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electrical_engineering_and_electronics_1:block07 [2025/10/17 22:20] – mexleadminelectrical_engineering_and_electronics_1:block07 [2025/11/05 03:08] (aktuell) – mexleadmin
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 <callout icon="fa fa-lightbulb-o" color="blue"> <callout icon="fa fa-lightbulb-o" color="blue">
   - Real sources are modeled by an **ideal source** plus **internal resistance** $R_{\rm i}$; the terminal voltage **drops under load**.     - Real sources are modeled by an **ideal source** plus **internal resistance** $R_{\rm i}$; the terminal voltage **drops under load**.  
-  - **Efficiency** $\eta$ compares *delivered* to *drawn* power. In the simple DC source–load case, $\displaystyle \eta=\frac{R_{\rm L}}{R_{\rm L}+R_{\rm i}}$ (dimensionless). High-efficiency design wants $R_{\rm L}\gg R_{\rm i}$.+  - **Efficiency** $\eta$ compares **delivered** to **drawn** power. In the simple DC source–load case, $\displaystyle \eta=\frac{R_{\rm L}}{R_{\rm L}+R_{\rm i}}$ (dimensionless). High-efficiency design wants $R_{\rm L}\gg R_{\rm i}$.
   - **Utilization rate** $\varepsilon$ compares delivered power to the **maximum** available from the ideal source: $\displaystyle \varepsilon=\frac{R_{\rm L}R_{\rm i}}{(R_{\rm L}+R_{\rm i})^2}$. It peaks at $R_{\rm L}=R_{\rm i}$ with $\varepsilon_{\max}=25~\%$. This is the **maximum power transfer** condition.    - **Utilization rate** $\varepsilon$ compares delivered power to the **maximum** available from the ideal source: $\displaystyle \varepsilon=\frac{R_{\rm L}R_{\rm i}}{(R_{\rm L}+R_{\rm i})^2}$. It peaks at $R_{\rm L}=R_{\rm i}$ with $\varepsilon_{\max}=25~\%$. This is the **maximum power transfer** condition. 
   - Different goals → different $R_{\rm L}$:   - Different goals → different $R_{\rm L}$:
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 Application: Application:
   - In __power engineering__ $\eta \rightarrow 100\%$ is often desired: We want the maximum power output with the lowest losses at the internal resistance of the source. Thus, the internal resistance of the source should be low compared to the load $R_{\rm L} \gg R_{\rm i} $.    - In __power engineering__ $\eta \rightarrow 100\%$ is often desired: We want the maximum power output with the lowest losses at the internal resistance of the source. Thus, the internal resistance of the source should be low compared to the load $R_{\rm L} \gg R_{\rm i} $. 
 +
 +==== The Utilization Rate ====
  
 The **utilization rate**  $\varepsilon$ describes the delivered power $P_{\rm out}$ concerning the maximum possible power $P_{\rm in, max}$ of the ideal source.  The **utilization rate**  $\varepsilon$ describes the delivered power $P_{\rm out}$ concerning the maximum possible power $P_{\rm in, max}$ of the ideal source. 
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             = {{R_{\rm L}}                                 \over {(R_{\rm L}+R_{\rm i})}}\cdot {{R_{\rm i}} \over {(R_{\rm L}+R_{\rm i})}}}              = {{R_{\rm L}}                                 \over {(R_{\rm L}+R_{\rm i})}}\cdot {{R_{\rm i}} \over {(R_{\rm L}+R_{\rm i})}}} 
 \end{align*} \end{align*}
- 
-==== The Utilization Rate ==== 
  
 In other applications, the **absolute maximum power** has to be taken from the source, without consideration of the losses via the internal resistance. This corresponds to the situation (2.) in <imgref imageNo14>. For this purpose, the internal resistance of the source and the load are matched. This case is called **{{https://en.wikipedia.org/wiki/Impedance_matching|impedance matching}}  ** (the impedance is up to for DC circuits equal to the resistance). The utilization rate here becomes maximum: $\varepsilon = 25~\%$. In other applications, the **absolute maximum power** has to be taken from the source, without consideration of the losses via the internal resistance. This corresponds to the situation (2.) in <imgref imageNo14>. For this purpose, the internal resistance of the source and the load are matched. This case is called **{{https://en.wikipedia.org/wiki/Impedance_matching|impedance matching}}  ** (the impedance is up to for DC circuits equal to the resistance). The utilization rate here becomes maximum: $\varepsilon = 25~\%$.
  
 Application: Application:
-  - In __communications engineering__ the impedance matching of the source (the antenna) and the load (the signal-acquiring microcontroller) uses resistors, capacitors, and inductors.  There, we want to get the maximum power out of an antenna. For this purpose, the internal resistance of the source (e.g., a receiver) and the load (e.g., the downstream evaluation) are matched. An example can be seen in this {{electrical_engineering_1:anp084a_en_-_impedance_matching_for_near_field_com.pdf#page=4|application note for near field communication}}.+  - In __communications engineering__ the impedance matching of the source (the antenna) and the load (the signal-acquiring microcontroller) uses resistors, capacitors, and inductors.  There, we want to get the maximum power out of an antenna. For this purpose, the internal resistance of the source (e.g., a receiver) and the load (e.g., the downstream evaluation) are matched. An example can be seen in this {{anp084a_en_-_impedance_matching_for_near_field_commu.pdf#page=4|application note for near field communication}}.
   - Furthermore, also for __photovoltaic cells__ one wants to get the maximum power out. In this case, the concept is often called **{{https://en.wikipedia.org/wiki/Maximum_power_point_tracking|Maximum Power Point Tracking (MPPT)}}  **   - Furthermore, also for __photovoltaic cells__ one wants to get the maximum power out. In this case, the concept is often called **{{https://en.wikipedia.org/wiki/Maximum_power_point_tracking|Maximum Power Point Tracking (MPPT)}}  **
  
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 A source has $U_0=9.0~\rm V$, $R_{\rm i}=1.0~\Omega$. A source has $U_0=9.0~\rm V$, $R_{\rm i}=1.0~\Omega$.
   - (a) Choose $R_{\rm L}=9.0~\Omega$. Compute $I_{\rm L}$, $U_{\rm L}$, $P_{\rm L}$, $\eta$, $\varepsilon$.   - (a) Choose $R_{\rm L}=9.0~\Omega$. Compute $I_{\rm L}$, $U_{\rm L}$, $P_{\rm L}$, $\eta$, $\varepsilon$.
-  - (b) Choose $R_{\rm L}=1.0~\Omega$. Repeat. Which choice maximizes $P_{\rm L}$? Which yields higher $\eta$?  +  - (b) Choose $R_{\rm L}=1.0~\Omega$. Repeat. \\ \\ Which choice maximizes $P_{\rm L}$? Which yields higher $\eta$?  
 **Strategy:** use the boxed formulas in this block; for (b) note $R_{\rm L}=R_{\rm i} \Rightarrow \eta=50~\%$.  **Strategy:** use the boxed formulas in this block; for (b) note $R_{\rm L}=R_{\rm i} \Rightarrow \eta=50~\%$. 
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