Series & Parallel Analysis#

The first simplification technique: reduce series and parallel combinations to equivalent single components. It’s the starting point for understanding most circuits and the first tool to reach for when making sense of a schematic.

Series Combinations#

Components in series carry the same current. Voltages add.

  • Resistors: R_total = R1 + R2 + R3 + …
  • Capacitors: 1/C_total = 1/C1 + 1/C2 + … (capacitance decreases — the smallest cap dominates)
  • Inductors: L_total = L1 + L2 + … (assuming no mutual coupling)

Why Series Capacitors Seem Backwards#

The series capacitor formula confuses everyone at first. Think of it this way: series capacitors are like making the dielectric thicker. A thicker dielectric means less capacitance. The series combination is always less than the smallest individual capacitor.

For two caps in series: C_total = (C1 × C2) / (C1 + C2)

Parallel Combinations#

Components in parallel share the same voltage. Currents add.

  • Resistors: 1/R_total = 1/R1 + 1/R2 + … (resistance decreases — the smallest R dominates)
  • Capacitors: C_total = C1 + C2 + C3 + … (capacitance adds directly)
  • Inductors: 1/L_total = 1/L1 + 1/L2 + … (assuming no mutual coupling)

For two resistors in parallel: R_total = (R1 × R2) / (R1 + R2)

Quick mental math: two equal resistors in parallel = half of one. A 10 kΩ in parallel with a 1 kΩ is approximately 0.9 kΩ — the smaller one dominates.

When Simplification Helps#

  • Calculating total load resistance — What does the power supply see? Reduce parallel loads to one equivalent resistance
  • Finding expected voltages — Combine resistors, then use Ohm’s law or voltage divider equations
  • Estimating filter behavior — Equivalent RC or LC values determine cutoff frequencies
  • Quick sanity checks — “Is this node about where expected?” often comes down to a quick series/parallel reduction

When Simplification Hides Problems#

  • Component tolerances disappear — When R1 + R2 is combined, individual tolerances are lost. The worst-case equivalent resistance depends on which way each tolerance goes, and simplification sweeps this under the rug
  • Power distribution is hidden — A single equivalent resistor doesn’t indicate how power is split among the actual components. Each real resistor has its own power dissipation and thermal limit
  • Parasitic behavior doesn’t simplify cleanly — Two capacitors in parallel don’t just add capacitance — they also combine their ESR (in parallel, reducing it) and their ESL (in parallel, reducing it). The simplified model of “bigger C” misses the improved high-frequency behavior that was the whole reason for paralleling them
  • Loading effects — Replacing a load with its equivalent resistance can mask the fact that adding another branch changes what the original components see. This is where voltage divider loading analysis takes over

Mixed Series-Parallel Networks#

Most circuits aren’t purely series or purely parallel. The approach:

  1. Identify the innermost series or parallel group
  2. Reduce it to a single equivalent element
  3. Repeat until simplification is no longer possible
  4. Apply KVL/KCL to whatever remains

Some networks (bridges, T-networks, pi-networks) can’t be fully reduced by series/parallel alone. These require node or loop analysis, or delta-wye transformations.

Tips#

  • Start from the innermost combinations and work outward
  • Two equal resistors in parallel = half the value of one — a useful mental shortcut
  • For quick estimates, the parallel combination of two very different values is approximately the smaller one

Caveats#

  • Series elements must carry the same current — If there’s a branch point between two elements, they’re not in series. This sounds obvious but causes errors in complex schematics
  • Parallel elements must share the same two nodes — If two elements connect between the same pair of nodes, they’re in parallel. If they don’t share both nodes, they’re not parallel, even if they look close on the schematic
  • Real components aren’t purely R, C, or L — A real capacitor in a series chain adds its ESR as a series resistance and its ESL as a series inductance. At some frequency, this changes the effective impedance significantly
  • Don’t over-simplify for AC — At AC, impedances are needed, not just resistances. A 1 kΩ resistor in parallel with a 100 nF cap has different equivalent impedance at every frequency. Series/parallel reduction still works, but with complex numbers

In Practice#

  • A measured resistance that doesn’t match the expected series/parallel combination suggests a component is out of tolerance, damaged, or there’s a parallel path not accounted for
  • Unexpected voltage at a node often means the assumed series/parallel model is missing a load or leakage path
  • Filter behavior that doesn’t match calculations may indicate parasitic elements (ESR, ESL, stray capacitance) that the simplified model ignores
Backward Circuit Analysis Techniques Voltage Dividers & Loading Forward