# What is wing inductance?

I am currently trying to understand the ground roll and climbing stage of normal aircraft. I came across this formula which is an expression of the airplanes drag:

$X_a = c_{{x}_a} S \frac{\rho V^2}{2}$

where: $c_{{x}_a} = c_{x0} + A c^2_{x_{a}}$

• S: wing area

• $\rho$: air density

• V: velocity

• $c_{x0}$: drag coefficient when the lifting force = 0

• $c_{x}$: drag coefficient

A is said to be the wing inductance. Could somebody explain what is meant by wing inductance?

• Are there typos in your formulae? Should the second one not be $c_{x_a}=c_{x_0}+A c_{z_a}^2$ where $c_{z_a}$ is the lift coefficient? Mar 3 '17 at 13:26
• @Christo nope, no mistakes this is the formula that I was provided. Unless there is a mistake in the docs I was given... Mar 4 '17 at 21:07

## 2 Answers

$$A = \frac{1}{\pi \cdot e \cdot AR}$$ where $\pi = 3.14159....$, $e$ is the Oswald efficiency factor, and $AR$ is aspect ratio of the wing.

• This answer would be better if it explained the physical significance of this number. Mar 3 '17 at 19:05
• It would indeed be nice if you could explain the physical significance of this number. Mar 4 '17 at 21:06

The drag of a conventional aircraft is $$D = \frac{1}{2}\rho V^2 S C_D$$ where $\rho$ is the air density, $V$ is the true airspeed, $S$ is the wing reference area, and $C_D$ is the drag coefficient.

The drag coeffcient can be approximated by $$C_D = C_{D_0} + \frac{C_L^2}{\pi eA\!R}$$ where $C_{D_0}$ is the zero-lift drag coefficient, $C_L$ is the lift coefficient, $\pi=3.1416\ldots$, $e$ is the span efficiency factor (also called Oswald's efficiency factor), and $A\!R$ is the aspect ratio of the wing. The term $$\frac{C_L^2}{\pi eA\!R}$$ is called the induced drag coefficient.

I suspect that there is a typo in the document where you obtained your formulae. The second formula should be $c_{x_a} = c_{x_o}+Ac_{z_a}^2$ corresponding to $C_D = C_{D_0} + C_L^2/(\pi e A\!R)$. In that case, the $A$ in your second formula corresponds to $1/(\pi e A\!R)$ as @Jared pointed out. The term "wing inductance" that your source uses is probably some variation of the term "induced drag" due to the wing.

You can read this Wikipedia article to find out more about "induced drag", also called "lift-induced drag".