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ee2:task_abh4vhlgczdbni37_with_calculation [2024/07/04 02:07]
mexleadmin angelegt 		ee2:task_abh4vhlgczdbni37_with_calculation [2024/07/04 02:36] (aktuell)
mexleadmin angelegt
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 #@HiddenBegin_HTML~abH4vhlGCZdBni37_11,Path~@# #@HiddenBegin_HTML~abH4vhlGCZdBni37_11,Path~@#
-The complex impedance $\underline{Z}$ for a resistive-inductive load (=$R$-$L$ series circuit) is given as  +  * The amplitude values $\hat{U}$, $\hat{I}$ are given directly by the coefficient of the cosine and sine functions 
-\begin{align*} +  * For the RMS values of sinusoidal functions the amplitudes have to be multiplied with ${{1}\over{2}}\sqrt{2}$ 
-\underline{Z}  &= {\rm j} \cdot X_L + R_{\rm M} \\ +#@HiddenEnd_HTML~abH4vhlGCZdBni37_11,Path~@#
-               &= {\rm j} \cdot 2\pi \cdot f \cdot L_{\rm M} + R_{\rm M} \\ +
-\end{align*}+
  
-The Pythagorean theorem can derive the absolute value: +#@HiddenBegin_HTML~abH4vhlGCZdBni37_12,Result~@# 
-\begin{align*} +Amplitude values: 
-|\underline{Z}|&= \sqrt{ (2\pi \cdot f \cdot L_{\rm M})^2 + R_{\rm M}^2 }\\ +  * $\hat{U} = 50{~\rm V}$  
-\end{align*}+  * $\hat{I} = 30{~\rm A}$
  
-#@HiddenEnd_HTML~abH4vhlGCZdBni37_11,Path~@#+RMS values: 
 +  * $U = 35.4{~\rm V}$  
 +  * $I = 21.2{~\rm A}$
  
-#@HiddenBegin_HTML~abH4vhlGCZdBni37_12,Result~@# 
-\begin{align*} 
-Z = \sqrt{ (2\pi \cdot f \cdot L_{\rm M})^2 + R_{\rm M}^2 } 
-\end{align*} 
 #@HiddenEnd_HTML~abH4vhlGCZdBni37_12,Result~@# #@HiddenEnd_HTML~abH4vhlGCZdBni37_12,Result~@#
  
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 #@HiddenBegin_HTML~abH4vhlGCZdBni37_21,Path~@# #@HiddenBegin_HTML~abH4vhlGCZdBni37_21,Path~@#
-The complex impedance $\underline{Z}$ for a resistive-inductive load (=$R$-$L$ series circuit) is given as +The frequency can be derived by the term in the sine function:
 \begin{align*} \begin{align*}
-\underline{Z}  &= {\rm j} \cdot X_L + R_{\rm M} \\ +\omega       &= 6000 {{1}\over{\rm s}} \\ 
-               &= {\rm j} \cdot 2\pi \cdot f \cdot L_{\rm M} + R_{\rm M} \\+2\pi \cdot f &= 6000 {{1}\over{\rm s}} \\ 
 +           f &= {{6000}\over{2\pi}} {{1}\over{\rm s}} \\ 
 +           f &= 954.93... ~\rm Hz \\
 \end{align*} \end{align*}
  
-The Pythagorean theorem can derive the absolute value:+For the phase $\varphi$, we have to subtract $\varphi_i $ from $\varphi_u$. \\ 
 +But to get these values, both the $u(t)$ and $i(t)$ need to have the same sinusoidal function! 
 +Therefore:  
 +  * $\varphi_i = 5$ 
 +  * $\varphi_u = 4 + {{\pi}\over{2}}$ 
 + 
 +By this we get for $\varphi$
 \begin{align*} \begin{align*}
-|\underline{Z}|&= \sqrt{ (2\pi \cdot f \cdot L_{\rm M})^2 + R_{\rm M}^2 }\\+\varphi &= \varphi_u - \varphi_i \\ 
 +        &=  4 + {{\pi}\over{2}} - 5  \\ 
 +        &= 2.14159...  \\
 \end{align*} \end{align*}
  
 +Converted in degree: 
 +\begin{align*}
 +\varphi &=  2.14159... \cdot {{360°}\over{2\pi}} \\
 +        &= 32.7042...°  \\
 +\end{align*}
 #@HiddenEnd_HTML~abH4vhlGCZdBni37_21,Path~@# #@HiddenEnd_HTML~abH4vhlGCZdBni37_21,Path~@#
  
 #@HiddenBegin_HTML~abH4vhlGCZdBni37_22,Result~@# #@HiddenBegin_HTML~abH4vhlGCZdBni37_22,Result~@#
-\begin{align*} +  * $f = 955 ~\rm Hz$ 
-Z = \sqrt{ (2\pi \cdot f \cdot L_{\rm M})^2 + R_{\rm M}^2 } +  * $\varphi = +32.7°$
-\end{align*}+
 #@HiddenEnd_HTML~abH4vhlGCZdBni37_22,Result~@# #@HiddenEnd_HTML~abH4vhlGCZdBni37_22,Result~@#
  
 c) Is the measured element resistive-capacitive or resistive-inductive? \\ The quantities are available in the consumer arrow system. (hard)   c) Is the measured element resistive-capacitive or resistive-inductive? \\ The quantities are available in the consumer arrow system. (hard)  
- 
-#@HiddenBegin_HTML~abH4vhlGCZdBni37_31,Path~@# 
-The complex impedance $\underline{Z}$ for a resistive-inductive load (=$R$-$L$ series circuit) is given as  
-\begin{align*} 
-\underline{Z}  &= {\rm j} \cdot X_L + R_{\rm M} \\ 
-               &= {\rm j} \cdot 2\pi \cdot f \cdot L_{\rm M} + R_{\rm M} \\ 
-\end{align*} 
- 
-The Pythagorean theorem can derive the absolute value: 
-\begin{align*} 
-|\underline{Z}|&= \sqrt{ (2\pi \cdot f \cdot L_{\rm M})^2 + R_{\rm M}^2 }\\ 
-\end{align*} 
- 
-#@HiddenEnd_HTML~abH4vhlGCZdBni37_31,Path~@# 
  
 #@HiddenBegin_HTML~abH4vhlGCZdBni37_32,Result~@# #@HiddenBegin_HTML~abH4vhlGCZdBni37_32,Result~@#
-\begin{align*} +The phase shift is positive - therefore, the element is resistive-inductive.
-Z = \sqrt{ (2\pi \cdot f \cdot L_{\rm M})^2 + R_{\rm M}^2 } +
-\end{align*}+
 #@HiddenEnd_HTML~abH4vhlGCZdBni37_32,Result~@# #@HiddenEnd_HTML~abH4vhlGCZdBni37_32,Result~@#
  
  
 #@TaskEnd_HTML@# #@TaskEnd_HTML@#