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This might be a dumb question, but with a circuit like this one:enter image description here

Are we allowed to add the resistances and capacitance in series to get a single resistor, capacitor, and inductor? Can elements in series be reduced despite components in between them? If not, how else would we be able to reduce this circuit so we can get the ideal form to apply equations?

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    \$\begingroup\$ It depends on what you are trying to answer. (Just a note: there is no power source in your diagram.) But if all you are asking is if it is possible, for some questions and some purposes, then the answer is yes -- so long as you perform the series operations correctly. (Series R adds, but series C is treated differently. But I suspect you know the differences.) But you still need to retain the different energy storage and dissipative bits. \$\endgroup\$ – jonk Feb 27 '18 at 1:15
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In your circuit the differential equation given by KVL is $$IR_1+\frac{1}{C_1}\int{I}dt+L\frac{dI}{dt}+R_2I+\frac{1}{C_2}\int{I}dt= 0$$ which can be shown to be equal to $$I(R_1+R_2)+(\frac{1}{C_1}+\frac{1}{C_2})\int{I}dt+L\frac{dI}{dt}= 0$$ which can be rewritten as: $$IR+\frac{1}{C}\int{I}dt+L\frac{dI}{dt}= 0$$ where $$R=R_1+R_2$$ is the equivalent series resistance and $$C=1/C_1+1/C_2$$ is the equivalent series capacitance in the circuit.

So yes, you can rewrite this in this manner and you will see the same differential equations governing the system's current. Thus, this shouldn't pose a problem in your analysis. The only concern I would have is if phase information might be lost when combining the capacitances, but I don't think that it will be.

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