8.2. Elementary circuits#
8.2.1. Series and Parallel Connections#
Series Connection. A series connection of linear passive components of the same type involves the same current passing through each component, \(i_n = i, \forall n=1:N\), and the total voltage difference between the “input terminal” of the first element and the “output terminal” of the last element being the sum of the voltage differences, \(v = \sum_{n=1:N} v_n\). Therefore:
For resistors in series, \(R_n\), the equivalent resistance is equal to the sum of the resistances:
\[v = \sum_n v_n = \sum_n \left( R_n \, i_n \right) = \left( \sum_n R_n \right) i = R_{series} \, i \qquad \rightarrow \qquad R_{series} = \sum_n R_n\]For capacitors in series, \(C_n\), the inverse of the equivalent capacitance is equal to the sum of the inverses of the capacitances:
\[\dfrac{d v}{dt} = \sum_n \dfrac{d v_n}{dt} = \sum_n \left( \frac{1}{C_n} \, i_n \right) = \left( \sum_n \frac{1}{C_n} \right) i = \dfrac{1}{C_{series}} \, i \qquad \rightarrow \qquad \frac{1}{C_{series}} = \sum_n \frac{1}{C_n}\]For inductors in series, \(L_n\), the equivalent inductance is equal to the sum of the inductances:
\[v = \sum_n v_n = \sum_n \left( L_n \, \dfrac{d i_n}{d t} \right) = \left( \sum_n L_n \right) \dfrac{d i}{dt} = L_{series} \, \dfrac{d i}{dt} \qquad \rightarrow \qquad L_{series} = \sum_n L_n\]
Consequently, the resistance and inductance of series-connected resistors and inductors are greater than the maximum resistance/inductance in the system; the equivalent capacitance of series-connected capacitors is less than the minimum capacitance of the capacitors in the system.
Parallel Connection. A parallel connection of linear passive components of the same type involves the same voltage difference across the terminals of each component, \(v_n = i, \forall n=1:N\), and the current through each component being generally different, with the sum of the currents equal to the current at the two extreme nodes of the connection, \(\sum_{n=1:N} i_n = i\). Therefore:
For resistors in parallel, \(R_n\), the inverse of the equivalent resistance is equal to the sum of the inverses of the resistances:
\[i = \sum_n i_n = \sum_n \left( \frac{1}{R_n} \, i_n \right) = \left( \sum_n \frac{1}{R_n} \right) i = \frac{1}{R_{\parallel}} \, i \qquad \rightarrow \qquad \frac{1}{R_{\parallel}} = \sum_n \frac{1}{R_n}\]For capacitors in parallel, \(C_n\), the equivalent capacitance is equal to the sum of the capacitances:
\[i = \sum_n i_n = \sum_n \left( C_n \, \dfrac{d v_n}{d t} \right) = \left( \sum_n C_n \right) \dfrac{d v}{dt} = C_{\parallel} \, \dfrac{d v}{dt} \qquad \rightarrow \qquad C_{\parallel} = \sum_n C_n\]For inductors in parallel, \(L_n\), the inverse of the equivalent inductance is equal to the sum of the inverses of the inductances:
\[\dfrac{d i}{dt} = \sum_n \dfrac{d i_n}{dt} = \sum_n \left( \frac{1}{L_n} \, v_n \right) = \left( \sum_n \frac{1}{L_n} \right) v = \dfrac{1}{L_{\parallel}} \, v \qquad \rightarrow \qquad \frac{1}{L_{\parallel}} = \sum_n \frac{1}{L_n}\]
Consequently, the resistance and inductance of parallel-connected resistors and inductors are less than the minimum resistance/inductance in the system; the equivalent capacitance of parallel-connected capacitors is greater than the maximum capacitance of the capacitors in the system.
8.2.2. Special Cases#
8.2.2.1. Open Circuit#
A circuit is open in the absence of a physical closure (with a wire) of a loop or behaves as such in the presence of a side through which the passage of electric current is impeded:
8.2.2.2. Short Circuit#
A short circuit occurs through a component with zero voltage drop:
If a short circuit occurs in an entire loop, it is traversed by infinite current—in a linear model that does not consider the limits of validity; in reality, non-linear effects occur much earlier, or sparks, explosions, or other destructive effects—often characterized by zero resistance. todo check the generality of this condition