect to $\alpha$ while the reference locations remain fixed:

$$0 =\frac{dc_{m,ref}}{d\alpha} +\frac{x_{ac}-x_{ref}}c \frac{dc_l}{d\alpha}.$$

Therefore

$$\boxed{ \frac{x_{ac}-x_{ref}}c =-\frac{dc_{m,ref}/d\alpha}{dc_l/d\alpha} }.$$

Equivalently, where $c_l$ varies monotonically with $\alpha$,

$$\boxed{ \frac{x_{ac}}c =\frac{x_{ref}}c-rac{dc_{m,ref}}{dc_l} }.$$

Unlike the center of pressure, the aerodynamic center is defined from a slope, not from the moment being zero.

Worked example: locating the aerodynamic center

Suppose aerodynamic data referenced to the leading edge give, over a small attached-flow angle range,

$$\frac{dc_l}{d\alpha}=6.0\ \mathrm{rad^{-1}},$$

and

$$\frac{dc_{m,LE}}{d\alpha}=-1.50\ \mathrm{rad^{-1}}.$$

With

$$x_{ref}=0,$$

we obtain

$$\frac{x_{ac}}c =-\frac{-1.50}{6.0} =\boxed{0.25}.$$

Thus the aerodynamic center lies at the quarter-chord point in this example.

Worked example: center of pressure at one operating point

Suppose, at one angle of attack,

$$c_l=0.80$$

and the quarter-chord pitching moment is

$$c_{m,c/4}=-0.05.$$

Then

$$\frac{x_{cp}}c =\frac14-\frac{-0.05}{0.80}$$

$$=0.25+0.0625 =\boxed{0.3125}.$$

The resultant lift acts at about $31.3%$ chord for that operating point.

If angle of attack changes, both $c_l$ and the center of pressure can change even when the moment about the aerodynamic center remains essentially constant.

Do not confuse the two locations

At the center of pressure,

$$\boxed{c_{m,cp}=0}$$

by definition.

At the aerodynamic center,

$$\boxed{\frac{dc_{m,ac}}{d\alpha}=0}$$

by definition, but

$$c_{m,ac}$$

can be nonzero.

A symmetric ideal thin airfoil is a special case where the quarter-chord moment is zero, so the aerodynamic center and center of pressure coincide at $c/4$ whenever lift is nonzero. A cambered airfoil generally has a nonzero quarter-chord moment, so its center of pressure moves with lift while its aerodynamic center remains approximately fixed within the linear attached-flow regime.

The aerodynamic center is therefore usually the more useful fixed reference for stability and aerodynamic modeling.