Circuit Configurations: Series and parallel circuits

Circuit Configurations: Series and parallel circuits

1. What They Are

• Series: components connected one after another, same current $I$ flows through all. Voltage divides among them. Single path.
• Parallel: components connected across same two nodes, same voltage $V$ across all. Current divides among them. Multiple paths.

Many circuits are series-parallel combinations.

2. Resistors in Series / Parallel

Resistors Series:

e.g., $R1+R2$.

Current same:

Voltage across $R_i$:

(voltage divider).

Resistors Parallel:

e.g., two:

Voltage same:

, current through $R_i$:

or

for two.

Power: series $P_{total}=Σ I²R_i$, parallel $P_{total}=Σ V²/R_i$.

3. Inductors and Capacitors Series/Parallel (dual)

• Inductors (no mutual): Series $L_{eq}=ΣL_i$ (like $R$), Parallel $1/L_{eq}=Σ1/L_i$.
• Capacitors: Opposite — Series $1/C_{eq}=Σ1/C_i$, Parallel $C_{eq}=ΣC_i$. (Because $Q=CV$, series $Q$ same).

4. Worked Examples

(a) 2Ω+3Ω series at 10V: R_eq=5Ω, I=2A, V2=4V, V3=6V ✓

(b) 2Ω||3Ω at 10V: R_eq=1.2Ω, I_total≈8.33A, I2=5A, I3≈3.33A ✓

(c) Identify: circuit with R1 in series with (R2||R3): R_eq = R1 + (R2R3/(R2+R3)) ✓

(d) Voltage divider: want 6V from 12V with R1=1k, R2=1k ⇒ V2=12·1/(1+1)=6V ✓ (requires load >> divider)

(e) Capacitors 2µF series with 3µF: C_eq=1/(1/2+1/3)=1.2µF

5. Exam Points

• Distinguish series (same $I$, $R$ adds) vs parallel (same $V$, $1/R$ adds) — 5 marks, very frequent.
• Compute $R_{eq}$ for series/parallel/combination — 5-7 marks, very frequent (common numerical).
• Voltage divider $V_i=V·R_i/ΣR$ and current divider $I_i=I·(other R/ sum)$ — 5 marks, very frequent (next topics but preview).
• $L/C$ series/parallel — 3-5 marks, frequent.