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electrical_engineering_1:introduction_in_alternating_current_technology [2023/03/27 09:30]
mexleadmin 		electrical_engineering_1:introduction_in_alternating_current_technology [2023/09/19 23:37]
mexleadmin
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-====== 6. Introduction to Alternating Current Technology ======+====== 6 Introduction to Alternating Current Technology ======
  
 Up to now, we had analyzed DC signals (chapters 1. -  4.) and abrupt voltage changes for (dis)charging capacitors (chapter 5.). In households, we use alternating voltage (AC) instead of a constant voltage (DC). This is due to at least three main facts Up to now, we had analyzed DC signals (chapters 1. -  4.) and abrupt voltage changes for (dis)charging capacitors (chapter 5.). In households, we use alternating voltage (AC) instead of a constant voltage (DC). This is due to at least three main facts
Zeile 542: Zeile 542:
   * $X = Z \sin \varphi$   * $X = Z \sin \varphi$
  
-==== 6.5.2 Application on pure Loads ====+value - and therefore a phasor - can simply ==== 6.5.2 Application on pure Loads ====
  
 With the complex impedance in mind, the <tabref tab01> can be expanded to:  With the complex impedance in mind, the <tabref tab01> can be expanded to: 
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 \\ \\ \\ \\
 The relationship between ${\rm j}$ and integral calculus should be clear:  The relationship between ${\rm j}$ and integral calculus should be clear: 
-  - The derivative of a sinusoidal value - and therefore a phasor - can simply be written as "$\cdot {\rm j}$",  which also means a phase shift of $+90°$: \\ ${{\rm d}\over{{\rm d}t}} {\rm e}^{{\rm j}(\omega t + \varphi_x)} = {\rm j} \cdot {\rm e}^{{\rm j}(\omega t + \varphi_x)}$ +  - The derivative of a sinusoidal value - and therefore a phasor - can simply be written as "$\cdot {\rm j}$", \\ which also means a phase shift of $+90°$: \\ \begin{align*}{{\rm d}\over{{\rm d}t}} {\rm e}^{{\rm j}(\omega t + \varphi_x)} = {\rm j} \cdot {\rm e}^{{\rm j}(\omega t + \varphi_x)}\end{align*} 
-  - The integral of a sinusoidal value - and therefore a phasor - can simply be written as "$\cdot (-{\rm j})$", which also means a phase shift of $-90°$.((in general, here the integration constant must be considered. This is however often be neglectable since only AC values (without a DC value) are considered.)) <WRAP>  +  - The integral of a sinusoidal value - and therefore a phasor - can simply be written as "$\cdot (-{\rm j})$", \\ which also means a phase shift of $-90°$.((in general, here the integration constant must be considered. This is however often neglectable since only AC values (without a DC value) are considered.)) <WRAP>  
-$                      \int {\rm e}^{{\rm j}(\omega t + \varphi_x)} +\begin{align*} 
 +                     \int {\rm e}^{{\rm j}(\omega t + \varphi_x)} 
   = {{1}\over{\rm j}} \cdot {\rm e}^{{\rm j}(\omega t + \varphi_x)}    = {{1}\over{\rm j}} \cdot {\rm e}^{{\rm j}(\omega t + \varphi_x)} 
-  =         - {\rm j} \cdot {\rm e}^{{\rm j}(\omega t + \varphi_x)}$+  =         - {\rm j} \cdot {\rm e}^{{\rm j}(\omega t + \varphi_x)} 
 +\end{align*}
 </WRAP> </WRAP>