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Beide Seiten der vorigen Revision Vorhergehende Überarbeitung Nächste Überarbeitung | Vorhergehende Überarbeitung | ||
electrical_engineering_1:non-ideal_sources_and_two_terminal_networks [2023/12/03 23:18] mexleadmin |
electrical_engineering_1:non-ideal_sources_and_two_terminal_networks [2023/12/04 00:22] (aktuell) mexleadmin |
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Zeile 455: | Zeile 455: | ||
</ | </ | ||
- | # | + | # |
Two heater resistors (both with $R_\rm L = 0.5 ~\Omega$) shall be supplied with two lithium-ion-batteries (both with $U_{\rm S} = 3.3 ~\rm V$, $R_{\rm i} = 0.1 ~\Omega$). | Two heater resistors (both with $R_\rm L = 0.5 ~\Omega$) shall be supplied with two lithium-ion-batteries (both with $U_{\rm S} = 3.3 ~\rm V$, $R_{\rm i} = 0.1 ~\Omega$). | ||
Zeile 476: | Zeile 476: | ||
\end{align*} | \end{align*} | ||
- | <callout type=" | + | As near the resulting equivalent internal resistance approaches the resulting equivalent load resistance, as higher the utilization rate $\varepsilon$ will be.\\ |
- | **As near the resulting equivalent internal resistance approaches the resulting equivalent load resistance, as higher the utilization rate $\varepsilon$ will be.\\ | + | Therefore, a series configuration of the batteries ($2 R_{\rm i} = 0.2~\Omega$) and a parallel configuration of the load (${{1}\over{2}} R_{\rm L}= 0.25~\Omega$) will have the highest output. |
- | Therefore, a series configuration of the batteries** ($2 R_{\rm i} = 0.2~\Omega$) | + | |
- | </ | + | |
- | + | ||
- | A detailed analysis is shown here | + | |
- | {{drawio> | + | |
# | # | ||
+ | |||
+ | # | ||
+ | The following configuration has the maximum output power. | ||
+ | |||
+ | {{drawio> | ||
+ | # | ||
+ | |||
3. What is the value of the maximum power $P_{\rm L ~max}$? | 3. What is the value of the maximum power $P_{\rm L ~max}$? | ||
+ | |||
+ | # | ||
+ | The maximum utilization rate is: | ||
+ | \begin{align*} | ||
+ | \varepsilon &= {{{{1}\over{2}} R_{\rm L} \cdot 2 R_{\rm i} } \over { ({{1}\over{2}} R_{\rm L} + 2 R_{\rm i} )^2}} \\ | ||
+ | &= { {0.25 ~\Omega | ||
+ | &= 24.6~\% | ||
+ | \end{align*} | ||
+ | |||
+ | Therefore, the maximum power is: | ||
+ | \begin{align*} | ||
+ | \varepsilon | ||
+ | \rightarrow P_{\rm out} &= \varepsilon | ||
+ | & | ||
+ | & | ||
+ | \end{align*} | ||
+ | |||
+ | # | ||
+ | |||
+ | # | ||
+ | \begin{align*} | ||
+ | P_{\rm out} = 26.8 W | ||
+ | \end{align*} | ||
+ | # | ||
4. Which circuit has the highest efficiency? | 4. Which circuit has the highest efficiency? | ||
+ | |||
+ | # | ||
+ | The highest efficiency $\eta$ is given when the output power compared to the input power is minimal. \\ | ||
+ | A parallel configuration of the batteries (${{1}\over{2}} R_{\rm i} = 0.05~\Omega$) and a series configuration of the load ($2 R_{\rm L}= 1.0~\Omega$) will have the highest efficiency. | ||
+ | # | ||
+ | |||
+ | # | ||
+ | {{drawio> | ||
+ | # | ||
5. What is the value of the highest efficiency? | 5. What is the value of the highest efficiency? | ||
+ | |||
+ | # | ||
+ | The efficiency $\eta$ is given as: | ||
+ | \begin{align*} | ||
+ | \eta &= { {2 R_{\rm L} }\over{ 2 R_{\rm L}+ {{1}\over{2}} R_{\rm i} }} \\ | ||
+ | &= { { 1.0~\Omega }\over{ 1.0~\Omega + 0.05~\Omega }} | ||
+ | \end{align*} | ||
+ | |||
+ | # | ||
+ | |||
+ | # | ||
+ | \begin{align*} | ||
+ | \eta = 95.2~\% | ||
+ | \end{align*} | ||
+ | # | ||
+ | \\ \\ | ||
+ | # | ||
+ | {{drawio> | ||
+ | |||
+ | # | ||
+ | |||
# | # |