Überlagerung von Sinusgrößen gleicher Frequenz
\( u(t) = u_1 (t) + u_2 (t) \)
|
\( u(t) = \hat{u} cos ( \omega t \phi_u ) \)
|
\( u(t) = \hat{u}_1 cos ( \omega t \phi_{u1} )
+ \hat{u}_2 cos ( \omega t \phi_{u2} ) \)
|
mit: \( cos( \alpha + \beta) = cos \alpha cos \beta - sin \alpha sin \beta \)
\( u(t) = \hat{u} ( cos \phi_u cos \omega t - sin \phi_u sin \omega t ) \)
\( u(t) = \hat{u}_1 ( cos \omega t cos \phi_{u1} - sin \phi_{u1} sin \omega t )
+ \hat{u}_2 ( cos \omega t cos \phi_{u2} - sin \phi_{u2} sin \omega t ) \)
\( u(t) = ( \hat{u}_1 cos \phi_{u1} + \hat{u}_2 cos \phi_{u2} ) cos \omega t )
- ( \hat{u}_1 sin \phi_{u1} ) sin \omega t ) \)
Koeffizientenvergleich für \( sin \omega t \) und \( cos \omega t \) :
(1) \( \hat{u} cos \phi_u = \hat{u}_1 cos \phi_{u1} + \hat{u}_2 cos \phi_{u2} \)
(2) \( \hat{u} sin \phi_u = \hat{u}_1 sin \phi_{u1} + \hat{u}_2 sin \phi_{u2} \)
Bestimmung des Winkels aus dem Quotienten \( \frac{(1)}{(2)} \):
\( \frac{\hat{u} sin \phi_u}{\hat{u} cos \phi_u}
= \frac{\hat{u}_1 sin \phi_{u1} + \hat{u}_2 sin \phi_{u2}}
{\hat{u}_1 cos \phi_{u1} + \hat{u}_2 cos \phi_{u2}} \)
\( \phi_u = arctan \left( \frac{\hat{u}_1 sin \phi_{u1} + \hat{u}_2 sin \phi_{u2}}
{\hat{u}_1 cos \phi_{u1} + \hat{u}_2 cos \phi_{u2}} \right) \)
Bestimmung der Amplitude aus \( (1)^2 + (2)^2 \)
\( \left( \hat{u} cos \phi_u \right)^{2} + \left( \hat{u} sin \phi_u \right)^{2}
= \left( \hat{u}_1 cos \phi_{u1} + \hat{u}_2 cos \phi_{u2} \right)^{2}
+ \left( \hat{u}_1 sin \phi_{u1} + \hat{u}_2 sin \phi_{u2} \right)^{2} \)
\( \hat{u} = \sqrt{ \left( \hat{u}_1 cos \phi_{u1} + \hat{u}_2 cos \phi_{u2} \right)^{2}
+ \left( \hat{u}_1 sin \phi_{u1} + \hat{u}_2 sin \phi_{u2} \right)^{2} } \)
\( \hat{u} = \sqrt{ \left( \hat{u}_1 cos \phi_{u1}\right)^{2} + \left( \hat{u}_2 cos \phi_{u2} \right)^{2}
+ 2 \hat{u}_1 cos \phi_{u1} \hat{u}_2 cos \phi_{u2}
+ \left( \hat{u}_1 sin \phi_{u1} \right)^{2} + \left( \hat{u}_2 sin \phi_{u2} \right)^{2}
+ 2 \hat{u}_1 sin \phi_{u1} \hat{u}_2 sin \phi_{u2} } \)
\( sin^{2} \phi + cos^{2} \phi = 1 \)
\( cos \alpha cos \beta + sin \alpha sin \beta = cos ( \alpha - \beta) \)
\( \hat{u} = \sqrt{ \left( \hat{u}_1 \right)^{2}
+ \left( \hat{u}_2 \right)^{2} + 2 \hat{u}_1 \hat{u}_2 cos(\phi_{u2} - \phi_{u1}) } \)
