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Capacitors in series sum reciprocals: 1/C_eq = 1/C₁ + 1/C₂ + ... Capacitors in parallel sum directly: C_eq = C₁ + C₂ + ... These relationships are opposite to resistor combinations. Series capacitors share total applied voltage and store equal charge; parallel capacitors share the same voltage and distribute total charge. Understanding capacitor networks is essential for filter design and timing circuits.
You know how resistors combine from series-parallel analysis — series resistors add directly, parallel resistors add reciprocals. Capacitors do the opposite at every step, and understanding *why* builds deeper intuition than memorizing the formulas.
Recall that capacitance is defined as C = Q/V: it measures how much charge is stored per volt of applied voltage. A large capacitor stores more charge at a given voltage; a small one stores less. For parallel capacitors, both devices are connected between the same two nodes — they share the same voltage V. Each stores its own charge: Q₁ = C₁V and Q₂ = C₂V. The total charge drawn from the source is Q_total = Q₁ + Q₂ = (C₁ + C₂)V. Since C_eq = Q_total/V, the parallel equivalent is C_eq = C₁ + C₂ — capacitances add directly. Physically, parallel capacitors act as one larger capacitor with combined plate area, which is why adding plates increases capacitance.
For series capacitors, the same current flows through all elements in sequence. This enforces equal charge on each capacitor: the charge that flows onto one plate of C₁ must equal the charge flowing off the adjacent plate of C₂ (those plates form an isolated conductor — charge cannot appear or disappear within it). So Q₁ = Q₂ = Q. But voltage distributes across each element: V_total = V₁ + V₂ = Q/C₁ + Q/C₂ = Q(1/C₁ + 1/C₂). Since V_total = Q/C_eq, we get 1/C_eq = 1/C₁ + 1/C₂ — the reciprocal rule. Physically, series capacitors act as one capacitor with the combined thickness of both dielectrics; more separation between opposite charges means less capacitance.
The inversion relative to resistors is not arbitrary — it follows directly from duality. Resistors in series share the same current, so resistance (voltage per unit current) adds. Capacitors in series share the same charge, so the quantity Q/V inverted — that is, 1/C — adds. This is the pattern: when a circuit variable is shared, the corresponding impedance-like quantity adds. In AC analysis, capacitive reactance X_C = 1/(ωC) behaves exactly like resistance: series reactances add and parallel reactances combine reciprocally, now consistent with resistors. The apparent inversion at DC is really the same rule applied to the underlying physical variable — charge instead of current — that capacitors share in series.