Describing Function Sinusoidal I/P

$$N(A,w) = \frac{A_{1}{(A,w)}}{A} e^{j\phi(A,w)}$$ (3.1)

$$= n_{p}+jn_{q}$$ (3.2)

$$n_{p} = \frac{1}{\pi A} \int^{2\pi}_{0}{y \sin \psi d\psi}$$ (3.3)

$$n_{q} = \frac{1}{\pi A} \int^{2\pi}_{0}{y \cos \psi d\psi}$$ (3.4)

Static non-linearity not depends upon $w$ Mulivalue function are known as memory functions For memory less static function $n_{q}=0$ Memoryless-static-odd $f(-x)=-f(x)$

$$n_{p} = \frac{4}{\pi A} \int^{\frac{\pi}{2}}_{0}y \sin \psi d \psi$$ (3.5)

$$N(A) = \frac{2j}{\pi A} \int^{\pi}_{0}y (A\sin \psi)e^{-j\psi} d\psi$$ (3.6)

$$N(A) = \frac{2j}{\pi A} \int^{\psi_1}_{0}{-De^{-j\psi} d\psi +\int^{\pi}_{\psi_{2}}{-De^{-j\psi} d\psi}}$$ (3.7)

$$N(A) = \frac{2D}{\pi A}[\cos \psi_2+\cos \psi_1-j(\sin \psi_2+\sin \psi_1)]$$ (3.8)

$$A\sin \psi_{1} = \delta(1-\epsilon)$$ (3.9)

Representing $\psi_{2}$ also

$$N(A) = \frac{2D}{\pi A}\sqrt{1-(\frac{\delta}{a})^2(1+\epsilon)^2}+\sqrt{1-(\frac{\delta}{a})^2(1-\epsilon)^2}-j\frac{2\delta}{A}$$ (3.10)