Series and parallel networks
resistors in seriesresistors in parallelcapacitors in seriescombining components
Combining resistors and capacitors two at a time — and the reason the two components follow exactly opposite rules.
Two Resistors in Series
Total resistance of two resistors connected end to end is simply their sum.
Two Resistors in Parallel
Total resistance of two resistors connected side by side: product over sum, always less than either branch.
Two Capacitors in Series
Combines two series capacitors into a total that is smaller than either one.
Two Capacitors in Parallel
Combines two parallel capacitors by simply adding their capacitances.
How they fit together
Resistors in series add; in parallel their reciprocals add. Capacitors do precisely the reverse — parallel capacitances add, series capacitances add as reciprocals. That inversion is not a quirk to memorise. Resistance opposes current and stacking obstacles in a row makes it worse, while capacitance stores charge and putting plates side by side gives you more area to store it in. Wire capacitors in series and you are effectively increasing the gap, which cuts capacitance.
Two sanity checks catch nearly every arithmetic slip. A parallel resistance must come out smaller than the smallest resistor in the group — adding another path can only make it easier for current to get through. A series capacitance must come out smaller than the smallest capacitor. If your answer sits between the two values, or above both, you have used the wrong rule for that component.