106 lines
3.7 KiB
Markdown
Executable File
106 lines
3.7 KiB
Markdown
Executable File
## Maskregeln och nodregeln
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*Maskregeln*: I varje sluten krets (maska) är summan av spänningarna lika med noll.
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*Nodregeln*: I varje nod är summan av inflödande strömmar lika med summan av utflödande strömmar.
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<note>
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Kirchhoffs lagar
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</note>
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---
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#### Lösningsgång
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* givet: $R_1 = R_2 = R_3 = \qty{10}{\kilo\ohm}$
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* givet: $I_3 = I_2 = \qty{1}{\milli\ampere}$
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* sökt: $U$
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<fragment>
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$R_{\mathrm{tot}} = R_1 + \frac{R_2 \cdot R_3}{R_2 + R_3} = \qty{10}{\kilo\ohm} + \frac{\qty{10}{\kilo\ohm} \cdot \qty{10}{\kilo\ohm}}{\qty{10}{\kilo\ohm} + \qty{10}{\kilo\ohm}} = \qty{15}{\kilo\ohm}$
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</fragment>
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<fragment>
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$I = I_2 + I_3 = \qty{1}{\milli\ampere} + \qty{1}{\milli\ampere} = \qty{2}{\milli\ampere}$
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</fragment>
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<fragment>
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$U = R_{\mathrm{tot}} \cdot I = \qty{15}{\kilo\ohm} \cdot \qty{2}{\milli\ampere} = \qty{30}{\volt}$
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</fragment>
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---
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#### Lösningsgång
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* givet: $R_1 = R_2 = R_3 = \qty{10}{\kilo\ohm}$
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* givet: $U=\qty{15}{\volt}$
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* sökt: $I_3$
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<fragment>
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$R_{\mathrm{tot}} = R_1 + \frac{R_2 \cdot R_3}{R_2 + R_3} = \qty{10}{\kilo\ohm} + \frac{\qty{10}{\kilo\ohm} \cdot \qty{10}{\kilo\ohm}}{\qty{10}{\kilo\ohm} + \qty{10}{\kilo\ohm}} = \qty{15}{\kilo\ohm}$
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</fragment>
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<fragment>
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$\frac{U_3}{U} = \frac{R_2 \parallel R_3}{R_{\mathrm{tot}}} \Rightarrow U_3 = \frac{R_2 \parallel R_3}{R_{\mathrm{tot}}} \cdot U = \frac{\qty{5}{\kilo\ohm}}{\qty{15}{\kilo\ohm}} \cdot \qty{15}{\volt} = \qty{5}{\volt}$
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</fragment>
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<fragment>
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$I_3 = \frac{U_3}{R_3} = \frac{\qty{5}{\volt}}{\qty{10}{\kilo\ohm}} = \qty{0,5}{\milli\ampere}$
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</fragment>
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---
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#### Lösningsgång
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* givet: $R_1 = R_2 = R_3 = \qty{10}{\kilo\ohm}$
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* givet: $U=\qty{15}{\volt}$
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* sökt: $P_2$
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<fragment>
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$\frac{U_2}{U} = \frac{R_2 \parallel R_3}{R_{\mathrm{tot}}} \Rightarrow U_2 = \frac{R_2 \parallel R_3}{R_{\mathrm{tot}}} \cdot U = \frac{\qty{5}{\kilo\ohm}}{\qty{15}{\kilo\ohm}} \cdot \qty{15}{\volt} = \qty{5}{\volt}$
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</fragment>
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<fragment>
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$P_2 = \frac{U_2^2}{R_2} = \frac{(\qty{5}{\volt})^2}{\qty{10}{\kilo\ohm}} = \qty{2,5}{\milli\watt}$
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</fragment>
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--- style="font-size: 0.7em;"
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#### Lösningsgång
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<left>
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* givet: $R = \qtyrange{0}{1}{\kilo\ohm}$
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* givet: $R_1 = \qty{200}{\ohm}$
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* givet: $R_2 = \qty{100}{\ohm}$
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* givet: $R_3 = \qty{200}{\ohm}$
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</left>
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<right>
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$R_{\mathrm{tot}} = R_1 + \frac{R_2 \cdot (R_3 + R)}{R_2 + (R_3 + R)}$
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</right>
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<fragment>
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Vid $R = \qty{0}{\ohm}$:
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$R_{\mathrm{tot}} = \qty{200}{\ohm} + \frac{\qty{100}{\ohm} \cdot (\qty{200}{\ohm} + \qty{0}{\ohm})}{\qty{100}{\ohm} + \qty{200}{\ohm} + \qty{0}{\ohm}} \approx \qty{267}{\ohm}$
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</fragment>
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<fragment>
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Vid $R = \qty{1}{\kilo\ohm}$:
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$R_{\mathrm{tot}} = \qty{200}{\ohm} + \frac{\qty{100}{\ohm} \cdot (\qty{200}{\ohm} + \qty{1}{\kilo\ohm})}{\qty{100}{\ohm} + \qty{200}{\ohm} +\qty{1}{\kilo\ohm}} \approx \qty{292}{\ohm}$
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</fragment>
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---
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#### Lösningsgång
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* givet: $R_1 = R_3 = \qty{2,2}{\kilo\ohm}$
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* givet: $R_2 = R_4 = \qty{220}{\ohm}$
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* sökt: $R_{\mathrm{tot}}$
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<fragment>
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$R_1 || R_3 + R_2 || R_4 = \qty{1100}{\ohm} + \qty{110}{\ohm} = \qty{1210}{\ohm}$
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</fragment>
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--- style="font-size: 0.7em;"
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#### Lösningsgång
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* givet: $R_1 = \qty{10}{\kilo\ohm}$
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* givet: $R_2 = \qty{2,2}{\kilo\ohm}$
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* givet: $R_L = \qty{8,2}{\kilo\ohm}$
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* givet: $U_B = \qty{12}{\volt}$
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* sökt: $U_2$
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<fragment>
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$\frac{U_2}{U_B} = \frac{R_2 \parallel R_L}{R_{\mathrm{tot}}}$
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$R_2 \parallel R_L = \frac{R_2 \cdot R_L}{R_2 + R_L} = \frac{\qty{2,2}{\kilo\ohm} \cdot \qty{8,2}{\kilo\ohm}}{\qty{2,2}{\kilo\ohm} + \qty{8,2}{\kilo\ohm}} = \qty{1,74}{\kilo\ohm}$
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$R_{\mathrm{tot}} = R_1 + R_2 \parallel R_L = \qty{10}{\kilo\ohm} + \qty{1,74}{\kilo\ohm} = \qty{11,74}{\kilo\ohm}$
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</fragment>
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<fragment>
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$U_2 = \frac{R_2 \parallel R_L}{R_{\mathrm{tot}}} \cdot U_B = \frac{\qty{1,74}{\kilo\ohm}}{\qty{11,74}{\kilo\ohm}} \cdot \qty{12}{\volt} \approx \qty{1,8}{\volt}$
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</fragment>
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