Ersatzwiderstände berechnen
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The following quantities appear in the problem:
elektrischer Widerstand \(R\) / Federkonstante \(D\) /
The following formulas must be used to solve the exercise:
\(\frac{1}{R} = \frac{1}{R_1} + \frac{1}{R_2} \quad \) \(\frac{1}{D} = \frac{1}{D_1} + \frac{1}{D_2} \quad \) \(R = R_1 + R_2 \quad \)
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Exercise:
Berechne die Ersatzwiderstände aller folg abgebildeten Schaltungen aus zwei drei oder vier Widerständen! center circuitikz %ST ROW START %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% scope draw to european resistor l^ohm ; draw to european resistor l^ohm .; scope scopexshiftcm draw -. to european resistor l^ohm -.; draw . to european resistor l^ohm .; draw -. to european resistor l^ohm -.; draw .---.; draw .---.; draw ---.; draw --.; scope scopexshiftcm yshiftcm draw . to european resistor l^ohm .; draw -. to european resistor l^ohm -.; draw .---.; draw .---.; draw ---.; draw to european resistor l^ohm ; draw to european resistor l^ohm ; scope %ND ROW START %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% scopexshiftcm yshift-.cm draw . to european resistor l^ohm .; draw -. to european resistor l^ohm -.; draw .---.; draw .---.; draw ---.; draw --.; scope scopexshiftcm yshift-.cm draw . to european resistor l^ohm .; draw . to european resistor l^ohm .; draw -. to european resistor l^ohm -.; draw .---.; draw .---.; draw ---.; draw --.; scope scopexshiftcm yshift-.cm draw . to european resistor l^ohm .; draw . to european resistor l^ohm .; draw -. to european resistor l^ohm -.; draw .---.; draw .---.; draw ---.; draw to european resistor l^ohm ; scope %RD ROW START %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% scopexshiftcm yshift-.cm draw to european resistor l^ohm ; draw - to european resistor l^ohm -; draw to european resistor l^ohm ; draw ---; draw ---; draw ---.; draw --.; scope scopexshiftcm yshift-.cm draw . to european resistor l^ohm .; draw . to european resistor l^ohm .; draw -. to european resistor l^ohm -.; draw -. to european resistor l^ohm -.; draw .---.; draw .---.; draw ---.; draw --.; scope scopexshift.cm yshift-.cm draw to european resistor l^ohm ; draw to european resistor l^ohm ; draw to european resistor l^ohm ; draw - to european resistor l^ohm -; draw ---; draw ---; draw ---.; draw --.; scope %TH ROW START %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% scopexshiftcm yshift-cm draw . to european resistor l^ohm .; draw -. to european resistor l^ohm -.; draw .---.; draw .---.; draw ---.; draw to european resistor l^ohm ; draw -.-. to european resistor l^ohm -.; draw ---.; draw ---.---.-.; draw -.-.----.; draw -.--.-.; scope scopexshift.cm yshift-.cm draw . to european resistor l^ohm .; draw . to european resistor l^ohm .; draw -. to european resistor l^ohm -.; draw -. to european resistor l^ohm -.; draw ----; draw ----; draw -.---.; draw -.---.; draw .---; draw .---; draw -.---.-.; draw -.--.-.; scope scopexshiftcm yshift-.cm draw to european resistor l^ohm ; draw - to european resistor l^ohm -; draw to european resistor l^ohm ; draw ---; draw ---; draw ---.; draw to european resistor l^ohm ; scope circuitikz center
Solution:
