Closure Audits

Series-Parallel Closure Audit

topology decides C_eq

Capacitor topology decides whether capacitance adds directly or through reciprocals; the equivalent charge then follows Q = C V.

Example

Capacitor topology decides whether capacitance adds directly or through reciprocals; the equivalent charge then follows Q = C V. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.

highlighted = computed this step

The same component kind has two different closures

Parallel capacitors share voltage; series capacitors share charge. The topology decides which arithmetic is honest.

parallel: add Cseries: add 1/C\text{parallel: add }C\qquad \text{series: add }1/C
Parallel topologyParallel capacitors share the same two buses.2 F4 F

Parallel closure adds capacitance, then charge

At 6 volts, each parallel row adds capacitance first and then closes charge with Q equals C V.

CaCbCeqQeq1 F2 F3 F18 C2 F4 F6 F36 C3 F5 F8 F48 C\begin{array}{c|c|c|c}C_a&C_b&C_{\text{eq}}&Q_{\text{eq}}\\1\ \text{F}&2\ \text{F}&3\ \text{F}&18\ \text{C}\\2\ \text{F}&4\ \text{F}&6\ \text{F}&36\ \text{C}\\3\ \text{F}&5\ \text{F}&8\ \text{F}&48\ \text{C}\\\end{array}
Parallel topologyThe middle table row is the checked parallel circuit.2 F4 F

Series closure uses reciprocal capacitance

A series chain is not a direct sum. The equivalent capacitance is smaller because the same charge must pass through both capacitors.

1Ceq=1Ca+1Cb\frac{1}{C_{\text{eq}}}=\frac{1}{C_a}+\frac{1}{C_b}
Series topologySeries capacitors form one chain.2 F3 F

Series closure keeps the fractional values visible

At the same 6 volts, series rows produce fractional equivalent capacitances and fractional equivalent charges. Those fractions are the audit, not rounding noise.

CaCbCeqQeq1 F2 F23 F4 C2 F3 F65 F365 C3 F4 F127 F727 C\begin{array}{c|c|c|c}C_a&C_b&C_{\text{eq}}&Q_{\text{eq}}\\1\ \text{F}&2\ \text{F}&\tfrac{2}{3}\ \text{F}&4\ \text{C}\\2\ \text{F}&3\ \text{F}&\tfrac{6}{5}\ \text{F}&\tfrac{36}{5}\ \text{C}\\3\ \text{F}&4\ \text{F}&\tfrac{12}{7}\ \text{F}&\tfrac{72}{7}\ \text{C}\\\end{array}
Series topologyThe middle table row is the checked series circuit.2 F3 F

Do not swap resistor rules into capacitor topology

For capacitors, parallel gives the larger storage number and series gives the smaller storage number. The diagram topology is part of the calculation, not a caption decoration.

Cparallel=Ca+CbCseries<Ca, CbC_{\text{parallel}} = C_a+C_b\qquad C_{\text{series}} < C_a,\ C_b
Series topologyThe series chain is the smaller-capacitance topology.2 F3 F
capacitors Compare parallel and series capacitor closures with exact equivalent charge rows.