Special time-domain models

AM modulated AC source

An AC voltage source in the time-domain is characterized by its frequency $f$, the initial phase $\phi$ and the amplitude $A$. During amplitude modulation the modulation level $M$ must be considered. The output voltage of the source is determined by the following equation.

$$V_1\left(t\right) - V_2\left(t\right) = \left(1 + M\cdot V_3\left(t\right)\right)\cdot A\cdot \sin{\left(\omega\cdot t + \phi\right)}$$ (6.103)

Figure 6.7: AM modulated AC source

The appropriate MNA matrix entries during the transient analysis decribing a simple linear operation can be written as

$$\begin{bmatrix}. & . & . & 1\\ . & . & . & -1\\ . & . & . & 0\\ 1... ..._3\left(t\right)\\ A\cdot \sin{\left(\omega\cdot t + \phi\right)} \end{bmatrix}$$ (6.104)

PM modulated AC source

The phase modulated AC source is also characterized by the frequency $f$, the amplidude $A$ and by an initial phase $\phi$. The output voltage in the time-domain is determinded by the following equation

$$V_1\left(t\right) - V_2\left(t\right) = A\cdot\sin{\left(\omega\cdot t + \phi + M\cdot V_3\left(t\right)\right)}$$ (6.105)

whereas $M$ denotes the modulation index and $V_3$ the modulating voltage.

Figure 6.8: PM modulated AC source

The component is non-linear in the frequency- as well in the time-domain. In order to prepare the source for subsequent Newton-Raphson iterations the derivative

$$g = \dfrac{\partial \left(V_1 - V_2\right)}{\partial V_3} = M\cdot A\cdot\cos{\left(\omega\cdot t + \phi + M\cdot V_3\right)}$$ (6.106)

is required. With this at hand the MNA matrix entries of the PM modulated AC voltage source during the transient analysis can be written as

$$\begin{bmatrix}. & . & . & +1\\ . & . & . & -1\\ . & . & . & 0\\ ... ...V_3 - A\cdot \sin{\left(\omega\cdot t + \phi + M\cdot V_3\right)} \end{bmatrix}$$ (6.107)