Die Reihenfolge der Widerstände ist jeweils von oben nach unten und bei zweien auf gleicher Höhe von links nach rechts. abcliste abc newqtyReohm newqtyRzohm % solqtyRR_+R_Ren+Rznohm al R Rf Re + Rz RTTT abc newqtyReohm newqtyRz.ohm newqtyRdohm % solqtyRfracR_R_R_+R_ + R_Ren*Rdn/Ren+Rdn+Rznohm al R R_ + R_ Rf fracReRdRe+Rd + Rz RTTT abc newqtyReohm newqtyRzohm newqtyRdohm newqtyRvohm % solqtyRfracR_R_R_+R_ + R_ + R_Ren*Rvn/Ren+Rvn+Rzn+Rdnohm al R R_ + R_ + R_ Rf fracReRvRe+Rv + Rz + Rd RTTT abc newqtyRe.ohm newqtyRzohm % solqtyRfracR_R_R_+R_Ren*Rzn/Ren+Rznohm al R Rf fracRe RzRe + Rz RTTT abc newqtyReohm newqtyRzohm newqtyRdohm % solqtyRfracR_+R_R_R_+R_+R_Ren+Rzn*Rdn/Ren+Rzn+Rdnohm al R fracR_R_R_ + R_ Rf fracqtyRe + Rz RdRe + Rz + Rd RTTT abc newqtyRe.ohm newqtyRzohm newqtyRdohm newqtyRvohm % solqtyRfracR_+R_R_R_+R_+R_ + R_Ren+Rzn*Rvn/Ren+Rzn+Rvn+Rdnohm al R R_ + R_ fracR_R_R_ + R_ + R_ Rf fracqtyRe + Rz RvRe + Rz + Rv + Rd RTTT abc newqtyRe.ohm newqtyRzohm newqtyRdohm % solqtyRfracR_R_R_R_R_ + R_R_ + R_R_/Ren+/Rzn+/Rdn**-ohm al R qtyfracR_ + fracR_ + fracR_^- Rf fracRe Rz RdRz Rd + Re Rd + Re Rz RTTT abc newqtyReohm newqtyRzohm newqtyRdohm newqtyRvohm % solqtyRfracR_+R_R_+R_R_+R_+R_+R_/Ren+Rzn + /Rdn+Rvn^-ohm al R fracR_R_R_ + R_ Rf fracqtyRe + Rz qtyRd+RvRe + Rz + Rd + Rv RTTT abc newqtyReohm newqtyRzohm newqtyRdohm newqtyRvohm % solqtyRqtyfracR_+R_ + fracR_ + fracR_^-/Ren+Rzn + /Rdn + /Rvn**-ohm al R qtyfracR_ + fracR_ + fracR_^- Rf qtyfracRe+Rz + fracRd + fracRv^- RTTT abc newqtyReohm newqtyRzohm newqtyRdohm newqtyRv.ohm % solqtyRedfracR_R_R_+R_Ren*Rdn/Ren+Rdnohm al R_ Redf fracR_R_R_+R_ RedTTTT solqtyRezdR_ + R_Redn+Rznohm al R_ Rezdf RedTTTT + Rz RezdTTTT solqtyRfracR_R_R_ + R_Rezdn*Rvn/Rezdn+Rvnohm al R Rf fracRezdTTTT RvRezdTTTT + Rv RTTT abc newqtyRe.ohm newqtyRzohm newqtyRdohm newqtyRvohm % solqtyRdvfracR_R_R_+R_Rdn*Rvn/Rdn+Rvnohm al R_ Rdvf fracRd RvRd + Rv Rdv solqtyRfracR_+R_R_R_+R_+R_Ren+Rzn*Rdvn/Ren+Rzn+Rdvnohm al R Rf fracqtyRe + Rz RdvRe + Rz + Rdv RTTT abc newqtyReohm newqtyRzohm newqtyRdohm newqtyRvohm % solqtyRqtyfracR_ + fracR_ + fracR_^- + R_/Ren+/Rzn+/Rvn**-+Rdnohm al R Rf qtyfracRe + fracRz + fracRv^- + Rd RTTT abcliste
Berechne die Ersatzwiderstände aller folg abgebildeten Schaltungen aus zwei drei oder vier Widerständen! center circuitikz %ST ROW START %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% scope draw to european resistor l^ohm ; draw to european resistor l^ohm .; scope scopexshiftcm draw -. to european resistor l^ohm -.; draw . to european resistor l^ohm .; draw -. to european resistor l^ohm -.; draw .---.; draw .---.; draw ---.; draw --.; scope scopexshiftcm yshiftcm draw . to european resistor l^ohm .; draw -. to european resistor l^ohm -.; draw .---.; draw .---.; draw ---.; draw to european resistor l^ohm ; draw to european resistor l^ohm ; scope %ND ROW START %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% scopexshiftcm yshift-.cm draw . to european resistor l^ohm .; draw -. to european resistor l^ohm -.; draw .---.; draw .---.; draw ---.; draw --.; scope scopexshiftcm yshift-.cm draw . to european resistor l^ohm .; draw . to european resistor l^ohm .; draw -. to european resistor l^ohm -.; draw .---.; draw .---.; draw ---.; draw --.; scope scopexshiftcm yshift-.cm draw . to european resistor l^ohm .; draw . to european resistor l^ohm .; draw -. to european resistor l^ohm -.; draw .---.; draw .---.; draw ---.; draw to european resistor l^ohm ; scope %RD ROW START %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% scopexshiftcm yshift-.cm draw to european resistor l^ohm ; draw - to european resistor l^ohm -; draw to european resistor l^ohm ; draw ---; draw ---; draw ---.; draw --.; scope scopexshiftcm yshift-.cm draw . to european resistor l^ohm .; draw . to european resistor l^ohm .; draw -. to european resistor l^ohm -.; draw -. to european resistor l^ohm -.; draw .