General describing function

$$Y = \sum A_{n}\sin \left( n\omega t+\phi _{n}\right)$$ (2.7)

$$\Rightarrow Y = \sum A_{n}\sin \left( n\omega t+\phi _{n}\right) .\sin \omega t$$ (2.8)

$$\int Y\sin \psi d\psi = \sum \int A_{n}\left[ \left( \sin n\psi \sin \psi \right) \cos \phi _{n}+\left( \cos n\psi \sin \psi \right) \sin \phi _{n}\right] d\psi$$ (2.9)

replacing $\omega t$ with $\psi$

$$A_{1}\cos \phi _{1} = \frac{1}{\pi }\int Y\sin \psi d\psi$$ (2.10)

$$A_{1}\sin \phi _{1} = \frac{1}{\pi }\int Y\cos \psi d\psi$$ (2.11)

$$N = \frac{A_{1}\cos \phi _{1}+jA_{1}\sin \phi _{1}}{A}=n_{p}+jn_{q}$$ (2.12)

where

$$n_{p} = \frac{1}{\pi A}\int Y\left( ,\right) \sin \psi d\psi$$ (2.13)

$$n_{q} = \frac{1}{\pi A}\int Y\left( ,\right) \cos \psi d\psi$$ (2.14)