AcademyCapacitors and Dielectrics
Academy
Series and Parallel Capacitors
Level 1 - Physics topic page in Capacitors and Dielectrics.
Principle
Capacitor networks reduce by matching the constraint: common voltage in parallel, common charge in series.
Notation
\(C_{\mathrm{eq}}\)
equivalent capacitance
\(\mathrm{F}\)
\(C_i\)
capacitance of capacitor i
\(\mathrm{F}\)
\(Q_i\)
charge magnitude on capacitor i
\(\mathrm{C}\)
\(V_i\)
voltage across capacitor i
\(\mathrm{V}\)
\(V\)
voltage across the network
\(\mathrm{V}\)
\(Q\)
charge supplied to the equivalent capacitor
\(\mathrm{C}\)
Method
Derivation 1: Parallel capacitors
Parallel capacitors share the same two nodes, so they have the same voltage.
Common voltage
\[V_i=V\]
Total charge
\[Q=\sum_i Q_i=\sum_i C_iV\]
Equivalent charge
\[Q=C_{\mathrm{eq}}V\]
Parallel rule
\[C_{\mathrm{eq}}=\sum_i C_i\]
Derivation 2: Series capacitors
Series capacitors carry the same charge because the intermediate conductor plates receive equal and opposite induced charge.
Common charge
\[Q_i=Q\]
Total voltage
\[V=\sum_i V_i=\sum_i \frac{Q}{C_i}\]
Equivalent voltage
\[V=\frac{Q}{C_{\mathrm{eq}}}\]
Series rule
\[\frac{1}{C_{\mathrm{eq}}}=\sum_i\frac{1}{C_i}\]
Derivation 3: Series voltage division
Once \(C_\{\\mathrm\{eq\}}\) is known, the common series charge is \(Q=C_\{\\mathrm\{eq\}}V\).
Capacitor voltage
\[V_i=\frac{Q}{C_i}\]
Substitute network charge
\[V_i=\frac{C_{\mathrm{eq}}}{C_i}V\]
Rules
These are the network reduction rules.
Parallel voltage
\[V_i=V\]
Parallel equivalent
\[C_{\mathrm{eq}}=\sum_i C_i\]
Series charge
\[Q_i=Q\]
Series equivalent
\[\frac{1}{C_{\mathrm{eq}}}=\sum_i\frac{1}{C_i}\]
Series voltage
\[V_i=\frac{C_{\mathrm{eq}}}{C_i}V\]
Examples
Question
Find the equivalent capacitance of
\[3.0\,\mu\mathrm{F}\]
and \[5.0\,\mu\mathrm{F}\]
in parallel.Answer
\[C_{\mathrm{eq}}=3.0+5.0=8.0\,\mu\mathrm{F}\]
Checks
- Parallel capacitors add to a larger capacitance than any one branch.
- Series equivalent capacitance is smaller than the smallest series capacitor.
- In series, voltages split inversely with capacitance.
- In parallel, charges split in proportion to capacitance.