---.; draw .---.; draw ---.; draw --.; scope scopexshift.cm yshift-.cm draw to european resistor l^ohm ; draw to european resistor l^ohm ; draw to european resistor l^ohm ; draw - to european resistor l^ohm -; draw ---; draw ---; draw ---.; draw --.; scope %TH ROW START %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% scopexshiftcm yshift-cm draw . to european resistor l^ohm .; draw -. to european resistor l^ohm -.; draw .---.; draw .---.; draw ---.; draw to european resistor l^ohm ; draw -.-. to european resistor l^ohm -.; draw ---.; draw ---.---.-.; draw -.-.----.; draw -.--.-.; scope scopexshift.cm yshift-.cm draw . to european resistor l^ohm .; draw . to european resistor l^ohm .; draw -. to european resistor l^ohm -.; draw -. to european resistor l^ohm -.; draw ----; draw ----; draw -.---.; draw -.---.; draw .---; draw .---; draw -.---.-.; draw -.--.-.; scope scopexshiftcm yshift-.cm draw to european resistor l^ohm ; draw - to european resistor l^ohm -; draw to european resistor l^ohm ; draw ---; draw ---; draw ---.; draw to european resistor l^ohm ; scope circuitikz center
Solution:
Die Reihenfolge der Widerstände ist jeweils von oben nach unten und bei zweien auf gleicher Höhe von links nach rechts. abcliste abc newqtyReohm newqtyRzohm % solqtyRR_+R_Ren+Rznohm al R Rf Re + Rz RTTT abc newqtyReohm newqtyRz.ohm newqtyRdohm % solqtyRfracR_R_R_+R_ + R_Ren*Rdn/Ren+Rdn+Rznohm al R R_ + R_ Rf fracReRdRe+Rd + Rz RTTT abc newqtyReohm newqtyRzohm newqtyRdohm newqtyRvohm % solqtyRfracR_R_R_+R_ + R_ + R_Ren*Rvn/Ren+Rvn+Rzn+Rdnohm al R R_ + R_ + R_ Rf fracReRvRe+Rv + Rz + Rd RTTT abc newqtyRe.ohm newqtyRzohm % solqtyRfracR_R_R_+R_Ren*Rzn/Ren+Rznohm al R Rf fracRe RzRe + Rz RTTT abc newqtyReohm newqtyRzohm newqtyRdohm % solqtyRfracR_+R_R_R_+R_+R_Ren+Rzn*Rdn/Ren+Rzn+Rdnohm al R fracR_R_R_ + R_ Rf fracqtyRe + Rz RdRe + Rz + Rd RTTT abc newqtyRe.ohm newqtyRzohm newqtyRdohm newqtyRvohm % solqtyRfracR_+R_R_R_+R_+R_ + R_Ren+Rzn*Rvn/Ren+Rzn+Rvn+Rdnohm al R R_ + R_ fracR_R_R_ + R_ + R_ Rf fracqtyRe + Rz RvRe + Rz + Rv + Rd RTTT abc newqtyRe.ohm newqtyRzohm newqtyRdohm % solqtyRfracR_R_R_R_R_ + R_R_ + R_R_/Ren+/Rzn+/Rdn**-ohm al R qtyfracR_ + fracR_ + fracR_^- Rf fracRe Rz RdRz Rd + Re Rd + Re Rz RTTT abc newqtyReohm newqtyRzohm newqtyRdohm newqtyRvohm % solqtyRfracR_+R_R_+R_R_+R_+R_+R_/Ren+Rzn + /Rdn+Rvn^-ohm al R fracR_R_R_ + R_ Rf fracqtyRe + Rz qtyRd+RvRe + Rz + Rd + Rv RTTT abc newqtyReohm newqtyRzohm newqtyRdohm newqtyRvohm % solqtyRqtyfracR_+R_ + fracR_ + fracR_^-/Ren+Rzn + /Rdn + /Rvn**-ohm al R qtyfracR_ + fracR_ + fracR_^- Rf qtyfracRe+Rz + fracRd + fracRv^- RTTT abc newqtyReohm newqtyRzohm newqtyRdohm newqtyRv.ohm % solqtyRedfracR_R_R_+R_Ren*Rdn/Ren+Rdnohm al R_ Redf fracR_R_R_+R_ RedTTTT solqtyRezdR_ + R_Redn+Rznohm al R_ Rezdf RedTTTT + Rz RezdTTTT solqtyRfracR_R_R_ + R_Rezdn*Rvn/Rezdn+Rvnohm al R Rf fracRezdTTTT RvRezdTTTT + Rv RTTT abc newqtyRe.ohm newqtyRzohm newqtyRdohm newqtyRvohm % solqtyRdvfracR_R_R_+R_Rdn*Rvn/Rdn+Rvnohm al R_ Rdvf fracRd RvRd + Rv Rdv solqtyRfracR_+R_R_R_+R_+R_Ren+Rzn*Rdvn/Ren+Rzn+Rdvnohm al R Rf fracqtyRe + Rz RdvRe + Rz + Rdv RTTT abc newqtyReohm newqtyRzohm newqtyRdohm newqtyRvohm % solqtyRqtyfracR_ + fracR_ + fracR_^- + R_/Ren+/Rzn+/Rvn**-+Rdnohm al R Rf qtyfracRe + fracRz + fracRv^- + Rd RTTT abcliste