Kirchhoff's current and voltage laws

Kirchhoff's current and voltage laws

1. Two Laws (Gustav Kirchhoff, 1845) — Conservation Laws for Circuits

KCL — Kirchhoff's Current Law (Current/Node Law): Sum of currents entering a node = sum leaving (conservation of charge). For node with currents $I_1...I_n$ (sign: entering +):

(algebraic sum at node zero).

KVL — Kirchhoff's Voltage Law (Voltage/Loop Law): Algebraic sum of voltages around any closed loop =0 (conservation of energy: net emf = drops). Going around loop, sum of rises = sum of drops:

or

Sign convention: traverse loop, voltage drop across $R$ in direction of traversal = $+IR$ (if traverse with current), emf rise if go from - to + of source.

2. Why They Work

• KCL: charge cannot accumulate at node (steady state) — follows continuity $∇·J = -∂ρ/∂t =0$.
• KVL: energy conserved — net work around loop zero for electrostatic field ($∮ E·dl=0$). Fails at high frequency where $dΦ_B/dt$ induces emf (use Maxwell-Faraday).

3. Worked Examples

(a) Node with I1=2A in, I2=3A in, I3=? out: I3=5A by KCL ✓

(b) Single loop: 10V source, R1=2Ω, R2=3Ω in series: KVL -10 + I·2 + I·3=0 ⇒ I=2A ✓

(c) Two loops sharing resistor: need simultaneous KVLs (mesh analysis next): e.g., loop1: 10 -I1·5 - (I1-I2)·10=0 ; loop2: -12 + (I2-I1)·10 + I2·8=0 ⇒ solve I1, I2 ✓

(d) KCL at supernode with 3 branches: I1= V/R1 etc. used in nodal analysis.

(e) Check: branched parallel resistors: KCL at top node gives I_total = I1+I2

4. Exam Points

• State KCL ($ΣI=0$) and KVL ($ΣV=0$) with sign — 5-7 marks, very frequent (almost certain to ask both statements).
• Apply to simple series/parallel and two-loop circuit to find $I$ — 7-10 marks, very frequent (the most common Kirchhoff numerical).
• Explain conservation basis (charge, energy) — 3-5 marks, frequent.
• Show KCL at node and KVL around mesh — 3 marks, frequent.