How to Calculate Series and Parallel Circuits
Learn how to analyze series and parallel resistor, capacitor, and inductor circuits. Includes formulas for total resistance, voltage, and current with worked examples.
Series Circuits: One Path for Current
In a series circuit, components are connected end-to-end so there is only one path for current to flow. The same current passes through every component: I_total = I1 = I2 = I3. Voltages, however, divide across the components in proportion to their resistance: V_total = V1 + V2 + V3. This voltage division is governed by Kirchhoff's Voltage Law (KVL), which states that the sum of voltage drops around any closed loop equals the source voltage.
Total Resistance in Series
For resistors in series, resistances add directly: R_total = R1 + R2 + R3 + .... A series string of 100 Ω, 220 Ω, and 470 Ω gives R_total = 790 Ω. The current through the circuit is then I = Vs / R_total. Each resistor's voltage drop is V = I × Rn, so the 470 Ω resistor drops I × 470 V, which is the largest share. Series resistors are commonly used as voltage dividers and current limiters.
Parallel Circuits: Multiple Paths for Current
In a parallel circuit, all components share the same two nodes, meaning the voltage across each branch is identical: V_total = V1 = V2 = V3. Current, however, splits between the branches inversely proportional to resistance: I_total = I1 + I2 + I3. This is Kirchhoff's Current Law (KCL) — the total current entering a node equals the total current leaving it. Parallel circuits are used when components must all receive the same voltage, such as outlets in a house.
Total Resistance in Parallel
For resistors in parallel, the reciprocal formula applies: 1/R_total = 1/R1 + 1/R2 + 1/R3. For two resistors, the product-over-sum shortcut is R_total = (R1 × R2)/(R1 + R2). Two 1 kΩ resistors in parallel give R_total = (1000 × 1000)/(1000 + 1000) = 500 Ω. The total parallel resistance is always less than the smallest individual resistor in the group. Three 300 Ω resistors in parallel give R_total = 100 Ω.
Mixed Series-Parallel Networks
Real circuits often combine both topologies. The method is to simplify from the inside out: identify parallel groups first, replace each with its equivalent resistance, then handle the remaining series connections. For example, if R2 and R3 are in parallel with each other, and that combination is in series with R1, first find R23 = (R2 × R3)/(R2 + R3), then R_total = R1 + R23. Redraw the simplified circuit at each step to avoid errors.
Capacitors and Inductors: The Opposite Rules
Capacitors follow the opposite combining rules from resistors: capacitors in parallel add (C_total = C1 + C2), while capacitors in series use the reciprocal formula (1/C_total = 1/C1 + 1/C2). Inductors behave exactly like resistors: inductors in series add (L_total = L1 + L2), and inductors in parallel use the reciprocal formula, provided no mutual inductance exists between the coils. Remembering these rules prevents the common mistake of applying resistor formulas to capacitors.
Using Node Voltage and Mesh Analysis for Complex Networks
When simple series-parallel reduction is not possible — for example, in a bridge (Wheatstone) circuit — use node voltage analysis or mesh analysis. Node voltage analysis assigns a variable to each node and writes KCL equations; mesh analysis assigns loop currents and writes KVL equations. Both yield a system of simultaneous equations solvable with matrix algebra or substitution. For circuit simulation, tools like SPICE automate this process for networks with hundreds of nodes.
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