Circuit Theory/Phasors/proof2


 * $$g(t)=G_m sin(\omega t)$$ starting point
 * $$g(t)=G_m cos(\omega t - \frac{\pi}{2})$$
 * $$g(t)=G_m \operatorname{Re}(e^{j(\omega t - \frac{\pi}{2})})$$
 * $$g(t)=G_m \operatorname{Re}(e^{-j*\frac{\pi}{2}}e^{j\omega t})$$
 * $$g(t)=\operatorname{Re}(G_m e^{-j*\frac{\pi}{2}}e^{j\omega t})$$
 * $$g(t)=\operatorname{Re}(\mathbb{G} e^{j\omega t})$$
 * $$\mathbb{G} = G_m e^{-j*\frac{\pi}{2}} = G_m(cos(-\frac{\pi}{2}) + j*sin(-\frac{\pi}{2})) = -jGm$$