AcademyDC Circuits
Academy
Kirchhoff Analysis
Level 1 - Physics topic page in DC Circuits.
Principle
Kirchhoff analysis turns a circuit into equations by applying conservation laws. Junction equations conserve charge. Loop equations conserve energy per unit charge.
Notation
\(I_j\)
current in branch j
\(\mathrm{A}\)
\(\mathcal E\)
ideal source emf
\(\mathrm{V}\)
\(R\)
resistance
\(\mathrm{\Omega}\)
\(\Delta V\)
potential change during a traversal
\(\mathrm{V}\)
\(V_a\)
electric potential at node a
\(\mathrm{V}\)
\(P\)
power delivered or dissipated
\(\mathrm{W}\)
Method
Derivation 1: Junction law
At an ideal node, charge cannot accumulate in steady state. The rate of charge entering equals the rate of charge leaving.
Charge conservation
\[\frac{dQ_{\mathrm{node}}}{dt}=0\]
Current balance
\[\sum I_{\mathrm{in}}=\sum I_{\mathrm{out}}\]
Algebraic form
\[\sum_j I_j=0\]
Derivation 2: Loop law
Electric potential is single valued in a lumped DC circuit. If you return to the starting point, the algebraic sum of all potential changes is zero.
Return to same potential
\[V_{\mathrm{final}}-V_{\mathrm{initial}}=0\]
Sum changes around loop
\[\sum_{\mathrm{loop}}\Delta V=0\]
Resistor traversal with current
\[\Delta V=-IR\]
Ideal source from minus to plus
\[\Delta V=+\mathcal E\]
Derivation 3: Solving with assumed directions
Current arrows are bookkeeping choices. If a solved current is negative, the physical current flows opposite the assumed arrow.
Rules
Kirchhoff junction law
\[\sum I_{\mathrm{in}}=\sum I_{\mathrm{out}}\]
Kirchhoff loop law
\[\sum_{\mathrm{loop}}\Delta V=0\]
Resistor drop with current
\[\Delta V=-IR\]
Resistor rise against current
\[\Delta V=+IR\]
Source rise
\[\Delta V=+\mathcal E\quad\text{from negative to positive terminal}\]
Examples
Question
A
\[12\,\mathrm{V}\]
source drives \[2.0\,\Omega\]
and \[4.0\,\Omega\]
in series. Write the loop equation and find \(I\).Answer
\[12-2I-4I=0\quad\Rightarrow\quad I=2.0\,\mathrm{A}\]
Checks
- Define current arrows before writing equations.
- Use one sign convention consistently around each loop.
- Every independent junction equation expresses charge conservation; every independent loop equation expresses energy conservation.
- A negative solved current is information, not an error.
- Check units: every term in a loop equation is a voltage, and every term in a junction equation is a current.